diff --git a/.github/workflows/master.yml b/.github/workflows/master.yml deleted file mode 100644 index 3d79c6aa..00000000 --- a/.github/workflows/master.yml +++ /dev/null @@ -1,94 +0,0 @@ -# This workflow will install Python dependencies, run tests and lint with a variety of Python versions -# For more information see: https://help.github.com/actions/language-and-framework-guides/using-python-with-github-actions - -name: build - -on: - push: - branches: [ master ] - -jobs: - unittest: - runs-on: ubuntu-latest - strategy: - matrix: - python-version: [3.9, "3.10", 3.11, 3.12] - steps: - - uses: actions/checkout@v2 - - name: Set up Python ${{ matrix.python-version }} - uses: actions/setup-python@v1 - with: - python-version: ${{ matrix.python-version }} - - name: Install dependencies - run: | - python -m pip install --upgrade pip - - name: Test with pytest - run: | - pip install .[dev] - pytest - codecov: - # If all tests pass: - # Run coverage and upload to codecov - needs: unittest - runs-on: ubuntu-latest - steps: - - uses: actions/checkout@v2 - - name: Set up Python 3.11 - uses: actions/setup-python@v1 - with: - python-version: 3.11 - - name: Install dependencies - run: | - python -m pip install --upgrade pip - - name: Run coverage - run: | - pip install -e .[dev] - pytest --cov=pgraph --cov-report xml:coverage.xml - coverage report - - name: upload coverage to Codecov - uses: codecov/codecov-action@v1 - with: - file: ./coverage.xml - verbose: true - sphinx: - # If the above worked: - # Build docs and upload to GH Pages - needs: unittest - runs-on: ubuntu-latest - steps: - - uses: actions/checkout@v2 - - name: Set up Python 3.11 - uses: actions/setup-python@v1 - with: - python-version: 3.11 - - name: Install dependencies - run: | - python -m pip install --upgrade pip - pip install .[docs] - pip install git+https://github.com/petercorke/sphinx-autorun.git - sudo apt-get install graphviz - - name: Build docs - run: | - cd docs - make html - # Tell GitHub not to use jekyll to compile the docs - touch build/html/.nojekyll - cd ../ - - name: Commit documentation changes - run: | - git clone https://github.com/petercorke/robotics-toolbox-python.git --branch gh-pages --single-branch gh-pages - cp -r docs/build/html/* gh-pages/ - cd gh-pages - git config --local user.email "action@github.com" - git config --local user.name "GitHub Action" - git add . - git commit -m "Update documentation" -a || true - # The above command will fail if no changes were present, so we ignore - # that. - - name: Push changes - uses: ad-m/github-push-action@master - with: - branch: gh-pages - directory: gh-pages - github_token: ${{ secrets.GITHUB_TOKEN }} - force: true diff --git a/.gitignore b/.gitignore index 16f919da..df40e8ae 100644 --- a/.gitignore +++ b/.gitignore @@ -1,9 +1,181 @@ -docs/build +# Vim +*.sw* + + +### VisualStudioCode ### +.vscode/* +#!.vscode/settings.json +#!.vscode/tasks.json +#!.vscode/launch.json +#!.vscode/extensions.json + +# Byte-compiled / optimized / DLL files +__pycache__/ +*.py[cod] +*$py.class + +# C extensions +*.so + +# Distribution / packaging +.Python +build/ +develop-eggs/ dist/ -*.code-workspace -__pycache__ -*.egg-info -.vscode/ -*.pdf -*.dot -*.mat +downloads/ +eggs/ +.eggs/ +lib/ +lib64/ +parts/ +sdist/ +var/ +wheels/ +pip-wheel-metadata/ +share/python-wheels/ +*.egg-info/ +.installed.cfg +*.egg +MANIFEST +wheelhouse/ + +# PyInstaller +# Usually these files are written by a python script from a template +# before PyInstaller builds the exe, so as to inject date/other infos into it. +*.manifest +*.spec + +# Installer logs +pip-log.txt +pip-delete-this-directory.txt + +# Unit test / coverage reports +htmlcov/ +.tox/ +.nox/ +.coverage +.coverage.* +.cache +nosetests.xml +coverage.xml +*.cover +.hypothesis/ +.pytest_cache/ + +# Translations +*.mo +*.pot + +# Django stuff: +*.log +local_settings.py +db.sqlite3 + +# Flask stuff: +instance/ +.webassets-cache + +# Scrapy stuff: +.scrapy + +# Sphinx documentation +docs/_build/ + +# PyBuilder +target/ + +# Jupyter Notebook +.ipynb_checkpoints + +# IPython +profile_default/ +ipython_config.py + +# pyenv +.python-version + +# pipenv +# According to pypa/pipenv#598, it is recommended to include Pipfile.lock in version control. + +# celery beat schedule file +celerybeat-schedule + +# SageMath parsed files +*.sage.py + +# Environments +.env +.venv +env/ +venv/ +ENV/ +env.bak/ +venv.bak/ + +# Spyder project settings +.spyderproject +.spyproject + +# Rope project settings +.ropeproject + +# mkdocs documentation +/site + +# mypy +.mypy_cache/ +.dmypy.json +dmypy.json + +# Pyre type checker +.pyre/ + +### ROS ### +devel/ +logs/ +bin/ +msg_gen/ +srv_gen/ +msg/*Action.msg +msg/*ActionFeedback.msg +msg/*ActionGoal.msg +msg/*ActionResult.msg +msg/*Feedback.msg +msg/*Goal.msg +msg/*Result.msg +msg/_*.py +build_isolated/ +devel_isolated/ + +# Generated by dynamic reconfigure +*.cfgc +/cfg/cpp/ +/cfg/*.py + +# Ignore generated docs +*.dox +*.wikidoc + +# eclipse stuff +.project +.cproject + +# qcreator stuff +CMakeLists.txt.user + +srv/_*.py +*.pcd +*.pyc +qtcreator-* +*.user + +/planning/cfg +/planning/docs +/planning/src + +# Catkin custom files +CATKIN_IGNORE + +### VisualStudioCode Patch ### +# Ignore all local history of files +.history diff --git a/.nojekyll b/.nojekyll new file mode 100644 index 00000000..e69de29b diff --git a/CHANGELOG.md b/CHANGELOG.md deleted file mode 100644 index 21e8f2cb..00000000 --- a/CHANGELOG.md +++ /dev/null @@ -1,5 +0,0 @@ -0.6.3 - -- runs with Numpy2.x -- changed build system to .toml file -- misc doco and changes \ No newline at end of file diff --git a/IK/ik.html b/IK/ik.html new file mode 100644 index 00000000..9b591b1b --- /dev/null +++ b/IK/ik.html @@ -0,0 +1,424 @@ + + + + + + + + + + + Inverse Kinematics — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

Inverse Kinematics

+

The Robotics Toolbox supports an extensive set of numerical inverse kinematics (IK) tools and we will demonstrate the different ways these IK tools can be interacted with in this document.

+

For a tutorial on numerical IK, see here.

+

Within the Toolbox, we have two sets of solvers: solvers written in C++ and solvers written in Python. However, within all of our solvers there are several common arguments:

+

Tep

+

Tep represent the desired end-effector pose.

+

A note on the semantics of the above variable:

+
    +
  • T represents an SE(3) (a homogeneous tranformation matrix in 3 dimensions, a 4x4 matrix)

  • +
  • e is short for end-effector referring to the end of the kinematic chain

  • +
  • p is short for prime or desired

  • +
  • Since there is no letter to the left of the T, the world or base reference frame is implied

  • +
+

Therefore, Tep refers to the desired end-effector pose in the base robot frame represented as an SE(3).

+

ilimit

+

The ilimit specifies how many iterations are allowed within a single search. After ilimit is reached, either, a new attempt is made or the IK solution has failed depending on slimit

+

slimit

+

The slimit specifies how many searches are allowed before the problem is deemed unsolvable. The maximum number of iterations allowed is therefore ilimit x slimit. By having slimit > 1, a global search is performed. Since finding a solution with numerical IK heavily depends on the initial choice of q0, performing a global search where slimit >> 1 will provide a far greater chance of success.

+

q0

+

q0 is the inital joint coordinate vector. If q0 is 1 dimensional (, n), then q0 is only used for the first attempt, after which a new random valid initial joint coordinate vector will be generated. If q0 is 2 dimensional (slimit, n), then the next vector within q0 will be used for the next search.

+

tol

+

tol sets the error tolerance before the solution is deemed successful. The error is typically set by some quadratic error function

+
+\[E = \frac{1}{2} \vec{e}^{\top} \mat{W}_e \vec{e}\]
+

where \(\vec{e} \in \mathbb{R}^6\) is the angle-axis error, and \(\mat{W}_e\) assigns weights to Cartesian degrees-of-freedom

+

mask

+

mask is a (,6) array that sets \(\mat{W}_e\) in error equation above. The vector has six elements that correspond to translation in X, Y and Z, and rotation about X, Y and Z respectively. The value can be 0 (for ignore) or above to assign a priority relative to other Cartesian DoF.

+

For the case where the manipulator has fewer than 6 DoF the solution space has more dimensions than can be spanned by the manipulator joint coordinates.

+

In this case we use the mask option where the mask vector specifies the Cartesian DOF that will be ignored in reaching a solution. The number of non-zero elements must equal the number of manipulator DOF.

+

For example when using a 3 DOF manipulator tool orientation might be unimportant, in which case use the option mask=[1, 1, 1, 0, 0, 0].

+

joint_limits

+

setting joint_limits = True will reject solutions with joint limit violations. Note that finding a solution with valid joint coordinates is likely to take longer than without.

+

Others

+

There are other arguments which may be unique to the solver, so check the documentation of the solver you wish to use for a complete list and explanation of arguments.

+
+

C++ Solvers

+

These solvers are written in high performance C++ and wrapped in Python methods. The methods are made available within the ETS and Robot classes. Being written in C++, these solvers are extraordinarily fast and typically take 30 to 90 µs. However, these solvers are hard to extend or modify.

+

These methods have been written purely for speed so they do not contain the niceties of the Python alternative. For example, if you give the incorrect length for the q0 vector, you could end up with a seg-fault or other undetermined behaviour. Therefore, when using these methods it is very important that you understand each of the parameters and the parameters passed are of the correct type and length.

+

The C++ solvers return a tuple with the following members:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

Element

Type

Description

q

ndarray

The joint coordinates of the solution. Note that these will not be valid if failed to find a solution

success

bool

True if a valid solution was found

iterations

int

How many iterations were performed

searches

int

How many searches were performed

residual

float

The final error value from the cost function

+

The C++ solvers can be identified as methods which start with ik_.

+

ETS C++ IK Methods

+ + + + + + + + + + + + +

ik_LM(Tep[, q0, ilimit, slimit, tol, mask, ...])

Fast levenberg-Marquadt Numerical Inverse Kinematics Solver

ik_GN(Tep[, q0, ilimit, slimit, tol, mask, ...])

Fast numerical inverse kinematics by Gauss-Newton optimization

ik_NR(Tep[, q0, ilimit, slimit, tol, mask, ...])

Fast numerical inverse kinematics using Newton-Raphson optimization

+

Robot C++ IK Methods

+ + + + + + + + + + + + +

ik_LM(Tep[, end, start, q0, ilimit, slimit, ...])

Fast levenberg-Marquadt Numerical Inverse Kinematics Solver

ik_GN(Tep[, end, start, q0, ilimit, slimit, ...])

Fast numerical inverse kinematics by Gauss-Newton optimization

ik_NR(Tep[, end, start, q0, ilimit, slimit, ...])

Fast numerical inverse kinematics using Newton-Raphson optimization

+

In the following example, we create a Panda robot and one of the fast IK solvers available within the Robot class.

+
>>> import roboticstoolbox as rtb
+>>> # Make a Panda robot
+>>> panda = rtb.models.Panda()
+>>> # Make a goal pose
+>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> # Solve the IK problem
+>>> panda.ik_LM(Tep)
+(array([ 0.5007, -0.3356, -0.4457, -2.1972, -0.1636,  1.9881,  0.9144]), 1, 16, 2, 9.976484070621097e-11)
+
+
+

In the following example, we create a Panda robot and and then get the ETS representation. Subsequently, we use one of the fast IK solvers available within the ETS class.

+
>>> import roboticstoolbox as rtb
+>>> # Make a Panda robot
+>>> panda = rtb.models.Panda()
+>>> # Get the ETS
+>>> ets = panda.ets()
+>>> # Make a goal pose
+>>> Tep = ets.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> # Solve the IK problem
+>>> ets.ik_LM(Tep)
+(array([ 0.5007, -0.3356, -0.4457, -2.1972, -0.1636,  1.9881,  0.9144]), 1, 16, 2, 9.976484070621097e-11)
+
+
+
+
+

Python Solvers

+

These solvers are Python classes which extend the abstract base class IKSolver and the solve() method returns an IKSolution. These solvers are slow and will typically take 100 - 1000 ms. However, these solvers are easy to extend and modify.

+

The Abstract Base Class

+ +

The IKSolver provides basic functionality for performing numerical IK. Superclasses can inherit this class and must implement the solve() method. Additionally a superclass redefine any other methods necessary such as error() to provide a custom error function.

+

The Solution DataClass

+ +

The IKSolution is a dataclasses.dataclass instance with the following members.

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

Element

Type

Description

q

ndarray

The joint coordinates of the solution. Note that these will not be valid if failed to find a solution

success

bool

True if a valid solution was found

iterations

int

How many iterations were performed

searches

int

How many searches were performed

residual

float

The final error value from the cost function

reason

str

The reason the IK problem failed if applicable

+

The Implemented IK Solvers

+

These solvers can be identified as a Class starting with IK_.

+ +

Example

+

In the following example, we create an IK Solver class and pass an ETS to it to solve the problem. This style may be preferable to experiments where you wish to compare the same solver on different robots.

+
>>> import roboticstoolbox as rtb
+>>> # Make a Panda robot
+>>> panda = rtb.models.Panda()
+>>> # Get the ETS of the Panda
+>>> ets = panda.ets()
+>>> # Make an IK solver
+>>> solver = rtb.IK_LM()
+>>> # Make a goal pose
+>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> # Solve the IK problem
+>>> solver.solve(ets, Tep)
+IKSolution(q=array([-2.006 ,  0.5697,  2.1978, -2.174 , -0.5032,  1.8822,  1.1792]), success=True, iterations=10, searches=1, residual=2.3488721776519882e-07, reason='Success')
+
+
+

Additionally, these Class based solvers have been implemented as methods within the ETS and Robot classes. The method names start with ikine_.

+

ETS Python IK Methods

+ + + + + + + + + + + + + + + +

ikine_LM(Tep[, q0, ilimit, slimit, tol, ...])

Levemberg-Marquadt Numerical Inverse Kinematics Solver

ikine_QP(Tep[, q0, ilimit, slimit, tol, ...])

Quadratic Programming Numerical Inverse Kinematics Solver

ikine_GN(Tep[, q0, ilimit, slimit, tol, ...])

Gauss-Newton Numerical Inverse Kinematics Solver

ikine_NR(Tep[, q0, ilimit, slimit, tol, ...])

Newton-Raphson Numerical Inverse Kinematics Solver

+

Robot Python IK Methods

+ + + + + + + + + + + + + + + +

ikine_LM(Tep[, end, start, q0, ilimit, ...])

Levenberg-Marquadt Numerical Inverse Kinematics Solver

ikine_QP(Tep[, end, start, q0, ilimit, ...])

Quadratic Programming Numerical Inverse Kinematics Solver

ikine_GN(Tep[, end, start, q0, ilimit, ...])

Gauss-Newton Numerical Inverse Kinematics Solver

ikine_NR(Tep[, end, start, q0, ilimit, ...])

Newton-Raphson Numerical Inverse Kinematics Solver

+

Example

+

In the following example, we create a Panda robot and one of the IK solvers available within the Robot class. This style is far more convenient than the above example.

+
>>> import roboticstoolbox as rtb
+>>> # Make a Panda robot
+>>> panda = rtb.models.Panda()
+>>> # Make a goal pose
+>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> # Solve the IK problem
+>>> panda.ikine_LM(Tep)
+IKSolution(q=array([-0.617 , -0.3565,  0.5461, -2.1954,  0.2088,  1.9805,  0.6208]), success=True, iterations=10, searches=1, residual=3.31895442979262e-07, reason='Success')
+
+
+

In the following example, we create a Panda robot and and then get the ETS representation. Subsequently, we use one of the IK solvers available within the ETS class.

+
>>> import roboticstoolbox as rtb
+>>> # Make a Panda robot
+>>> panda = rtb.models.Panda()
+>>> # Get the ETS
+>>> ets = panda.ets()
+>>> # Make a goal pose
+>>> Tep = ets.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> # Solve the IK problem
+>>> ets.ikine_LM(Tep)
+IKSolution(q=array([-1.396 , -0.8222,  1.0591, -2.1449,  0.7496,  1.7232,  0.2073]), success=True, iterations=11, searches=1, residual=7.265959113355526e-10, reason='Success')
+
+
+
+
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/ik_gn.html b/IK/ik_gn.html new file mode 100644 index 00000000..e87083f3 --- /dev/null +++ b/IK/ik_gn.html @@ -0,0 +1,271 @@ + + + + + + + + + + + IK_GN - Gauss-Newton Numerical IK — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

IK_GN - Gauss-Newton Numerical IK

+
+
+class roboticstoolbox.robot.IK.IK_GN(name='IK Solver', ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, seed=None, pinv=False, kq=0.0, km=0.0, ps=0.0, pi=0.3, **kwargs)[source]
+

Bases: IKSolver

+

Gauss-Newton Numerical Inverse Kinematics Solver

+

A class which provides functionality to perform numerical inverse kinematics (IK) +using the Gauss-Newton method. See step method for mathematical description.

+
+

Note

+

When using this class with redundant robots (>6 DoF), pinv must be set to True

+
+
+
Parameters:
+
    +
  • name (str) – The name of the IK algorithm

  • +
  • ilimit (int) – How many iterations are allowed within a search before a new search +is started

  • +
  • slimit (int) – How many searches are allowed before being deemed unsuccessful

  • +
  • tol (float) – Maximum allowed residual error E

  • +
  • mask (Union[ndarray, List[float], Tuple[float], Set[float], None]) – A 6 vector which assigns weights to Cartesian degrees-of-freedom +error priority

  • +
  • joint_limits (bool) – Reject solutions with joint limit violations

  • +
  • seed (Optional[int]) – A seed for the private RNG used to generate random joint coordinate +vectors

  • +
  • pinv (bool) – If True, will use the psuedoinverse in the step method instead of +the normal inverse

  • +
  • kq (float) – The gain for joint limit avoidance. Setting to 0.0 will remove this +completely from the solution

  • +
  • km (float) – The gain for maximisation. Setting to 0.0 will remove this completely +from the solution

  • +
  • ps (float) – The minimum angle/distance (in radians or metres) in which the joint is +allowed to approach to its limit

  • +
  • pi (Union[ndarray, float]) – The influence angle/distance (in radians or metres) in null space motion +becomes active

  • +
+
+
+

Examples

+

The following example gets the ets of a panda robot object, instantiates +the IK_GN solver class using default parameters, makes a goal pose Tep, +and then solves for the joint coordinates which result in the pose Tep +using the solve method.

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda().ets()
+>>> solver = rtb.IK_GN(pinv=True)
+>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> solver.solve(panda, Tep)
+IKSolution(q=array([ 1.2886,  1.4862, -2.1934, -2.0771,  1.244 ,  1.1468, -0.096 ]), success=True, iterations=53, searches=4, residual=9.796561164911834e-10, reason='Success')
+
+
+

Notes

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian. When the +the problem is solvable, it converges very quickly.

+

This class supports null-space motion to assist with maximising manipulability and +avoiding joint limits. These are enabled by setting kq and km to non-zero values.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
IKSolver

An abstract super class for numerical IK solvers

+
+
IK_NR

Implements IKSolver using the Newton-Raphson method

+
+
IK_LM

Implements IKSolver using the Levemberg-Marquadt method

+
+
IK_QP

Implements IKSolver using a quadratic programming approach

+
+
+
+
+

Changed in version 1.0.3: Added the Gauss-Newton IK solver class

+
+
+ +

Methods

+ + + + + + + + + + + + +

step(ets, Tep, q)

Performs a single iteration of the Gauss-Newton optimisation method

solve(ets, Tep[, q0])

Solves the IK problem

error(Te, Tep)

Calculates the error between Te and Tep

+

Private Methods

+ + + + + + + + + +

_random_q(ets[, i])

Generate a random valid joint configuration using a private RNG

_check_jl(ets, q)

Checks if the joints are within their respective limits

+
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/ik_lm.html b/IK/ik_lm.html new file mode 100644 index 00000000..42f1f75d --- /dev/null +++ b/IK/ik_lm.html @@ -0,0 +1,271 @@ + + + + + + + + + + + IK_LM - Levemberg-Marquadt Numerical IK — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

IK_LM - Levemberg-Marquadt Numerical IK

+
+
+class roboticstoolbox.robot.IK.IK_LM(name='IK Solver', ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, seed=None, k=1.0, method='chan', kq=0.0, km=0.0, ps=0.0, pi=0.3, **kwargs)[source]
+

Bases: IKSolver

+

Levemberg-Marquadt Numerical Inverse Kinematics Solver

+

A class which provides functionality to perform numerical inverse kinematics (IK) +using the Levemberg-Marquadt method. See step method for mathematical description.

+
+
Parameters:
+
    +
  • name (str) – The name of the IK algorithm

  • +
  • ilimit (int) – How many iterations are allowed within a search before a new search +is started

  • +
  • slimit (int) – How many searches are allowed before being deemed unsuccessful

  • +
  • tol (float) – Maximum allowed residual error E

  • +
  • mask (Union[ndarray, List[float], Tuple[float], Set[float], None]) – A 6 vector which assigns weights to Cartesian degrees-of-freedom +error priority

  • +
  • joint_limits (bool) – Reject solutions with joint limit violations

  • +
  • seed (Optional[int]) – A seed for the private RNG used to generate random joint coordinate +vectors

  • +
  • k (float) – Sets the gain value for the damping matrix Wn in the step method. See +notes

  • +
  • method – One of “chan”, “sugihara” or “wampler”. Defines which method is used +to calculate the damping matrix Wn in the step method

  • +
  • kq (float) – The gain for joint limit avoidance. Setting to 0.0 will remove this +completely from the solution

  • +
  • km (float) – The gain for maximisation. Setting to 0.0 will remove this completely +from the solution

  • +
  • ps (float) – The minimum angle/distance (in radians or metres) in which the joint is +allowed to approach to its limit

  • +
  • pi (Union[ndarray, float]) – The influence angle/distance (in radians or metres) in null space motion +becomes active

  • +
+
+
+

Examples

+

The following example gets the ets of a panda robot object, instantiates +the IK_LM solver class using default parameters, makes a goal pose Tep, +and then solves for the joint coordinates which result in the pose Tep +using the solve method.

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda().ets()
+>>> solver = rtb.IK_LM()
+>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> solver.solve(panda, Tep)
+IKSolution(q=array([-1.1097, -0.5518,  0.9276, -2.176 ,  0.483 ,  1.8918,  0.4071]), success=True, iterations=13, searches=1, residual=1.7220037459970422e-10, reason='Success')
+
+
+

Notes

+

The value for the k kwarg will depend on the method chosen and the arm you are +using. Use the following as a rough guide chan, k = 1.0 - 0.01, +wampler, k = 0.01 - 0.0001, and sugihara, k = 0.1 - 0.0001

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian.

+

This class supports null-space motion to assist with maximising manipulability and +avoiding joint limits. These are enabled by setting kq and km to non-zero values.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
IKSolver

An abstract super class for numerical IK solvers

+
+
IK_NR

Implements the IKSolver class using the Newton-Raphson method

+
+
IK_GN

Implements the IKSolver class using the Gauss-Newton method

+
+
IK_QP

Implements the IKSolver class using a quadratic programming approach

+
+
+
+
+

Changed in version 1.0.3: Added the Levemberg-Marquadt IK solver class

+
+
+ +

Methods

+ + + + + + + + + + + + +

step(ets, Tep, q)

Performs a single iteration of the Levenberg-Marquadt optimisation

solve(ets, Tep[, q0])

Solves the IK problem

error(Te, Tep)

Calculates the error between Te and Tep

+

Private Methods

+ + + + + + + + + +

_random_q(ets[, i])

Generate a random valid joint configuration using a private RNG

_check_jl(ets, q)

Checks if the joints are within their respective limits

+
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/ik_nr.html b/IK/ik_nr.html new file mode 100644 index 00000000..61582dfe --- /dev/null +++ b/IK/ik_nr.html @@ -0,0 +1,272 @@ + + + + + + + + + + + IK_NR - Newton-Raphson Numerical IK — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

IK_NR - Newton-Raphson Numerical IK

+
+
+class roboticstoolbox.robot.IK.IK_NR(name='IK Solver', ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, seed=None, pinv=False, kq=0.0, km=0.0, ps=0.0, pi=0.3, **kwargs)[source]
+

Bases: IKSolver

+

Newton-Raphson Numerical Inverse Kinematics Solver

+

A class which provides functionality to perform numerical inverse kinematics (IK) +using the Newton-Raphson method. See step method for mathematical description.

+
+

Note

+

When using this class with redundant robots (>6 DoF), pinv must be set to True

+
+
+
Parameters:
+
    +
  • name (str) – The name of the IK algorithm

  • +
  • ilimit (int) – How many iterations are allowed within a search before a new search +is started

  • +
  • slimit (int) – How many searches are allowed before being deemed unsuccessful

  • +
  • tol (float) – Maximum allowed residual error E

  • +
  • mask (Union[ndarray, List[float], Tuple[float], Set[float], None]) – A 6 vector which assigns weights to Cartesian degrees-of-freedom +error priority

  • +
  • joint_limits (bool) – Reject solutions with joint limit violations

  • +
  • seed (Optional[int]) – A seed for the private RNG used to generate random joint coordinate +vectors

  • +
  • pinv (bool) – If True, will use the psuedoinverse in the step method instead of +the normal inverse

  • +
  • kq (float) – The gain for joint limit avoidance. Setting to 0.0 will remove this +completely from the solution

  • +
  • km (float) – The gain for maximisation. Setting to 0.0 will remove this completely +from the solution

  • +
  • ps (float) – The minimum angle/distance (in radians or metres) in which the joint is +allowed to approach to its limit

  • +
  • pi (Union[ndarray, float]) – The influence angle/distance (in radians or metres) in null space motion +becomes active

  • +
+
+
+

Examples

+

The following example gets the ets of a panda robot object, instantiates +the IK_NR solver class using default parameters, makes a goal pose Tep, +and then solves for the joint coordinates which result in the pose Tep +using the solve method.

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda().ets()
+>>> solver = rtb.IK_NR(pinv=True)
+>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> solver.solve(panda, Tep)
+IKSolution(q=array([-2.3853,  0.3909,  2.4784, -2.1923, -0.27  ,  1.9673,  0.9981]), success=True, iterations=188, searches=12, residual=2.42191743196601e-10, reason='Success')
+
+
+

Notes

+

When using the NR method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian. When the +the problem is solvable, it converges very quickly. However, this method frequently +fails to converge on the goal.

+

This class supports null-space motion to assist with maximising manipulability and +avoiding joint limits. These are enabled by setting kq and km to non-zero values.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
IKSolver

An abstract super class for numerical IK solvers

+
+
IK_GN

Implements the IKSolver class using the Gauss-Newton method

+
+
IK_LM

Implements the IKSolver class using the Levemberg-Marquadt method

+
+
IK_QP

Implements the IKSolver class using a quadratic programming approach

+
+
+
+
+

Changed in version 1.0.3: Added the Newton-Raphson IK solver class

+
+
+ +

Methods

+ + + + + + + + + + + + +

step(ets, Tep, q)

Performs a single iteration of the Newton-Raphson optimisation method

solve(ets, Tep[, q0])

Solves the IK problem

error(Te, Tep)

Calculates the error between Te and Tep

+

Private Methods

+ + + + + + + + + +

_random_q(ets[, i])

Generate a random valid joint configuration using a private RNG

_check_jl(ets, q)

Checks if the joints are within their respective limits

+
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/ik_qp.html b/IK/ik_qp.html new file mode 100644 index 00000000..726e86f6 --- /dev/null +++ b/IK/ik_qp.html @@ -0,0 +1,264 @@ + + + + + + + + + + + IK_QP - Numerical IK with Quadratic Programming — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

IK_QP - Numerical IK with Quadratic Programming

+
+
+class roboticstoolbox.robot.IK.IK_QP(name='IK Solver', ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, seed=None, kj=0.01, ks=1.0, kq=0.0, km=0.0, ps=0.0, pi=0.3, **kwargs)[source]
+

Bases: IKSolver

+

Quadratic Progamming Numerical Inverse Kinematics Solver

+

A class which provides functionality to perform numerical inverse kinematics (IK) +using a quadratic progamming approach. See step method for mathematical +description.

+
+
Parameters:
+
    +
  • name (str) – The name of the IK algorithm

  • +
  • ilimit (int) – How many iterations are allowed within a search before a new search +is started

  • +
  • slimit (int) – How many searches are allowed before being deemed unsuccessful

  • +
  • tol (float) – Maximum allowed residual error E

  • +
  • mask (Union[ndarray, List[float], Tuple[float], Set[float], None]) – A 6 vector which assigns weights to Cartesian degrees-of-freedom +error priority

  • +
  • joint_limits (bool) – Reject solutions with joint limit violations

  • +
  • seed (Optional[int]) – A seed for the private RNG used to generate random joint coordinate +vectors

  • +
  • kj – A gain for joint velocity norm minimisation

  • +
  • ks – A gain which adjusts the cost of slack (intentional error)

  • +
  • kq (float) – The gain for joint limit avoidance. Setting to 0.0 will remove this +completely from the solution

  • +
  • km (float) – The gain for maximisation. Setting to 0.0 will remove this completely +from the solution

  • +
  • ps (float) – The minimum angle/distance (in radians or metres) in which the joint is +allowed to approach to its limit

  • +
  • pi (Union[ndarray, float]) – The influence angle/distance (in radians or metres) in null space motion +becomes active

  • +
+
+
Raises:
+

ImportError – If the package qpsolvers is not installed

+
+
+

Examples

+

The following example gets the ets of a panda robot object, instantiates +the IK_QP solver class using default parameters, makes a goal pose Tep, +and then solves for the joint coordinates which result in the pose Tep +using the solve method.

+
Traceback (most recent call last):
+  File "<input>", line 1, in <module>
+NameError: name 'solver' is not defined
+
+
+

Notes

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian. When the +the problem is solvable, it converges very quickly.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
IKSolver

An abstract super class for numerical IK solvers

+
+
IK_NR

Implements IKSolver class using the Newton-Raphson method

+
+
IK_GN

Implements IKSolver class using the Gauss-Newton method

+
+
IK_LM

Implements IKSolver class using the Levemberg-Marquadt method

+
+
+
+
+

Changed in version 1.0.3: Added the Quadratic Programming IK solver class

+
+
+ +

Methods

+ + + + + + + + + + + + +

step(ets, Tep, q)

Performs a single iteration of the Gauss-Newton optimisation method

solve(ets, Tep[, q0])

Solves the IK problem

error(Te, Tep)

Calculates the error between Te and Tep

+

Private Methods

+ + + + + + + + + +

_random_q(ets[, i])

Generate a random valid joint configuration using a private RNG

_check_jl(ets, q)

Checks if the joints are within their respective limits

+
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/iksolution.html b/IK/iksolution.html new file mode 100644 index 00000000..297dc567 --- /dev/null +++ b/IK/iksolution.html @@ -0,0 +1,204 @@ + + + + + + + + + + + IKSolution - as IK Solution dataclass — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

IKSolution - as IK Solution dataclass

+
+
+class roboticstoolbox.robot.IK.IKSolution(q, success, iterations=0, searches=0, residual=0.0, reason='')[source]
+

A dataclass for representing an IK solution

+
+
+q
+

The joint coordinates of the solution (ndarray). Note that these +will not be valid if failed to find a solution

+
+ +
+
+success
+

True if a valid solution was found

+
+ +
+
+iterations
+

How many iterations were performed

+
+ +
+
+searches
+

How many searches were performed

+
+ +
+
+residual
+

The final error value from the cost function

+
+ +
+
+reason
+

The reason the IK problem failed if applicable

+
+ +
+

Changed in version 1.0.3: Added IKSolution dataclass to replace the IKsolution named tuple

+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/iksolver.html b/IK/iksolver.html new file mode 100644 index 00000000..06631728 --- /dev/null +++ b/IK/iksolver.html @@ -0,0 +1,230 @@ + + + + + + + + + + + IKSolver - IK Solver Abstract Base Class — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

IKSolver - IK Solver Abstract Base Class

+
+
+class roboticstoolbox.robot.IK.IKSolver(name='IK Solver', ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, seed=None)[source]
+

Bases: ABC

+

An abstract super class for numerical inverse kinematics (IK)

+

This class provides basic functionality to perform numerical IK. Superclasses +can inherit this class and implement the solve method and redefine any other +methods necessary.

+
+
Parameters:
+
    +
  • name (str) – The name of the IK algorithm

  • +
  • ilimit (int) – How many iterations are allowed within a search before a new search +is started

  • +
  • slimit (int) – How many searches are allowed before being deemed unsuccessful

  • +
  • tol (float) – Maximum allowed residual error E

  • +
  • mask (Union[ndarray, List[float], Tuple[float], Set[float], None]) – A 6 vector which assigns weights to Cartesian degrees-of-freedom +error priority

  • +
  • joint_limits (bool) – Reject solutions with joint limit violations

  • +
  • seed (Optional[int]) – A seed for the private RNG used to generate random joint coordinate +vectors

  • +
+
+
+
+

See also

+
+
IK_NR

Implements this class using the Newton-Raphson method

+
+
IK_GN

Implements this class using the Gauss-Newton method

+
+
IK_LM

Implements this class using the Levemberg-Marquadt method

+
+
IK_QP

Implements this class using a quadratic programming approach

+
+
+
+
+

Changed in version 1.0.3: Added the abstract super class IKSolver

+
+
+ +

Methods

+ + + + + + + + + + + + +

step(ets, Tep, q)

Abstract step method

solve(ets, Tep[, q0])

Solves the IK problem

error(Te, Tep)

Calculates the error between Te and Tep

+

Private Methods

+ + + + + + + + + +

_random_q(ets[, i])

Generate a random valid joint configuration using a private RNG

_check_jl(ets, q)

Checks if the joints are within their respective limits

+
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.ERobot.ERobot.ik_gn.html b/IK/stubs/roboticstoolbox.robot.ERobot.ERobot.ik_gn.html new file mode 100644 index 00000000..37916ae8 --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.ERobot.ERobot.ik_gn.html @@ -0,0 +1,126 @@ + + + + + + + + + + + ERobot.ik_gn — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+
    +
  • + +
  • +
  • +
+
+
+
+
+ +
+

ERobot.ik_gn

+
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.ERobot.ERobot.ik_lm_chan.html b/IK/stubs/roboticstoolbox.robot.ERobot.ERobot.ik_lm_chan.html new file mode 100644 index 00000000..2204a12b --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.ERobot.ERobot.ik_lm_chan.html @@ -0,0 +1,126 @@ + + + + + + + + + + + ERobot.ik_lm_chan — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+
    +
  • + +
  • +
  • +
+
+
+
+
+ +
+

ERobot.ik_lm_chan

+
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.ERobot.ERobot.ik_lm_sugihara.html b/IK/stubs/roboticstoolbox.robot.ERobot.ERobot.ik_lm_sugihara.html new file mode 100644 index 00000000..6b114664 --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.ERobot.ERobot.ik_lm_sugihara.html @@ -0,0 +1,126 @@ + + + + + + + + + + + ERobot.ik_lm_sugihara — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+
    +
  • + +
  • +
  • +
+
+
+
+
+ +
+

ERobot.ik_lm_sugihara

+
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.ERobot.ERobot.ik_lm_wampler.html b/IK/stubs/roboticstoolbox.robot.ERobot.ERobot.ik_lm_wampler.html new file mode 100644 index 00000000..61340bf9 --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.ERobot.ERobot.ik_lm_wampler.html @@ -0,0 +1,126 @@ + + + + + + + + + + + ERobot.ik_lm_wampler — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+
    +
  • + +
  • +
  • +
+
+
+
+
+ +
+

ERobot.ik_lm_wampler

+
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.ERobot.ERobot.ik_nr.html b/IK/stubs/roboticstoolbox.robot.ERobot.ERobot.ik_nr.html new file mode 100644 index 00000000..fe7af5ed --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.ERobot.ERobot.ik_nr.html @@ -0,0 +1,126 @@ + + + + + + + + + + + ERobot.ik_nr — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+
    +
  • + +
  • +
  • +
+
+
+
+
+ +
+

ERobot.ik_nr

+
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.ERobot.ERobot.ikine_LM.html b/IK/stubs/roboticstoolbox.robot.ERobot.ERobot.ikine_LM.html new file mode 100644 index 00000000..54755238 --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.ERobot.ERobot.ikine_LM.html @@ -0,0 +1,253 @@ + + + + + + + + + + + ERobot.ikine_LM — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+
    +
  • + +
  • +
  • +
+
+
+
+
+ +
+

ERobot.ikine_LM

+
+
+ERobot.ikine_LM(Tep, end=None, start=None, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, seed=None, k=1.0, method='chan', kq=0.0, km=0.0, ps=0.0, pi=0.3, **kwargs)
+

Levenberg-Marquadt Numerical Inverse Kinematics Solver

+

A method which provides functionality to perform numerical inverse kinematics (IK) +using the Levemberg-Marquadt method.

+

See the Inverse Kinematics Docs Page for more details and for a +tutorial on numerical IK, see here.

+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose

  • +
  • end (Union[str, Link, Gripper, None]) – the link considered as the end-effector

  • +
  • start (Union[str, Link, Gripper, None]) – the link considered as the base frame, defaults to the robots’s base frame

  • +
  • q0 (Union[ndarray, List[float], Tuple[float], Set[float], None]) – The initial joint coordinate vector

  • +
  • ilimit (int) – How many iterations are allowed within a search before a new search +is started

  • +
  • slimit (int) – How many searches are allowed before being deemed unsuccessful

  • +
  • tol (float) – Maximum allowed residual error E

  • +
  • mask (Union[ndarray, List[float], Tuple[float], Set[float], None]) – A 6 vector which assigns weights to Cartesian degrees-of-freedom +error priority

  • +
  • joint_limits (bool) – Reject solutions with joint limit violations

  • +
  • seed (Optional[int]) – A seed for the private RNG used to generate random joint coordinate +vectors

  • +
  • k (float) – Sets the gain value for the damping matrix Wn in the next iteration. See +synopsis

  • +
  • method (Literal['chan', 'wampler', 'sugihara']) – One of “chan”, “sugihara” or “wampler”. Defines which method is used +to calculate the damping matrix Wn in the step method

  • +
  • kq (float) – The gain for joint limit avoidance. Setting to 0.0 will remove this +completely from the solution

  • +
  • km (float) – The gain for maximisation. Setting to 0.0 will remove this completely +from the solution

  • +
  • ps (float) – The minimum angle/distance (in radians or metres) in which the joint is +allowed to approach to its limit

  • +
  • pi (Union[ndarray, float]) – The influence angle/distance (in radians or metres) in null space motion +becomes active

  • +
+
+
+

Synopsis

+

The operation is defined by the choice of the method kwarg.

+

The step is deined as

+
+\[\begin{split}\vec{q}_{k+1} +&= +\vec{q}_k + +\left( + \mat{A}_k +\right)^{-1} +\bf{g}_k \\ +% +\mat{A}_k +&= +{\mat{J}(\vec{q}_k)}^\top +\mat{W}_e \ +{\mat{J}(\vec{q}_k)} ++ +\mat{W}_n\end{split}\]
+

where \(\mat{W}_n = \text{diag}(\vec{w_n})(\vec{w_n} \in \mathbb{R}^n_{>0})\) is a +diagonal damping matrix. The damping matrix ensures that \(\mat{A}_k\) is +non-singular and positive definite. The performance of the LM method largely depends +on the choice of \(\mat{W}_n\).

+

Chan’s Method

+

Chan proposed

+
+\[\mat{W}_n += +λ E_k \mat{1}_n\]
+

where λ is a constant which reportedly does not have much influence on performance. +Use the kwarg k to adjust the weighting term λ.

+

Sugihara’s Method

+

Sugihara proposed

+
+\[\mat{W}_n += +E_k \mat{1}_n + \text{diag}(\hat{\vec{w}}_n)\]
+

where \(\hat{\vec{w}}_n \in \mathbb{R}^n\), \(\hat{w}_{n_i} = l^2 \sim 0.01 l^2\), +and \(l\) is the length of a typical link within the manipulator. We provide the +variable k as a kwarg to adjust the value of \(w_n\).

+

Wampler’s Method

+

Wampler proposed \(\vec{w_n}\) to be a constant. This is set through the k kwarg.

+

Examples

+

The following example makes a panda robot object, makes a goal +pose Tep, and then solves for the joint coordinates which result in the pose +Tep using the ikine_LM method.

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda()
+>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> panda.ikine_LM(Tep)
+IKSolution(q=array([-1.3026, -0.7154,  1.0297, -2.1574,  0.6525,  1.7953,  0.2787]), success=True, iterations=12, searches=1, residual=5.201036387515119e-10, reason='Success')
+
+
+

Notes

+

The value for the k kwarg will depend on the method chosen and the arm you are +using. Use the following as a rough guide chan, k = 1.0 - 0.01, +wampler, k = 0.01 - 0.0001, and sugihara, k = 0.1 - 0.0001

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian.

+

This class supports null-space motion to assist with maximising manipulability and +avoiding joint limits. These are enabled by setting kq and km to non-zero values.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
IK_LM

An IK Solver class which implements the Levemberg Marquadt optimisation technique

+
+
ikine_NR

Implements the IK_NR class as a method within the Robot class

+
+
ikine_GN

Implements the IK_GN class as a method within the Robot class

+
+
ikine_QP

Implements the IK_QP class as a method within the Robot class

+
+
+
+
+

Changed in version 1.0.4: Added the Levemberg-Marquadt IK solver method on the Robot class

+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_GN.html b/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_GN.html new file mode 100644 index 00000000..03c40155 --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_GN.html @@ -0,0 +1,258 @@ + + + + + + + + + + + ETS.ik_GN — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

ETS.ik_GN

+
+
+ETS.ik_GN(Tep, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, pinv=True, pinv_damping=0.0)[source]
+

Fast numerical inverse kinematics by Gauss-Newton optimization

+

sol = ets.ik_GN(Tep) are the joint coordinates (n) corresponding +to the robot end-effector pose Tep which is an SE3 or ndarray object. +This method can be used for robots with any number of degrees of freedom. This +is a fast solver implemented in C++.

+

See the Inverse Kinematics Docs Page for more details and for a +tutorial on numerical IK, see here.

+
+

Note

+

When using this method with redundant robots (>6 DoF), pinv must be set to True

+
+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose or pose trajectory

  • +
  • q0 (Optional[ndarray]) – initial joint configuration (default to random valid joint +configuration contrained by the joint limits of the robot)

  • +
  • ilimit (int) – maximum number of iterations per search

  • +
  • slimit (int) – maximum number of search attempts

  • +
  • tol (float) – final error tolerance

  • +
  • mask (Optional[ndarray]) – a mask vector which weights the end-effector error priority. +Corresponds to translation in X, Y and Z and rotation about X, Y and Z +respectively

  • +
  • joint_limits (bool) – constrain the solution to being within the joint limits of +the robot (reject solution with invalid joint configurations and perfrom +another search up to the slimit)

  • +
  • pinv (int) – Use the psuedo-inverse instad of the normal matrix inverse

  • +
  • pinv_damping (float) – Damping factor for the psuedo-inverse

  • +
+
+
Return type:
+

Tuple[ndarray, int, int, int, float]

+
+
Returns:
+

    +
  • sol – tuple (q, success, iterations, searches, residual)

  • +
  • The return value sol is a tuple with elements

  • +
  • ============ ========== ===============================================

  • +
  • Element Type Description

  • +
  • ============ ========== ===============================================

  • +
  • q ndarray(n) joint coordinates in units of radians or metres

  • +
  • success int whether a solution was found

  • +
  • iterations int total number of iterations

  • +
  • searches int total number of searches

  • +
  • residual float final value of cost function

  • +
  • ============ ========== ===============================================

  • +
  • If success == 0 the q values will be valid numbers, but the

  • +
  • solution will be in error. The amount of error is indicated by

  • +
  • the residual.

  • +
+

+
+
+

Synopsis

+

Each iteration uses the Gauss-Newton optimisation method

+
+\[\begin{split}\vec{q}_{k+1} &= \vec{q}_k + +\left( +{\mat{J}(\vec{q}_k)}^\top +\mat{W}_e \ +{\mat{J}(\vec{q}_k)} +\right)^{-1} +\bf{g}_k \\ +\bf{g}_k &= +{\mat{J}(\vec{q}_k)}^\top +\mat{W}_e +\vec{e}_k\end{split}\]
+

where \(\mat{J} = {^0\mat{J}}\) is the base-frame manipulator Jacobian. If +\(\mat{J}(\vec{q}_k)\) is non-singular, and \(\mat{W}_e = \mat{1}_n\), then +the above provides the pseudoinverse solution. However, if \(\mat{J}(\vec{q}_k)\) +is singular, the above can not be computed and the GN solution is infeasible.

+

Examples

+

The following example gets the ets of a panda robot object, makes a goal +pose Tep, and then solves for the joint coordinates which result in the pose +Tep using the ikine_GN method.

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda().ets()
+>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> panda.ik_GN(Tep)
+(array([-1.0805, -0.5328,  0.9086, -2.1781,  0.4612,  1.9018,  0.4239]), 1, 510, 32, 2.803306327008683e-09)
+
+
+

Notes

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
ik_NR

A fast numerical inverse kinematics solver using Newton-Raphson optimisation

+
+
ik_GN

A fast numerical inverse kinematics solver using Gauss-Newton optimisation

+
+
+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_LM.html b/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_LM.html new file mode 100644 index 00000000..407ad04d --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_LM.html @@ -0,0 +1,267 @@ + + + + + + + + + + + ETS.ik_LM — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

ETS.ik_LM

+
+
+ETS.ik_LM(Tep, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, k=1.0, method='chan')[source]
+

Fast levenberg-Marquadt Numerical Inverse Kinematics Solver

+

A method which provides functionality to perform numerical inverse kinematics (IK) +using the Levemberg-Marquadt method. This +is a fast solver implemented in C++.

+

See the Inverse Kinematics Docs Page for more details and for a +tutorial on numerical IK, see here.

+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose

  • +
  • q0 (Optional[ndarray]) – The initial joint coordinate vector

  • +
  • ilimit (int) – How many iterations are allowed within a search before a new search +is started

  • +
  • slimit (int) – How many searches are allowed before being deemed unsuccessful

  • +
  • tol (float) – Maximum allowed residual error E

  • +
  • mask (Optional[ndarray]) – A 6 vector which assigns weights to Cartesian degrees-of-freedom +error priority

  • +
  • joint_limits (bool) – Reject solutions with joint limit violations

  • +
  • seed – A seed for the private RNG used to generate random joint coordinate +vectors

  • +
  • k (float) – Sets the gain value for the damping matrix Wn in the next iteration. See +synopsis

  • +
  • method (Literal['chan', 'wampler', 'sugihara']) – One of “chan”, “sugihara” or “wampler”. Defines which method is used +to calculate the damping matrix Wn in the step method

  • +
+
+
Return type:
+

Tuple[ndarray, int, int, int, float]

+
+
+

Synopsis

+

The operation is defined by the choice of the method kwarg.

+

The step is deined as

+
+\[\begin{split}\vec{q}_{k+1} +&= +\vec{q}_k + +\left( + \mat{A}_k +\right)^{-1} +\bf{g}_k \\ +% +\mat{A}_k +&= +{\mat{J}(\vec{q}_k)}^\top +\mat{W}_e \ +{\mat{J}(\vec{q}_k)} ++ +\mat{W}_n\end{split}\]
+

where \(\mat{W}_n = \text{diag}(\vec{w_n})(\vec{w_n} \in \mathbb{R}^n_{>0})\) is a +diagonal damping matrix. The damping matrix ensures that \(\mat{A}_k\) is +non-singular and positive definite. The performance of the LM method largely depends +on the choice of \(\mat{W}_n\).

+

Chan’s Method

+

Chan proposed

+
+\[\mat{W}_n += +λ E_k \mat{1}_n\]
+

where λ is a constant which reportedly does not have much influence on performance. +Use the kwarg k to adjust the weighting term λ.

+

Sugihara’s Method

+

Sugihara proposed

+
+\[\mat{W}_n += +E_k \mat{1}_n + \text{diag}(\hat{\vec{w}}_n)\]
+

where \(\hat{\vec{w}}_n \in \mathbb{R}^n\), \(\hat{w}_{n_i} = l^2 \sim 0.01 l^2\), +and \(l\) is the length of a typical link within the manipulator. We provide the +variable k as a kwarg to adjust the value of \(w_n\).

+

Wampler’s Method

+

Wampler proposed \(\vec{w_n}\) to be a constant. This is set through the k kwarg.

+

Examples

+

The following example gets the ets of a panda robot object, makes a goal +pose Tep, and then solves for the joint coordinates which result in the pose +Tep using the ikine_LM method.

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda().ets()
+>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> panda.ikine_LM(Tep)
+IKSolution(q=array([-1.8981,  0.6575,  2.1381, -2.1641, -0.5958,  1.8315,  1.2497]), success=True, iterations=43, searches=4, residual=1.9606232723378817e-09, reason='Success')
+
+
+

Notes

+

The value for the k kwarg will depend on the method chosen and the arm you are +using. Use the following as a rough guide chan, k = 1.0 - 0.01, +wampler, k = 0.01 - 0.0001, and sugihara, k = 0.1 - 0.0001

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian.

+

This class supports null-space motion to assist with maximising manipulability and +avoiding joint limits. These are enabled by setting kq and km to non-zero values.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
ik_NR

A fast numerical inverse kinematics solver using Newton-Raphson optimisation

+
+
ik_GN

A fast numerical inverse kinematics solver using Gauss-Newton optimisation

+
+
+
+
+

Changed in version 1.0.4: Merged the Levemberg-Marquadt IK solvers into the ik_LM method

+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_NR.html b/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_NR.html new file mode 100644 index 00000000..e03579e5 --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_NR.html @@ -0,0 +1,244 @@ + + + + + + + + + + + ETS.ik_NR — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

ETS.ik_NR

+
+
+ETS.ik_NR(Tep, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, pinv=True, pinv_damping=0.0)[source]
+

Fast numerical inverse kinematics using Newton-Raphson optimization

+

sol = ets.ik_NR(Tep) are the joint coordinates (n) corresponding +to the robot end-effector pose Tep which is an SE3 or ndarray object. +This method can be used for robots with any number of degrees of freedom. This +is a fast solver implemented in C++.

+

See the Inverse Kinematics Docs Page for more details and for a +tutorial on numerical IK, see here.

+
+

Note

+

When using this method with redundant robots (>6 DoF), pinv must be set to True

+
+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose or pose trajectory

  • +
  • q0 (Optional[ndarray]) – initial joint configuration (default to random valid joint +configuration contrained by the joint limits of the robot)

  • +
  • ilimit (int) – maximum number of iterations per search

  • +
  • slimit (int) – maximum number of search attempts

  • +
  • tol (float) – final error tolerance

  • +
  • mask (Optional[ndarray]) – a mask vector which weights the end-effector error priority. +Corresponds to translation in X, Y and Z and rotation about X, Y and Z +respectively

  • +
  • joint_limits (bool) – constrain the solution to being within the joint limits of +the robot (reject solution with invalid joint configurations and perfrom +another search up to the slimit)

  • +
  • pinv (int) – Use the psuedo-inverse instad of the normal matrix inverse

  • +
  • pinv_damping (float) – Damping factor for the psuedo-inverse

  • +
+
+
Return type:
+

Tuple[ndarray, int, int, int, float]

+
+
Returns:
+

    +
  • sol – tuple (q, success, iterations, searches, residual)

  • +
  • The return value sol is a tuple with elements

  • +
  • ============ ========== ===============================================

  • +
  • Element Type Description

  • +
  • ============ ========== ===============================================

  • +
  • q ndarray(n) joint coordinates in units of radians or metres

  • +
  • success int whether a solution was found

  • +
  • iterations int total number of iterations

  • +
  • searches int total number of searches

  • +
  • residual float final value of cost function

  • +
  • ============ ========== ===============================================

  • +
  • If success == 0 the q values will be valid numbers, but the

  • +
  • solution will be in error. The amount of error is indicated by

  • +
  • the residual.

  • +
+

+
+
+

Synopsis

+

Each iteration uses the Newton-Raphson optimisation method

+
+\[\vec{q}_{k+1} = \vec{q}_k + {^0\mat{J}(\vec{q}_k)}^{-1} \vec{e}_k\]
+

Examples

+

The following example gets the ets of a panda robot object, makes a goal +pose Tep, and then solves for the joint coordinates which result in the pose +Tep using the ikine_GN method.

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda().ets()
+>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> panda.ik_NR(Tep)
+(array([-1.0805, -0.5328,  0.9086, -2.1781,  0.4612,  1.9018,  0.4239]), 1, 489, 32, 2.8033063270234757e-09)
+
+
+

Notes

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
ik_LM

A fast numerical inverse kinematics solver using Levenberg-Marquadt optimisation

+
+
ik_GN

A fast numerical inverse kinematics solver using Gauss-Newton optimisation

+
+
+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_gn.html b/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_gn.html new file mode 100644 index 00000000..c810d493 --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_gn.html @@ -0,0 +1,234 @@ + + + + + + + + + + + ETS.ik_GN — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+
    +
  • + +
  • +
  • +
+
+
+
+
+ +
+

ETS.ik_GN

+
+
+ETS.ik_GN(Tep, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, pinv=True, pinv_damping=0.0)[source]
+

Fast numerical inverse kinematics by Gauss-Newton optimization

+

sol = ets.ik_GN(Tep) are the joint coordinates (n) corresponding +to the robot end-effector pose Tep which is an SE3 or ndarray object. +This method can be used for robots with any number of degrees of freedom. This +is a fast solver implemented in C++.

+

See the Inverse Kinematics Docs Page for more details and for a +tutorial on numerical IK, see here.

+
+

Note

+

When using this method with redundant robots (>6 DoF), pinv must be set to True

+
+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose or pose trajectory

  • +
  • q0 (Optional[ndarray]) – initial joint configuration (default to random valid joint +configuration contrained by the joint limits of the robot)

  • +
  • ilimit (int) – maximum number of iterations per search

  • +
  • slimit (int) – maximum number of search attempts

  • +
  • tol (float) – final error tolerance

  • +
  • mask (Optional[ndarray]) – a mask vector which weights the end-effector error priority. +Corresponds to translation in X, Y and Z and rotation about X, Y and Z +respectively

  • +
  • joint_limits (bool) – constrain the solution to being within the joint limits of +the robot (reject solution with invalid joint configurations and perfrom +another search up to the slimit)

  • +
  • pinv (int) – Use the psuedo-inverse instad of the normal matrix inverse

  • +
  • pinv_damping (float) – Damping factor for the psuedo-inverse

  • +
+
+
Return type:
+

Tuple[ndarray, int, int, int, float]

+
+
Returns:
+

    +
  • sol – tuple (q, success, iterations, searches, residual)

  • +
  • The return value sol is a tuple with elements

  • +
  • ============ ========== ===============================================

  • +
  • Element Type Description

  • +
  • ============ ========== ===============================================

  • +
  • q ndarray(n) joint coordinates in units of radians or metres

  • +
  • success int whether a solution was found

  • +
  • iterations int total number of iterations

  • +
  • searches int total number of searches

  • +
  • residual float final value of cost function

  • +
  • ============ ========== ===============================================

  • +
  • If success == 0 the q values will be valid numbers, but the

  • +
  • solution will be in error. The amount of error is indicated by

  • +
  • the residual.

  • +
+

+
+
+

Synopsis

+

Each iteration uses the Gauss-Newton optimisation method

+
+\[\begin{split}\vec{q}_{k+1} &= \vec{q}_k + +\left( +{\mat{J}(\vec{q}_k)}^\top +\mat{W}_e \ +{\mat{J}(\vec{q}_k)} +\right)^{-1} +\bf{g}_k \\ +\bf{g}_k &= +{\mat{J}(\vec{q}_k)}^\top +\mat{W}_e +\vec{e}_k\end{split}\]
+

where \(\mat{J} = {^0\mat{J}}\) is the base-frame manipulator Jacobian. If +\(\mat{J}(\vec{q}_k)\) is non-singular, and \(\mat{W}_e = \mat{1}_n\), then +the above provides the pseudoinverse solution. However, if \(\mat{J}(\vec{q}_k)\) +is singular, the above can not be computed and the GN solution is infeasible.

+

Examples

+

The following example gets the ets of a panda robot object, makes a goal +pose Tep, and then solves for the joint coordinates which result in the pose +Tep using the ikine_GN method.

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda().ets()
+>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> panda.ik_GN(Tep)
+(array([-1.0805, -0.5328,  0.9086, -2.1781,  0.4612,  1.9018,  0.4239]), 1, 510, 32, 2.803306327008683e-09)
+
+
+

Notes

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
ik_NR

A fast numerical inverse kinematics solver using Newton-Raphson optimisation

+
+
ik_GN

A fast numerical inverse kinematics solver using Gauss-Newton optimisation

+
+
+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_lm_chan.html b/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_lm_chan.html new file mode 100644 index 00000000..3b7cf1e1 --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_lm_chan.html @@ -0,0 +1,126 @@ + + + + + + + + + + + ETS.ik_lm_chan — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+
    +
  • + +
  • +
  • +
+
+
+
+
+ +
+

ETS.ik_lm_chan

+
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_lm_sugihara.html b/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_lm_sugihara.html new file mode 100644 index 00000000..eccfe232 --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_lm_sugihara.html @@ -0,0 +1,126 @@ + + + + + + + + + + + ETS.ik_lm_sugihara — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+
    +
  • + +
  • +
  • +
+
+
+
+
+ +
+

ETS.ik_lm_sugihara

+
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_lm_wampler.html b/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_lm_wampler.html new file mode 100644 index 00000000..e047f761 --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_lm_wampler.html @@ -0,0 +1,126 @@ + + + + + + + + + + + ETS.ik_lm_wampler — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+
    +
  • + +
  • +
  • +
+
+
+
+
+ +
+

ETS.ik_lm_wampler

+
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_nr.html b/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_nr.html new file mode 100644 index 00000000..c6592864 --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_nr.html @@ -0,0 +1,220 @@ + + + + + + + + + + + ETS.ik_NR — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+
    +
  • + +
  • +
  • +
+
+
+
+
+ +
+

ETS.ik_NR

+
+
+ETS.ik_NR(Tep, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, pinv=True, pinv_damping=0.0)[source]
+

Fast numerical inverse kinematics using Newton-Raphson optimization

+

sol = ets.ik_NR(Tep) are the joint coordinates (n) corresponding +to the robot end-effector pose Tep which is an SE3 or ndarray object. +This method can be used for robots with any number of degrees of freedom. This +is a fast solver implemented in C++.

+

See the Inverse Kinematics Docs Page for more details and for a +tutorial on numerical IK, see here.

+
+

Note

+

When using this method with redundant robots (>6 DoF), pinv must be set to True

+
+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose or pose trajectory

  • +
  • q0 (Optional[ndarray]) – initial joint configuration (default to random valid joint +configuration contrained by the joint limits of the robot)

  • +
  • ilimit (int) – maximum number of iterations per search

  • +
  • slimit (int) – maximum number of search attempts

  • +
  • tol (float) – final error tolerance

  • +
  • mask (Optional[ndarray]) – a mask vector which weights the end-effector error priority. +Corresponds to translation in X, Y and Z and rotation about X, Y and Z +respectively

  • +
  • joint_limits (bool) – constrain the solution to being within the joint limits of +the robot (reject solution with invalid joint configurations and perfrom +another search up to the slimit)

  • +
  • pinv (int) – Use the psuedo-inverse instad of the normal matrix inverse

  • +
  • pinv_damping (float) – Damping factor for the psuedo-inverse

  • +
+
+
Return type:
+

Tuple[ndarray, int, int, int, float]

+
+
Returns:
+

    +
  • sol – tuple (q, success, iterations, searches, residual)

  • +
  • The return value sol is a tuple with elements

  • +
  • ============ ========== ===============================================

  • +
  • Element Type Description

  • +
  • ============ ========== ===============================================

  • +
  • q ndarray(n) joint coordinates in units of radians or metres

  • +
  • success int whether a solution was found

  • +
  • iterations int total number of iterations

  • +
  • searches int total number of searches

  • +
  • residual float final value of cost function

  • +
  • ============ ========== ===============================================

  • +
  • If success == 0 the q values will be valid numbers, but the

  • +
  • solution will be in error. The amount of error is indicated by

  • +
  • the residual.

  • +
+

+
+
+

Synopsis

+

Each iteration uses the Newton-Raphson optimisation method

+
+\[\vec{q}_{k+1} = \vec{q}_k + {^0\mat{J}(\vec{q}_k)}^{-1} \vec{e}_k\]
+

Examples

+

The following example gets the ets of a panda robot object, makes a goal +pose Tep, and then solves for the joint coordinates which result in the pose +Tep using the ikine_GN method.

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda().ets()
+>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> panda.ik_NR(Tep)
+(array([-1.0805, -0.5328,  0.9086, -2.1781,  0.4612,  1.9018,  0.4239]), 1, 489, 32, 2.8033063270234757e-09)
+
+
+

Notes

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
ik_LM

A fast numerical inverse kinematics solver using Levenberg-Marquadt optimisation

+
+
ik_GN

A fast numerical inverse kinematics solver using Gauss-Newton optimisation

+
+
+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.ETS.ETS.ikine_GN.html b/IK/stubs/roboticstoolbox.robot.ETS.ETS.ikine_GN.html new file mode 100644 index 00000000..df8441af --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.ETS.ETS.ikine_GN.html @@ -0,0 +1,258 @@ + + + + + + + + + + + ETS.ikine_GN — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

ETS.ikine_GN

+
+
+ETS.ikine_GN(Tep, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, seed=None, pinv=False, kq=0.0, km=0.0, ps=0.0, pi=0.3, **kwargs)[source]
+

Gauss-Newton Numerical Inverse Kinematics Solver

+

A method which provides functionality to perform numerical inverse kinematics (IK) +using the Gauss-Newton method.

+

See the Inverse Kinematics Docs Page for more details and for a +tutorial on numerical IK, see here.

+
+

Note

+

When using this method with redundant robots (>6 DoF), pinv must be set to True

+
+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose

  • +
  • q0 (Union[ndarray, List[float], Tuple[float], Set[float], None]) – The initial joint coordinate vector

  • +
  • ilimit (int) – How many iterations are allowed within a search before a new search +is started

  • +
  • slimit (int) – How many searches are allowed before being deemed unsuccessful

  • +
  • tol (float) – Maximum allowed residual error E

  • +
  • mask (Union[ndarray, List[float], Tuple[float], Set[float], None]) – A 6 vector which assigns weights to Cartesian degrees-of-freedom +error priority

  • +
  • joint_limits (bool) – Reject solutions with joint limit violations

  • +
  • seed (Optional[int]) – A seed for the private RNG used to generate random joint coordinate +vectors

  • +
  • pinv (bool) – If True, will use the psuedoinverse in the step method instead of +the normal inverse

  • +
  • kq (float) – The gain for joint limit avoidance. Setting to 0.0 will remove this +completely from the solution

  • +
  • km (float) – The gain for maximisation. Setting to 0.0 will remove this completely +from the solution

  • +
  • ps (float) – The minimum angle/distance (in radians or metres) in which the joint is +allowed to approach to its limit

  • +
  • pi (Union[ndarray, float]) – The influence angle/distance (in radians or metres) in null space motion +becomes active

  • +
+
+
+

Synopsis

+

Each iteration uses the Gauss-Newton optimisation method

+
+\[\begin{split}\vec{q}_{k+1} &= \vec{q}_k + +\left( +{\mat{J}(\vec{q}_k)}^\top +\mat{W}_e \ +{\mat{J}(\vec{q}_k)} +\right)^{-1} +\bf{g}_k \\ +\bf{g}_k &= +{\mat{J}(\vec{q}_k)}^\top +\mat{W}_e +\vec{e}_k\end{split}\]
+

where \(\mat{J} = {^0\mat{J}}\) is the base-frame manipulator Jacobian. If +\(\mat{J}(\vec{q}_k)\) is non-singular, and \(\mat{W}_e = \mat{1}_n\), then +the above provides the pseudoinverse solution. However, if \(\mat{J}(\vec{q}_k)\) +is singular, the above can not be computed and the GN solution is infeasible.

+

Examples

+

The following example gets the ets of a panda robot object, makes a goal +pose Tep, and then solves for the joint coordinates which result in the pose +Tep using the ikine_GN method.

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda().ets()
+>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> panda.ikine_GN(Tep)
+IKSolution(q=array([-0.8553, -0.5978,  2.3022, -1.3476, -2.6294,  2.498 ,  0.9897]), success=False, iterations=100, searches=100, residual=0.0, reason='iteration and search limit reached, 100 numpy.LinAlgError encountered')
+
+
+

Notes

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian.

+

This class supports null-space motion to assist with maximising manipulability and +avoiding joint limits. These are enabled by setting kq and km to non-zero values.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
IK_NR

An IK Solver class which implements the Newton-Raphson optimisation technique

+
+
ikine_LM

Implements the IK_LM class as a method within the ETS class

+
+
ikine_NR

Implements the IK_NR class as a method within the ETS class

+
+
ikine_QP

Implements the IK_QP class as a method within the ETS class

+
+
+
+
+

Changed in version 1.0.4: Added the Gauss-Newton IK solver method on the ETS class

+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.ETS.ETS.ikine_LM.html b/IK/stubs/roboticstoolbox.robot.ETS.ETS.ikine_LM.html new file mode 100644 index 00000000..1a12acf1 --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.ETS.ETS.ikine_LM.html @@ -0,0 +1,283 @@ + + + + + + + + + + + ETS.ikine_LM — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

ETS.ikine_LM

+
+
+ETS.ikine_LM(Tep, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, seed=None, k=1.0, method='chan', kq=0.0, km=0.0, ps=0.0, pi=0.3, **kwargs)[source]
+

Levemberg-Marquadt Numerical Inverse Kinematics Solver

+

A method which provides functionality to perform numerical inverse kinematics (IK) +using the Levemberg-Marquadt method.

+

See the Inverse Kinematics Docs Page for more details and for a +tutorial on numerical IK, see here.

+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose

  • +
  • q0 (Union[ndarray, List[float], Tuple[float], Set[float], None]) – The initial joint coordinate vector

  • +
  • ilimit (int) – How many iterations are allowed within a search before a new search +is started

  • +
  • slimit (int) – How many searches are allowed before being deemed unsuccessful

  • +
  • tol (float) – Maximum allowed residual error E

  • +
  • mask (Union[ndarray, List[float], Tuple[float], Set[float], None]) – A 6 vector which assigns weights to Cartesian degrees-of-freedom +error priority

  • +
  • joint_limits (bool) – Reject solutions with joint limit violations

  • +
  • seed (Optional[int]) – A seed for the private RNG used to generate random joint coordinate +vectors

  • +
  • k (float) – Sets the gain value for the damping matrix Wn in the next iteration. See +synopsis

  • +
  • method (Literal['chan', 'wampler', 'sugihara']) – One of “chan”, “sugihara” or “wampler”. Defines which method is used +to calculate the damping matrix Wn in the step method

  • +
  • kq (float) – The gain for joint limit avoidance. Setting to 0.0 will remove this +completely from the solution

  • +
  • km (float) – The gain for maximisation. Setting to 0.0 will remove this completely +from the solution

  • +
  • ps (float) – The minimum angle/distance (in radians or metres) in which the joint is +allowed to approach to its limit

  • +
  • pi (Union[ndarray, float]) – The influence angle/distance (in radians or metres) in null space motion +becomes active

  • +
+
+
+

Synopsis

+

The operation is defined by the choice of the method kwarg.

+

The step is deined as

+
+\[\begin{split}\vec{q}_{k+1} +&= +\vec{q}_k + +\left( + \mat{A}_k +\right)^{-1} +\bf{g}_k \\ +% +\mat{A}_k +&= +{\mat{J}(\vec{q}_k)}^\top +\mat{W}_e \ +{\mat{J}(\vec{q}_k)} ++ +\mat{W}_n\end{split}\]
+

where \(\mat{W}_n = \text{diag}(\vec{w_n})(\vec{w_n} \in \mathbb{R}^n_{>0})\) is a +diagonal damping matrix. The damping matrix ensures that \(\mat{A}_k\) is +non-singular and positive definite. The performance of the LM method largely depends +on the choice of \(\mat{W}_n\).

+

Chan’s Method

+

Chan proposed

+
+\[\mat{W}_n += +λ E_k \mat{1}_n\]
+

where λ is a constant which reportedly does not have much influence on performance. +Use the kwarg k to adjust the weighting term λ.

+

Sugihara’s Method

+

Sugihara proposed

+
+\[\mat{W}_n += +E_k \mat{1}_n + \text{diag}(\hat{\vec{w}}_n)\]
+

where \(\hat{\vec{w}}_n \in \mathbb{R}^n\), \(\hat{w}_{n_i} = l^2 \sim 0.01 l^2\), +and \(l\) is the length of a typical link within the manipulator. We provide the +variable k as a kwarg to adjust the value of \(w_n\).

+

Wampler’s Method

+

Wampler proposed \(\vec{w_n}\) to be a constant. This is set through the k kwarg.

+

Examples

+

The following example gets the ets of a panda robot object, makes a goal +pose Tep, and then solves for the joint coordinates which result in the pose +Tep using the ikine_LM method.

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda().ets()
+>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> panda.ikine_LM(Tep)
+IKSolution(q=array([-1.0134, -0.4938,  0.8625, -2.1822,  0.4142,  1.9214,  0.4602]), success=True, iterations=8, searches=1, residual=1.871393860120283e-08, reason='Success')
+
+
+

Notes

+

The value for the k kwarg will depend on the method chosen and the arm you are +using. Use the following as a rough guide chan, k = 1.0 - 0.01, +wampler, k = 0.01 - 0.0001, and sugihara, k = 0.1 - 0.0001

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian.

+

This class supports null-space motion to assist with maximising manipulability and +avoiding joint limits. These are enabled by setting kq and km to non-zero values.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
IK_LM

An IK Solver class which implements the Levemberg Marquadt optimisation technique

+
+
ikine_NR

Implements the IK_NR class as a method within the ETS class

+
+
ikine_GN

Implements the IK_GN class as a method within the ETS class

+
+
ikine_QP

Implements the IK_QP class as a method within the ETS class

+
+
+
+
+

Changed in version 1.0.4: Added the Levemberg-Marquadt IK solver method on the ETS class

+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.ETS.ETS.ikine_NR.html b/IK/stubs/roboticstoolbox.robot.ETS.ETS.ikine_NR.html new file mode 100644 index 00000000..619a0fe2 --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.ETS.ETS.ikine_NR.html @@ -0,0 +1,244 @@ + + + + + + + + + + + ETS.ikine_NR — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

ETS.ikine_NR

+
+
+ETS.ikine_NR(Tep, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, seed=None, pinv=False, kq=0.0, km=0.0, ps=0.0, pi=0.3, **kwargs)[source]
+

Newton-Raphson Numerical Inverse Kinematics Solver

+

A method which provides functionality to perform numerical inverse kinematics (IK) +using the Newton-Raphson method.

+

See the Inverse Kinematics Docs Page for more details and for a +tutorial on numerical IK, see here.

+
+

Note

+

When using this method with redundant robots (>6 DoF), pinv must be set to True

+
+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose

  • +
  • q0 (Union[ndarray, List[float], Tuple[float], Set[float], None]) – The initial joint coordinate vector

  • +
  • ilimit (int) – How many iterations are allowed within a search before a new search +is started

  • +
  • slimit (int) – How many searches are allowed before being deemed unsuccessful

  • +
  • tol (float) – Maximum allowed residual error E

  • +
  • mask (Union[ndarray, List[float], Tuple[float], Set[float], None]) – A 6 vector which assigns weights to Cartesian degrees-of-freedom +error priority

  • +
  • joint_limits (bool) – Reject solutions with joint limit violations

  • +
  • seed (Optional[int]) – A seed for the private RNG used to generate random joint coordinate +vectors

  • +
  • pinv (bool) – If True, will use the psuedoinverse in the step method instead of +the normal inverse

  • +
  • kq (float) – The gain for joint limit avoidance. Setting to 0.0 will remove this +completely from the solution

  • +
  • km (float) – The gain for maximisation. Setting to 0.0 will remove this completely +from the solution

  • +
  • ps (float) – The minimum angle/distance (in radians or metres) in which the joint is +allowed to approach to its limit

  • +
  • pi (Union[ndarray, float]) – The influence angle/distance (in radians or metres) in null space motion +becomes active

  • +
+
+
+

Synopsis

+

Each iteration uses the Newton-Raphson optimisation method

+
+\[\vec{q}_{k+1} = \vec{q}_k + {^0\mat{J}(\vec{q}_k)}^{-1} \vec{e}_k\]
+

Examples

+

The following example gets the ets of a panda robot object, makes a goal +pose Tep, and then solves for the joint coordinates which result in the pose +Tep using the ikine_NR method.

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda().ets()
+>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> panda.ikine_NR(Tep)
+IKSolution(q=array([ 1.2619, -1.1522,  0.1673, -1.007 ,  2.5686,  0.6884,  2.4083]), success=False, iterations=100, searches=100, residual=0.0, reason='iteration and search limit reached, 100 numpy.LinAlgError encountered')
+
+
+

Notes

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian.

+

This class supports null-space motion to assist with maximising manipulability and +avoiding joint limits. These are enabled by setting kq and km to non-zero values.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
IK_NR

An IK Solver class which implements the Newton-Raphson optimisation technique

+
+
ikine_LM

Implements the IK_LM class as a method within the ETS class

+
+
ikine_GN

Implements the IK_GN class as a method within the ETS class

+
+
ikine_QP

Implements the IK_QP class as a method within the ETS class

+
+
+
+
+

Changed in version 1.0.4: Added the Newton-Raphson IK solver method on the ETS class

+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.ETS.ETS.ikine_QP.html b/IK/stubs/roboticstoolbox.robot.ETS.ETS.ikine_QP.html new file mode 100644 index 00000000..c8aad187 --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.ETS.ETS.ikine_QP.html @@ -0,0 +1,287 @@ + + + + + + + + + + + ETS.ikine_QP — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

ETS.ikine_QP

+
+
+ETS.ikine_QP(Tep, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, seed=None, kj=1.0, ks=1.0, kq=0.0, km=0.0, ps=0.0, pi=0.3, **kwargs)[source]
+

Quadratic Programming Numerical Inverse Kinematics Solver

+

A method that provides functionality to perform numerical inverse kinematics +(IK) using a quadratic progamming approach.

+

See the Inverse Kinematics Docs Page for more details and for a +tutorial on numerical IK, see here.

+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose

  • +
  • q0 (Union[ndarray, List[float], Tuple[float], Set[float], None]) – The initial joint coordinate vector

  • +
  • ilimit (int) – How many iterations are allowed within a search before a new search +is started

  • +
  • slimit (int) – How many searches are allowed before being deemed unsuccessful

  • +
  • tol (float) – Maximum allowed residual error E

  • +
  • mask (Union[ndarray, List[float], Tuple[float], Set[float], None]) – A 6 vector which assigns weights to Cartesian degrees-of-freedom +error priority

  • +
  • joint_limits (bool) – Reject solutions with joint limit violations

  • +
  • seed (Optional[int]) – A seed for the private RNG used to generate random joint coordinate +vectors

  • +
  • kj – A gain for joint velocity norm minimisation

  • +
  • ks – A gain which adjusts the cost of slack (intentional error)

  • +
  • kq (float) – The gain for joint limit avoidance. Setting to 0.0 will remove this +completely from the solution

  • +
  • km (float) – The gain for maximisation. Setting to 0.0 will remove this completely +from the solution

  • +
  • ps (float) – The minimum angle/distance (in radians or metres) in which the joint is +allowed to approach to its limit

  • +
  • pi (Union[ndarray, float]) – The influence angle/distance (in radians or metres) in null space motion +becomes active

  • +
+
+
Raises:
+

ImportError – If the package qpsolvers is not installed

+
+
+

Synopsis

+

Each iteration uses the following approach

+
+\[\vec{q}_{k+1} = \vec{q}_{k} + \dot{\vec{q}}.\]
+

where the QP is defined as

+
+\[\begin{split}\min_x \quad f_o(\vec{x}) &= \frac{1}{2} \vec{x}^\top \mathcal{Q} \vec{x}+ \mathcal{C}^\top \vec{x}, \\ +\text{subject to} \quad \mathcal{J} \vec{x} &= \vec{\nu}, \\ +\mathcal{A} \vec{x} &\leq \mathcal{B}, \\ +\vec{x}^- &\leq \vec{x} \leq \vec{x}^+\end{split}\]
+

with

+
+\[\begin{split}\vec{x} &= +\begin{pmatrix} + \dvec{q} \\ \vec{\delta} +\end{pmatrix} \in \mathbb{R}^{(n+6)} \\ +\mathcal{Q} &= +\begin{pmatrix} + \lambda_q \mat{1}_{n} & \mathbf{0}_{6 \times 6} \\ \mathbf{0}_{n \times n} & \lambda_\delta \mat{1}_{6} +\end{pmatrix} \in \mathbb{R}^{(n+6) \times (n+6)} \\ +\mathcal{J} &= +\begin{pmatrix} + \mat{J}(\vec{q}) & \mat{1}_{6} +\end{pmatrix} \in \mathbb{R}^{6 \times (n+6)} \\ +\mathcal{C} &= +\begin{pmatrix} + \mat{J}_m \\ \bf{0}_{6 \times 1} +\end{pmatrix} \in \mathbb{R}^{(n + 6)} \\ +\mathcal{A} &= +\begin{pmatrix} + \mat{1}_{n \times n + 6} \\ +\end{pmatrix} \in \mathbb{R}^{(l + n) \times (n + 6)} \\ +\mathcal{B} &= +\eta +\begin{pmatrix} + \frac{\rho_0 - \rho_s} + {\rho_i - \rho_s} \\ + \vdots \\ + \frac{\rho_n - \rho_s} + {\rho_i - \rho_s} +\end{pmatrix} \in \mathbb{R}^{n} \\ +\vec{x}^{-, +} &= +\begin{pmatrix} + \dvec{q}^{-, +} \\ + \vec{\delta}^{-, +} +\end{pmatrix} \in \mathbb{R}^{(n+6)},\end{split}\]
+

where \(\vec{\delta} \in \mathbb{R}^6\) is the slack vector, +\(\lambda_\delta \in \mathbb{R}^+\) is a gain term which adjusts the +cost of the norm of the slack vector in the optimiser, +\(\dvec{q}^{-,+}\) are the minimum and maximum joint velocities, and +\(\dvec{\delta}^{-,+}\) are the minimum and maximum slack velocities.

+

Examples

+

The following example gets the ets of a panda robot object, makes a goal +pose Tep, and then solves for the joint coordinates which result in the pose +Tep using the ikine_QP method.

+
ImportError: the package qpsolvers is required for this class. 
+Install using 'pip install qpsolvers'
+
+
+

Notes

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian.

+

This class supports null-space motion to assist with maximising manipulability and +avoiding joint limits. These are enabled by setting kq and km to non-zero values.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
IK_NR

An IK Solver class which implements the Newton-Raphson optimisation technique

+
+
ikine_LM

Implements the IK_LM class as a method within the ETS class

+
+
ikine_GN

Implements the IK_GN class as a method within the ETS class

+
+
ikine_NR

Implements the IK_NR class as a method within the ETS class

+
+
+
+
+

Changed in version 1.0.4: Added the Quadratic Programming IK solver method on the ETS class

+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.IK.IKSolution.html b/IK/stubs/roboticstoolbox.robot.IK.IKSolution.html new file mode 100644 index 00000000..ee56aa52 --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.IK.IKSolution.html @@ -0,0 +1,208 @@ + + + + + + + + + + + roboticstoolbox.robot.IK.IKSolution — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+
    +
  • + +
  • +
  • +
+
+
+
+
+ +
+

roboticstoolbox.robot.IK.IKSolution

+
+
+class roboticstoolbox.robot.IK.IKSolution(q, success, iterations=0, searches=0, residual=0.0, reason='')[source]
+

A dataclass for representing an IK solution

+
+
+q
+

The joint coordinates of the solution (ndarray). Note that these +will not be valid if failed to find a solution

+
+ +
+
+success
+

True if a valid solution was found

+
+ +
+
+iterations
+

How many iterations were performed

+
+ +
+
+searches
+

How many searches were performed

+
+ +
+
+residual
+

The final error value from the cost function

+
+ +
+
+reason
+

The reason the IK problem failed if applicable

+
+ +
+

Changed in version 1.0.3: Added IKSolution dataclass to replace the IKsolution named tuple

+
+
+
+__init__(q, success, iterations=0, searches=0, residual=0.0, reason='')
+
+ +

Methods

+ + + + + + +

__init__(q, success[, iterations, searches, ...])

+

Attributes

+ + + + + + + + + + + + + + + + + + + + + +

iterations

reason

residual

searches

q

success

+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.IK.IKSolver._check_jl.html b/IK/stubs/roboticstoolbox.robot.IK.IKSolver._check_jl.html new file mode 100644 index 00000000..a05f6eb2 --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.IK.IKSolver._check_jl.html @@ -0,0 +1,181 @@ + + + + + + + + + + + IKSolver._check_jl — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

IKSolver._check_jl

+
+
+IKSolver._check_jl(ets, q)[source]
+

Checks if the joints are within their respective limits

+
+
Parameters:
+
    +
  • ets (ETS) – the ETS

  • +
  • q (ndarray) – the current joint coordinate vector

  • +
+
+
Return type:
+

True if joints within feasible limits otherwise False

+
+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.IK.IKSolver._random_q.html b/IK/stubs/roboticstoolbox.robot.IK.IKSolver._random_q.html new file mode 100644 index 00000000..c23a37a9 --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.IK.IKSolver._random_q.html @@ -0,0 +1,187 @@ + + + + + + + + + + + IKSolver._random_q — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

IKSolver._random_q

+
+
+IKSolver._random_q(ets, i=1)[source]
+

Generate a random valid joint configuration using a private RNG

+

Generates a random q vector within the joint limits defined by +ets.qlim.

+
+
Parameters:
+
    +
  • ets (ETS) – The ETS representing the manipulators kinematics

  • +
  • i (int) – number of configurations to generate

  • +
+
+
Returns:
+

An i x n ndarray of random valid joint configurations, where n +is the number of joints in the ets

+
+
Return type:
+

q

+
+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.IK.IKSolver.error.html b/IK/stubs/roboticstoolbox.robot.IK.IKSolver.error.html new file mode 100644 index 00000000..933a67fd --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.IK.IKSolver.error.html @@ -0,0 +1,196 @@ + + + + + + + + + + + IKSolver.error — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

IKSolver.error

+
+
+IKSolver.error(Te, Tep)[source]
+

Calculates the error between Te and Tep

+

Calculates the engle axis error between current end-effector pose Te and +the desired end-effector pose Tep. Also calulates the quadratic error E +which is weighted by the diagonal matrix We.

+
+\[E = \frac{1}{2} \vec{e}^{\top} \mat{W}_e \vec{e}\]
+

where \(\vec{e} \in \mathbb{R}^6\) is the angle-axis error.

+
+
Parameters:
+
    +
  • Te (ndarray) – The current end-effector pose

  • +
  • Tep (ndarray) – The desired end-effector pose

  • +
+
+
Return type:
+

Tuple[ndarray, float]

+
+
Returns:
+

    +
  • e – angle-axis error (6 vector)

  • +
  • E – The quadratic error weighted by We

  • +
+

+
+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.IK.IKSolver.html b/IK/stubs/roboticstoolbox.robot.IK.IKSolver.html new file mode 100644 index 00000000..1bee6d33 --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.IK.IKSolver.html @@ -0,0 +1,189 @@ + + + + + + + + + + + roboticstoolbox.robot.IK.IKSolver — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+
    +
  • + +
  • +
  • +
+
+
+
+
+ +
+

roboticstoolbox.robot.IK.IKSolver

+
+
+class roboticstoolbox.robot.IK.IKSolver(name='IK Solver', ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, seed=None)[source]
+

An abstract super class for numerical inverse kinematics (IK)

+

This class provides basic functionality to perform numerical IK. Superclasses +can inherit this class and implement the solve method and redefine any other +methods necessary.

+
+
Parameters:
+
    +
  • name (str) – The name of the IK algorithm

  • +
  • ilimit (int) – How many iterations are allowed within a search before a new search +is started

  • +
  • slimit (int) – How many searches are allowed before being deemed unsuccessful

  • +
  • tol (float) – Maximum allowed residual error E

  • +
  • mask (Union[ndarray, List[float], Tuple[float], Set[float], None]) – A 6 vector which assigns weights to Cartesian degrees-of-freedom +error priority

  • +
  • joint_limits (bool) – Reject solutions with joint limit violations

  • +
  • seed (Optional[int]) – A seed for the private RNG used to generate random joint coordinate +vectors

  • +
+
+
+
+

See also

+
+
IK_NR

Implements this class using the Newton-Raphson method

+
+
IK_GN

Implements this class using the Gauss-Newton method

+
+
IK_LM

Implements this class using the Levemberg-Marquadt method

+
+
IK_QP

Implements this class using a quadratic programming approach

+
+
+
+
+

Changed in version 1.0.3: Added the abstract super class IKSolver

+
+
+
+__init__(name='IK Solver', ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, seed=None)[source]
+
+ +

Methods

+ + + + + + + + + + + + + + + +

__init__([name, ilimit, slimit, tol, mask, ...])

error(Te, Tep)

Calculates the error between Te and Tep

solve(ets, Tep[, q0])

Solves the IK problem

step(ets, Tep, q)

Abstract step method

+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.IK.IKSolver.solve.html b/IK/stubs/roboticstoolbox.robot.IK.IKSolver.solve.html new file mode 100644 index 00000000..bf1c0854 --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.IK.IKSolver.solve.html @@ -0,0 +1,197 @@ + + + + + + + + + + + IKSolver.solve — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

IKSolver.solve

+
+
+IKSolver.solve(ets, Tep, q0=None)[source]
+

Solves the IK problem

+

This method will attempt to solve the IK problem and obtain joint coordinates +which result the the end-effector pose Tep.

+
+
Parameters:
+
+
+
Return type:
+

IKSolution

+
+
Returns:
+

    +
  • q – The joint coordinates of the solution (ndarray). Note that these +will not be valid if failed to find a solution

  • +
  • success – True if a valid solution was found

  • +
  • iterations – How many iterations were performed

  • +
  • searches – How many searches were performed

  • +
  • residual – The final error value from the cost function

  • +
  • jl_valid – True if q is inbounds of the robots joint limits

  • +
  • reason – The reason the IK problem failed if applicable

  • +
+

+
+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.IK.IKSolver.step.html b/IK/stubs/roboticstoolbox.robot.IK.IKSolver.step.html new file mode 100644 index 00000000..034c5625 --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.IK.IKSolver.step.html @@ -0,0 +1,194 @@ + + + + + + + + + + + IKSolver.step — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

IKSolver.step

+
+
+abstract IKSolver.step(ets, Tep, q)[source]
+

Abstract step method

+

Superclasses will implement this method to perform a step of the +implemented IK algorithm

+
+
Parameters:
+
    +
  • ets (ETS) – The ETS representing the manipulators kinematics

  • +
  • Tep (ndarray) – The desired end-effector pose

  • +
  • q (ndarray) – The current joint coordinate vector

  • +
+
+
Raises:
+

numpy.LinAlgError – If a step is impossible due to a linear algebra error

+
+
Return type:
+

Tuple[float, ndarray]

+
+
Returns:
+

    +
  • E – The new error value

  • +
  • q – The new joint coordinate vector

  • +
+

+
+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.IK.IK_GN._check_jl.html b/IK/stubs/roboticstoolbox.robot.IK.IK_GN._check_jl.html new file mode 100644 index 00000000..0c4a02dc --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.IK.IK_GN._check_jl.html @@ -0,0 +1,181 @@ + + + + + + + + + + + IK_GN._check_jl — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

IK_GN._check_jl

+
+
+IK_GN._check_jl(ets, q)
+

Checks if the joints are within their respective limits

+
+
Parameters:
+
    +
  • ets (ETS) – the ETS

  • +
  • q (ndarray) – the current joint coordinate vector

  • +
+
+
Return type:
+

True if joints within feasible limits otherwise False

+
+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.IK.IK_GN._random_q.html b/IK/stubs/roboticstoolbox.robot.IK.IK_GN._random_q.html new file mode 100644 index 00000000..98d8578e --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.IK.IK_GN._random_q.html @@ -0,0 +1,187 @@ + + + + + + + + + + + IK_GN._random_q — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

IK_GN._random_q

+
+
+IK_GN._random_q(ets, i=1)
+

Generate a random valid joint configuration using a private RNG

+

Generates a random q vector within the joint limits defined by +ets.qlim.

+
+
Parameters:
+
    +
  • ets (ETS) – The ETS representing the manipulators kinematics

  • +
  • i (int) – number of configurations to generate

  • +
+
+
Returns:
+

An i x n ndarray of random valid joint configurations, where n +is the number of joints in the ets

+
+
Return type:
+

q

+
+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.IK.IK_GN.error.html b/IK/stubs/roboticstoolbox.robot.IK.IK_GN.error.html new file mode 100644 index 00000000..9ef6dd71 --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.IK.IK_GN.error.html @@ -0,0 +1,196 @@ + + + + + + + + + + + IK_GN.error — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

IK_GN.error

+
+
+IK_GN.error(Te, Tep)
+

Calculates the error between Te and Tep

+

Calculates the engle axis error between current end-effector pose Te and +the desired end-effector pose Tep. Also calulates the quadratic error E +which is weighted by the diagonal matrix We.

+
+\[E = \frac{1}{2} \vec{e}^{\top} \mat{W}_e \vec{e}\]
+

where \(\vec{e} \in \mathbb{R}^6\) is the angle-axis error.

+
+
Parameters:
+
    +
  • Te (ndarray) – The current end-effector pose

  • +
  • Tep (ndarray) – The desired end-effector pose

  • +
+
+
Return type:
+

Tuple[ndarray, float]

+
+
Returns:
+

    +
  • e – angle-axis error (6 vector)

  • +
  • E – The quadratic error weighted by We

  • +
+

+
+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.IK.IK_GN.solve.html b/IK/stubs/roboticstoolbox.robot.IK.IK_GN.solve.html new file mode 100644 index 00000000..1e44a73b --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.IK.IK_GN.solve.html @@ -0,0 +1,197 @@ + + + + + + + + + + + IK_GN.solve — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

IK_GN.solve

+
+
+IK_GN.solve(ets, Tep, q0=None)
+

Solves the IK problem

+

This method will attempt to solve the IK problem and obtain joint coordinates +which result the the end-effector pose Tep.

+
+
Parameters:
+
+
+
Return type:
+

IKSolution

+
+
Returns:
+

    +
  • q – The joint coordinates of the solution (ndarray). Note that these +will not be valid if failed to find a solution

  • +
  • success – True if a valid solution was found

  • +
  • iterations – How many iterations were performed

  • +
  • searches – How many searches were performed

  • +
  • residual – The final error value from the cost function

  • +
  • jl_valid – True if q is inbounds of the robots joint limits

  • +
  • reason – The reason the IK problem failed if applicable

  • +
+

+
+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.IK.IK_GN.step.html b/IK/stubs/roboticstoolbox.robot.IK.IK_GN.step.html new file mode 100644 index 00000000..18154878 --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.IK.IK_GN.step.html @@ -0,0 +1,211 @@ + + + + + + + + + + + IK_GN.step — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

IK_GN.step

+
+
+IK_GN.step(ets, Tep, q)[source]
+

Performs a single iteration of the Gauss-Newton optimisation method

+

The next step is defined as

+
+\[\begin{split}\vec{q}_{k+1} &= \vec{q}_k + +\left( +{\mat{J}(\vec{q}_k)}^\top +\mat{W}_e \ +{\mat{J}(\vec{q}_k)} +\right)^{-1} +\bf{g}_k \\ +\bf{g}_k &= +{\mat{J}(\vec{q}_k)}^\top +\mat{W}_e +\vec{e}_k\end{split}\]
+

where \(\mat{J} = {^0\mat{J}}\) is the base-frame manipulator Jacobian. If +\(\mat{J}(\vec{q}_k)\) is non-singular, and \(\mat{W}_e = \mat{1}_n\), then +the above provides the pseudoinverse solution. However, if \(\mat{J}(\vec{q}_k)\) +is singular, the above can not be computed and the GN solution is infeasible.

+
+
Parameters:
+
    +
  • ets (ETS) – The ETS representing the manipulators kinematics

  • +
  • Tep (ndarray) – The desired end-effector pose

  • +
  • q (ndarray) – The current joint coordinate vector

  • +
+
+
Raises:
+

numpy.LinAlgError – If a step is impossible due to a linear algebra error

+
+
Return type:
+

Tuple[float, ndarray]

+
+
Returns:
+

    +
  • E – The new error value

  • +
  • q – The new joint coordinate vector

  • +
+

+
+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.IK.IK_LM._check_jl.html b/IK/stubs/roboticstoolbox.robot.IK.IK_LM._check_jl.html new file mode 100644 index 00000000..f875316d --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.IK.IK_LM._check_jl.html @@ -0,0 +1,181 @@ + + + + + + + + + + + IK_LM._check_jl — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

IK_LM._check_jl

+
+
+IK_LM._check_jl(ets, q)
+

Checks if the joints are within their respective limits

+
+
Parameters:
+
    +
  • ets (ETS) – the ETS

  • +
  • q (ndarray) – the current joint coordinate vector

  • +
+
+
Return type:
+

True if joints within feasible limits otherwise False

+
+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.IK.IK_LM._random_q.html b/IK/stubs/roboticstoolbox.robot.IK.IK_LM._random_q.html new file mode 100644 index 00000000..15d3bc49 --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.IK.IK_LM._random_q.html @@ -0,0 +1,187 @@ + + + + + + + + + + + IK_LM._random_q — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

IK_LM._random_q

+
+
+IK_LM._random_q(ets, i=1)
+

Generate a random valid joint configuration using a private RNG

+

Generates a random q vector within the joint limits defined by +ets.qlim.

+
+
Parameters:
+
    +
  • ets (ETS) – The ETS representing the manipulators kinematics

  • +
  • i (int) – number of configurations to generate

  • +
+
+
Returns:
+

An i x n ndarray of random valid joint configurations, where n +is the number of joints in the ets

+
+
Return type:
+

q

+
+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.IK.IK_LM.error.html b/IK/stubs/roboticstoolbox.robot.IK.IK_LM.error.html new file mode 100644 index 00000000..5253516b --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.IK.IK_LM.error.html @@ -0,0 +1,196 @@ + + + + + + + + + + + IK_LM.error — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

IK_LM.error

+
+
+IK_LM.error(Te, Tep)
+

Calculates the error between Te and Tep

+

Calculates the engle axis error between current end-effector pose Te and +the desired end-effector pose Tep. Also calulates the quadratic error E +which is weighted by the diagonal matrix We.

+
+\[E = \frac{1}{2} \vec{e}^{\top} \mat{W}_e \vec{e}\]
+

where \(\vec{e} \in \mathbb{R}^6\) is the angle-axis error.

+
+
Parameters:
+
    +
  • Te (ndarray) – The current end-effector pose

  • +
  • Tep (ndarray) – The desired end-effector pose

  • +
+
+
Return type:
+

Tuple[ndarray, float]

+
+
Returns:
+

    +
  • e – angle-axis error (6 vector)

  • +
  • E – The quadratic error weighted by We

  • +
+

+
+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.IK.IK_LM.solve.html b/IK/stubs/roboticstoolbox.robot.IK.IK_LM.solve.html new file mode 100644 index 00000000..9ac253b7 --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.IK.IK_LM.solve.html @@ -0,0 +1,197 @@ + + + + + + + + + + + IK_LM.solve — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

IK_LM.solve

+
+
+IK_LM.solve(ets, Tep, q0=None)
+

Solves the IK problem

+

This method will attempt to solve the IK problem and obtain joint coordinates +which result the the end-effector pose Tep.

+
+
Parameters:
+
+
+
Return type:
+

IKSolution

+
+
Returns:
+

    +
  • q – The joint coordinates of the solution (ndarray). Note that these +will not be valid if failed to find a solution

  • +
  • success – True if a valid solution was found

  • +
  • iterations – How many iterations were performed

  • +
  • searches – How many searches were performed

  • +
  • residual – The final error value from the cost function

  • +
  • jl_valid – True if q is inbounds of the robots joint limits

  • +
  • reason – The reason the IK problem failed if applicable

  • +
+

+
+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.IK.IK_LM.step.html b/IK/stubs/roboticstoolbox.robot.IK.IK_LM.step.html new file mode 100644 index 00000000..d6584cd9 --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.IK.IK_LM.step.html @@ -0,0 +1,232 @@ + + + + + + + + + + + IK_LM.step — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

IK_LM.step

+
+
+IK_LM.step(ets, Tep, q)[source]
+

Performs a single iteration of the Levenberg-Marquadt optimisation

+

The operation is defined by the choice of method when instantiating the class.

+

The next step is deined as

+
+\[\begin{split}\vec{q}_{k+1} +&= +\vec{q}_k + +\left( + \mat{A}_k +\right)^{-1} +\bf{g}_k \\ +% +\mat{A}_k +&= +{\mat{J}(\vec{q}_k)}^\top +\mat{W}_e \ +{\mat{J}(\vec{q}_k)} ++ +\mat{W}_n\end{split}\]
+

where \(\mat{W}_n = \text{diag}(\vec{w_n})(\vec{w_n} \in \mathbb{R}^n_{>0})\) is a +diagonal damping matrix. The damping matrix ensures that \(\mat{A}_k\) is +non-singular and positive definite. The performance of the LM method largely depends +on the choice of \(\mat{W}_n\).

+

Chan’s Method

+

Chan proposed

+
+\[\mat{W}_n += +λ E_k \mat{1}_n\]
+

where λ is a constant which reportedly does not have much influence on performance. +Use the kwarg k to adjust the weighting term λ.

+

Sugihara’s Method

+

Sugihara proposed

+
+\[\mat{W}_n += +E_k \mat{1}_n + \text{diag}(\hat{\vec{w}}_n)\]
+

where \(\hat{\vec{w}}_n \in \mathbb{R}^n\), \(\hat{w}_{n_i} = l^2 \sim 0.01 l^2\), +and \(l\) is the length of a typical link within the manipulator. We provide the +variable k as a kwarg to adjust the value of \(w_n\).

+

Wampler’s Method

+

Wampler proposed \(\vec{w_n}\) to be a constant. This is set through the k kwarg.

+
+
Parameters:
+
    +
  • ets (ETS) – The ETS representing the manipulators kinematics

  • +
  • Tep (ndarray) – The desired end-effector pose

  • +
  • q (ndarray) – The current joint coordinate vector

  • +
+
+
Raises:
+

numpy.LinAlgError – If a step is impossible due to a linear algebra error

+
+
Returns:
+

    +
  • E – The new error value

  • +
  • q – The new joint coordinate vector

  • +
+

+
+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.IK.IK_NR._check_jl.html b/IK/stubs/roboticstoolbox.robot.IK.IK_NR._check_jl.html new file mode 100644 index 00000000..2ac8a54d --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.IK.IK_NR._check_jl.html @@ -0,0 +1,181 @@ + + + + + + + + + + + IK_NR._check_jl — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

IK_NR._check_jl

+
+
+IK_NR._check_jl(ets, q)
+

Checks if the joints are within their respective limits

+
+
Parameters:
+
    +
  • ets (ETS) – the ETS

  • +
  • q (ndarray) – the current joint coordinate vector

  • +
+
+
Return type:
+

True if joints within feasible limits otherwise False

+
+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.IK.IK_NR._random_q.html b/IK/stubs/roboticstoolbox.robot.IK.IK_NR._random_q.html new file mode 100644 index 00000000..5ae8c65c --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.IK.IK_NR._random_q.html @@ -0,0 +1,187 @@ + + + + + + + + + + + IK_NR._random_q — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

IK_NR._random_q

+
+
+IK_NR._random_q(ets, i=1)
+

Generate a random valid joint configuration using a private RNG

+

Generates a random q vector within the joint limits defined by +ets.qlim.

+
+
Parameters:
+
    +
  • ets (ETS) – The ETS representing the manipulators kinematics

  • +
  • i (int) – number of configurations to generate

  • +
+
+
Returns:
+

An i x n ndarray of random valid joint configurations, where n +is the number of joints in the ets

+
+
Return type:
+

q

+
+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.IK.IK_NR.error.html b/IK/stubs/roboticstoolbox.robot.IK.IK_NR.error.html new file mode 100644 index 00000000..ad8ca6cc --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.IK.IK_NR.error.html @@ -0,0 +1,196 @@ + + + + + + + + + + + IK_NR.error — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

IK_NR.error

+
+
+IK_NR.error(Te, Tep)
+

Calculates the error between Te and Tep

+

Calculates the engle axis error between current end-effector pose Te and +the desired end-effector pose Tep. Also calulates the quadratic error E +which is weighted by the diagonal matrix We.

+
+\[E = \frac{1}{2} \vec{e}^{\top} \mat{W}_e \vec{e}\]
+

where \(\vec{e} \in \mathbb{R}^6\) is the angle-axis error.

+
+
Parameters:
+
    +
  • Te (ndarray) – The current end-effector pose

  • +
  • Tep (ndarray) – The desired end-effector pose

  • +
+
+
Return type:
+

Tuple[ndarray, float]

+
+
Returns:
+

    +
  • e – angle-axis error (6 vector)

  • +
  • E – The quadratic error weighted by We

  • +
+

+
+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.IK.IK_NR.solve.html b/IK/stubs/roboticstoolbox.robot.IK.IK_NR.solve.html new file mode 100644 index 00000000..a60bca5e --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.IK.IK_NR.solve.html @@ -0,0 +1,197 @@ + + + + + + + + + + + IK_NR.solve — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

IK_NR.solve

+
+
+IK_NR.solve(ets, Tep, q0=None)
+

Solves the IK problem

+

This method will attempt to solve the IK problem and obtain joint coordinates +which result the the end-effector pose Tep.

+
+
Parameters:
+
+
+
Return type:
+

IKSolution

+
+
Returns:
+

    +
  • q – The joint coordinates of the solution (ndarray). Note that these +will not be valid if failed to find a solution

  • +
  • success – True if a valid solution was found

  • +
  • iterations – How many iterations were performed

  • +
  • searches – How many searches were performed

  • +
  • residual – The final error value from the cost function

  • +
  • jl_valid – True if q is inbounds of the robots joint limits

  • +
  • reason – The reason the IK problem failed if applicable

  • +
+

+
+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.IK.IK_NR.step.html b/IK/stubs/roboticstoolbox.robot.IK.IK_NR.step.html new file mode 100644 index 00000000..1290b309 --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.IK.IK_NR.step.html @@ -0,0 +1,196 @@ + + + + + + + + + + + IK_NR.step — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

IK_NR.step

+
+
+IK_NR.step(ets, Tep, q)[source]
+

Performs a single iteration of the Newton-Raphson optimisation method

+
+\[\vec{q}_{k+1} = \vec{q}_k + {^0\mat{J}(\vec{q}_k)}^{-1} \vec{e}_k\]
+
+
Parameters:
+
    +
  • ets (ETS) – The ETS representing the manipulators kinematics

  • +
  • Tep (ndarray) – The desired end-effector pose

  • +
  • q (ndarray) – The current joint coordinate vector

  • +
+
+
Raises:
+

numpy.LinAlgError – If a step is impossible due to a linear algebra error

+
+
Return type:
+

Tuple[float, ndarray]

+
+
Returns:
+

    +
  • E – The new error value

  • +
  • q – The new joint coordinate vector

  • +
+

+
+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.IK.IK_QP._check_jl.html b/IK/stubs/roboticstoolbox.robot.IK.IK_QP._check_jl.html new file mode 100644 index 00000000..fa059103 --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.IK.IK_QP._check_jl.html @@ -0,0 +1,181 @@ + + + + + + + + + + + IK_QP._check_jl — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

IK_QP._check_jl

+
+
+IK_QP._check_jl(ets, q)
+

Checks if the joints are within their respective limits

+
+
Parameters:
+
    +
  • ets (ETS) – the ETS

  • +
  • q (ndarray) – the current joint coordinate vector

  • +
+
+
Return type:
+

True if joints within feasible limits otherwise False

+
+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.IK.IK_QP._random_q.html b/IK/stubs/roboticstoolbox.robot.IK.IK_QP._random_q.html new file mode 100644 index 00000000..fbf508f4 --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.IK.IK_QP._random_q.html @@ -0,0 +1,187 @@ + + + + + + + + + + + IK_QP._random_q — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

IK_QP._random_q

+
+
+IK_QP._random_q(ets, i=1)
+

Generate a random valid joint configuration using a private RNG

+

Generates a random q vector within the joint limits defined by +ets.qlim.

+
+
Parameters:
+
    +
  • ets (ETS) – The ETS representing the manipulators kinematics

  • +
  • i (int) – number of configurations to generate

  • +
+
+
Returns:
+

An i x n ndarray of random valid joint configurations, where n +is the number of joints in the ets

+
+
Return type:
+

q

+
+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.IK.IK_QP.error.html b/IK/stubs/roboticstoolbox.robot.IK.IK_QP.error.html new file mode 100644 index 00000000..5932f5cb --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.IK.IK_QP.error.html @@ -0,0 +1,196 @@ + + + + + + + + + + + IK_QP.error — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

IK_QP.error

+
+
+IK_QP.error(Te, Tep)
+

Calculates the error between Te and Tep

+

Calculates the engle axis error between current end-effector pose Te and +the desired end-effector pose Tep. Also calulates the quadratic error E +which is weighted by the diagonal matrix We.

+
+\[E = \frac{1}{2} \vec{e}^{\top} \mat{W}_e \vec{e}\]
+

where \(\vec{e} \in \mathbb{R}^6\) is the angle-axis error.

+
+
Parameters:
+
    +
  • Te (ndarray) – The current end-effector pose

  • +
  • Tep (ndarray) – The desired end-effector pose

  • +
+
+
Return type:
+

Tuple[ndarray, float]

+
+
Returns:
+

    +
  • e – angle-axis error (6 vector)

  • +
  • E – The quadratic error weighted by We

  • +
+

+
+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.IK.IK_QP.solve.html b/IK/stubs/roboticstoolbox.robot.IK.IK_QP.solve.html new file mode 100644 index 00000000..f31d5758 --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.IK.IK_QP.solve.html @@ -0,0 +1,197 @@ + + + + + + + + + + + IK_QP.solve — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

IK_QP.solve

+
+
+IK_QP.solve(ets, Tep, q0=None)
+

Solves the IK problem

+

This method will attempt to solve the IK problem and obtain joint coordinates +which result the the end-effector pose Tep.

+
+
Parameters:
+
+
+
Return type:
+

IKSolution

+
+
Returns:
+

    +
  • q – The joint coordinates of the solution (ndarray). Note that these +will not be valid if failed to find a solution

  • +
  • success – True if a valid solution was found

  • +
  • iterations – How many iterations were performed

  • +
  • searches – How many searches were performed

  • +
  • residual – The final error value from the cost function

  • +
  • jl_valid – True if q is inbounds of the robots joint limits

  • +
  • reason – The reason the IK problem failed if applicable

  • +
+

+
+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.IK.IK_QP.step.html b/IK/stubs/roboticstoolbox.robot.IK.IK_QP.step.html new file mode 100644 index 00000000..3ee80193 --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.IK.IK_QP.step.html @@ -0,0 +1,244 @@ + + + + + + + + + + + IK_QP.step — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

IK_QP.step

+
+
+IK_QP.step(ets, Tep, q)[source]
+

Performs a single iteration of the Gauss-Newton optimisation method

+

The next step is defined as

+
+\[\vec{q}_{k+1} = \vec{q}_{k} + \dot{\vec{q}}.\]
+

where the QP is defined as

+
+\[\begin{split}\min_x \quad f_o(\vec{x}) &= \frac{1}{2} \vec{x}^\top \mathcal{Q} \vec{x}+ \mathcal{C}^\top \vec{x}, \\ +\text{subject to} \quad \mathcal{J} \vec{x} &= \vec{\nu}, \\ +\mathcal{A} \vec{x} &\leq \mathcal{B}, \\ +\vec{x}^- &\leq \vec{x} \leq \vec{x}^+\end{split}\]
+

with

+
+\[\begin{split}\vec{x} &= +\begin{pmatrix} + \dvec{q} \\ \vec{\delta} +\end{pmatrix} \in \mathbb{R}^{(n+6)} \\ +\mathcal{Q} &= +\begin{pmatrix} + \lambda_q \mat{1}_{n} & \mathbf{0}_{6 \times 6} \\ \mathbf{0}_{n \times n} & \lambda_\delta \mat{1}_{6} +\end{pmatrix} \in \mathbb{R}^{(n+6) \times (n+6)} \\ +\mathcal{J} &= +\begin{pmatrix} + \mat{J}(\vec{q}) & \mat{1}_{6} +\end{pmatrix} \in \mathbb{R}^{6 \times (n+6)} \\ +\mathcal{C} &= +\begin{pmatrix} + \mat{J}_m \\ \bf{0}_{6 \times 1} +\end{pmatrix} \in \mathbb{R}^{(n + 6)} \\ +\mathcal{A} &= +\begin{pmatrix} + \mat{1}_{n \times n + 6} \\ +\end{pmatrix} \in \mathbb{R}^{(l + n) \times (n + 6)} \\ +\mathcal{B} &= +\eta +\begin{pmatrix} + \frac{\rho_0 - \rho_s} + {\rho_i - \rho_s} \\ + \vdots \\ + \frac{\rho_n - \rho_s} + {\rho_i - \rho_s} +\end{pmatrix} \in \mathbb{R}^{n} \\ +\vec{x}^{-, +} &= +\begin{pmatrix} + \dvec{q}^{-, +} \\ + \vec{\delta}^{-, +} +\end{pmatrix} \in \mathbb{R}^{(n+6)},\end{split}\]
+

where \(\vec{\delta} \in \mathbb{R}^6\) is the slack vector, +\(\lambda_\delta \in \mathbb{R}^+\) is a gain term which adjusts the +cost of the norm of the slack vector in the optimiser, +\(\dvec{q}^{-,+}\) are the minimum and maximum joint velocities, and +\(\dvec{\delta}^{-,+}\) are the minimum and maximum slack velocities.

+
+
Parameters:
+
    +
  • ets (ETS) – The ETS representing the manipulators kinematics

  • +
  • Tep (ndarray) – The desired end-effector pose

  • +
  • q (ndarray) – The current joint coordinate vector

  • +
+
+
Raises:
+

numpy.LinAlgError – If a step is impossible due to a linear algebra error

+
+
Return type:
+

Tuple[float, ndarray]

+
+
Returns:
+

    +
  • E – The new error value

  • +
  • q – The new joint coordinate vector

  • +
+

+
+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.Robot.Robot.ik_GN.html b/IK/stubs/roboticstoolbox.robot.Robot.Robot.ik_GN.html new file mode 100644 index 00000000..dd7e99c2 --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.Robot.Robot.ik_GN.html @@ -0,0 +1,260 @@ + + + + + + + + + + + Robot.ik_GN — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

Robot.ik_GN

+
+
+Robot.ik_GN(Tep, end=None, start=None, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, pinv=True, pinv_damping=0.0)
+

Fast numerical inverse kinematics by Gauss-Newton optimization

+

sol = ets.ik_GN(Tep) are the joint coordinates (n) corresponding +to the robot end-effector pose Tep which is an SE3 or ndarray object. +This method can be used for robots with any number of degrees of freedom. This +is a fast solver implemented in C++.

+

See the Inverse Kinematics Docs Page for more details and for a +tutorial on numerical IK, see here.

+
+

Note

+

When using this method with redundant robots (>6 DoF), pinv must be set to True

+
+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose or pose trajectory

  • +
  • end (Union[str, Link, Gripper, None]) – the link considered as the end-effector

  • +
  • start (Union[str, Link, Gripper, None]) – the link considered as the base frame, defaults to the robots’s base frame

  • +
  • q0 (Optional[ndarray]) – initial joint configuration (default to random valid joint +configuration contrained by the joint limits of the robot)

  • +
  • ilimit (int) – maximum number of iterations per search

  • +
  • slimit (int) – maximum number of search attempts

  • +
  • tol (float) – final error tolerance

  • +
  • mask (Optional[ndarray]) – a mask vector which weights the end-effector error priority. +Corresponds to translation in X, Y and Z and rotation about X, Y and Z +respectively

  • +
  • joint_limits (bool) – constrain the solution to being within the joint limits of +the robot (reject solution with invalid joint configurations and perfrom +another search up to the slimit)

  • +
  • pinv (int) – Use the psuedo-inverse instad of the normal matrix inverse

  • +
  • pinv_damping (float) – Damping factor for the psuedo-inverse

  • +
+
+
Return type:
+

Tuple[ndarray, int, int, int, float]

+
+
Returns:
+

    +
  • sol – tuple (q, success, iterations, searches, residual)

  • +
  • The return value sol is a tuple with elements

  • +
  • ============ ========== ===============================================

  • +
  • Element Type Description

  • +
  • ============ ========== ===============================================

  • +
  • q ndarray(n) joint coordinates in units of radians or metres

  • +
  • success int whether a solution was found

  • +
  • iterations int total number of iterations

  • +
  • searches int total number of searches

  • +
  • residual float final value of cost function

  • +
  • ============ ========== ===============================================

  • +
  • If success == 0 the q values will be valid numbers, but the

  • +
  • solution will be in error. The amount of error is indicated by

  • +
  • the residual.

  • +
+

+
+
+

Synopsis

+

Each iteration uses the Gauss-Newton optimisation method

+
+\[\begin{split}\vec{q}_{k+1} &= \vec{q}_k + +\left( +{\mat{J}(\vec{q}_k)}^\top +\mat{W}_e \ +{\mat{J}(\vec{q}_k)} +\right)^{-1} +\bf{g}_k \\ +\bf{g}_k &= +{\mat{J}(\vec{q}_k)}^\top +\mat{W}_e +\vec{e}_k\end{split}\]
+

where \(\mat{J} = {^0\mat{J}}\) is the base-frame manipulator Jacobian. If +\(\mat{J}(\vec{q}_k)\) is non-singular, and \(\mat{W}_e = \mat{1}_n\), then +the above provides the pseudoinverse solution. However, if \(\mat{J}(\vec{q}_k)\) +is singular, the above can not be computed and the GN solution is infeasible.

+

Examples

+

The following example gets a panda robot object, makes a goal +pose Tep, and then solves for the joint coordinates which result in the pose +Tep using the ikine_GN method.

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda()
+>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> panda.ik_GN(Tep)
+(array([-1.0805, -0.5328,  0.9086, -2.1781,  0.4612,  1.9018,  0.4239]), 1, 510, 32, 2.803306327008683e-09)
+
+
+

Notes

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
ik_NR

A fast numerical inverse kinematics solver using Newton-Raphson optimisation

+
+
ik_GN

A fast numerical inverse kinematics solver using Gauss-Newton optimisation

+
+
+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.Robot.Robot.ik_LM.html b/IK/stubs/roboticstoolbox.robot.Robot.Robot.ik_LM.html new file mode 100644 index 00000000..ca77468f --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.Robot.Robot.ik_LM.html @@ -0,0 +1,269 @@ + + + + + + + + + + + Robot.ik_LM — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

Robot.ik_LM

+
+
+Robot.ik_LM(Tep, end=None, start=None, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, k=1.0, method='chan')
+

Fast levenberg-Marquadt Numerical Inverse Kinematics Solver

+

A method which provides functionality to perform numerical inverse kinematics (IK) +using the Levemberg-Marquadt method. This +is a fast solver implemented in C++.

+

See the Inverse Kinematics Docs Page for more details and for a +tutorial on numerical IK, see here.

+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose

  • +
  • end (Union[str, Link, Gripper, None]) – the link considered as the end-effector

  • +
  • start (Union[str, Link, Gripper, None]) – the link considered as the base frame, defaults to the robots’s base frame

  • +
  • q0 (Optional[ndarray]) – The initial joint coordinate vector

  • +
  • ilimit (int) – How many iterations are allowed within a search before a new search +is started

  • +
  • slimit (int) – How many searches are allowed before being deemed unsuccessful

  • +
  • tol (float) – Maximum allowed residual error E

  • +
  • mask (Optional[ndarray]) – A 6 vector which assigns weights to Cartesian degrees-of-freedom +error priority

  • +
  • joint_limits (bool) – Reject solutions with joint limit violations

  • +
  • seed – A seed for the private RNG used to generate random joint coordinate +vectors

  • +
  • k (float) – Sets the gain value for the damping matrix Wn in the next iteration. See +synopsis

  • +
  • method (Literal['chan', 'wampler', 'sugihara']) – One of “chan”, “sugihara” or “wampler”. Defines which method is used +to calculate the damping matrix Wn in the step method

  • +
+
+
Return type:
+

Tuple[ndarray, int, int, int, float]

+
+
+

Synopsis

+

The operation is defined by the choice of the method kwarg.

+

The step is deined as

+
+\[\begin{split}\vec{q}_{k+1} +&= +\vec{q}_k + +\left( + \mat{A}_k +\right)^{-1} +\bf{g}_k \\ +% +\mat{A}_k +&= +{\mat{J}(\vec{q}_k)}^\top +\mat{W}_e \ +{\mat{J}(\vec{q}_k)} ++ +\mat{W}_n\end{split}\]
+

where \(\mat{W}_n = \text{diag}(\vec{w_n})(\vec{w_n} \in \mathbb{R}^n_{>0})\) is a +diagonal damping matrix. The damping matrix ensures that \(\mat{A}_k\) is +non-singular and positive definite. The performance of the LM method largely depends +on the choice of \(\mat{W}_n\).

+

Chan’s Method

+

Chan proposed

+
+\[\mat{W}_n += +λ E_k \mat{1}_n\]
+

where λ is a constant which reportedly does not have much influence on performance. +Use the kwarg k to adjust the weighting term λ.

+

Sugihara’s Method

+

Sugihara proposed

+
+\[\mat{W}_n += +E_k \mat{1}_n + \text{diag}(\hat{\vec{w}}_n)\]
+

where \(\hat{\vec{w}}_n \in \mathbb{R}^n\), \(\hat{w}_{n_i} = l^2 \sim 0.01 l^2\), +and \(l\) is the length of a typical link within the manipulator. We provide the +variable k as a kwarg to adjust the value of \(w_n\).

+

Wampler’s Method

+

Wampler proposed \(\vec{w_n}\) to be a constant. This is set through the k kwarg.

+

Examples

+

The following example makes a panda robot object, makes a goal +pose Tep, and then solves for the joint coordinates which result in the pose +Tep using the ikine_LM method.

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda()
+>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> panda.ikine_LM(Tep)
+IKSolution(q=array([-1.176 , -0.6001,  0.9677, -2.1706,  0.5363,  1.8653,  0.3663]), success=True, iterations=39, searches=4, residual=4.525953207786055e-11, reason='Success')
+
+
+

Notes

+

The value for the k kwarg will depend on the method chosen and the arm you are +using. Use the following as a rough guide chan, k = 1.0 - 0.01, +wampler, k = 0.01 - 0.0001, and sugihara, k = 0.1 - 0.0001

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian.

+

This class supports null-space motion to assist with maximising manipulability and +avoiding joint limits. These are enabled by setting kq and km to non-zero values.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
ik_NR

A fast numerical inverse kinematics solver using Newton-Raphson optimisation

+
+
ik_GN

A fast numerical inverse kinematics solver using Gauss-Newton optimisation

+
+
+
+
+

Changed in version 1.0.4: Merged the Levemberg-Marquadt IK solvers into the ik_LM method

+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.Robot.Robot.ik_NR.html b/IK/stubs/roboticstoolbox.robot.Robot.Robot.ik_NR.html new file mode 100644 index 00000000..4be40916 --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.Robot.Robot.ik_NR.html @@ -0,0 +1,246 @@ + + + + + + + + + + + Robot.ik_NR — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

Robot.ik_NR

+
+
+Robot.ik_NR(Tep, end=None, start=None, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, pinv=True, pinv_damping=0.0)
+

Fast numerical inverse kinematics using Newton-Raphson optimization

+

sol = ets.ik_NR(Tep) are the joint coordinates (n) corresponding +to the robot end-effector pose Tep which is an SE3 or ndarray object. +This method can be used for robots with any number of degrees of freedom. This +is a fast solver implemented in C++.

+

See the Inverse Kinematics Docs Page for more details and for a +tutorial on numerical IK, see here.

+
+

Note

+

When using this method with redundant robots (>6 DoF), pinv must be set to True

+
+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose or pose trajectory

  • +
  • end (Union[str, Link, Gripper, None]) – the link considered as the end-effector

  • +
  • start (Union[str, Link, Gripper, None]) – the link considered as the base frame, defaults to the robots’s base frame

  • +
  • q0 (Optional[ndarray]) – initial joint configuration (default to random valid joint +configuration contrained by the joint limits of the robot)

  • +
  • ilimit (int) – maximum number of iterations per search

  • +
  • slimit (int) – maximum number of search attempts

  • +
  • tol (float) – final error tolerance

  • +
  • mask (Optional[ndarray]) – a mask vector which weights the end-effector error priority. +Corresponds to translation in X, Y and Z and rotation about X, Y and Z +respectively

  • +
  • joint_limits (bool) – constrain the solution to being within the joint limits of +the robot (reject solution with invalid joint configurations and perfrom +another search up to the slimit)

  • +
  • pinv (int) – Use the psuedo-inverse instad of the normal matrix inverse

  • +
  • pinv_damping (float) – Damping factor for the psuedo-inverse

  • +
+
+
Return type:
+

Tuple[ndarray, int, int, int, float]

+
+
Returns:
+

    +
  • sol – tuple (q, success, iterations, searches, residual)

  • +
  • The return value sol is a tuple with elements

  • +
  • ============ ========== ===============================================

  • +
  • Element Type Description

  • +
  • ============ ========== ===============================================

  • +
  • q ndarray(n) joint coordinates in units of radians or metres

  • +
  • success int whether a solution was found

  • +
  • iterations int total number of iterations

  • +
  • searches int total number of searches

  • +
  • residual float final value of cost function

  • +
  • ============ ========== ===============================================

  • +
  • If success == 0 the q values will be valid numbers, but the

  • +
  • solution will be in error. The amount of error is indicated by

  • +
  • the residual.

  • +
+

+
+
+

Synopsis

+

Each iteration uses the Newton-Raphson optimisation method

+
+\[\vec{q}_{k+1} = \vec{q}_k + {^0\mat{J}(\vec{q}_k)}^{-1} \vec{e}_k\]
+

Examples

+

The following example gets a panda robot object, makes a goal +pose Tep, and then solves for the joint coordinates which result in the pose +Tep using the ikine_GN method.

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda()
+>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> panda.ik_NR(Tep)
+(array([-1.0805, -0.5328,  0.9086, -2.1781,  0.4612,  1.9018,  0.4239]), 1, 489, 32, 2.8033063270234757e-09)
+
+
+

Notes

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
ik_LM

A fast numerical inverse kinematics solver using Levenberg-Marquadt optimisation

+
+
ik_GN

A fast numerical inverse kinematics solver using Gauss-Newton optimisation

+
+
+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.Robot.Robot.ikine_GN.html b/IK/stubs/roboticstoolbox.robot.Robot.Robot.ikine_GN.html new file mode 100644 index 00000000..55c90fd8 --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.Robot.Robot.ikine_GN.html @@ -0,0 +1,260 @@ + + + + + + + + + + + Robot.ikine_GN — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

Robot.ikine_GN

+
+
+Robot.ikine_GN(Tep, end=None, start=None, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, seed=None, pinv=False, kq=0.0, km=0.0, ps=0.0, pi=0.3, **kwargs)
+

Gauss-Newton Numerical Inverse Kinematics Solver

+

A method which provides functionality to perform numerical inverse kinematics (IK) +using the Gauss-Newton method.

+

See the Inverse Kinematics Docs Page for more details and for a +tutorial on numerical IK, see here.

+
+

Note

+

When using this method with redundant robots (>6 DoF), pinv must be set to True

+
+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose

  • +
  • end (Union[str, Link, Gripper, None]) – the link considered as the end-effector

  • +
  • start (Union[str, Link, Gripper, None]) – the link considered as the base frame, defaults to the robots’s base frame

  • +
  • q0 (Union[ndarray, List[float], Tuple[float], Set[float], None]) – The initial joint coordinate vector

  • +
  • ilimit (int) – How many iterations are allowed within a search before a new search +is started

  • +
  • slimit (int) – How many searches are allowed before being deemed unsuccessful

  • +
  • tol (float) – Maximum allowed residual error E

  • +
  • mask (Union[ndarray, List[float], Tuple[float], Set[float], None]) – A 6 vector which assigns weights to Cartesian degrees-of-freedom +error priority

  • +
  • joint_limits (bool) – Reject solutions with joint limit violations

  • +
  • seed (Optional[int]) – A seed for the private RNG used to generate random joint coordinate +vectors

  • +
  • pinv (bool) – If True, will use the psuedoinverse in the step method instead of +the normal inverse

  • +
  • kq (float) – The gain for joint limit avoidance. Setting to 0.0 will remove this +completely from the solution

  • +
  • km (float) – The gain for maximisation. Setting to 0.0 will remove this completely +from the solution

  • +
  • ps (float) – The minimum angle/distance (in radians or metres) in which the joint is +allowed to approach to its limit

  • +
  • pi (Union[ndarray, float]) – The influence angle/distance (in radians or metres) in null space motion +becomes active

  • +
+
+
+

Synopsis

+

Each iteration uses the Gauss-Newton optimisation method

+
+\[\begin{split}\vec{q}_{k+1} &= \vec{q}_k + +\left( +{\mat{J}(\vec{q}_k)}^\top +\mat{W}_e \ +{\mat{J}(\vec{q}_k)} +\right)^{-1} +\bf{g}_k \\ +\bf{g}_k &= +{\mat{J}(\vec{q}_k)}^\top +\mat{W}_e +\vec{e}_k\end{split}\]
+

where \(\mat{J} = {^0\mat{J}}\) is the base-frame manipulator Jacobian. If +\(\mat{J}(\vec{q}_k)\) is non-singular, and \(\mat{W}_e = \mat{1}_n\), then +the above provides the pseudoinverse solution. However, if \(\mat{J}(\vec{q}_k)\) +is singular, the above can not be computed and the GN solution is infeasible.

+

Examples

+

The following example gets a panda robot object, makes a goal +pose Tep, and then solves for the joint coordinates which result in the pose +Tep using the ikine_GN method.

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda()
+>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> panda.ikine_GN(Tep)
+IKSolution(q=array([ 2.2611, -1.4514, -2.8337, -0.6221, -2.6177,  1.1555, -2.4071]), success=False, iterations=100, searches=100, residual=0.0, reason='iteration and search limit reached, 100 numpy.LinAlgError encountered')
+
+
+

Notes

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian.

+

This class supports null-space motion to assist with maximising manipulability and +avoiding joint limits. These are enabled by setting kq and km to non-zero values.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
IK_NR

An IK Solver class which implements the Newton-Raphson optimisation technique

+
+
ikine_LM

Implements the IK_LM class as a method within the ETS class

+
+
ikine_NR

Implements the IK_NR class as a method within the ETS class

+
+
ikine_QP

Implements the IK_QP class as a method within the ETS class

+
+
+
+
+

Changed in version 1.0.4: Added the Gauss-Newton IK solver method on the Robot class

+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.Robot.Robot.ikine_LM.html b/IK/stubs/roboticstoolbox.robot.Robot.Robot.ikine_LM.html new file mode 100644 index 00000000..1917772c --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.Robot.Robot.ikine_LM.html @@ -0,0 +1,285 @@ + + + + + + + + + + + Robot.ikine_LM — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

Robot.ikine_LM

+
+
+Robot.ikine_LM(Tep, end=None, start=None, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, seed=None, k=1.0, method='chan', kq=0.0, km=0.0, ps=0.0, pi=0.3, **kwargs)
+

Levenberg-Marquadt Numerical Inverse Kinematics Solver

+

A method which provides functionality to perform numerical inverse kinematics (IK) +using the Levemberg-Marquadt method.

+

See the Inverse Kinematics Docs Page for more details and for a +tutorial on numerical IK, see here.

+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose

  • +
  • end (Union[str, Link, Gripper, None]) – the link considered as the end-effector

  • +
  • start (Union[str, Link, Gripper, None]) – the link considered as the base frame, defaults to the robots’s base frame

  • +
  • q0 (Union[ndarray, List[float], Tuple[float], Set[float], None]) – The initial joint coordinate vector

  • +
  • ilimit (int) – How many iterations are allowed within a search before a new search +is started

  • +
  • slimit (int) – How many searches are allowed before being deemed unsuccessful

  • +
  • tol (float) – Maximum allowed residual error E

  • +
  • mask (Union[ndarray, List[float], Tuple[float], Set[float], None]) – A 6 vector which assigns weights to Cartesian degrees-of-freedom +error priority

  • +
  • joint_limits (bool) – Reject solutions with joint limit violations

  • +
  • seed (Optional[int]) – A seed for the private RNG used to generate random joint coordinate +vectors

  • +
  • k (float) – Sets the gain value for the damping matrix Wn in the next iteration. See +synopsis

  • +
  • method (Literal['chan', 'wampler', 'sugihara']) – One of “chan”, “sugihara” or “wampler”. Defines which method is used +to calculate the damping matrix Wn in the step method

  • +
  • kq (float) – The gain for joint limit avoidance. Setting to 0.0 will remove this +completely from the solution

  • +
  • km (float) – The gain for maximisation. Setting to 0.0 will remove this completely +from the solution

  • +
  • ps (float) – The minimum angle/distance (in radians or metres) in which the joint is +allowed to approach to its limit

  • +
  • pi (Union[ndarray, float]) – The influence angle/distance (in radians or metres) in null space motion +becomes active

  • +
+
+
+

Synopsis

+

The operation is defined by the choice of the method kwarg.

+

The step is deined as

+
+\[\begin{split}\vec{q}_{k+1} +&= +\vec{q}_k + +\left( + \mat{A}_k +\right)^{-1} +\bf{g}_k \\ +% +\mat{A}_k +&= +{\mat{J}(\vec{q}_k)}^\top +\mat{W}_e \ +{\mat{J}(\vec{q}_k)} ++ +\mat{W}_n\end{split}\]
+

where \(\mat{W}_n = \text{diag}(\vec{w_n})(\vec{w_n} \in \mathbb{R}^n_{>0})\) is a +diagonal damping matrix. The damping matrix ensures that \(\mat{A}_k\) is +non-singular and positive definite. The performance of the LM method largely depends +on the choice of \(\mat{W}_n\).

+

Chan’s Method

+

Chan proposed

+
+\[\mat{W}_n += +λ E_k \mat{1}_n\]
+

where λ is a constant which reportedly does not have much influence on performance. +Use the kwarg k to adjust the weighting term λ.

+

Sugihara’s Method

+

Sugihara proposed

+
+\[\mat{W}_n += +E_k \mat{1}_n + \text{diag}(\hat{\vec{w}}_n)\]
+

where \(\hat{\vec{w}}_n \in \mathbb{R}^n\), \(\hat{w}_{n_i} = l^2 \sim 0.01 l^2\), +and \(l\) is the length of a typical link within the manipulator. We provide the +variable k as a kwarg to adjust the value of \(w_n\).

+

Wampler’s Method

+

Wampler proposed \(\vec{w_n}\) to be a constant. This is set through the k kwarg.

+

Examples

+

The following example makes a panda robot object, makes a goal +pose Tep, and then solves for the joint coordinates which result in the pose +Tep using the ikine_LM method.

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda()
+>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> panda.ikine_LM(Tep)
+IKSolution(q=array([ 0.3059, -0.3126, -0.2739, -2.199 , -0.0959,  1.9959,  0.8611]), success=True, iterations=8, searches=1, residual=4.99812874879148e-07, reason='Success')
+
+
+

Notes

+

The value for the k kwarg will depend on the method chosen and the arm you are +using. Use the following as a rough guide chan, k = 1.0 - 0.01, +wampler, k = 0.01 - 0.0001, and sugihara, k = 0.1 - 0.0001

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian.

+

This class supports null-space motion to assist with maximising manipulability and +avoiding joint limits. These are enabled by setting kq and km to non-zero values.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
IK_LM

An IK Solver class which implements the Levemberg Marquadt optimisation technique

+
+
ikine_NR

Implements the IK_NR class as a method within the Robot class

+
+
ikine_GN

Implements the IK_GN class as a method within the Robot class

+
+
ikine_QP

Implements the IK_QP class as a method within the Robot class

+
+
+
+
+

Changed in version 1.0.4: Added the Levemberg-Marquadt IK solver method on the Robot class

+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.Robot.Robot.ikine_NR.html b/IK/stubs/roboticstoolbox.robot.Robot.Robot.ikine_NR.html new file mode 100644 index 00000000..e0b38bda --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.Robot.Robot.ikine_NR.html @@ -0,0 +1,246 @@ + + + + + + + + + + + Robot.ikine_NR — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

Robot.ikine_NR

+
+
+Robot.ikine_NR(Tep, end=None, start=None, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, seed=None, pinv=False, kq=0.0, km=0.0, ps=0.0, pi=0.3, **kwargs)
+

Newton-Raphson Numerical Inverse Kinematics Solver

+

A method which provides functionality to perform numerical inverse kinematics (IK) +using the Newton-Raphson method.

+

See the Inverse Kinematics Docs Page for more details and for a +tutorial on numerical IK, see here.

+
+

Note

+

When using this method with redundant robots (>6 DoF), pinv must be set to True

+
+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose

  • +
  • end (Union[str, Link, Gripper, None]) – the link considered as the end-effector

  • +
  • start (Union[str, Link, Gripper, None]) – the link considered as the base frame, defaults to the robots’s base frame

  • +
  • q0 (Union[ndarray, List[float], Tuple[float], Set[float], None]) – The initial joint coordinate vector

  • +
  • ilimit (int) – How many iterations are allowed within a search before a new search +is started

  • +
  • slimit (int) – How many searches are allowed before being deemed unsuccessful

  • +
  • tol (float) – Maximum allowed residual error E

  • +
  • mask (Union[ndarray, List[float], Tuple[float], Set[float], None]) – A 6 vector which assigns weights to Cartesian degrees-of-freedom +error priority

  • +
  • joint_limits (bool) – Reject solutions with joint limit violations

  • +
  • seed (Optional[int]) – A seed for the private RNG used to generate random joint coordinate +vectors

  • +
  • pinv (bool) – If True, will use the psuedoinverse in the step method instead of +the normal inverse

  • +
  • kq (float) – The gain for joint limit avoidance. Setting to 0.0 will remove this +completely from the solution

  • +
  • km (float) – The gain for maximisation. Setting to 0.0 will remove this completely +from the solution

  • +
  • ps (float) – The minimum angle/distance (in radians or metres) in which the joint is +allowed to approach to its limit

  • +
  • pi (Union[ndarray, float]) – The influence angle/distance (in radians or metres) in null space motion +becomes active

  • +
+
+
+

Synopsis

+

Each iteration uses the Newton-Raphson optimisation method

+
+\[\vec{q}_{k+1} = \vec{q}_k + {^0\mat{J}(\vec{q}_k)}^{-1} \vec{e}_k\]
+

Examples

+

The following example gets a panda robot object, makes a goal +pose Tep, and then solves for the joint coordinates which result in the pose +Tep using the ikine_NR method.

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda()
+>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> panda.ikine_NR(Tep)
+IKSolution(q=array([-2.5448,  0.0635, -0.6093, -2.5018, -1.2853,  1.3894, -0.2102]), success=False, iterations=100, searches=100, residual=0.0, reason='iteration and search limit reached, 100 numpy.LinAlgError encountered')
+
+
+

Notes

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian.

+

This class supports null-space motion to assist with maximising manipulability and +avoiding joint limits. These are enabled by setting kq and km to non-zero values.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
IK_NR

An IK Solver class which implements the Newton-Raphson optimisation technique

+
+
ikine_LM

Implements the IK_LM class as a method within the ETS class

+
+
ikine_GN

Implements the IK_GN class as a method within the ETS class

+
+
ikine_QP

Implements the IK_QP class as a method within the ETS class

+
+
+
+
+

Changed in version 1.0.4: Added the Newton-Raphson IK solver method on the Robot class

+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/IK/stubs/roboticstoolbox.robot.Robot.Robot.ikine_QP.html b/IK/stubs/roboticstoolbox.robot.Robot.Robot.ikine_QP.html new file mode 100644 index 00000000..82442e96 --- /dev/null +++ b/IK/stubs/roboticstoolbox.robot.Robot.Robot.ikine_QP.html @@ -0,0 +1,291 @@ + + + + + + + + + + + Robot.ikine_QP — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

Robot.ikine_QP

+
+
+Robot.ikine_QP(Tep, end=None, start=None, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, seed=None, kj=1.0, ks=1.0, kq=0.0, km=0.0, ps=0.0, pi=0.3, **kwargs)
+

Quadratic Programming Numerical Inverse Kinematics Solver

+

A method that provides functionality to perform numerical inverse kinematics +(IK) using a quadratic progamming approach.

+

See the Inverse Kinematics Docs Page for more details and for a +tutorial on numerical IK, see here.

+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose

  • +
  • end (Union[str, Link, Gripper, None]) – the link considered as the end-effector

  • +
  • start (Union[str, Link, Gripper, None]) – the link considered as the base frame, defaults to the robots’s base frame

  • +
  • q0 (Union[ndarray, List[float], Tuple[float], Set[float], None]) – The initial joint coordinate vector

  • +
  • ilimit (int) – How many iterations are allowed within a search before a new search +is started

  • +
  • slimit (int) – How many searches are allowed before being deemed unsuccessful

  • +
  • tol (float) – Maximum allowed residual error E

  • +
  • mask (Union[ndarray, List[float], Tuple[float], Set[float], None]) – A 6 vector which assigns weights to Cartesian degrees-of-freedom +error priority

  • +
  • joint_limits (bool) – Reject solutions with joint limit violations

  • +
  • seed (Optional[int]) – A seed for the private RNG used to generate random joint coordinate +vectors

  • +
  • kj – A gain for joint velocity norm minimisation

  • +
  • ks – A gain which adjusts the cost of slack (intentional error)

  • +
  • kq (float) – The gain for joint limit avoidance. Setting to 0.0 will remove this +completely from the solution

  • +
  • km (float) – The gain for maximisation. Setting to 0.0 will remove this completely +from the solution

  • +
  • ps (float) – The minimum angle/distance (in radians or metres) in which the joint is +allowed to approach to its limit

  • +
  • pi (Union[ndarray, float]) – The influence angle/distance (in radians or metres) in null space motion +becomes active

  • +
+
+
Raises:
+

ImportError – If the package qpsolvers is not installed

+
+
+

Synopsis

+

Each iteration uses the following approach

+
+\[\vec{q}_{k+1} = \vec{q}_{k} + \dot{\vec{q}}.\]
+

where the QP is defined as

+
+\[\begin{split}\min_x \quad f_o(\vec{x}) &= \frac{1}{2} \vec{x}^\top \mathcal{Q} \vec{x}+ \mathcal{C}^\top \vec{x}, \\ +\text{subject to} \quad \mathcal{J} \vec{x} &= \vec{\nu}, \\ +\mathcal{A} \vec{x} &\leq \mathcal{B}, \\ +\vec{x}^- &\leq \vec{x} \leq \vec{x}^+\end{split}\]
+

with

+
+\[\begin{split}\vec{x} &= +\begin{pmatrix} + \dvec{q} \\ \vec{\delta} +\end{pmatrix} \in \mathbb{R}^{(n+6)} \\ +\mathcal{Q} &= +\begin{pmatrix} + \lambda_q \mat{1}_{n} & \mathbf{0}_{6 \times 6} \\ \mathbf{0}_{n \times n} & \lambda_\delta \mat{1}_{6} +\end{pmatrix} \in \mathbb{R}^{(n+6) \times (n+6)} \\ +\mathcal{J} &= +\begin{pmatrix} + \mat{J}(\vec{q}) & \mat{1}_{6} +\end{pmatrix} \in \mathbb{R}^{6 \times (n+6)} \\ +\mathcal{C} &= +\begin{pmatrix} + \mat{J}_m \\ \bf{0}_{6 \times 1} +\end{pmatrix} \in \mathbb{R}^{(n + 6)} \\ +\mathcal{A} &= +\begin{pmatrix} + \mat{1}_{n \times n + 6} \\ +\end{pmatrix} \in \mathbb{R}^{(l + n) \times (n + 6)} \\ +\mathcal{B} &= +\eta +\begin{pmatrix} + \frac{\rho_0 - \rho_s} + {\rho_i - \rho_s} \\ + \vdots \\ + \frac{\rho_n - \rho_s} + {\rho_i - \rho_s} +\end{pmatrix} \in \mathbb{R}^{n} \\ +\vec{x}^{-, +} &= +\begin{pmatrix} + \dvec{q}^{-, +} \\ + \vec{\delta}^{-, +} +\end{pmatrix} \in \mathbb{R}^{(n+6)},\end{split}\]
+

where \(\vec{\delta} \in \mathbb{R}^6\) is the slack vector, +\(\lambda_\delta \in \mathbb{R}^+\) is a gain term which adjusts the +cost of the norm of the slack vector in the optimiser, +\(\dvec{q}^{-,+}\) are the minimum and maximum joint velocities, and +\(\dvec{\delta}^{-,+}\) are the minimum and maximum slack velocities.

+

Examples

+

The following example gets a panda robot object, makes a goal +pose Tep, and then solves for the joint coordinates which result in the pose +Tep using the ikine_QP method.

+
  File "/opt/hostedtoolcache/Python/3.7.17/x64/lib/python3.7/site-packages/roboticstoolbox/robot/IK.py", line 1331, in __init__
+    "the package qpsolvers is required for this class. \nInstall using 'pip"
+ImportError: the package qpsolvers is required for this class. 
+Install using 'pip install qpsolvers'
+
+
+

Notes

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian.

+

This class supports null-space motion to assist with maximising manipulability and +avoiding joint limits. These are enabled by setting kq and km to non-zero values.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
IK_NR

An IK Solver class which implements the Newton-Raphson optimisation technique

+
+
ikine_LM

Implements the IK_LM class as a method within the ETS class

+
+
ikine_GN

Implements the IK_GN class as a method within the ETS class

+
+
ikine_NR

Implements the IK_NR class as a method within the ETS class

+
+
+
+
+

Changed in version 1.0.4: Added the Quadratic Programming IK solver method on the Robot class

+
+
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/LICENSE b/LICENSE index e37a7bbf..bbc9077e 100644 --- a/LICENSE +++ b/LICENSE @@ -1,6 +1,6 @@ MIT License -Copyright (c) 2020 Peter Corke +Copyright (c) 2020 jhavl Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal diff --git a/Makefile b/Makefile deleted file mode 100644 index 053382ef..00000000 --- a/Makefile +++ /dev/null @@ -1,43 +0,0 @@ -.FORCE: - -BLUE=\033[0;34m -BLACK=\033[0;30m - -help: - @echo "$(BLUE) make test - run all unit tests" - @echo " make coverage - run unit tests and coverage report" - @echo " make docs - build Sphinx documentation" - @echo " make docupdate - upload Sphinx documentation to GitHub pages" - @echo " make dist - build dist files" - @echo " make upload - upload to PyPI" - @echo " make clean - remove dist and docs build files" - @echo " make help - this message$(BLACK)" - -test: - python -m unittest - -coverage: - coverage run --omit=\*/test_\* -m unittest - coverage report - -docs: .FORCE - (cd docs; make html) - open docs/build/html/index.html - -docupdate: docs - git clone https://github.com/petercorke/pgraph-python.git --branch gh-pages --single-branch /tmp/gh-pages - cp -r docs/build/html/. /tmp/gh-pages - (cd /tmp/gh-pages; git add .; git commit -m "rebuilt docs"; git push origin gh-pages) - rm -rf /tmp/gh-pages - -dist: .FORCE - #$(MAKE) test - python -m build - -upload: .FORCE - twine upload dist/* - -clean: .FORCE - (cd docs; make clean) - -rm -r *.egg-info - -rm -r dist diff --git a/PGraph.html b/PGraph.html new file mode 100644 index 00000000..27a0590b --- /dev/null +++ b/PGraph.html @@ -0,0 +1,1363 @@ + + + + + + + + + PGraph module — Simple graph functionality for Python documentation + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

PGraph module

+
+
+class PGraph.DGraph(arg=None, metric=None, heuristic=None, verbose=False)[source]
+

Bases: PGraph

+

Class for directed graphs

+
Inheritance diagram of DGraph
+ + + +
+
+add_vertex(coord=None, name=None)[source]
+

Add vertex to directed graph

+
+
Parameters:
+
    +
  • coord (array-like, optional) – coordinate for an embedded graph, defaults to None

  • +
  • name (str, optional) – vertex name, defaults to “#i”

  • +
+
+
Returns:
+

new vertex

+
+
Return type:
+

DVertex

+
+
+
    +
  • g.add_vertex() creates a new vertex with optional coord and +name.

  • +
  • g.add_vertex(v) takes an instance or subclass of DVertex and adds +it to the graph

  • +
+
+ +
+
+classmethod vertex_copy(vertex)[source]
+
+ +
+ +
+
+class PGraph.DVertex(coord=None, name=None)[source]
+

Bases: Vertex

+

Vertex subclass for directed graphs

+

This class can be inherited to provide user objects with graph capability.

+
Inheritance diagram of DVertex
+ + + +
+
+connect(other, **kwargs)[source]
+

Connect two vertices with an edge

+
+
Parameters:
+
    +
  • dest (Vertex subclass) – The vertex to connect to

  • +
  • edge (Edge subclass, optional) – Use this as the edge object, otherwise a new Edge +object is created from the vertices being connected, +and the cost and edge parameters, defaults to None

  • +
  • cost (float, optional) – the cost to traverse this edge, defaults to None

  • +
  • data (Any, optional) – reference to arbitrary data associated with the edge, +defaults to None

  • +
+
+
Raises:
+

TypeError – vertex types are different subclasses

+
+
Returns:
+

the edge connecting the vertices

+
+
Return type:
+

Edge

+
+
+

v1.connect(v2) connects vertex v1 to vertex v2.

+
+

Note

+
    +
  • If the vertices subclass UVertex the edge is undirected, and if +they subclass DVertex the edge is directed.

  • +
  • Vertices must both be of the same Vertex subclass

  • +
+
+
+
Seealso:
+

Edge()

+
+
+
+ +
+
+remove()[source]
+
+ +
+ +
+
+class PGraph.Edge(v1=None, v2=None, cost=None, data=None)[source]
+

Bases: object

+

Edge class

+

Is used to represent directed directed and undirected edges.

+

Each edge has: +- cost cost of traversing this edge, required for planning methods +- data reference to arbitrary data associated with the edge +- v1 first vertex, start vertex for a directed edge +- v2 second vertex, end vertex for a directed edge

+
+

Note

+
    +
  • An undirected graph is created by having a single edge object in the +edgelist of _each_ vertex.

  • +
  • This class can be inherited to provide user objects with graph capability.

  • +
  • Inheritance is an alternative to providing arbitrary user data.

  • +
+
+

An Edge points to a pair of vertices. At connect time the vertices +get references back to the Edge object.

+

graph.add_edge(v1, v2) calls v1.connect(v2)

+
+
+connect(v1, v2)[source]
+

Add edge to the graph

+
+
Parameters:
+
    +
  • v1 (Vertex subclass) – start of the edge

  • +
  • v2 (Vertex subclass) – end of the edge

  • +
+
+
+

The edge is added to the graph and connects vertices v1 and v2.

+
+

Note

+

The vertices must already be added to the graph.

+
+
+ +
+
+property endpoints
+
+ +
+
+next(vertex)[source]
+

Return other end of an edge

+
+
Parameters:
+

vertex (Vertex subclass) – one vertex on the edge

+
+
Raises:
+

ValueErrorvertex is not on the edge

+
+
Returns:
+

the other vertex on the edge

+
+
Return type:
+

Vertex subclass

+
+
+

e.next(v1) is the vertex at the other end of edge e, ie. the +vertex that is not v1.

+
+ +
+
+vertices()[source]
+
+ +
+ +
+
+class PGraph.PGraph(arg=None, metric=None, heuristic=None, verbose=False)[source]
+

Bases: ABC

+
+
+classmethod Adjacency(A, coords=None, names=None)[source]
+

Create graph from adjacency matrix

+
+
Parameters:
+
    +
  • A (ndarray(N,N)) – adjacency matrix

  • +
  • coords (ndarray(N,M), optional) – coordinates of vertices, defaults to None

  • +
  • names (list(N) of str, optional) – names of vertices, defaults to None

  • +
+
+
Returns:
+

[description]

+
+
Return type:
+

[type]

+
+
+

Create a directed or undirected graph where non-zero elements A[i,j] +correspond to edges from vertex i to vertex j.

+
+

Warning

+

For undirected graph A should be symmetric but this +is not checked. Only the upper triangular part is used.

+
+
+ +
+
+classmethod Dict(d, reverse=False)[source]
+

Create graph from parent/child dictionary

+
+
Parameters:
+
    +
  • d (dict) – dictionary that maps from Vertex subclass to Vertex subclass

  • +
  • reverse (bool, optional) – reverse link direction, defaults to False

  • +
+
+
Returns:
+

graph

+
+
Return type:
+

UGraph or DGraph

+
+
+

Behaves like a constructor for a DGraph or UGraph from a +dictionary that maps vertices to parents. From this information it +can create a tree graph.

+

By default parent vertices are linked their children. If reverse is +True then children are linked to their parents.

+
+ +
+
+Laplacian()[source]
+

Laplacian matrix for the graph

+
+
Returns:
+

Laplacian matrix

+
+
Return type:
+

NumPy ndarray

+
+
+

g.Laplacian() is the Laplacian matrix (NxN) of the graph where N +is the number of vertices.

+
+

Note

+
    +
  • Laplacian is always positive-semidefinite.

  • +
  • Laplacian has at least one zero eigenvalue.

  • +
  • +
    The number of zero-valued eigenvalues is the number of connected

    components in the graph.

    +
    +
    +
  • +
+
+
+
Seealso:
+

adjacency() incidence() degree()

+
+
+
+ +
+
+add_edge(v1, v2, **kwargs)[source]
+

Add an edge to the graph (superclass method)

+
+
Parameters:
+
    +
  • v1 (Vertex subclass) – first vertex (start if a directed graph)

  • +
  • v2 (Vertex subclass) – second vertex (end if a directed graph)

  • +
  • kwargs – optional arguments to pass to Vertex.connect

  • +
+
+
Returns:
+

edge

+
+
Return type:
+

Edge

+
+
+

Create an edge between a vertex pair and adds it to the graph.

+

This is a graph centric way of creating an edge. The +alternative is the connect method of a vertex.

+
+
Seealso:
+

Edge.connect() Vertex.connect()

+
+
+
+ +
+
+add_vertex(vertex, name=None)[source]
+

Add a vertex to the graph (superclass method)

+
+
Parameters:
+
    +
  • vertex (Vertex subclass) – vertex to add

  • +
  • name (str) – name of vertex

  • +
+
+
+

G.add_vertex(v) add vertex v to the graph G.

+

If the vertex has no name and name is None give it a default name +#N where N is a consecutive integer.

+

The vertex is placed into a dictionary with a key equal to its name.

+
+ +
+
+adjacency()[source]
+

Adjacency matrix of graph

+
+
Returns:
+

adjacency matrix

+
+
Return type:
+

ndarray(N,N)

+
+
+

The elements of the adjacency matrix [i,j] are 1 if vertex i is +connected to vertex j, else 0.

+
+

Note

+
    +
  • vertices are numbered in their order of creation. A vertex index +can be resolved to a vertex reference by graph[i].

  • +
  • for an undirected graph the matrix is symmetric

  • +
  • Eigenvalues of A are real and are known as the spectrum of the graph.

  • +
  • The element A[i,j] can be considered the number of walks of length one +edge from vertex i to vertex j (either zero or one).

  • +
  • If Ak = A ** k the element Ak[i,j] is the number of +walks of length k from vertex i to vertex j.

  • +
+
+
+
Seealso:
+

Laplacian() incidence() degree()

+
+
+
+ +
+
+average_degree()[source]
+

Average degree of the graph

+
+
Returns:
+

average degree

+
+
Return type:
+

float

+
+
+

Average degree is \(2 E / N\) for an undirected graph and +\(E / N\) for a directed graph where \(E\) is the total number of +edges and \(N\) is the number of vertices.

+
+ +
+
+closest(coord)[source]
+

Vertex closest to point

+
+
Parameters:
+

coord (ndarray(n)) – coordinates of a point

+
+
Returns:
+

closest vertex

+
+
Return type:
+

Vertex subclass

+
+
+

Returns the vertex closest to the given point. Distance is computed +according to the graph’s metric.

+
+
Seealso:
+

metric()

+
+
+
+ +
+
+component(c)[source]
+

All vertices in specified graph component

+

graph.component(c) is a list of all vertices in graph component c.

+
+ +
+
+connectivity(vertices=None)[source]
+

Graph connectivity

+
+
Returns:
+

a list with the number of edges per vertex

+
+
Return type:
+

list

+
+
+

The average vertex connectivity is:

+
mean(g.connectivity())
+
+
+

and the minimum vertex connectivity is:

+
min(g.connectivity())
+
+
+
+ +
+
+copy()[source]
+

Deepcopy of graph

+
+
Parameters:
+

g (PGraph) – A graph

+
+
Returns:
+

deep copy

+
+
Return type:
+

PGraph

+
+
+
+ +
+
+degree()[source]
+

Degree matrix of graph

+
+
Returns:
+

degree matrix

+
+
Return type:
+

ndarray(N,N)

+
+
+

This is a diagonal matrix where element [i,i] is the number +of edges connected to vertex id i.

+
+
Seealso:
+

adjacency() incidence() laplacian()

+
+
+
+ +
+
+distance()[source]
+

Distance matrix of graph

+
+
Returns:
+

distance matrix

+
+
Return type:
+

ndarray(n,n)

+
+
+

The elements of the distance matrix D[i,j] is the edge cost of moving +from vertex i to vertex j. It is zero if the vertices are not +connected.

+
+ +
+
+dotfile(filename=None, direction=None)[source]
+

Export graph as a GraphViz dot file

+
+
Parameters:
+

filename (str, optional) – filename to save graph to, defaults to None

+
+
+

g.dotfile() creates the specified file which contains the +GraphViz code to represent the embedded graph. By default output +is to the console.

+
+

Note

+
    +
  • The graph is undirected if it is a subclass of UGraph

  • +
  • The graph is directed if it is a subclass of DGraph

  • +
  • Use neato rather than dot to get the embedded layout

  • +
+
+
+

Note

+

If filename is a file object then the file will not +be closed after the GraphViz model is written.

+
+
+
Seealso:
+

showgraph()

+
+
+
+ +
+
+edges()[source]
+

Get all edges in graph (superclass method)

+
+
Returns:
+

All edges in the graph

+
+
Return type:
+

list of Edge references

+
+
+

We can iterate over all edges in the graph by:

+
for e in g.edges():
+    print(e)
+
+
+
+

Note

+

The edges() of a Vertex is a list of all edges connected +to that vertex.

+
+
+
Seealso:
+

Vertex.edges()

+
+
+
+ +
+
+property heuristic
+

Get the heuristic distance metric for graph

+
+
Returns:
+

heuristic distance metric

+
+
Return type:
+

callable

+
+
+

This is a function of a vector and returns a scalar.

+
+ +
+
+highlight_edge(edge, scale=2, color='r', alpha=0.5)[source]
+

Highlight an edge in the graph

+
+
Parameters:
+
    +
  • edge (Edge subclass) – The edge to highlight

  • +
  • scale (float, optional) – Overwrite with a line this much bigger than the original, +defaults to 1.5

  • +
  • color (str, optional) – Overwrite with a line in this color, defaults to ‘r’

  • +
+
+
+
+ +
+
+highlight_path(path, block=False, **kwargs)[source]
+

Highlight a path through the graph

+
+
Parameters:
+
    +
  • path ([type]) – [description]

  • +
  • block (bool, optional) – [description], defaults to True

  • +
+
+
+

The vertices and edges along the path are overwritten with a different +size/width and color.

+
+
Seealso:
+

highlight_vertex() highlight_edge()

+
+
+
+ +
+
+highlight_vertex(vertex, scale=2, color='r', alpha=0.5)[source]
+

Highlight a vertex in the graph

+
+
Parameters:
+
    +
  • edge (Vertex subclass) – The vertex to highlight

  • +
  • scale (float, optional) – Overwrite with a line this much bigger than the original, +defaults to 1.5

  • +
  • color (str, optional) – Overwrite with a line in this color, defaults to ‘r’

  • +
+
+
+
+ +
+
+incidence()[source]
+

Incidence matrix of graph

+
+
Returns:
+

incidence matrix

+
+
Return type:
+

ndarray(n,ne)

+
+
+

The elements of the incidence matrix I[i,j] are 1 if vertex i is +connected to edge j, else 0.

+
+

Note

+
    +
  • vertices are numbered in their order of creation. A vertex index +can be resolved to a vertex reference by graph[i].

  • +
  • edges are numbered in the order they appear in graph.edges().

  • +
+
+
+
Seealso:
+

Laplacian() adjacency() degree()

+
+
+
+ +
+
+iscyclic()[source]
+
+ +
+
+property metric
+

Get the distance metric for graph

+
+
Returns:
+

distance metric

+
+
Return type:
+

callable

+
+
+

This is a function of a vector and returns a scalar.

+
+ +
+
+property n
+

Number of vertices

+
+
Returns:
+

Number of vertices

+
+
Return type:
+

int

+
+
+
+ +
+
+property nc
+

Number of components

+
+
Returns:
+

Number of components

+
+
Return type:
+

int

+
+
+
+

Note

+
    +
  • Components are labeled from 0 to g.nc-1.

  • +
  • A graph coloring algorithm is run if the graph connectivity +has changed.

  • +
+
+
+

Note

+

A lazy approach is used, and if a connectivity changing +operation has been performed since the last call, the graph +coloring algorithm is run which is potentially expensive for +a large graph.

+
+
+ +
+
+property ne
+

Number of edges

+
+
Returns:
+

Number of vertices

+
+
Return type:
+

int

+
+
+
+ +
+
+path_Astar(S, G, verbose=False, summary=False)[source]
+

A* search for path

+
+
Parameters:
+
    +
  • S (Vertex subclass) – start vertex

  • +
  • G (Vertex subclass) – goal vertex

  • +
+
+
Returns:
+

list of vertices from S to G inclusive, path length, tree

+
+
Return type:
+

list of Vertex subclass, float, dict

+
+
+

Returns a list of vertices that form a path from vertex S to +vertex G if possible, otherwise return None.

+

The search tree is returned as dict that maps a vertex to its parent.

+

The heuristic is the distance metric of the graph, which defaults to +Euclidean distance.

+
+
Seealso:
+

heuristic()

+
+
+
+ +
+
+path_BFS(S, G, verbose=False, summary=False)[source]
+

Breadth-first search for path

+
+
Parameters:
+
    +
  • S (Vertex subclass) – start vertex

  • +
  • G (Vertex subclass) – goal vertex

  • +
+
+
Returns:
+

list of vertices from S to G inclusive, path length

+
+
Return type:
+

list of Vertex subclass, float

+
+
+

Returns a list of vertices that form a path from vertex S to +vertex G if possible, otherwise return None.

+
+ +
+
+path_UCS(S, G, verbose=False, summary=False)[source]
+

Uniform cost search for path

+
+
Parameters:
+
    +
  • S (Vertex subclass) – start vertex

  • +
  • G (Vertex subclass) – goal vertex

  • +
+
+
Returns:
+

list of vertices from S to G inclusive, path length, tree

+
+
Return type:
+

list of Vertex subclass, float, dict

+
+
+

Returns a list of vertices that form a path from vertex S to +vertex G if possible, otherwise return None.

+

The search tree is returned as dict that maps a vertex to its parent.

+

The heuristic is the distance metric of the graph, which defaults to +Euclidean distance.

+
+ +
+
+plot(colorcomponents=True, force2d=False, vopt={}, eopt={}, text={}, block=False, grid=True, ax=None)[source]
+

Plot the graph

+
+
Parameters:
+
    +
  • vopt (dict, optional) – vertex format, defaults to 12pt o-marker

  • +
  • eopt (dict, optional) – edge format, defaults to None

  • +
  • text (False or dict, optional) – text label format, defaults to None

  • +
  • colorcomponents – color vertices and edges by component, defaults to None

  • +
  • block (bool, optional) – block until figure is dismissed, defaults to True

  • +
+
+
+

The graph is plotted using matplotlib.

+

If colorcomponents is True then each component is assigned a unique +color. vertex and edge cannot include a color keyword.

+

If text is a dict it is used to format text labels for the vertices +which are the vertex names. If text is None default formatting is +used. If text is False no labels are added.

+
+ +
+
+remove(x)[source]
+

Remove element from graph (superclass method)

+
+
Parameters:
+

x (Edge or Vertex subclass) – element to remove from graph

+
+
Raises:
+

TypeError – unknown type

+
+
+

The edge or vertex is removed, and all references and lists are +updated.

+
+

Warning

+

The connectivity of the network may be changed.

+
+
+ +
+
+samecomponent(v1, v2)[source]
+

Test if vertices belong to same graph component

+
+
Parameters:
+
    +
  • v1 (Vertex subclass) – vertex

  • +
  • v2 (Vertex subclass) – vertex

  • +
+
+
Returns:
+

true if vertices belong to same graph component

+
+
Return type:
+

bool

+
+
+

Test whether vertices belong to the same component. For a:

+
    +
  • directed graph this implies a path between them

  • +
  • undirected graph there is not necessarily a path between them

  • +
+
+ +
+
+show()[source]
+
+ +
+
+showgraph(**kwargs)[source]
+

Display graph in a browser tab

+
+
Parameters:
+

kwargs – arguments passed to dotfile()

+
+
+

g.showgraph() renders and displays the graph in a browser tab. The +graph is exported in GraphViz format, rendered to +PDF using dot and then displayed in a browser tab.

+
+
Seealso:
+

dotfile()

+
+
+
+ +
+ +
+
+class PGraph.UGraph(arg=None, metric=None, heuristic=None, verbose=False)[source]
+

Bases: PGraph

+

Class for undirected graphs

+
Inheritance diagram of UGraph
+ + + +
+
+add_vertex(coord=None, name=None)[source]
+

Add vertex to undirected graph

+
+
Parameters:
+
    +
  • coord (array-like, optional) – coordinate for an embedded graph, defaults to None

  • +
  • name (str, optional) – vertex name, defaults to “#i”

  • +
+
+
Returns:
+

new vertex

+
+
Return type:
+

UVertex

+
+
+
    +
  • g.add_vertex() creates a new vertex with optional coord and +name.

  • +
  • g.add_vertex(v) takes an instance or subclass of UVertex and adds +it to the graph

  • +
+
+ +
+
+classmethod vertex_copy(vertex)[source]
+
+ +
+ +
+
+class PGraph.UVertex(coord=None, name=None)[source]
+

Bases: Vertex

+

Vertex subclass for undirected graphs

+

This class can be inherited to provide user objects with graph capability.

+
Inheritance diagram of UVertex
+ + + +
+
+connect(other, **kwargs)[source]
+

Connect two vertices with an edge

+
+
Parameters:
+
    +
  • dest (Vertex subclass) – The vertex to connect to

  • +
  • edge (Edge subclass, optional) – Use this as the edge object, otherwise a new Edge +object is created from the vertices being connected, +and the cost and edge parameters, defaults to None

  • +
  • cost (float, optional) – the cost to traverse this edge, defaults to None

  • +
  • data (Any, optional) – reference to arbitrary data associated with the edge, +defaults to None

  • +
+
+
Raises:
+

TypeError – vertex types are different subclasses

+
+
Returns:
+

the edge connecting the vertices

+
+
Return type:
+

Edge

+
+
+

v1.connect(v2) connects vertex v1 to vertex v2.

+
+

Note

+
    +
  • If the vertices subclass UVertex the edge is undirected, and if +they subclass DVertex the edge is directed.

  • +
  • Vertices must both be of the same Vertex subclass

  • +
+
+
+
Seealso:
+

Edge()

+
+
+
+ +
+ +
+
+class PGraph.Vertex(coord=None, name=None)[source]
+

Bases: object

+

Superclass for vertices of directed and non-directed graphs.

+
+
Each vertex has:
    +
  • name

  • +
  • label an int indicating which graph component contains it

  • +
  • _edgelist a list of edge objects that connect this vertex to others

  • +
  • coord the coordinate in an embedded graph (optional)

  • +
+
+
+
+
+closest()[source]
+
+ +
+
+connect(dest, edge=None, cost=None, data=None)[source]
+

Connect two vertices with an edge

+
+
Parameters:
+
    +
  • dest (Vertex subclass) – The vertex to connect to

  • +
  • edge (Edge subclass, optional) – Use this as the edge object, otherwise a new Edge +object is created from the vertices being connected, +and the cost and edge parameters, defaults to None

  • +
  • cost (float, optional) – the cost to traverse this edge, defaults to None

  • +
  • data (Any, optional) – reference to arbitrary data associated with the edge, +defaults to None

  • +
+
+
Raises:
+

TypeError – vertex types are different subclasses

+
+
Returns:
+

the edge connecting the vertices

+
+
Return type:
+

Edge

+
+
+

v1.connect(v2) connects vertex v1 to vertex v2.

+
+

Note

+
    +
  • If the vertices subclass UVertex the edge is undirected, and if +they subclass DVertex the edge is directed.

  • +
  • Vertices must both be of the same Vertex subclass

  • +
+
+
+
Seealso:
+

Edge()

+
+
+
+ +
+
+copy(cls=None)[source]
+
+ +
+
+property degree
+

Degree of vertex

+
+
Returns:
+

degree of the vertex

+
+
Return type:
+

int

+
+
+

Returns the number of edges connected to the vertex.

+
+

Note

+

For a DGraph only outgoing edges are considered.

+
+
+
Seealso:
+

edges()

+
+
+
+ +
+
+distance(coord)[source]
+

Distance from vertex to point

+
+
Parameters:
+

coord (ndarray(n) or Vertex) – coordinates of the point

+
+
Returns:
+

distance

+
+
Return type:
+

float

+
+
+

Distance is computed according to the graph’s metric.

+
+
Seealso:
+

metric()

+
+
+
+ +
+
+edges()[source]
+

All outgoing edges of vertex

+
+
Returns:
+

List of all edges leaving this vertex

+
+
Return type:
+

list of Edge

+
+
+
+

Note

+
    +
  • For a directed graph the edges are those leaving this vertex

  • +
  • +
    For a non-directed graph the edges are those leaving or entering

    this vertex

    +
    +
    +
  • +
+
+
+ +
+
+edgeto(dest)[source]
+

Get edge connecting vertex to specific neighbour

+
+
Parameters:
+

dest (Vertex subclass) – a neigbouring vertex

+
+
Raises:
+

ValueErrordest is not a neighbour

+
+
Returns:
+

the edge from this vertex to dest

+
+
Return type:
+

Edge

+
+
+
+

Note

+
    +
  • For a directed graph dest must be at the arrow end of the edge

  • +
+
+
+ +
+
+heuristic_distance(v2)[source]
+
+ +
+
+incidences()[source]
+

Neighbours and edges of a vertex

+

v.incidences() is a generator that returns a list of incidences, +tuples of (vertex, edge) for all neighbours of the vertex v.

+
+

Note

+

For a directed graph the edges are those leaving this vertex

+
+
+ +
+
+isneighbour(vertex)[source]
+

Test if vertex is a neigbour

+
+
Parameters:
+

vertex (Vertex subclass) – vertex reference

+
+
Returns:
+

true if a neighbour

+
+
Return type:
+

bool

+
+
+

For a directed graph this is true only if the edge is from self to +vertex.

+
+ +
+
+neighbors()[source]
+

Neighbors of a vertex

+

v.neighbors() is a list of neighbors of this vertex.

+
+

Note

+

For a directed graph the neighbours are those on edges leaving this vertex

+
+
+ +
+
+neighbours()[source]
+

Neighbours of a vertex

+

v.neighbours() is a list of neighbours of this vertex.

+
+

Note

+

For a directed graph the neighbours are those on edges leaving this vertex

+
+
+ +
+
+property x
+

The x-coordinate of an embedded vertex

+
+
Returns:
+

The x-coordinate

+
+
Return type:
+

float

+
+
+
+ +
+
+property y
+

The y-coordinate of an embedded vertex

+
+
Returns:
+

The y-coordinate

+
+
Return type:
+

float

+
+
+
+ +
+
+property z
+

The z-coordinate of an embedded vertex

+
+
Returns:
+

The z-coordinate

+
+
Return type:
+

float

+
+
+
+ +
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/README.md b/README.md deleted file mode 100644 index ecc553ee..00000000 --- a/README.md +++ /dev/null @@ -1,138 +0,0 @@ -# PGraph: graphs for Python - -[![A Python Robotics Package](https://raw.githubusercontent.com/petercorke/robotics-toolbox-python/master/.github/svg/py_collection.min.svg)](https://github.com/petercorke/robotics-toolbox-python) -[![Powered by Spatial Maths](https://raw.githubusercontent.com/petercorke/spatialmath-python/master/.github/svg/sm_powered.min.svg)](https://github.com/petercorke/spatialmath-python) -[![QUT Centre for Robotics Open Source](https://github.com/qcr/qcr.github.io/raw/master/misc/badge.svg)](https://qcr.github.io) - -[![PyPI version fury.io](https://badge.fury.io/py/pgraph-python.svg)](https://pypi.python.org/pypi/pgraph-python/) -[![PyPI pyversions](https://img.shields.io/pypi/pyversions/pgraph-python)](https://pypi.python.org/pypi/pgraph-python/) -[![GitHub license](https://img.shields.io/github/license/Naereen/StrapDown.js.svg)](https://github.com/petercorke/pgraph-python/blob/master/LICENSE) - -[![Build Status](https://github.com/petercorke/pgraph-python/actions/workflows/master.yml/badge.svg)](https://github.com/petercorke/pgraph-python/actions?query=workflow%3Abuild) -[![Coverage](https://codecov.io/gh/petercorke/pgraph-python/branch/master/graph/badge.svg)](https://codecov.io/gh/petercorke/pgraph-python) -![pypi downloads](https://img.shields.io/pypi/dw/pgraph-python) - -- [GitHub repository](https://github.com/petercorke/pgraph-python) -- [Wiki (examples and details)](https://github.com/petercorke/pgraph-python/wiki) -- [Documentation](https://petercorke.github.io/pgraph-python) -- [Changelog](https://github.com/petercorke/pgraph-python/blob/master/CHANGELOG.md) -- Dependencies: [`numpy`](https://numpy.org) [`spatialmath`](https://github.com/bdaiinstitute/spatialmath-python) - - -This Python package allows the manipulation of directed and non-directed graphs. Also supports embedded graphs. It is suitable for graphs with thousands of nodes. - -![road network](https://github.com/petercorke/pgraph-python/raw/master/examples/roads.png) - -``` -from pgraph import * -import json - -# load places and routes -with open('places.json', 'r') as f: - places = json.loads(f.read()) -with open('routes.json', 'r') as f: - routes = json.loads(f.read()) - -# build the graph -g = UGraph() - -for name, info in places.items(): - g.add_vertex(name=name, coord=info["utm"]) - -for route in routes: - g.add_edge(route[0], route[1], cost=route[2]) - -# plan a path from Hughenden to Brisbane -p = g.path_Astar('Hughenden', 'Brisbane') -g.plot(block=False) # plot it -g.highlight_path(p) # overlay the path -``` - -### Properties and methods of the graph -Graphs belong to the class `UGraph` or `DGraph` for undirected or directed graphs respectively. The graph is essentially a container for the vertices. - -- `g.add_vertex()` add a vertex -- `g.n` the number of vertices -- `g` is an iterator over vertices, can be used as `for vertex in g:` -- `g[i]` reference a vertex by its index or name - - *** -- `g.add_edge()` connect two vertices -- `g.edges()` all edges in the graph -- `g.plot()` plots the vertices and edges -- `g.nc` the number of graph components, 1 if fully connected -- `g.component(v)` the component that vertex `v` belongs to - - *** -- `g.path_BFS()` breadth-first search -- `g.path_Astar()` A* search - - *** -- `g.adjacency()` adjacency matrix -- `g.Laplacian()` Laplacian matrix -- `g.incidence()` incidence matrix - -### Properties and methods of a vertex -Vertices belong to the class `UVertex` (for undirected graphs) or `DVertex` (for directed graphs), which are each subclasses of `Vertex`. - -- `v.coord` the coordinate vector for embedded graph (optional) -- `v.name` the name of the vertex (optional) -- `v.neighbours()` is a list of the neighbouring vertices -- `v1.samecomponent(v2)` predicate for vertices belonging to the same component - -Vertices can be named and referenced by name. - -### Properties and methods of an edge -Edges are instances of the class `Edge`. -Edges are not referenced by the graph object, each edge references a pair of vertices, and the vertices reference the edges. For a directed graph only the start vertex of an edge references the edge object, whereas for an undirected graph both vertices reference the edge object. - -- `e.cost` cost of edge for planning methods -- `e.next(v)` vertex on edge `e` that is not `v` -- `e.v1`, `e.v2` the two vertices that define the edge `e` - -## Modifying a graph - -- `g.remove(v)` remove vertex `v` -- `e.remove()` remove edge `e` - -## Subclasing pgraph classes - -Consider a user class `Foo` that we would like to connect using a graph _overlay_, ie. -instances of `Foo` becomes vertices in a graph. - -- Have it subclass either `DVertex` or `UVertex` depending on graph type -- Then place instances of `Foo` into the graph using `add_vertex` and create edges as required - -``` -class Foo(UVertex): - # foo stuff goes here - -f1 = Foo(...) -f2 = Foo(...) - -g = UGraph() # create a new undirected graph -g.add_vertex(f1) -g.add_vertex(f2) - -f1.connect(f2, cost=3) -for f in f1.neighbours(): - # say hi to the neighbours -``` - -## Under the hood - -The key objects and their interactions are shown below. - -![data structures](https://github.com/petercorke/pgraph-python/raw/master/docs/source/datastructures.png) - -## MATLAB version - -This is a re-engineered version of [PGraph.m](https://github.com/petercorke/spatialmath-matlab/blob/master/PGraph.m) which ships as part of the [Spatial Math Toolbox for MATLAB](https://github.com/petercorke/spatialmath-matlab). This class is used to support bundle adjustment, pose-graph SLAM and various planners such as PRM, RRT and Lattice. - -The Python version was designed from the start to work with directed and undirected graphs, whereas directed graphs were a late addition to the MATLAB version. Semantics are similar but not identical. 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b/_images/redcar.png new file mode 100644 index 00000000..ae3716ff Binary files /dev/null and b/_images/redcar.png differ diff --git a/_images/swift.png b/_images/swift.png new file mode 100644 index 00000000..29bd0ad7 Binary files /dev/null and b/_images/swift.png differ diff --git a/_images/transforms2d.png b/_images/transforms2d.png new file mode 100644 index 00000000..930dbe70 Binary files /dev/null and b/_images/transforms2d.png differ diff --git a/_images/transforms3d.png b/_images/transforms3d.png new file mode 100644 index 00000000..b210dd8f Binary files /dev/null and b/_images/transforms3d.png differ diff --git a/_images/vp_ss_0000-2.png b/_images/vp_ss_0000-2.png new file mode 100644 index 00000000..d218f55a Binary files /dev/null and b/_images/vp_ss_0000-2.png differ diff --git a/_modules/PGraph.html b/_modules/PGraph.html new file mode 100644 index 00000000..a2b94c2e --- /dev/null +++ b/_modules/PGraph.html @@ -0,0 +1,2189 @@ + + + + + + + + PGraph — Simple graph functionality for Python documentation + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +

Source code for PGraph

+from abc import ABC
+import sys
+import numpy as np
+import matplotlib.pyplot as plt
+import copy
+from collections.abc import Iterable
+import tempfile
+import subprocess
+import webbrowser
+
+from spatialmath.base.graphics import axes_logic
+
+
+
+[docs] +class PGraph(ABC): + + def __init__(self, arg=None, metric=None, heuristic=None, verbose=False): + # we use a list and a dict, the list respects the order of adding + self._vertexlist = [] + self._vertexdict = {} + self._edgelist = set() + self._verbose = verbose + self._ncomponents = 0 + self._connectivitychange = False + if metric is None: + self.metric = "L2" + else: + self.metric = metric + if heuristic is None: + self.heuristic = self.metric + else: + self.heuristic = heuristic + + def __str__(self): + s = f"{self.__class__.__name__}: {self.n} {'vertex' if self.n==1 else 'vertices'}, {self.ne} edge{'s'[:self.ne^1]}, {self.nc} component{'s'[:self.nc^1]}" + return s + + def __repr__(self): + return str(self) + +
+[docs] + @classmethod + def Dict(cls, d, reverse=False): + """ + Create graph from parent/child dictionary + + :param d: dictionary that maps from ``Vertex`` subclass to ``Vertex`` subclass + :type d: dict + :param reverse: reverse link direction, defaults to False + :type reverse: bool, optional + :return: graph + :rtype: UGraph or DGraph + + Behaves like a constructor for a ``DGraph`` or ``UGraph`` from a + dictionary that maps vertices to parents. From this information it + can create a tree graph. + + By default parent vertices are linked their children. If ``reverse`` is + True then children are linked to their parents. + """ + + g = cls() + + for vertex, parent in d.items(): + if isinstance(vertex, str): + vertex_name = vertex + else: + vertex_name = vertex.name + + if vertex_name in g: + vertex = g[vertex_name] + else: + vertex = g.add_vertex(UVertex(), name=vertex_name) + + if isinstance(parent, str): + parent_name = parent + else: + parent_name = parent.name + if parent_name in g: + parent = g[parent_name] + else: + parent = g.add_vertex(UVertex(), name=parent_name) + + if reverse: + g.add_edge(vertex, parent) + else: + g.add_edge(parent, vertex) + + return g
+ + +
+[docs] + @classmethod + def Adjacency(cls, A, coords=None, names=None): + """ + Create graph from adjacency matrix + + :param A: adjacency matrix + :type A: ndarray(N,N) + :param coords: coordinates of vertices, defaults to None + :type coords: ndarray(N,M), optional + :param names: names of vertices, defaults to None + :type names: list(N) of str, optional + + :return: [description] + :rtype: [type] + + Create a directed or undirected graph where non-zero elements ``A[i,j]`` + correspond to edges from vertex ``i`` to vertex ``j``. + + .. warning:: For undirected graph ``A`` should be symmetric but this + is not checked. Only the upper triangular part is used. + """ + + if A.shape[0] != A.shape[1]: + raise ValueError("Adjacency matrix must be square") + if names is not None and len(names) != A.shape[0]: + raise ValueError("length of names must match dimension of adjacency matrix") + if coords is not None and coords.shape[0] != A.shape[0]: + raise ValueError("coords must have same number of rows as adjacency matrix") + + g = cls() + + name = None + coord = None + for i in range(A.shape[0]): + if names is not None: + name = names[i] + if coords is not None: + coord = coords[i, :] + g.add_vertex(name=name, coord=coord) + + if isinstance(g, UGraph): + # undirected graph + for i in range(A.shape[0]): + for j in range(i + 1, A.shape[1]): + if A[i, j] > 0: + g[i].connect(g[j], cost=A[i, j]) + else: + # directed graph + for i in range(A.shape[0]): + for j in range(A.shape[1]): + if A[i, j] > 0: + if i == j: + raise ValueError("loops in graph not supported") + g[i].connect(g[j], cost=A[i, j]) + + return g
+ + +
+[docs] + def copy(self): + """ + Deepcopy of graph + + :param g: A graph + :type g: PGraph + :return: deep copy + :rtype: PGraph + """ + return copy.deepcopy(self)
+ + +
+[docs] + def add_vertex(self, vertex, name=None): + """ + Add a vertex to the graph (superclass method) + + :param vertex: vertex to add + :type vertex: Vertex subclass + :param name: name of vertex + :type name: str + + ``G.add_vertex(v)`` add vertex ``v`` to the graph ``G``. + + If the vertex has no name and ``name`` is None give it a default name + ``#N`` where ``N`` is a consecutive integer. + + The vertex is placed into a dictionary with a key equal to its name. + """ + if name is None: + name = vertex.name + if name is None: + name = f"#{len(self._vertexlist)}" + vertex.name = name + self._vertexlist.append(vertex) + self._vertexdict[vertex.name] = vertex + if self._verbose: + print(f"New vertex {vertex.name}: {vertex.coord}") + vertex._graph = self + self._connectivitychange = True + return vertex
+ + +
+[docs] + def add_edge(self, v1, v2, **kwargs): + """ + Add an edge to the graph (superclass method) + + :param v1: first vertex (start if a directed graph) + :type v1: Vertex subclass + :param v2: second vertex (end if a directed graph) + :type v2: Vertex subclass + :param kwargs: optional arguments to pass to ``Vertex.connect`` + :return: edge + :rtype: Edge + + Create an edge between a vertex pair and adds it to the graph. + + This is a graph centric way of creating an edge. The + alternative is the ``connect`` method of a vertex. + + :seealso: :meth:`Edge.connect` :meth:`Vertex.connect` + """ + if isinstance(v1, str): + v1 = self[v1] + elif not isinstance(v1, Vertex): + raise TypeError("v1 must be Vertex subclass or string name") + if isinstance(v2, str): + v2 = self[v2] + elif not isinstance(v2, Vertex): + raise TypeError("v2 must be Vertex subclass or string name") + + if self._verbose: + print(f"New edge from {v1.name} to {v2.name}") + return v1.connect(v2, **kwargs)
+ + +
+[docs] + def remove(self, x): + """ + Remove element from graph (superclass method) + + :param x: element to remove from graph + :type x: Edge or Vertex subclass + :raises TypeError: unknown type + + The edge or vertex is removed, and all references and lists are + updated. + + .. warning:: The connectivity of the network may be changed. + """ + if isinstance(x, Edge): + # remove an edge + + # remove edge from the edgelist of connected vertices + x.v1._edgelist.remove(x) + x.v2._edgelist.remove(x) + + # indicate that connectivity has changed + x.v1._connectivitychange = True + x.v2._connectivitychange = True + self._connectivitychange = True + + # remove references to the vertices + x.v1 = None + x.v2 = None + + # remove from list of all edges + self._edgelist.remove(x) + + elif isinstance(x, Vertex): + # remove a vertex + + # remove all edges of this vertex + for edge in copy.copy(x._edgelist): + self.remove(edge) + + # remove from list and dict of all edges + self._vertexlist.remove(x) + del self._vertexdict[x.name] + else: + raise TypeError("expecting Edge or Vertex")
+ + +
+[docs] + def show(self): + print("vertices:") + for v in self._vertexlist: + print(" " + str(v)) + print("edges:") + for e in self._edgelist: + print(" " + str(e))
+ + + @property + def n(self): + """ + Number of vertices + + :return: Number of vertices + :rtype: int + """ + return len(self._vertexdict) + + @property + def ne(self): + """ + Number of edges + + :return: Number of vertices + :rtype: int + """ + return len(self._edgelist) + + @property + def nc(self): + """ + Number of components + + :return: Number of components + :rtype: int + + .. note:: + + - Components are labeled from 0 to ``g.nc-1``. + - A graph coloring algorithm is run if the graph connectivity + has changed. + + .. note:: A lazy approach is used, and if a connectivity changing + operation has been performed since the last call, the graph + coloring algorithm is run which is potentially expensive for + a large graph. + """ + self._graphcolor() + return self._ncomponents + + def _metricfunc(self, metric): + + def L1(v): + return np.linalg.norm(v, 1) + + def L2(v): + return np.linalg.norm(v) + + def SE2(v): + # wrap angle to range [-pi, pi) + v[2] = (v[2] + np.pi) % (2 * np.pi) - np.pi + return np.linalg.norm(v) + + if callable(metric): + return metric + elif isinstance(metric, str): + if metric == "L1": + return L1 + elif metric == "L2": + return L2 + elif metric == "SE2": + return SE2 + else: + raise ValueError("unknown metric") + + @property + def metric(self): + """ + Get the distance metric for graph + + :return: distance metric + :rtype: callable + + This is a function of a vector and returns a scalar. + """ + return self._metric + + @metric.setter + def metric(self, metric): + r""" + Set the distance metric for graph + + :param metric: distance metric + :type metric: callable or str + + This is a function of a vector and returns a scalar. It can be + user defined function or a string: + + - 'L1' is the norm :math:`L_1 = \Sigma_i | v_i |` + - 'L2' is the norm :math:`L_2 = \sqrt{ \Sigma_i v_i^2}` + - 'SE2' is a mixed norm for vectors :math:`(x, y, \theta)` and + is :math:`\sqrt{x^2 + y^2 + \bar{\theta}^2}` where :math:`\bar{\theta}` + is :math:`\theta` wrapped to the interval :math:`[-\pi, \pi)` + + The metric is used by :meth:`closest` and :meth:`distance` + """ + self._metric = self._metricfunc(metric) + + @property + def heuristic(self): + """ + Get the heuristic distance metric for graph + + :return: heuristic distance metric + :rtype: callable + + This is a function of a vector and returns a scalar. + """ + return self._heuristic + + @heuristic.setter + def heuristic(self, heuristic): + r""" + Set the heuristic distance metric for graph + + :param metric: heuristic distance metric + :type metric: callable or str + + This is a function of a vector and returns a scalar. It can be + user defined function or a string: + + - 'L1' is the norm :math:`L_1 = \Sigma_i | v_i |` + - 'L2' is the norm :math:`L_2 = \sqrt{ \Sigma_i v_i^2}` + - 'SE2' is a mixed norm for vectors :math:`(x, y, \theta)` and + is :math:`\sqrt{x^2 + y^2 + \bar{\theta}^2}` where :math:`\bar{\theta}` + is :math:`\theta` wrapped to the interval :math:`[-\pi, \pi)` + + The heuristic distance is only used by the A* planner :meth:`path_Astar`. + """ + self._heuristic = self._metricfunc(heuristic) + + def __repr__(self): + s = [] + for vertex in self: + ss = f"{vertex.name} at {vertex.coord}" + if vertex.label is not None: + ss += " component={vertex.label}" + s.append(ss) + return "\n".join(s) + + def __getitem__(self, i): + """ + Get vertex (superclass method) + + :param i: vertex description + :type i: int or str + :return: the referenced vertex + :rtype: Vertex subclass + + Retrieve a vertex by index or name: + + -``g[i]`` is the i'th vertex in the graph. This reflects the order of + addition to the graph. + -``g[s]`` is vertex named ``s`` + -``g[v]`` is ``v`` where ``v`` is a ``Vertex`` subclass + + This method also supports iteration over the vertices in a graph:: + + for v in g: + print(v) + + will iterate over all the vertices. + """ + if isinstance(i, int): + return self._vertexlist[i] + elif isinstance(i, str): + return self._vertexdict[i] + elif isinstance(i, Vertex): + return i + + def __contains__(self, item): + """ + Test if vertex in graph + + :param item: vertex or name of vertex + :type item: Vertex subclass or str + :return: true if vertex exists in the graph + :rtype: bool + + - ``'name' in graph`` is true if a vertex named ``'name'`` exists in the + graph. + - ``v in graph`` is true if the vertex reference ``v`` exists in the + graph. + + """ + if isinstance(item, str): + return item in self._vertexdict + elif isinstance(item, Vertex): + return item in self._vertexdict.values() + +
+[docs] + def closest(self, coord): + """ + Vertex closest to point + + :param coord: coordinates of a point + :type coord: ndarray(n) + :return: closest vertex + :rtype: Vertex subclass + + Returns the vertex closest to the given point. Distance is computed + according to the graph's metric. + + :seealso: :meth:`metric` + """ + min_dist = np.inf + min_which = None + + for vertex in self: + d = self.metric(vertex.coord - coord) + if d < min_dist: + min_dist = d + min_which = vertex + + return min_which, min_dist
+ + +
+[docs] + def edges(self): + """ + Get all edges in graph (superclass method) + + :return: All edges in the graph + :rtype: list of Edge references + + We can iterate over all edges in the graph by:: + + for e in g.edges(): + print(e) + + .. note:: The ``edges()`` of a Vertex is a list of all edges connected + to that vertex. + + :seealso: :meth:`Vertex.edges` + """ + return self._edgelist
+ + +
+[docs] + def plot( + self, + colorcomponents=True, + force2d=False, + vopt={}, + eopt={}, + text={}, + block=False, + grid=True, + ax=None, + ): + """ + Plot the graph + + :param vopt: vertex format, defaults to 12pt o-marker + :type vopt: dict, optional + :param eopt: edge format, defaults to None + :type eopt: dict, optional + :param text: text label format, defaults to None + :type text: False or dict, optional + :param colorcomponents: color vertices and edges by component, defaults to None + :type color: bool, optional + :param block: block until figure is dismissed, defaults to True + :type block: bool, optional + + The graph is plotted using matplotlib. + + If ``colorcomponents`` is True then each component is assigned a unique + color. ``vertex`` and ``edge`` cannot include a color keyword. + + If ``text`` is a dict it is used to format text labels for the vertices + which are the vertex names. If ``text`` is None default formatting is + used. If ``text`` is False no labels are added. + """ + vopt = {**dict(marker="o", markersize=12), **vopt} + eopt = {**dict(linewidth=3), **eopt} + + if colorcomponents: + color = plt.cm.coolwarm(np.linspace(0, 1, self.nc)) + + if len(self[0].coord) == 2 or force2d: + # 2D plotting + if ax is None: + ax = axes_logic(ax, 2) + for c in range(self.nc): + # for each component + for vertex in self.component(c): + if text is not False: + ax.text(vertex.x, vertex.y, " " + vertex.name, **text) + if colorcomponents: + ax.plot(vertex.x, vertex.y, color=color[c, :], **vopt) + for v in vertex.neighbours(): + ax.plot( + [vertex.x, v.x], + [vertex.y, v.y], + color=color[c, :], + **eopt, + ) + else: + ax.plot(vertex.x, vertex.y, **vopt) + for v in vertex.neighbours(): + ax.plot([vertex.x, v.x], [vertex.y, v.y], **eopt) + else: + # 3D or higher plotting, just do (x, y, z) + if ax is None: + ax = axes_logic(ax, 3) + for c in range(self.nc): + # for each component + for vertex in self.component(c): + if text is not False: + ax.text( + vertex.x, vertex.y, vertex.z, " " + vertex.name, **text + ) + if colorcomponents: + ax.plot( + vertex.x, + vertex.y, + vertex.z, + **{**dict(color=color[c, :]), **vopt}, + ) + for v in vertex.neighbours(): + ax.plot( + [vertex.x, v.x], + [vertex.y, v.y], + [vertex.z, v.z], + **{**dict(color=color[c, :]), **eopt}, + ) + else: + ax.plot(vertex.x, vertex.y, **vopt) + for v in vertex.neighbours(): + ax.plot( + [vertex.x, v.x], + [vertex.y, v.y], + [vertex.z, v.z], + **eopt, + ) + # if nc > 1: + # # add a colorbar + # plt.colorbar() + ax.grid(grid) + plt.show(block=block)
+ + +
+[docs] + def highlight_path(self, path, block=False, **kwargs): + """ + Highlight a path through the graph + + :param path: [description] + :type path: [type] + :param block: [description], defaults to True + :type block: bool, optional + + The vertices and edges along the path are overwritten with a different + size/width and color. + + :seealso: :meth:`highlight_vertex` :meth:`highlight_edge` + """ + for i in range(len(path)): + if i < len(path) - 1: + e = path[i].edgeto(path[i + 1]) + self.highlight_edge(e, **kwargs) + self.highlight_vertex(path[i], **kwargs) + plt.show(block=block)
+ + +
+[docs] + def highlight_edge(self, edge, scale=2, color="r", alpha=0.5): + """ + Highlight an edge in the graph + + :param edge: The edge to highlight + :type edge: Edge subclass + :param scale: Overwrite with a line this much bigger than the original, + defaults to 1.5 + :type scale: float, optional + :param color: Overwrite with a line in this color, defaults to 'r' + :type color: str, optional + """ + p1 = edge.v1 + p2 = edge.v2 + plt.plot( + [p1.x, p2.x], [p1.y, p2.y], color=color, linewidth=3 * scale, alpha=alpha + )
+ + +
+[docs] + def highlight_vertex(self, vertex, scale=2, color="r", alpha=0.5): + """ + Highlight a vertex in the graph + + :param edge: The vertex to highlight + :type edge: Vertex subclass + :param scale: Overwrite with a line this much bigger than the original, + defaults to 1.5 + :type scale: float, optional + :param color: Overwrite with a line in this color, defaults to 'r' + :type color: str, optional + """ + if isinstance(vertex, Iterable): + for n in vertex: + if isinstance(n, str): + n = self[n] + plt.plot(n.x, n.y, "o", color=color, markersize=12 * scale, alpha=alpha) + else: + plt.plot( + vertex.x, vertex.y, "o", color=color, markersize=12 * scale, alpha=alpha + )
+ + +
+[docs] + def dotfile(self, filename=None, direction=None): + """ + Export graph as a GraphViz dot file + + :param filename: filename to save graph to, defaults to None + :type filename: str, optional + + ``g.dotfile()`` creates the specified file which contains the + `GraphViz <https://graphviz.org>`_ code to represent the embedded graph. By default output + is to the console. + + .. note:: + + - The graph is undirected if it is a subclass of ``UGraph`` + - The graph is directed if it is a subclass of ``DGraph`` + - Use ``neato`` rather than dot to get the embedded layout + + .. note:: If ``filename`` is a file object then the file will *not* + be closed after the GraphViz model is written. + + :seealso: :func:`showgraph` + """ + + if filename is None: + f = sys.stdout + elif isinstance(filename, str): + f = open(filename, "w") + else: + f = filename + + if isinstance(self, DGraph): + print("digraph {", file=f) + else: + print("graph {", file=f) + + if direction is not None: + print(f"rankdir = {direction}", file=f) + + # add the vertices including name and position + for vertex in self: + if vertex.coord is None: + print(' "{:s}"'.format(vertex.name), file=f) + else: + print( + ' "{:s}" [pos="{:.5g},{:.5g}"]'.format( + vertex.name, vertex.coord[0], vertex.coord[1] + ), + file=f, + ) + print(file=f) + # add the edges + for e in self.edges(): + if isinstance(self, DGraph): + print(' "{:s}" -> "{:s}"'.format(e.v1.name, e.v2.name), file=f) + else: + print(' "{:s}" -- "{:s}"'.format(e.v1.name, e.v2.name), file=f) + + print("}", file=f) + + if filename is None or isinstance(filename, str): + f.close() # noqa
+ + +
+[docs] + def showgraph(self, **kwargs): + """ + Display graph in a browser tab + + :param kwargs: arguments passed to :meth:`dotfile` + + ``g.showgraph()`` renders and displays the graph in a browser tab. The + graph is exported in `GraphViz <https://graphviz.org>`_ format, rendered to + PDF using ``dot`` and then displayed in a browser tab. + + :seealso: :func:`dotfile` + """ + # create the temporary dotfile + dotfile = tempfile.TemporaryFile(mode="w") + self.dotfile(dotfile, **kwargs) + + # rewind the dot file, create PDF file in the filesystem, run dot + dotfile.seek(0) + pdffile = tempfile.NamedTemporaryFile(suffix=".pdf", delete=False) + result = subprocess.run("dot -Tpdf", shell=True, stdin=dotfile, stdout=pdffile) + + if result.returncode == 0: + # dot ran happily + # open the PDF file in browser (hopefully portable), then cleanup + webbrowser.open(f"file://{pdffile.name}")
+ + # time.sleep(1) + # os.remove(pdffile.name) + +
+[docs] + def iscyclic(self): + pass
+ + +
+[docs] + def average_degree(self): + r""" + Average degree of the graph + + :return: average degree + :rtype: float + + Average degree is :math:`2 E / N` for an undirected graph and + :math:`E / N` for a directed graph where :math:`E` is the total number of + edges and :math:`N` is the number of vertices. + + """ + if isinstance(self, DGraph): + return len(self.edges()) / self.n + elif isinstance(self, UGraph): + return 2 * len(self.edges()) / self.n
+ + + # --------------------------------------------------------------------------- # + + # MATRIX REPRESENTATIONS + +
+[docs] + def Laplacian(self): + """ + Laplacian matrix for the graph + + :return: Laplacian matrix + :rtype: NumPy ndarray + + ``g.Laplacian()`` is the Laplacian matrix (NxN) of the graph where N + is the number of vertices. + + .. note:: + + - Laplacian is always positive-semidefinite. + - Laplacian has at least one zero eigenvalue. + - The number of zero-valued eigenvalues is the number of connected + components in the graph. + + :seealso: :meth:`adjacency` :meth:`incidence` :meth:`degree` + """ + return self.degree() - (self.adjacency() > 0)
+ + +
+[docs] + def connectivity(self, vertices=None): + """ + Graph connectivity + + :return: a list with the number of edges per vertex + :rtype: list + + The average vertex connectivity is:: + + mean(g.connectivity()) + + and the minimum vertex connectivity is:: + + min(g.connectivity()) + """ + + c = [] + if vertices is None: + vertices = self + for n in vertices: + c.append(len(n._edgelist)) + return c
+ + +
+[docs] + def degree(self): + """ + Degree matrix of graph + + :return: degree matrix + :rtype: ndarray(N,N) + + This is a diagonal matrix where element ``[i,i]`` is the number + of edges connected to vertex id ``i``. + + :seealso: :meth:`adjacency` :meth:`incidence` :meth:`laplacian` + """ + + return np.diag(self.connectivity())
+ + +
+[docs] + def adjacency(self): + """ + Adjacency matrix of graph + + :returns: adjacency matrix + :rtype: ndarray(N,N) + + The elements of the adjacency matrix ``[i,j]`` are 1 if vertex ``i`` is + connected to vertex ``j``, else 0. + + .. note:: + + - vertices are numbered in their order of creation. A vertex index + can be resolved to a vertex reference by ``graph[i]``. + - for an undirected graph the matrix is symmetric + - Eigenvalues of ``A`` are real and are known as the spectrum of the graph. + - The element ``A[i,j]`` can be considered the number of walks of length one + edge from vertex ``i`` to vertex ``j`` (either zero or one). + - If ``Ak = A ** k`` the element ``Ak[i,j]`` is the number of + walks of length ``k`` from vertex ``i`` to vertex ``j``. + + :seealso: :meth:`Laplacian` :meth:`incidence` :meth:`degree` + """ + # create a dict mapping vertex to an id + vdict = {} + for i, vert in enumerate(self): + vdict[vert] = i + + A = np.zeros((self.n, self.n)) + for vertex in self: + for n in vertex.neighbours(): + A[vdict[vertex], vdict[n]] = 1 + return A
+ + +
+[docs] + def incidence(self): + """ + Incidence matrix of graph + + :returns: incidence matrix + :rtype: ndarray(n,ne) + + The elements of the incidence matrix ``I[i,j]`` are 1 if vertex ``i`` is + connected to edge ``j``, else 0. + + .. note:: + + - vertices are numbered in their order of creation. A vertex index + can be resolved to a vertex reference by ``graph[i]``. + - edges are numbered in the order they appear in ``graph.edges()``. + + :seealso: :meth:`Laplacian` :meth:`adjacency` :meth:`degree` + """ + edges = self.edges() + I = np.zeros((self.n, len(edges))) + + # create a dict mapping edge to an id + edict = {} + for i, edge in enumerate(edges): + edict[edge] = i + + for i, vertex in enumerate(self): + for i, e in enumerate(vertex.edges()): + I[i, edict[e]] = 1 + + return I
+ + +
+[docs] + def distance(self): + """ + Distance matrix of graph + + :return: distance matrix + :rtype: ndarray(n,n) + + The elements of the distance matrix ``D[i,j]`` is the edge cost of moving + from vertex ``i`` to vertex ``j``. It is zero if the vertices are not + connected. + """ + # create a dict mapping vertex to an id + vdict = {} + for i, vert in enumerate(self): + vdict[vert] = i + + A = np.zeros((self.n, self.n)) + for v1 in self: + for v2, edge in v1.incidences(): + A[vdict[v1], vdict[v2]] = edge.cost + return A
+ + + # GRAPH COMPONENTS + +
+[docs] + def component(self, c): + """ + All vertices in specified graph component + + ``graph.component(c)`` is a list of all vertices in graph component ``c``. + """ + self._graphcolor() # ensure labels are uptodate + return [v for v in self if v.label == c]
+ + +
+[docs] + def samecomponent(self, v1, v2): + """ + Test if vertices belong to same graph component + + :param v1: vertex + :type v1: Vertex subclass + :param v2: vertex + :type v2: Vertex subclass + :return: true if vertices belong to same graph component + :rtype: bool + + Test whether vertices belong to the same component. For a: + + - directed graph this implies a path between them + - undirected graph there is not necessarily a path between them + """ + self._graphcolor() # ensure labels are uptodate + + return v1.label == v2.label
+ + + # def remove(self, v): + # # remove edges from neighbour's edge list + # for e in v.edges(): + # next = e.next(v) + # next._edgelist.remove(e) + # next._connectivitychange = True + + # # remove references from the graph + # self._vertexlist.remove(v) + # for key, value in self._vertexdict.items(): + # if value is v: + # del self._vertexdict[key] + # break + + # v._edgelist = [] # remove all references to edges + # --------------------------------------------------------------------------- # + +
+[docs] + def path_BFS(self, S, G, verbose=False, summary=False): + """ + Breadth-first search for path + + :param S: start vertex + :type S: Vertex subclass + :param G: goal vertex + :type G: Vertex subclass + :return: list of vertices from S to G inclusive, path length + :rtype: list of Vertex subclass, float + + Returns a list of vertices that form a path from vertex ``S`` to + vertex ``G`` if possible, otherwise return None. + + """ + if isinstance(S, str): + S = self[S] + elif not isinstance(S, Vertex): + raise TypeError("start must be Vertex subclass or string name") + if isinstance(G, str): + G = self[G] + elif not isinstance(S, Vertex): + raise TypeError("goal must be Vertex subclass or string name") + + # we use lists not sets since the order is instructive in verbose + # mode, really need ordered sets... + frontier = [S] + explored = [] + parent = {} + done = False + + while frontier: + if verbose: + print() + print("FRONTIER:", ", ".join([v.name for v in frontier])) + print("EXPLORED:", ", ".join([v.name for v in explored])) + + x = frontier.pop(0) + if verbose: + print(" expand", x.name) + + # expand the vertex + for n in x.neighbours(): + if n is G: + if verbose: + print(" goal", n.name, "reached") + parent[n] = x + done = True + break + if n not in frontier and n not in explored: + # add it to the frontier + frontier.append(n) + if verbose: + print(" add", n.name, "to the frontier") + parent[n] = x + if done: + break + explored.append(x) + if verbose: + print(" move", x.name, "to the explored list") + else: + # no path + return None + + # reconstruct the path from start to goal + x = G + path = [x] + length = 0 + + while x is not S: + p = parent[x] + length += x.edgeto(p).cost + path.insert(0, p) + x = p + + if summary or verbose: + print( + f"{len(explored)} vertices explored, {len(frontier)} remaining on the frontier" + ) + + return path, length
+ + +
+[docs] + def path_UCS(self, S, G, verbose=False, summary=False): + """ + Uniform cost search for path + + :param S: start vertex + :type S: Vertex subclass + :param G: goal vertex + :type G: Vertex subclass + :return: list of vertices from S to G inclusive, path length, tree + :rtype: list of Vertex subclass, float, dict + + Returns a list of vertices that form a path from vertex ``S`` to + vertex ``G`` if possible, otherwise return None. + + The search tree is returned as dict that maps a vertex to its parent. + + The heuristic is the distance metric of the graph, which defaults to + Euclidean distance. + """ + if isinstance(S, str): + S = self[S] + elif not isinstance(S, Vertex): + raise TypeError("start must be Vertex subclass or string name") + if isinstance(G, str): + G = self[G] + elif not isinstance(S, Vertex): + raise TypeError("goal must be Vertex subclass or string name") + + frontier = [S] + explored = [] + parent = {} + f = {S: 0} # evaluation function + + while frontier: + if verbose: + print() + print( + "FRONTIER:", ", ".join([f"{v.name}({f[v]:.0f})" for v in frontier]) + ) + print("EXPLORED:", ", ".join([v.name for v in explored])) + + i = np.argmin([f[n] for n in frontier]) # minimum f in frontier + x = frontier.pop(i) + if verbose: + print(" expand", x.name) + if x is G: + break + # expand the vertex + for n, e in x.incidences(): + fnew = f[x] + e.cost + if n not in frontier and n not in explored: + # add it to the frontier + parent[n] = x + f[n] = fnew + frontier.append(n) + if verbose: + print(" add", n.name, "to the frontier") + + elif n in frontier: + # neighbour is already in the frontier + # cost of path via x is lower that previous, reparent it + if fnew < f[n]: + if verbose: + print( + f" reparent {n.name}: cost {fnew} via {x.name} is less than cost {f[n]} via {parent[n].name}, change parent from {parent[n].name} to {x.name} " + ) + f[n] = fnew + parent[n] = x + + explored.append(x) + if verbose: + print(" move", x.name, "to the explored list") + else: + # no path + return None + + # reconstruct the path from start to goal + x = G + path = [x] + length = 0 + + while x is not S: + p = parent[x] + length += p.edgeto(x).cost + path.insert(0, p) + x = p + + parent_names = {} + for v, p in parent.items(): + parent_names[v.name] = p.name + + if summary or verbose: + print( + f"{len(explored)} vertices explored, {len(frontier)} remaining on the frontier" + ) + + return path, length, parent_names
+ + +
+[docs] + def path_Astar(self, S, G, verbose=False, summary=False): + """ + A* search for path + + :param S: start vertex + :type S: Vertex subclass + :param G: goal vertex + :type G: Vertex subclass + :return: list of vertices from S to G inclusive, path length, tree + :rtype: list of Vertex subclass, float, dict + + Returns a list of vertices that form a path from vertex ``S`` to + vertex ``G`` if possible, otherwise return None. + + The search tree is returned as dict that maps a vertex to its parent. + + The heuristic is the distance metric of the graph, which defaults to + Euclidean distance. + + :seealso: :meth:`heuristic` + """ + if isinstance(S, str): + S = self[S] + elif not isinstance(S, Vertex): + raise TypeError("start must be Vertex subclass or string name") + if isinstance(G, str): + G = self[G] + elif not isinstance(S, Vertex): + raise TypeError("goal must be Vertex subclass or string name") + + frontier = [S] + explored = [] + parent = {} + g = {S: 0} # cost to come + f = {S: 0} # evaluation function + + while frontier: + if verbose: + print() + print( + "FRONTIER:", ", ".join([f"{v.name}({f[v]:.0f})" for v in frontier]) + ) + print("EXPLORED:", ", ".join([v.name for v in explored])) + + i = np.argmin([f[n] for n in frontier]) # minimum f in frontier + x = frontier.pop(i) + if verbose: + print(" expand", x.name) + if x is G: + break + # expand the vertex + for n, e in x.incidences(): + if n not in frontier and n not in explored: + # add it to the frontier + frontier.append(n) + parent[n] = x + g[n] = g[x] + e.cost # update cost to come + f[n] = g[n] + n.heuristic_distance(G) # heuristic + if verbose: + print(" add", n.name, "to the frontier") + elif n in frontier: + # neighbour is already in the frontier + gnew = g[x] + e.cost + if gnew < g[n]: + # cost of path via x is lower that previous, reparent it + if verbose: + print( + f" reparent {n.name}: cost {gnew} via {x.name} is less than cost {g[n]} via {parent[n].name}, change parent from {parent[n].name} to {x.name} " + ) + g[n] = gnew + f[n] = g[n] + n.heuristic_distance(G) # heuristic + + parent[n] = x # reparent + + explored.append(x) + if verbose: + print(" move", x.name, "to the explored list") + + else: + # no path + return None + + # reconstruct the path from start to goal + x = G + path = [x] + length = 0 + + while x is not S: + p = parent[x] + length += p.edgeto(x).cost + path.insert(0, p) + x = p + + parent_names = {} + for v, p in parent.items(): + parent_names[v.name] = p.name + + if summary or verbose: + print( + f"{len(explored)} vertices explored, {len(frontier)} remaining on the frontier" + ) + + return path, length, parent_names
+
+ + + +# -------------------------------------------------------------------------- # + + +
+[docs] +class UGraph(PGraph): + """ + Class for undirected graphs + + .. inheritance-diagram:: UGraph + """ + +
+[docs] + def add_vertex(self, coord=None, name=None): + """ + Add vertex to undirected graph + + :param coord: coordinate for an embedded graph, defaults to None + :type coord: array-like, optional + :param name: vertex name, defaults to "#i" + :type name: str, optional + :return: new vertex + :rtype: UVertex + + - ``g.add_vertex()`` creates a new vertex with optional ``coord`` and + ``name``. + - ``g.add_vertex(v)`` takes an instance or subclass of UVertex and adds + it to the graph + """ + if isinstance(coord, UVertex): + vertex = coord + else: + vertex = UVertex(coord) + super().add_vertex(vertex, name=name) + return vertex
+ + +
+[docs] + @classmethod + def vertex_copy(self, vertex): + return DVertex(coord=vertex.coord, name=vertex.name)
+ + + def _graphcolor(self): + """ + Color the graph + + Performs a depth-first labeling operation, assigning the ``label`` + attribute of every vertex with a sequential integer starting from 0. + + This method checks the ``_connectivitychange`` attribute of all vertices + and if any are True it will perform the coloring operation. This flag + is set True by any operation that adds or removes a vertex or edge. + + :seealso: :meth:`nc` + """ + if self._connectivitychange or any([n._connectivitychange for n in self]): + + # color the graph + + # clear all the labels + for vertex in self: + vertex.label = None + vertex._connectivitychange = False + + lastlabel = None + for label in range(self.n): + assignment = False + for v in self: + # find first vertex with no label + if v.label is None: + # do BFS + q = [v] # initialize frontier + while len(q) > 0: + v = q.pop() # expand v + v.label = label + for n in v.neighbours(): + if n.label is None: + q.append(n) + lastlabel = label + assignment = True + break + if not assignment: + break + + self._ncomponents = lastlabel + 1
+ + + +
+[docs] +class DGraph(PGraph): + """ + Class for directed graphs + + .. inheritance-diagram:: DGraph + """ + +
+[docs] + def add_vertex(self, coord=None, name=None): + """ + Add vertex to directed graph + + :param coord: coordinate for an embedded graph, defaults to None + :type coord: array-like, optional + :param name: vertex name, defaults to "#i" + :type name: str, optional + :return: new vertex + :rtype: DVertex + + - ``g.add_vertex()`` creates a new vertex with optional ``coord`` and + ``name``. + - ``g.add_vertex(v)`` takes an instance or subclass of DVertex and adds + it to the graph + """ + if isinstance(coord, Vertex): + vertex = coord + else: + vertex = DVertex(coord=coord, name=name) + super().add_vertex(vertex, name=name) + return vertex
+ + +
+[docs] + @classmethod + def vertex_copy(self, vertex): + return DVertex(coord=vertex.coord, name=vertex.name)
+ + + def _graphcolor(self): + """ + Color the graph + + Performs a depth-first labeling operation, assigning the ``label`` + attribute of every vertex with a sequential integer starting from 0. + + This method checks the ``_connectivitychange`` attribute of all vertices + and if any are True it will perform the coloring operation. This flag + is set True by any operation that adds or removes a vertex or edge. + + :seealso: :meth:`nc` + """ + if self._connectivitychange or any([n._connectivitychange for n in self]): + + # color the graph + + # clear all the labels + for vertex in self: + vertex.label = None + vertex._connectivitychange = False + + # initial labeling pass + merge = {} + nextlabel = 1 + for v in self: + if v.label is None: + # no label, try to inherit one from a neighbour + for n in v.neighbours(): + if n.label is not None: + # neighbour has a label + v.label = n.label + break + + if v.label is None: + # still not labeled, assign a new label + v.label = nextlabel + nextlabel += 1 + + # now look for clashes + for n in v.neighbours(): + if n.label is None: + # neighbour has no label, give it this one + n.label = v.label + elif v.label != n.label: + # label clash, note it for merging + merge[n.label] = v.label + + # merge labels and find unique labels + unique = set() + for v in self: + while v.label in merge: + v.label = merge[v.label] + unique.add(v.label) + + final = {u: i for i, u in enumerate(unique)} + for v in self: + v.label = final[v.label] + + self._ncomponents = len(unique)
+ + + +# ========================================================================== # + + +
+[docs] +class Edge: + """ + Edge class + + Is used to represent directed directed and undirected edges. + + Each edge has: + - ``cost`` cost of traversing this edge, required for planning methods + - ``data`` reference to arbitrary data associated with the edge + - ``v1`` first vertex, start vertex for a directed edge + - ``v2`` second vertex, end vertex for a directed edge + + .. note:: + + - An undirected graph is created by having a single edge object in the + edgelist of _each_ vertex. + - This class can be inherited to provide user objects with graph capability. + - Inheritance is an alternative to providing arbitrary user data. + + An Edge points to a pair of vertices. At ``connect`` time the vertices + get references back to the Edge object. + + ``graph.add_edge(v1, v2)`` calls ``v1.connect(v2)`` + """ + +
+[docs] + def __init__(self, v1=None, v2=None, cost=None, data=None): + """ + Create an edge object + + :param v1: start of the edge, defaults to None + :type v1: Vertex subclass, optional + :param v2: end of the edge, defaults to None + :type v2: Vertex subclass, optional + :param cost: edge cost, defaults to None + :type cost: any, optional + :param data: edge data, defaults to None + :type data: any, optional + + Creates an edge but does not connect it to the vertices or add it to the + graph. + + If vertices are given, and have associated coordinates, the edge cost + will be computed according to the distance measure associated with the + graph. + + ``data`` is a way of associating any object with the edge, its value + can be found as the ``.data`` attribute of the edge. An alternative + approach is to subclass the ``Edge`` class. + + .. note:: To compute edge cost from the vertices, the vertices must have + been added to the graph. + + :seealso: :meth:`Edge.connect` :meth:`Vertex.connect` + """ + self.v1 = v1 + self.v2 = v2 + + self.data = data + + # try to compute edge cost as metric distance if not given + if cost is not None: + self.cost = cost + elif not (v1 is None or v1.coord is None or v2 is None or v2.coord is None): + self.cost = v1._graph.metric(v1.coord - v2.coord) + else: + self.cost = None
+ + + def __repr__(self): + return str(self) + + def __str__(self): + + s = f"Edge{{{self.v1} -- {self.v2}, cost={self.cost:.4g}}}" + if self.data is not None: + s += f" data={self.data}" + return s + +
+[docs] + def connect(self, v1, v2): + """ + Add edge to the graph + + :param v1: start of the edge + :type v1: Vertex subclass + :param v2: end of the edge + :type v2: Vertex subclass + + The edge is added to the graph and connects vertices ``v1`` and ``v2``. + + .. note:: The vertices must already be added to the graph. + """ + + if v1._graph is None: + raise ValueError("vertex v1 does not belong to a graph") + if v2._graph is None: + raise ValueError("vertex v2 does not belong to a graph") + if not v1._graph is v2._graph: + raise ValueError("vertices must belong to the same graph") + + # connect edge to its vertices + self.v1 = v1 + self.v2 = v2 + + # tell the vertices to add edge to their edgelists as appropriate for + # DGraph or UGraph + v1.connect(v2, edge=self)
+ + +
+[docs] + def next(self, vertex): + """ + Return other end of an edge + + :param vertex: one vertex on the edge + :type vertex: Vertex subclass + :raises ValueError: ``vertex`` is not on the edge + :return: the other vertex on the edge + :rtype: Vertex subclass + + ``e.next(v1)`` is the vertex at the other end of edge ``e``, ie. the + vertex that is not ``v1``. + """ + + if self.v1 is vertex: + return self.v2 + elif self.v2 is vertex: + return self.v1 + else: + raise ValueError("shouldnt happen")
+ + +
+[docs] + def vertices(self): + raise DeprecationWarning("use endpoints instead")
+ + + @property + def endpoints(self): + return [self.v1, self.v2]
+ + + # def remove(self): + # """ + # Remove edge from graph + + # ``e.remove()`` removes ``e`` from the graph, but does not delete the + # edge object. + # """ + # # remove this edge from the edge list of both end vertices + # if self in self.v1._edgelist: + # self.v1._edgelist.remove(self) + # if self in self.v2._edgelist: + # self.v2._edgelist.remove(self) + + # # indicate that connectivity has changed + # self.v1._connectivitychange = True + # self.v2._connectivitychange = True + + # # remove references to the vertices + # self.v1 = None + # self.v2 = None + + +# ========================================================================== # + + +
+[docs] +class Vertex: + """ + Superclass for vertices of directed and non-directed graphs. + + Each vertex has: + - ``name`` + - ``label`` an int indicating which graph component contains it + - ``_edgelist`` a list of edge objects that connect this vertex to others + - ``coord`` the coordinate in an embedded graph (optional) + """ + + def __init__(self, coord=None, name=None): + self._edgelist = [] + if coord is None: + self.coord = None + else: + self.coord = np.r_[coord] + self.name = name + self.label = None + self._connectivitychange = True + self._edgelist = [] + self._graph = None # reference to owning graph + # print('Vertex init', type(self)) + + def __str__(self): + return f"[{self.name:s}]" + + def __repr__(self): + if self.coord is None: + coord = "?" + else: + coord = ", ".join([f"{x:.4g}" for x in self.coord]) + return f"{self.__class__.__name__}[{self.name:s}, coord=({coord})]" + +
+[docs] + def copy(self, cls=None): + if cls is not None: + return cls.vertex_copy(self) + else: + return self.__class__(coord=self.coord, name=self.name)
+ + +
+[docs] + def neighbours(self): + """ + Neighbours of a vertex + + ``v.neighbours()`` is a list of neighbours of this vertex. + + .. note:: For a directed graph the neighbours are those on edges leaving this vertex + """ + return [e.next(self) for e in self._edgelist]
+ + +
+[docs] + def neighbors(self): + """ + Neighbors of a vertex + + ``v.neighbors()`` is a list of neighbors of this vertex. + + .. note:: For a directed graph the neighbours are those on edges leaving this vertex + """ + return [e.next(self) for e in self._edgelist]
+ + +
+[docs] + def isneighbour(self, vertex): + """ + Test if vertex is a neigbour + + :param vertex: vertex reference + :type vertex: Vertex subclass + :return: true if a neighbour + :rtype: bool + + For a directed graph this is true only if the edge is from ``self`` to + ``vertex``. + """ + return vertex in [e.next(self) for e in self._edgelist]
+ + +
+[docs] + def incidences(self): + """ + Neighbours and edges of a vertex + + ``v.incidences()`` is a generator that returns a list of incidences, + tuples of (vertex, edge) for all neighbours of the vertex ``v``. + + .. note:: For a directed graph the edges are those leaving this vertex + """ + return [(e.next(self), e) for e in self._edgelist]
+ + +
+[docs] + def connect(self, dest, edge=None, cost=None, data=None): + """ + Connect two vertices with an edge + + :param dest: The vertex to connect to + :type dest: ``Vertex`` subclass + :param edge: Use this as the edge object, otherwise a new ``Edge`` + object is created from the vertices being connected, + and the ``cost`` and ``edge`` parameters, defaults to None + :type edge: ``Edge`` subclass, optional + :param cost: the cost to traverse this edge, defaults to None + :type cost: float, optional + :param data: reference to arbitrary data associated with the edge, + defaults to None + :type data: Any, optional + :raises TypeError: vertex types are different subclasses + :return: the edge connecting the vertices + :rtype: Edge + + ``v1.connect(v2)`` connects vertex ``v1`` to vertex ``v2``. + + .. note:: + + - If the vertices subclass ``UVertex`` the edge is undirected, and if + they subclass ``DVertex`` the edge is directed. + - Vertices must both be of the same ``Vertex`` subclass + + :seealso: :meth:`Edge` + """ + + if not dest.__class__.__bases__[0] is self.__class__.__bases__[0]: + raise TypeError("must connect vertices of same type") + elif isinstance(edge, Edge): + e = edge + else: + e = Edge(self, dest, cost=cost, data=data) + + self._graph._edgelist.add(e) + self._graph._connectivitychange = True + self._connectivitychange = True + + return e
+ + +
+[docs] + def edgeto(self, dest): + """ + Get edge connecting vertex to specific neighbour + + :param dest: a neigbouring vertex + :type dest: ``Vertex`` subclass + :raises ValueError: ``dest`` is not a neighbour + :return: the edge from this vertex to ``dest`` + :rtype: Edge + + .. note:: + + - For a directed graph ``dest`` must be at the arrow end of the edge + """ + for n, e in self.incidences(): + if n is dest: + return e + raise ValueError("dest is not a neighbour")
+ + +
+[docs] + def edges(self): + """ + All outgoing edges of vertex + + :return: List of all edges leaving this vertex + :rtype: list of Edge + + .. note:: + + - For a directed graph the edges are those leaving this vertex + - For a non-directed graph the edges are those leaving or entering + this vertex + """ + return self._edgelist
+ + +
+[docs] + def heuristic_distance(self, v2): + return self._graph.heuristic(self.coord - v2.coord)
+ + +
+[docs] + def distance(self, coord): + """ + Distance from vertex to point + + :param coord: coordinates of the point + :type coord: ndarray(n) or Vertex + :return: distance + :rtype: float + + Distance is computed according to the graph's metric. + + :seealso: :meth:`metric` + """ + if isinstance(coord, Vertex): + coord = coord.coord + return self._graph.metric(self.coord - coord)
+ + + @property + def degree(self): + """ + Degree of vertex + + :return: degree of the vertex + :rtype: int + + Returns the number of edges connected to the vertex. + + .. note:: For a ``DGraph`` only outgoing edges are considered. + + :seealso: :meth:`edges` + """ + return len(self.edges()) + + @property + def x(self): + """ + The x-coordinate of an embedded vertex + + :return: The x-coordinate + :rtype: float + """ + return self.coord[0] + + @property + def y(self): + """ + The y-coordinate of an embedded vertex + + :return: The y-coordinate + :rtype: float + """ + return self.coord[1] + + @property + def z(self): + """ + The z-coordinate of an embedded vertex + + :return: The z-coordinate + :rtype: float + """ + return self.coord[2] + +
+[docs] + def closest(self): + return self._graph.closest(self.coord)
+
+ + + +
+[docs] +class UVertex(Vertex): + """ + Vertex subclass for undirected graphs + + This class can be inherited to provide user objects with graph capability. + + + .. inheritance-diagram:: UVertex + + """ + +
+[docs] + def connect(self, other, **kwargs): + + if isinstance(other, Vertex): + e = super().connect(other, **kwargs) + elif isinstance(other, Edge): + e = super().connect(edge=other) + else: + raise TypeError("bad argument") + + # e = super().connect(other, **kwargs) + + self._edgelist.append(e) + other._edgelist.append(e) + self._graph._edgelist.add(e) + + return e
+
+ + + +
+[docs] +class DVertex(Vertex): + """ + Vertex subclass for directed graphs + + This class can be inherited to provide user objects with graph capability. + + .. inheritance-diagram:: DVertex + + """ + +
+[docs] + def connect(self, other, **kwargs): + if isinstance(other, Vertex): + e = super().connect(other, **kwargs) + elif isinstance(other, Edge): + e = super().connect(edge=other) + else: + raise TypeError("bad argument") + + self._edgelist.append(e) + return e
+ + +
+[docs] + def remove(self): + self._edgelist = None # remove all references to edges
+
+ + + +if __name__ == "__main__": + + g = UGraph() + print(g) + + for i in range(10): + g.add_vertex() + + g.add_edge(g[0], g[1]) + + print(g) +
+ +
+
+ +
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+ + + + + + + \ No newline at end of file diff --git a/_modules/collections.html b/_modules/collections.html new file mode 100644 index 00000000..fb5a7c03 --- /dev/null +++ b/_modules/collections.html @@ -0,0 +1,1519 @@ + + + + + + + + + + collections — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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+ + + + + + + + + + + + + + + \ No newline at end of file diff --git a/_modules/index.html b/_modules/index.html new file mode 100644 index 00000000..1d2b6243 --- /dev/null +++ b/_modules/index.html @@ -0,0 +1,118 @@ + + + + + + + + Overview: module code — Simple graph functionality for Python documentation + + + + + + + + + + + + + + + + +
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All modules for which code is available

+ + +
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/backend/PyPlot/PyPlot.html b/_modules/roboticstoolbox/backend/PyPlot/PyPlot.html new file mode 100644 index 00000000..eb1671a6 --- /dev/null +++ b/_modules/roboticstoolbox/backend/PyPlot/PyPlot.html @@ -0,0 +1,564 @@ + + + + + + + + + + roboticstoolbox.backend.PyPlot.PyPlot — Robotics Toolbox for Python 0.3.7 + documentation + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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+ + + + + + + + + + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/backend/PyPlot/PyPlot2.html b/_modules/roboticstoolbox/backend/PyPlot/PyPlot2.html new file mode 100644 index 00000000..502a2c43 --- /dev/null +++ b/_modules/roboticstoolbox/backend/PyPlot/PyPlot2.html @@ -0,0 +1,428 @@ + + + + + + + + + + roboticstoolbox.backend.PyPlot.PyPlot2 — Robotics Toolbox for Python 0.3.7 + documentation + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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+ + + + + + + + + + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/backend/Swift/Swift.html b/_modules/roboticstoolbox/backend/Swift/Swift.html new file mode 100644 index 00000000..676329c5 --- /dev/null +++ b/_modules/roboticstoolbox/backend/Swift/Swift.html @@ -0,0 +1,574 @@ + + + + + + + + + + roboticstoolbox.backend.Swift.Swift — Robotics Toolbox for Python 0.3.7 + documentation + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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+ + + + + + + + + + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/backend/VPython/VPython.html b/_modules/roboticstoolbox/backend/VPython/VPython.html new file mode 100644 index 00000000..7caa8080 --- /dev/null +++ b/_modules/roboticstoolbox/backend/VPython/VPython.html @@ -0,0 +1,583 @@ + + + + + + + + + + roboticstoolbox.backend.VPython.VPython — Robotics Toolbox for Python 0.3.7 + documentation + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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+ + + + + + + + + + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/backends/PyPlot/PyPlot.html b/_modules/roboticstoolbox/backends/PyPlot/PyPlot.html new file mode 100644 index 00000000..a0091f6d --- /dev/null +++ b/_modules/roboticstoolbox/backends/PyPlot/PyPlot.html @@ -0,0 +1,728 @@ + + + + + + roboticstoolbox.backends.PyPlot.PyPlot — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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+ +

Source code for roboticstoolbox.backends.PyPlot.PyPlot

+#!/usr/bin/env python
+"""
+@author Jesse Haviland
+"""
+import time
+import roboticstoolbox as rp
+import numpy as np
+from roboticstoolbox.backends.Connector import Connector
+
+from roboticstoolbox.backends.PyPlot.RobotPlot import RobotPlot
+from roboticstoolbox.backends.PyPlot.EllipsePlot import EllipsePlot, ShapePlot
+from spatialmath.base.argcheck import getvector
+from spatialgeometry import Shape
+
+# from roboticstoolbox.tools import Ticker
+
+_mpl = False
+_pil = None
+
+try:
+    import matplotlib
+    import matplotlib.pyplot as plt
+
+    # from mpl_toolkits.mplot3d import Axes3D
+    from matplotlib.widgets import Slider
+
+    matplotlib.rcParams["pdf.fonttype"] = 42
+    matplotlib.rcParams["ps.fonttype"] = 42
+    plt.style.use("ggplot")
+    matplotlib.rcParams["font.size"] = 7
+    matplotlib.rcParams["lines.linewidth"] = 0.5
+    matplotlib.rcParams["xtick.major.size"] = 1.5
+    matplotlib.rcParams["ytick.major.size"] = 1.5
+    matplotlib.rcParams["axes.labelpad"] = 1
+    plt.rc("grid", linestyle="-", color="#dbdbdb")
+    _mpl = True
+except ImportError:  # pragma nocover
+    pass
+
+
+
[docs]class PyPlot(Connector): + """ + Graphical backend using matplotlib + + matplotlib is a common and highly portable graphics library for Python, + but has relatively limited 3D capability. + + Example: + + .. code-block:: python + :linenos: + + import roboticstoolbox as rtb + + robot = rtb.models.DH.Panda() # create a robot + + pyplot = rtb.backends.PyPlot() # create a PyPlot backend + pyplot.add(robot) # add the robot to the backend + robot.q = robot.qz # set the robot configuration + pyplot.step() # update the backend and graphical view + + .. note:: PyPlot is the default backend, and ``robot.plot(q)`` effectively + performs lines 7-8 above. + + """ + + def __init__(self): + + super(PyPlot, self).__init__() + self.robots = [] + self.ellipses = [] + self.shapes = [] + + if not _mpl: # pragma nocover + raise ImportError( + "\n\nYou do not have matplotlib installed, do:\n" + "pip install matplotlib\n\n" + ) + + def __repr__(self): + s = "" + for robot in self.robots: + s += f" robot: {robot.name}\n" + for ellipse in self.ellipses: + s += f" ellipse: {ellipse}\n" + for shape in self.shapes: + s += f" shape: {shape}\n" + + if s == "": + return f"PyPlot3D backend, t = {self.sim_time}, empty scene" + else: + return f"PyPlot3D backend, t = {self.sim_time}, scene:\n" + s + +
[docs] def launch(self, name=None, fig=None, limits=None, **kwargs): + """ + Launch a graphical interface + + ```env = launch()``` creates a blank 3D matplotlib figure and returns + a reference to the backend. + """ + + super().launch() + + self.limits = limits + if limits is not None: + self.limits = getvector(limits, 6) + + projection = "ortho" + labels = ["X", "Y", "Z"] + + if name is None: + name = "Robotics Toolbox for Python" + + if fig is None: + self.fig = plt.figure(name) + else: + self.fig = fig + self.fig.canvas.manager.set_window_title(name) + + self.fig.subplots_adjust(left=-0.09, bottom=0, top=1, right=0.99) + + # Create a 3D axes + self.ax = self.fig.add_subplot(111, projection="3d", proj_type=projection) + self.ax.set_facecolor("white") + self.ax.figure.canvas.manager.set_window_title( + f"Robotics Toolbox for Python (Figure {self.ax.figure.number})" + ) + + self.ax.set_xbound(-0.5, 0.5) + self.ax.set_ybound(-0.5, 0.5) + self.ax.set_zbound(0.0, 0.5) + + self.ax.set_xlabel(labels[0]) + self.ax.set_ylabel(labels[1]) + self.ax.set_zlabel(labels[2]) + + if limits is not None: + self.ax.set_xlim3d([limits[0], limits[1]]) + self.ax.set_ylim3d([limits[2], limits[3]]) + self.ax.set_zlim3d([limits[4], limits[5]]) + + self.limits = limits + + # disable the display of value under cursor + # self.ax.format_coord = lambda x, y: '' + + # add time display in top-right corner + self.timer = plt.figtext(0.85, 0.95, "") + + if _isnotebook(): + plt.ion() + self.fig.canvas.draw() + else: + plt.ion() + plt.show() + + self.sim_time = 0
+ + # # Set the signal handler and a 0.1 second plot updater + # signal.signal(signal.SIGALRM, self._plot_handler) + # signal.setitimer(signal.ITIMER_REAL, 0.1, 0.1) + # TODO still need to finish this, and get Jupyter animation working + +
[docs] def step(self, dt=0.05): + """ + Update the graphical scene + + :param dt: time step in seconds, defaults to 50 (0.05 s) + :type dt: int, optional + + ``env.step(args)`` triggers an update of the 3D scene in the matplotlib + window referenced by ``env``. + + .. note:: + + - Each robot in the scene is updated based on + their control type (position, velocity, acceleration, or torque). + - Upon acting, the other three of the four control types will be + updated in the internal state of the robot object. + - The control type is defined by the robot object, and not all + robot objects support all control types. + - Execution is blocked for the specified interval + + """ + + super().step() + + # update the robot's state + for rpl in self.robots: + robot = rpl.robot + + if rpl.readonly or robot.control_mode == "p": + pass # pragma: no cover + elif robot.control_mode == "v": + for i in range(robot.n): + robot.q[i] += robot.qd[i] * (dt) + elif robot.control_mode == "a": # pragma: no cover + pass + else: # pragma: no cover + raise ValueError( + "Invalid robot.control_type. Must be one of 'p', 'v', or 'a'" + ) + + # plt.ioff() + + # update all ellipses + for ellipse in self.ellipses: + ellipse.draw() + + # update all shapes + for shape in self.shapes: + shape.draw() + + # update all robots + for robot in self.robots: + robot.draw() + + self._set_axes_equal() + + # update time and display it on plot + if self.sim_time > 0: + self.timer.set_text(f"t = {self.sim_time:.2f}") + self.sim_time += dt + + # plt.ion() + + if _isnotebook(): + plt.draw() + self.fig.canvas.draw() + time.sleep(dt) + else: + plt.draw() + plt.pause(dt)
+ +
[docs] def reset(self): + """ + Reset the graphical scene + + ``env.reset()`` triggers a reset of the 3D scene in the matplotlib + window referenced by ``env``. It is restored to the original state + defined by ``launch()``. + """ + # TODO what does this actually do for matplotlib?? + + super().reset()
+ +
[docs] def restart(self): + """ + Restart the graphics display + + ``env.restart()`` triggers a restart of the matplotlib view referenced + by ``env``. It is closed and relaunched to the original state defined + by ``launch()``. + + """ + # TODO what does this actually do for matplotlib?? + + super().restart()
+ +
[docs] def close(self): + """ + ``env.close()`` gracefully closes the matplotlib window + referenced by ``env``. + """ + # TODO what does this actually do for matplotlib?? + + super().close() + + # signal.setitimer(signal.ITIMER_REAL, 0) + plt.close(self.fig)
+ + # + # Methods to interface with the robots created in other environments + # + +
[docs] def add( + self, + ob, + readonly=False, + display=True, + jointaxes=True, + jointlabels=False, + eeframe=True, + shadow=True, + name=True, + options=None, + ): + """ + Add a robot to the graphical scene + + :param ob: The object to add to the plot (robot or ellipse) + :type ob: DHRobot or EllipsePlot + :param readonly: Do not update the state of the object + (i.e. display not simulate), defaults to False + :type readonly: bool, optional + :param display: Display the object, defaults to True + :type display: bool, optional + :param jointaxes: Show the joint axes of the robot with arrows, + defaults to True + :type jointaxes: bool, optional + :param eeframe: Show the end-effector frame of the robot, + defaults to True + :type eeframe: bool, optional + :param shadow: Display a shadow of the robot on the x-y gound plane, + defaults to True + :type shadow: bool, optional + :param name: Display the name of the robot, defaults to True + :type name: bool, optional + + ``id = env.add(robot)`` adds the ``robot`` to the graphical + environment. + + .. note:: + + - ``robot`` must be of an appropriate class. + - Adds the robot object to a list of robots which will be updated + when the ``step()`` method is called. + + """ + + super().add() + + if ( + isinstance(ob, rp.DHRobot) + or isinstance(ob, rp.ERobot) + or isinstance(ob, rp.Robot) + ): + self.robots.append( + RobotPlot( + ob, + self, + readonly, + display, + jointaxes, + jointlabels, + eeframe, + shadow, + name, + options, + ) + ) + self.robots[-1].draw() + id = len(self.robots) + + elif isinstance(ob, EllipsePlot): + ob.ax = self.ax + self.ellipses.append(ob) + self.ellipses[-1].draw() + id = len(self.ellipses) + + elif isinstance(ob, Shape): + # recreate the shape using matplotlib + self.shapes.append(ShapePlot(ob)) + self.shapes[-1].draw(ax=self.ax) + id = len(self.shapes) + + plt.draw() # matplotlib refresh + plt.show(block=False) + + self._set_axes_equal() + + return id
+ +
[docs] def remove(self, id): + """ + Remove a robot or shape from the graphical scene + + :param id: The id of the robot to remove. Can be either the DHLink or + GraphicalRobot + :type id: class:`~roboticstoolbox.robot.DHRobot.DHRobot`, + class:`roboticstoolbox.backends.VPython.graphics_robot.GraphicalRobot` + :param fig_num: The canvas index to delete the robot from, defaults to + the initial one + :type fig_num: int, optional + :raises ValueError: Figure number must be between 0 and total number + of canvases + :raises TypeError: Input must be a DHLink or GraphicalRobot + + ``env.remove(robot)`` removes the ``robot`` from the graphical + environment. + """ + # TODO should be an id to remove? + + super().remove()
+ +
[docs] def hold(self): # pragma: no cover + """ + hold() keeps the plot open i.e. stops the plot from closing once + the main script has finished. + + """ + + # signal.setitimer(signal.ITIMER_REAL, 0) + plt.ioff() + + # keep stepping the environment while figure is open + while True: + if not plt.fignum_exists(self.fig.number): + break + self.step()
+ +
[docs] def getframe(self): + global _pil + + if _pil is None: + try: + import PIL + + _pil = PIL.Image.frombytes + except ImportError: # pragma nocover + pass + + if _pil is None: + raise RuntimeError( + "to save movies PIL must be installed:\npip3 install PIL" + ) + + # make the background white, looks better than grey stipple + self.ax.w_xaxis.set_pane_color((1.0, 1.0, 1.0, 1.0)) + self.ax.w_yaxis.set_pane_color((1.0, 1.0, 1.0, 1.0)) + self.ax.w_zaxis.set_pane_color((1.0, 1.0, 1.0, 1.0)) + plt.gcf().canvas.draw() + + # render the frame and save as a PIL image in the list + canvas = self.fig.canvas + return _pil("RGB", canvas.get_width_height(), canvas.tostring_rgb())
+ + # + # Private methods + # + + # def _plot_handler(self, sig, frame): + # try: + # plt.pause(0.001) + # except(AttributeError): + # pass + + def _set_axes_equal(self): + """ + Make axes of 3D plot have equal scale so that spheres appear as + spheres, cubes as cubes, etc.. This is one possible solution to + Matplotlib's ax.set_aspect('equal') and ax.axis('equal') not + working for 3D. + + """ + + if self.limits is not None: + return + + self.ax.autoscale(enable=True, axis="both", tight=False) + + x_limits = self.ax.get_xlim3d() + y_limits = self.ax.get_ylim3d() + z_limits = self.ax.get_zlim3d() + + x_range = abs(x_limits[1] - x_limits[0]) + x_middle = np.mean(x_limits) + y_range = abs(y_limits[1] - y_limits[0]) + y_middle = np.mean(y_limits) + z_range = abs(z_limits[1] - z_limits[0]) + z_middle = np.mean(z_limits) + + # The plot bounding box is a sphere in the sense of the infinity + # norm, hence I call half the max range the plot radius. + plot_radius = 0.5 * max([x_range, y_range, z_range]) + + self.ax.set_xlim3d([x_middle - plot_radius, x_middle + plot_radius]) + self.ax.set_ylim3d([y_middle - plot_radius, y_middle + plot_radius]) + self.ax.set_zlim3d([z_middle - plot_radius, z_middle + plot_radius]) + + def _add_teach_panel(self, robot, q): + + if _isnotebook(): + raise RuntimeError("cannot use teach panel under Jupyter") + + fig = self.fig + + # Add text to the plots + def text_trans(text, q): # pragma: no cover + T = robot.fkine(q) + t = np.round(T.t, 3) + r = np.round(T.rpy("deg"), 3) + text[0].set_text("x: {0}".format(t[0])) + text[1].set_text("y: {0}".format(t[1])) + text[2].set_text("z: {0}".format(t[2])) + text[3].set_text("r: {0}".format(r[0])) + text[4].set_text("p: {0}".format(r[1])) + text[5].set_text("y: {0}".format(r[2])) + + # Update the self state in mpl and the text + def update(val, text, robot): # pragma: no cover + for j in range(robot.n): + if robot.isrevolute(j): + robot.q[j] = np.radians(self.sjoint[j].val) + else: + robot.q[j] = self.sjoint[j].val + text_trans(text, robot.q) + + fig.subplots_adjust(left=0.25) + text = [] + + x1 = 0.04 + x2 = 0.22 + yh = 0.04 + ym = 0.5 - (robot.n * yh) / 2 + 0.17 / 2 + + self.axjoint = [] + self.sjoint = [] + + qlim = robot.todegrees(robot.qlim) + + # if np.all(qlim == 0): # pragma: no cover + # qlim[0, :] = -180 + # qlim[1, :] = 180 + + # Set the pose text + T = robot.fkine(q) + t = np.round(T.t, 3) + r = np.round(T.rpy(), 3) + + fig.text( + 0.02, + 1 - ym + 0.25, + "End-effector Pose", + fontsize=9, + weight="bold", + color="#4f4f4f", + ) + text.append( + fig.text( + 0.03, 1 - ym + 0.20, "x: {0}".format(t[0]), fontsize=9, color="#2b2b2b" + ) + ) + text.append( + fig.text( + 0.03, 1 - ym + 0.16, "y: {0}".format(t[1]), fontsize=9, color="#2b2b2b" + ) + ) + text.append( + fig.text( + 0.03, 1 - ym + 0.12, "z: {0}".format(t[2]), fontsize=9, color="#2b2b2b" + ) + ) + text.append( + fig.text( + 0.15, 1 - ym + 0.20, "r: {0}".format(r[0]), fontsize=9, color="#2b2b2b" + ) + ) + text.append( + fig.text( + 0.15, 1 - ym + 0.16, "p: {0}".format(r[1]), fontsize=9, color="#2b2b2b" + ) + ) + text.append( + fig.text( + 0.15, 1 - ym + 0.12, "y: {0}".format(r[2]), fontsize=9, color="#2b2b2b" + ) + ) + fig.text( + 0.02, + 1 - ym + 0.06, + "Joint angles", + fontsize=9, + weight="bold", + color="#4f4f4f", + ) + + for j in range(robot.n): + ymin = (1 - ym) - j * yh + ax = fig.add_axes([x1, ymin, x2, 0.03], facecolor="#dbdbdb") + self.axjoint.append(ax) + + if robot.isrevolute(j): + slider = Slider( + ax, "q" + str(j), qlim[0, j], qlim[1, j], np.degrees(q[j]), "% .1f°" + ) + else: + slider = Slider( + ax, "q" + str(j), qlim[0, j], qlim[1, j], robot.q[j], "% .1f" + ) + + slider.on_changed(lambda x: update(x, text, robot)) + self.sjoint.append(slider) + robot.q = q + self.step()
+ + +def _isnotebook(): + """ + Determine if code is being run from a Jupyter notebook + + ``_isnotebook`` is True if running Jupyter notebook, else False + + :references: + + - https://stackoverflow.com/questions/15411967/how-can-i-check-if-code- + is-executed-in-the-ipython-notebook/39662359#39662359 + """ + try: + shell = get_ipython().__class__.__name__ + if shell == "ZMQInteractiveShell": + return True # Jupyter notebook or qtconsole + elif shell == "TerminalInteractiveShell": + return False # Terminal running IPython + else: + return False # Other type (?) + except NameError: + return False # Probably standard Python interpreter +
+ +
+
+ +
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/backends/PyPlot/PyPlot2.html b/_modules/roboticstoolbox/backends/PyPlot/PyPlot2.html new file mode 100644 index 00000000..a3639bb2 --- /dev/null +++ b/_modules/roboticstoolbox/backends/PyPlot/PyPlot2.html @@ -0,0 +1,505 @@ + + + + + + roboticstoolbox.backends.PyPlot.PyPlot2 — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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+ +

Source code for roboticstoolbox.backends.PyPlot.PyPlot2

+#!/usr/bin/env python
+"""
+@author Jesse Haviland
+"""
+
+import roboticstoolbox as rp
+import numpy as np
+from roboticstoolbox.backends.Connector import Connector
+from roboticstoolbox.backends.PyPlot.RobotPlot2 import RobotPlot2
+from roboticstoolbox.backends.PyPlot.EllipsePlot import EllipsePlot
+import time
+from spatialmath import SE2
+
+_mpl = False
+
+try:
+    import matplotlib
+    import matplotlib.pyplot as plt
+    from matplotlib.widgets import Slider
+
+    matplotlib.rcParams["pdf.fonttype"] = 42
+    matplotlib.rcParams["ps.fonttype"] = 42
+    plt.style.use("ggplot")
+    matplotlib.rcParams["font.size"] = 7
+    matplotlib.rcParams["lines.linewidth"] = 0.5
+    matplotlib.rcParams["xtick.major.size"] = 1.5
+    matplotlib.rcParams["ytick.major.size"] = 1.5
+    matplotlib.rcParams["axes.labelpad"] = 1
+    plt.rc("grid", linestyle="-", color="#dbdbdb")
+    _mpl = True
+except ImportError:  # pragma nocover
+    pass
+
+
+
[docs]class PyPlot2(Connector): + def __init__(self): + + super(PyPlot2, self).__init__() + self.robots = [] + self.ellipses = [] + self.sim_time = 0 + + if not _mpl: # pragma nocover + raise ImportError( + "\n\nYou do not have matplotlib installed, do:\n" + "pip install matplotlib\n\n" + ) + + def __repr__(self): + s = "" + for robot in self.robots: + s += f" robot: {robot.name}\n" + for ellipse in self.ellipses: + s += f" ellipse: {ellipse}\n" + + if s == "": + return f"PyPlot2D backend, t = {self.sim_time}, empty scene" + else: + return f"PyPlot2D backend, t = {self.sim_time}, scene:\n" + s + +
[docs] def launch(self, name=None, limits=None, **kwargs): + """ + env = launch() launchs a blank 2D matplotlib figure + + """ + + super().launch() + + labels = ["X", "Y"] + + if name is not None and not _isnotebook(): + # jupyter does weird stuff when figures have the same name + self.fig = plt.figure() + else: + self.fig = plt.figure() + + # Create a 2D axes + self.ax = self.fig.add_subplot(1, 1, 1) + self.ax.set_facecolor("white") + plt.get_current_fig_manager().set_window_title( + f"Robotics Toolbox for Python (Figure {self.ax.figure.number})" + ) + + self.ax.set_xbound(-0.5, 0.5) + self.ax.set_ybound(-0.5, 0.5) + + self.ax.set_xlabel(labels[0]) + self.ax.set_ylabel(labels[1]) + + self.ax.autoscale(enable=True, axis="both", tight=False) + + if limits is not None: + self.ax.set_xlim([limits[0], limits[1]]) + self.ax.set_ylim([limits[2], limits[3]]) + + self.ax.axis("equal") + + plt.ion() + plt.show()
+ + # Set the signal handler and a 0.1 second plot updater + # signal.signal(signal.SIGALRM, self._plot_handler) + # signal.setitimer(signal.ITIMER_REAL, 0.1, 0.1) + +
[docs] def step(self, dt=0.05): + """ + state = step(args) triggers the external program to make a time step + of defined time updating the state of the environment as defined by + the robot's actions. + + The will go through each robot in the list and make them act based on + their control type (position, velocity, acceleration, or torque). Upon + acting, the other three of the four control types will be updated in + the internal state of the robot object. The control type is defined + by the robot object, and not all robot objects support all control + types. + + """ + + super().step() + + self._step_robots(dt) + + # plt.ioff() + self._draw_ellipses() + self._draw_robots() + # plt.ion() + + if _isnotebook(): + plt.draw() + self.fig.canvas.draw() + time.sleep(dt) + else: + plt.draw() + plt.pause(dt) + + self._update_robots()
+ +
[docs] def reset(self): + """ + state = reset() triggers the external program to reset to the + original state defined by launch + + """ + + super().reset()
+ +
[docs] def restart(self): + """ + state = restart() triggers the external program to close and relaunch + to thestate defined by launch + + """ + + super().restart()
+ +
[docs] def close(self): + """ + close() closes the plot + + """ + + super().close() + + # signal.setitimer(signal.ITIMER_REAL, 0) + plt.close(self.fig)
+ + # + # Methods to interface with the robots created in other environemnts + # + +
[docs] def add(self, ob, readonly=False, display=True, eeframe=True, name=False, **kwargs): + """ + id = add(robot) adds the robot to the external environment. robot must + be of an appropriate class. This adds a robot object to a list of + robots which will act upon the step() method being called. + + """ + + super().add() + + if isinstance(ob, rp.Robot2): + self.robots.append(RobotPlot2(ob, self, readonly, display, eeframe, name)) + self.robots[len(self.robots) - 1].draw() + + elif isinstance(ob, EllipsePlot): + ob.ax = self.ax + self.ellipses.append(ob) + self.ellipses[len(self.ellipses) - 1].draw2()
+ +
[docs] def remove(self): + """ + id = remove(robot) removes the robot to the external environment. + + """ + + super().remove() # ???
+ +
[docs] def hold(self): # pragma: no cover + """ + hold() keeps the plot open i.e. stops the plot from closing once + the main script has finished. + + """ + + # signal.setitimer(signal.ITIMER_REAL, 0) + plt.ioff() + + # keep stepping the environment while figure is open + while True: + if not plt.fignum_exists(self.fig.number): + break + self.step()
+ + # + # Private methods + # + + def _step_robots(self, dt): + + for rpl in self.robots: + robot = rpl.robot + + if rpl.readonly or robot.control_type == "p": + pass # pragma: no cover + + elif robot.control_type == "v": + + for i in range(robot.n): + robot.q[i] += robot.qd[i] * (dt / 1000) + + elif robot.control_type == "a": # pragma: no cover + pass + + else: # pragma: no cover + # Should be impossible to reach + raise ValueError( + "Invalid robot.control_type. Must be one of 'p', 'v', or 'a'" + ) + + def _update_robots(self): + pass + + def _draw_robots(self): + + for i in range(len(self.robots)): + self.robots[i].draw() + + def _draw_ellipses(self): + + for i in range(len(self.ellipses)): + self.ellipses[i].draw2() + + # def _plot_handler(self, sig, frame): + # plt.pause(0.001) + + def _add_teach_panel(self, robot, q): + """ + Add a teach panel + + :param robot: Robot being taught + :type robot: ERobot class + :param q: inital joint angles in radians + :type q: array_like(n) + """ + fig = self.fig + + # Add text to the plots + def text_trans(text, q): # pragma: no cover + # update displayed robot pose value + T = robot.fkine(q, end=robot.ee_links[0]) + t = np.round(T.t, 3) + r = np.round(T.theta(), 3) + text[0].set_text("x: {0}".format(t[0])) + text[1].set_text("y: {0}".format(t[1])) + text[2].set_text("yaw: {0}".format(r)) + + # Update the self state in mpl and the text + def update(val, text, robot): # pragma: no cover + for j in range(robot.n): + if robot.isrevolute(j): + robot.q[j] = np.radians(self.sjoint[j].val) + else: + robot.q[j] = self.sjoint[j].val + text_trans(text, robot.q) + + fig.subplots_adjust(left=0.38) + text = [] + + x1 = 0.04 + x2 = 0.22 + yh = 0.04 + ym = 0.5 - (robot.n * yh) / 2 + 0.17 / 2 + + self.axjoint = [] + self.sjoint = [] + + qlim = robot.todegrees(robot.qlim) + + # Set the pose text + # if multiple EE, display only the first one + T = SE2(robot.fkine(q, end=robot.ee_links[0])) + t = np.round(T.t, 3) + r = np.round(T.theta(), 3) + + # TODO maybe put EE name in here, possible issue with DH robot + # TODO maybe display pose of all EEs, layout hassles though + + if robot.nbranches == 0: + header = "End-effector Pose" + else: + header = "End-effector #0 Pose" + fig.text( + 0.02, 1 - ym + 0.25, header, fontsize=9, weight="bold", color="#4f4f4f" + ) + text.append( + fig.text( + 0.03, 1 - ym + 0.20, "x: {0}".format(t[0]), fontsize=9, color="#2b2b2b" + ) + ) + text.append( + fig.text( + 0.03, 1 - ym + 0.16, "y: {0}".format(t[1]), fontsize=9, color="#2b2b2b" + ) + ) + text.append( + fig.text( + 0.15, 1 - ym + 0.20, "yaw: {0}".format(r), fontsize=9, color="#2b2b2b" + ) + ) + fig.text( + 0.02, + 1 - ym + 0.06, + "Joint angles", + fontsize=9, + weight="bold", + color="#4f4f4f", + ) + + for j in range(robot.n): + # for each joint + ymin = (1 - ym) - j * yh + self.axjoint.append(fig.add_axes([x1, ymin, x2, 0.03], facecolor="#dbdbdb")) + + if robot.isrevolute(j): + slider = Slider( + self.axjoint[j], + "q" + str(j), + qlim[0, j], + qlim[1, j], + np.degrees(q[j]), + "% .1f°", + ) + else: + slider = Slider( + self.axjoint[j], "q" + str(j), qlim[0, j], qlim[1, j], q[j], "% .1f" + ) + + slider.on_changed(lambda x: update(x, text, robot)) + self.sjoint.append(slider) + robot.q = q + self.step()
+ + +def _isnotebook(): + """ + Determine if code is being run from a Jupyter notebook + + ``_isnotebook`` is True if running Jupyter notebook, else False + + :references: + + - https://stackoverflow.com/questions/15411967/how-can-i-check-if-code- + is-executed-in-the-ipython-notebook/39662359#39662359 + """ + try: + shell = get_ipython().__class__.__name__ + if shell == "ZMQInteractiveShell": + return True # Jupyter notebook or qtconsole + elif shell == "TerminalInteractiveShell": + return False # Terminal running IPython + else: + return False # Other type (?) + except NameError: + return False # Probably standard Python interpreter +
+ +
+
+ +
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/backends/Swift/Swift.html b/_modules/roboticstoolbox/backends/Swift/Swift.html new file mode 100644 index 00000000..8ca5ed80 --- /dev/null +++ b/_modules/roboticstoolbox/backends/Swift/Swift.html @@ -0,0 +1,1158 @@ + + + + + + + + + + roboticstoolbox.backends.Swift.Swift — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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+ + + + + + + + + + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/backends/VPython/VPython.html b/_modules/roboticstoolbox/backends/VPython/VPython.html new file mode 100644 index 00000000..ec1b90d2 --- /dev/null +++ b/_modules/roboticstoolbox/backends/VPython/VPython.html @@ -0,0 +1,593 @@ + + + + + + + + + + roboticstoolbox.backends.VPython.VPython — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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+ + + + + + + + + + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/mobile/Animations.html b/_modules/roboticstoolbox/mobile/Animations.html new file mode 100644 index 00000000..9375eda3 --- /dev/null +++ b/_modules/roboticstoolbox/mobile/Animations.html @@ -0,0 +1,592 @@ + + + + + + roboticstoolbox.mobile.Animations — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.mobile.Animations

+"""
+@Author: Peter Corke, original MATLAB code and Python version
+@Author: Kristian Gibson, initial MATLAB port
+"""
+from abc import ABC
+from math import pi, atan2
+import numpy as np
+
+# from scipy import integrate, linalg, interpolate
+from pathlib import Path
+import matplotlib.pyplot as plt
+from matplotlib import patches, colors
+import matplotlib.transforms as mtransforms
+
+from spatialmath import SE2, base
+from roboticstoolbox import rtb_load_data
+
+
+
[docs]class VehicleAnimationBase(ABC): + """ + Abstract base class to support animation of a vehicle in a Matplotlib plot + + There are three concrete subclasses: + + - ``VehicleMarker`` animates a Matplotlib marker (shows position only) + - ``VehiclePolygon`` animates a polygon shape (outline or filled), including predefined shapes (shows position and orientation) + - ``VehicleIcon`` animates an image (shows position and orientation) + + An instance ``a`` of these classes can be used in three different ways, firstly:: + + a = VehiclePolygon("car", color="red") + a.add() + + adds an instance of the animation shape to the plot and subsequent calls + to:: + + a.update(q) + + will animate it with the configuration given by ``q``. + + Secondly, an instance can be passed to a Vehicle subclass object to make an animation + during simulation:: + + a = VehiclePolygon("car", color="red") + veh = Bicycle(animation=a) + + Thirdly:: + + a = VehiclePolygon("car", color="red") + a.plot(q) + + adds an instance of the animation shape to the plot with the specified + configuration. It cannot be moved, but the method does return a reference + to the Matplotlib object added to the plot. + + """ + + def __init__(self): + self._object = None + self._ax = None + + def add(self, ax=None, **kwargs): + """ + Add vehicle animation to the current plot + + :param ax: Axis to add to, defaults to current axis + :type ax: Axes, optional + :param kwargs: additional arguments passed to Matplotlib :meth:`~matplotlib.axes.Axes.plot`, which + override arguments given to the constructor. + + A reference to the animation object is kept, and it will be deleted + from the plot when the ``VehicleAnimation`` object is garbage collected. + + The animation is not displayed until :meth:`update` is called. + + :seealso: :meth:`update` + """ + if ax is None: + self._ax = plt.gca() + else: + self._ax = ax + + self._add(**kwargs) + + def update(self, q): + """ + Update the vehicle animation (superclass) + + :param q: vehicle position or configuration + :type q: array_like(2) or array_like(3) + + The graphical depiction of the vehicle position or configuration is updated. + + :seealso: :meth:`add` + """ + self._update(q) + + def plot(self, q, **kwargs): + """ + Add vehicle to the current plot (superclass) + + :param q: vehicle position or configuration + :type q: array_like(2) or array_like(3) + :param kwargs: additional arguments passed to Matplotlib :meth:`~matplotlib.axes.Axes.plot`, which + override arguments given to the constructor. + :return: reference to Matplotlib object + + The animation object is rendered into the current axes. + """ + self.add(**kwargs) + self.update(q) + return self._object + + def __del__(self): + + if self._object is not None: + self._object.remove()
+ + +# ========================================================================= # +
[docs]class VehicleMarker(VehicleAnimationBase): +
[docs] def __init__(self, **kwargs): + """ + Create graphical animation of vehicle as a Matplotlib marker + + :param kwargs: additional arguments passed to Matplotlib :meth:`~matplotlib.axes.Axes.plot`. + :return: animation object + :rtype: VehicleAnimation + + Creates an object that can be passed to a ``Vehicle`` subclass to depict + the moving robot as a simple Matplotlib marker during simulation. + + The default marker is a red filled circle with a white outline. + + For example, to animate a simulation with a blue square marker:: + + a = VehicleMarker(marker="s", markerfacecolor="b") + veh = Bicycle(driver=RandomPath(10), animation=a) + veh.run() + + .. note:: A marker can only indicate vehicle position, not orientation. + + :seealso: :func:`~Vehicle` + """ + super().__init__() + if len(kwargs) == 0: + kwargs = { + "marker": "o", + "markerfacecolor": "r", + "markeredgecolor": "w", + "markersize": 12, + } + self._args = kwargs
+ + def _update(self, x): + self._object.set_xdata(x[0]) + self._object.set_ydata(x[1]) + + def _add(self, x=None, **kwargs): + if x is None: + x = (0, 0) + self._object = plt.plot(x[0], x[1], **{**self._args, **kwargs})[0]
+ + +# ========================================================================= # + + +
[docs]class VehiclePolygon(VehicleAnimationBase): +
[docs] def __init__(self, shape="car", scale=1, **kwargs): + """ + Create graphical animation of vehicle as a polygon + + :param shape: polygon shape as vertices or a predefined shape, defaults to "car" + :type shape: ndarray(2,n) or str + :param scale: Length of the vehicle on the plot, defaults to 1 + :type scale: float + :param kwargs: additional arguments passed to Matplotlib :class:`~matplotlib.patches.Polygon` such as + ``color`` (face+edge), ``alpha``, ``facecolor``, ``edgecolor``, + ``linewidth`` etc. + :raises ValueError: unknown shape name + :raises TypeError: bad shape argument + :return: animation object + :rtype: VehiclePolygon + + Creates an object that can be passed to a ``Vehicle`` subclass to + depict the moving robot as a polygon during simulation. + + For example, to animate a simulation with a red filled car-shaped polygon:: + + a = VehiclePolygon("car", color="r") + veh = Bicycle(driver=RandomPath(10), animation=a) + veh.run() + + ``shape`` can be: + + * ``"car"`` a rectangle with chamfered front corners + * ``"triangle"`` an isocles triangle pointing in the forward direction + * an 2xN NumPy array of vertices, does not have to be closed. + + The polygon is scaled to an image with a length of ``scale`` in the + vehicle x-direction, in the units of the plot. + + :seealso: :func:`~Vehicle` :class:`matplotlib.patches.Polygon` + """ + super().__init__() + if isinstance(shape, str): + + h = 0.3 + t = 0.8 # start of head taper + c = 0.5 # centre x coordinate + w = 1 # width in x direction + + if isinstance(shape, str): + if shape == "car": + self._coords = np.array( + [ + [-c, h], + [t - c, h], + [w - c, 0], + [t - c, -h], + [-c, -h], + ] + ).T + elif shape == "triangle": + self._coords = np.array( + [ + [-c, h], + [w, 0], + [-c, -h], + ] + ).T + else: + raise ValueError("unknown vehicle shape name") + + elif isinstance(shape, np.ndarray) and shape.shape[1] == 2: + self._coords = shape + else: + raise TypeError("unknown shape argument") + self._coords *= scale + self._args = kwargs
+ + def _add(self, **kwargs): + # color is fillcolor + edgecolor + # facecolor if None is default + self._ax = plt.gca() + self._object = patches.Polygon(self._coords.T, **{**self._args, **kwargs}) + self._ax.add_patch(self._object) + + def _update(self, x): + + if self._object is not None: + # if animation is initialized + xy = SE2(x) * self._coords + self._object.set_xy(xy.T)
+ + +# ========================================================================= # + + +
[docs]class VehicleIcon(VehicleAnimationBase): +
[docs] def __init__(self, filename, origin=None, scale=1, rotation=0): + """ + Create graphical animation of vehicle as an image icon + + :param filename: Standard icon name or a path to an image + :type filename: str + :param origin: Origin of the vehicle coordinate frame, defaults to centre + :type origin: array_like(2) + :param scale: Length of the vehicle on the plot, defaults to 1 + :type scale: float + :param rotation: Vehicle icon heading in degrees, defaults to 0 + :type rotation: float + :raises ValueError: Icon file not found + :return: animation object + :rtype: VehicleAnimation + + Creates an object that can be passed to a ``Vehicle`` subclass to + depict the moving robot as an image icon during simulation. The image + is translated and rotated to represent the vehicle configuration. + + The car is scaled to an image with a horizontal length (width) of + ``scale`` in the units of the plot. By default the image is assumed to + contain a car parallel to the x-axis and facing right. If the vehicle + is facing upward set ``rotation`` to 90. + + The vehicle rotates about its ``origin`` which is expressed in terms of + normalized coordinates in the range 0 to 1. By default it is in the + middle of the icon image, (0.2, 0.5) moves it toward the back of the + vehicle, (0.8, 0.5) moves it toward the front of the vehicle. + + ``filename`` can be an included image: + + * ``"greycar"`` a grey and white car (top view) + * ``"redcar"`` a red car (top view) + * ``"piano"`` a piano (top view) + + or the path to an image file, including extension. + + The included images are: + + .. image:: ../../rtb-data/rtbdata/data/greycar.png + :width: 200px + :align: center + :alt: "greycar" + + .. image:: ../../rtb-data/rtbdata/data/redcar.png + :width: 300px + :align: center + :alt: "redcar" + + .. image:: ../../rtb-data/rtbdata/data/piano.png + :width: 200px + :align: center + :alt: "piano" + + For example, to animate a simulation with the red car icon:: + + a = VehicleIcon("redcar", scale=2) + veh = Bicycle(driver=RandomPath(10), animation=a) + veh.run(animation=a) + + .. note:: The standard icons are provided in the package ``rtb-data`` + + :seealso: :class:`Vehicle` + """ + super().__init__() + if "." not in filename: + try: + # try the default folder first + image = rtb_load_data( + Path("data") / Path(filename + ".png"), plt.imread + ) + except FileNotFoundError: + raise ValueError(f"{filename} is not a provided icon") + else: + try: + image = plt.imread(filename) + except FileNotFoundError: + raise ValueError(f"icon file {filename} not found") + + self._rotation = rotation + self._image = image + + # figure size of bounding box the image will fill in data coordinates + if origin is None: + origin = [0.5, 0.5] + self._origin = origin + + if image.shape[0] >= image.shape[1]: + # width >= height + self._width = scale + self._height = scale * image.shape[1] / image.shape[0] + else: + # width < height + self._height = scale + self._width = scale * image.shape[0] / image.shape[1]
+ + def _add(self, ax=None, **kwargs): + def imshow_affine(ax, z, *args, **kwargs): + im = ax.imshow(z, *args, **kwargs) + x1, x2, y1, y2 = im.get_extent() + # im._image_skew_coordinate = (x2, y1) + return im + + self._ax = plt.gca() + extent = [ + -self._origin[0] * self._height, + (1 - self._origin[0]) * self._height, + -self._origin[1] * self._width, + (1 - self._origin[1]) * self._width, + ] + self._ax = plt.gca() + + args = {} + if "color" in kwargs and self._image.ndim == 2: + color = kwargs["color"] + del kwargs["color"] + rgb = colors.to_rgb(color) + cmapdata = { + "red": [(0.0, 0.0, 0.0), (1.0, rgb[0], 0.0)], + "green": [(0.0, 0.0, 0.0), (1.0, rgb[1], 0.0)], + "blue": [(0.0, 0.0, 0.0), (1.0, rgb[2], 0.0)], + } + cmap = colors.LinearSegmentedColormap("linear", segmentdata=cmapdata, N=256) + args = {"cmap": cmap} + elif self._image.ndim == 2: + args = {"cmap": "gray"} + if "zorder" not in kwargs: + args["zorder"] = 3 + + self._object = imshow_affine( + self._ax, + self._image, + interpolation="none", + extent=extent, + clip_on=True, + **{**kwargs, **args}, + ) + + def _update(self, x): + + # center_x = self._width // 2 + # center_y = self._height // 2 + center_x = 0 + center_y = 0 + + T = ( + mtransforms.Affine2D() + .rotate_deg_around(center_x, center_y, np.degrees(x[2]) - self._rotation) + .translate(x[0], x[1]) + + self._ax.transData + ) + self._object.set_transform(T)
+ + +if __name__ == "__main__": + from math import pi + + from roboticstoolbox import Bicycle, RandomPath + + V = np.diag(np.r_[0.02, 0.5 * pi / 180] ** 2) + + v = VehiclePolygon(facecolor="None", edgecolor="k") + # v = VehicleIcon('greycar2', scale=2, rotation=90) + + veh = Bicycle(covar=V, animation=v, control=RandomPath(10), verbose=False) + print(veh) + + odo = veh.step([1, 0.3]) + print(odo) + + print(veh.x) + + print(veh.f([0, 0, 0], odo)) + + def control(v, t, x): + goal = (6, 6) + goal_heading = atan2(goal[1] - x[1], goal[0] - x[0]) + d_heading = base.angdiff(goal_heading, x[2]) + v.stopif(base.norm(x[0:2] - goal) < 0.1) + + return (1, d_heading) + + veh.control = RandomPath(10) + p = veh.run(20, animate=True) + # plt.show() + print(p) + + veh.plot_xyt() + plt.show(block=True) + # veh.plot(p) + + # t, x = veh.path(5, u=control) + # print(t) + + # fig, ax = plt.subplots() + + # ax.set_xlim(-5, 5) + # ax.set_ylim(-5, 5) + + # v = VehicleAnimation.Polygon(shape='triangle', maxdim=0.1, color='r') + # v = VehicleAnimation.Icon('car3.png', maxdim=2, centre=[0.3, 0.5]) + # v = VehicleAnimation.Icon('/Users/corkep/Dropbox/code/robotics-toolbox-python/roboticstoolbox/data/car1.png', maxdim=2, centre=[0.3, 0.5]) + # v = VehicleAnimation.icon('car3.png', maxdim=2, centre=[0.3, 0.5]) + # v = VehicleAnimation.marker() + # v.start() + # plt.grid(True) + # # plt.axis('equal') + + # for theta in np.linspace(0, 2 * np.pi, 100): + # v.update([0, 0, theta]) + # plt.pause(0.1) +
+ +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/mobile/Bug2.html b/_modules/roboticstoolbox/mobile/Bug2.html new file mode 100644 index 00000000..50f04e3d --- /dev/null +++ b/_modules/roboticstoolbox/mobile/Bug2.html @@ -0,0 +1,575 @@ + + + + + + roboticstoolbox.mobile.Bug2 — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +

Source code for roboticstoolbox.mobile.Bug2

+"""
+Python Bug Planner
+@Author: Peter Corke, original MATLAB code and Python version
+@Author: Kristian Gibson, initial MATLAB port
+"""
+# from numpy import disp
+# from scipy import integrate
+
+# from spatialmath.pose2d import SE2
+from spatialmath import base
+from spatialmath.base.animate import *
+from scipy.ndimage import *
+
+# from matplotlib import cm
+import matplotlib.pyplot as plt
+
+# from matplotlib import animation
+from roboticstoolbox.mobile.PlannerBase import PlannerBase
+
+
+
[docs]class Bug2(PlannerBase): + """ + Construct a Bug2 reactive navigator + + :param occgrid: occupancy grid + :type occgrid: :class:`OccGrid` instance or ndarray(N,M) + :param kwargs: common arguments for :class:`PlannerBase` superclass + :return: Bug2 reactive navigator + :rtype: Bug2 instance + + Creates an object which simulates an automaton, capable of omnidirectional + motion, finding a path across an occupancy grid using only a bump sensor. + + :reference: "Path-Planning Strategies for a Point Mobile Automaton Moving + Amidst Unknown Obstacles of Arbitrary Shape", Lumelsky and Stepanov, + Algorithmica (1987)2, pp.403-430 + + .. note:: This class is not a planner, even though it subclasses + :class:`PlannerBase`. It can produce very inefficient paths. + + :author: Kristian Gibson and Peter Corke, based on MATLAB version by Peter Corke + :seealso: :class:`PlannerBase` + """ + + def __init__(self, **kwargs): + super().__init__(ndims=2, **kwargs) + + self.H = [] # hit points + self._j = 0 + self._step = 1 + self._m_line = None + self._edge = None + self._k = None + + @property + def m_line(self): + """ + Get m-line + + :return: m-line in homogeneous form + :rtype: ndarray(3) + + This is the m-line computed for the last :meth:`run`. + """ + return self._m_line + + # override query method of base class +
[docs] def run( + self, + start=None, + goal=None, + animate=False, + pause=0.001, + trail=True, + movie=None, + **kwargs, + ): + """ + Find a path using Bug2 reactive navigation algorithm + + :param start: starting position + :type start: array_like(2) + :param goal: goal position + :type goal: array_like(2) + :param animate: show animation of robot moving over the map, + defaults to False + :type animate: bool, optional + :param movie: save animation as a movie, defaults to None. Is either + name of movie or a tuple (filename, frame interval) + :type movie: str or tuple(str, float), optional + :param trail: show the path followed by the robot, defaults to True + :type current: bool, optional + :return: path from ``start`` to ``goal`` + :rtype: ndarray(2,N) + + Compute the path from ``start`` to ``goal`` assuming the robot is capable + of 8-way motion from its current cell to the next. + + .. note:: ``start`` and ``goal`` are given as (x,y) coordinates in the + occupancy grid map, not as matrix row and column coordinates. + + :seealso: :meth:`Bug2.plot` + """ + + # make sure start and goal are set and valid + # super().query(start=start, goal=goal, dtype=np.int, **kwargs) + # make sure start and goal are set and valid + self.start = self.validate_endpoint(start, dtype=int) + self.goal = self.validate_endpoint(goal, dtype=int) + + # compute the m-line + # create homogeneous representation of the line + # line*[x y 1]' = 0 + self._m_line = hom_line(self.start, self.goal) + + if movie is not None: + animate = True + + if animate: + self.plot() + self.plot_m_line() + plt.pause(0.05) + + # movie = MovieWriter(movie) + + robot = self.start + self._step = 1 + path = robot + h = None + + (trail_line,) = plt.plot(0, 0, "y.", label="robot path") + (trailHead,) = plt.plot(0, 0, "ko", zorder=10) + + # iterate using the next() method until we reach the goal + while True: + if animate: + trailHead.set_data(robot[0], robot[1]) + if trail: + trail_line.set_data(path.T) + + if pause > 0: + plt.pause(pause) + # plt.draw() + # plt.show(block=False) + # plt.gcf().canvas.flush_events() + + # movie.add() + + # move to next point on path + robot = self.next(robot) + + # # have we been here before, ie. in a loop + # if any([all(robot == x) for x in path]): + # raise RuntimeError('trapped') + + # are we there yet? + if robot is None: + break + else: + path = np.vstack((path, robot)) + + # movie.done() + if animate: + trailHead.remove() + + return path
+ +
[docs] def plot_m_line(self, ls=None): + """ + Plot m-line + + :param ls: linestyle, defaults to ``"k--"`` + :type ls: str, optional + + Plots the m-line on the current axes. + """ + if ls is None: + ls = "k--" + + x_min, x_max = plt.gca().get_xlim() + y_min, y_max = plt.gca().get_ylim() + if self._m_line[1] == 0: + # handle the case that the line is vertical + plt.plot( + [self._start[0], self._start[0]], [y_min, y_max], "k--", label="m-line" + ) + else: + # regular line + x = np.array([[x_min, 1], [x_max, 1]]) + y = -x @ np.r_[self._m_line[0], self._m_line[2]] + y = y / self._m_line[1] + plt.plot([x_min, x_max], y, ls, zorder=10, label="m-line")
+ + def next(self, position): + """ + One step of the finite state automaton + + :param position: current robot position + :type position: ndarray(2) + :raises RuntimeError: robot is trapped + :return: next robot position + :rtype: ndarray(2) + """ + + l = None + y = None + + if all(self._goal == position): + return None # we have arrived + + if self._step == 1: + # Step 1. Move along the M-line toward the goal + self.message(f"{position}: moving along the M-line (step 1)") + # motion on line toward goal + d = self._goal - position + if abs(d[0]) > abs(d[1]): + # line slope less than 45 deg + dx = 1 if d[0] >= 0 else -1 # np.sign(d[0]) + l = self._m_line + y = -((position[0] + dx) * l[0] + l[2]) / l[1] + dy = int(round(y - position[1])) + else: + # line slope greater than 45 deg + dy = 1 if d[1] >= 0 else -1 # np.sign(d[1]) + l = self._m_line + x = -((position[1] + dy) * l[1] + l[2]) / l[0] + dx = int(round(x - position[0])) + + # detect if next step is an obstacle + if self.isoccupied(position + np.r_[dx, dy]): + self.message(f" {position}: obstacle at {position + np.r_[dx, dy]}") + self.H.append(position) # save hit point + self._step = 2 # transition to step 2 + self.message(f" {position}: change to step 2") + + # get a list of all the points around the obstacle + self._edge, _ = edgelist(self.occgrid.grid == 0, position) + self._k = 0 + else: + n = position + np.array([dx, dy]) + + if self._step == 2: + # Step 2. Move around the obstacle until we reach a point + # on the M-line closer than when we started. + + self.message(f"{position}: moving around the obstacle (step 2)") + if self._k < len(self._edge): + n = self._edge[self._k] # next edge point + else: + # we are at the end of the list of edge points, we + # are back where we started. Step 2.c test. + plt.show(block=True) + raise RuntimeError("robot is trapped") + + # are we on the M-line now ? + if abs(np.inner(np.r_[position, 1], self._m_line)) <= 0.5: + self.message(f" {position}: crossed the M-line") + + # are we closer than when we encountered the obstacle? + if base.norm(position - self._goal) < base.norm( + self.H[-1] - self._goal + ): + self._step = 1 # transition to step 1 + self.message(f" {position}: change to step 1") + return n + + # no, keep going around + self._k += 1 + + return n + + def query(self): + raise NotImplementedError("This class has no query method") + + def plan(self): + raise NotImplementedError("This class has no plan method")
+ + +# Ported from Peter Corke's edgelist function found: +# https://github.com/petercorke/toolbox-common-matlab/blob/master/edgelist.m + + +# these are directions of 8-neighbours in a clockwise direction +# fmt: off +_dirs = np.array([ + [-1, 0], + [-1, 1], + [ 0, 1], + [ 1, 1], + [ 1, 0], + [ 1, -1], + [ 0, -1], + [-1, -1], + ]) +# fmt: on +def edgelist(im, p, direction=1): + """ + Find edge of a region + + :param im: binary image + :type im: ndarray(h,w,int) + :param p: initial point + :type p: array_like(2) + :param direction: direction to traverse region, +1 clockwise [default], -1 + counter-clockwise + :type direction: int, optional + :raises ValueError: initial point is not on the edge + :raises RuntimeError: not able to find path to the goal + :return: edge list, direction vector list + :rtype: tuple of lists + + ``edge, dirs = edgelist(im, seed)`` is the boundary/contour/edge of a region + in the binary image ``im``. ``seed=[X,Y]`` is the coordinate of a point on + the edge of the region of interest, but belonging to the region. + + ``edge`` is a list of coordinates (2) of edge pixels of a region in theThe + elements of the edgelist are NumPy ndarray(2). + + ``dirs`` is a list of integers representing the direction of the edge from + the corresponding point in ``edge`` to the next point in ``edge``. The + integers in the range 0 to 7 represent directions: W SW S SE E NW N NW + respectively. + + - Coordinates are given and returned assuming the matrix is an image, so the + indices are always in the form (x,y) or (column,row). + - ``im` is a binary image where 0 is assumed to be background, non-zero + is an object. + - ``p`` must be a point on the edge of the region. + - The initial point is always the first and last element of the returned edgelist. + - 8-direction chain coding can give incorrect results when used with + blobs founds using 4-way connectivty. + + :Reference: + + - METHODS TO ESTIMATE AREAS AND PERIMETERS OF BLOB-LIKE OBJECTS: A COMPARISON + Luren Yang, Fritz Albregtsen, Tor Lgnnestad and Per Grgttum + IAPR Workshop on Machine Vision Applications Dec. 13-15, 1994, Kawasaki + + """ + p = base.getvector(p, 2, dtype=int) + if direction > 0: + neighbours = np.arange(start=0, stop=8, step=1) + else: + neighbours = np.arange(start=7, stop=-1, step=-1) + + try: + pix0 = im[p[1], p[0]] # color of pixel we start at + except: + raise ValueError("specified coordinate is not within image") + + q = adjacent_point(im, p, pix0) + + # find an adjacent point outside the blob + if q is None: + raise ValueError("no neighbour outside the blob") + + d = None + e = [p] # initialize the edge list + dir = [] # initialize the direction list + p0 = None + + while True: + # find which direction is Q + dq = q - p + for kq in range(0, 8): + # get index of neighbour's direction in range [1,8] + if np.all(dq == _dirs[kq]): + break + + # now test for directions relative to Q + for j in neighbours: + # get index of neighbour's direction in range [1,8] + k = (j + kq) % 8 + # if k > 7: + # k = k - 7 + + # compute coordinate of the k'th neighbour + nk = p + _dirs[k] + + try: + if im[nk[1], nk[0]] == pix0: + # if this neighbour is in the blob it is the next edge pixel + p = nk + break + except: + raise ValueError("Something went wrong calculating edgelist") + + q = nk + + dir.append(k) + # check if we are back where we started + if p0 is None: + p0 = p # make a note of where we started + else: + if all(p == p0): + break + + # keep going, add this point to the edgelist + e.append(p) + + return e, dir + + +# Ported from Peter Corke's adjacent_point function found: +# https://github.com/petercorke/toolbox-common-matlab/blob/master/edgelist.m + + +def adjacent_point(im, seed, pix0): + """Find adjacent point + + :param im: occupancy grid + :type im: ndarray(m,n) + :param seed: initial point + :type seed: ndarray(2) + :param pix0: value of occupancy grid at ``seed`` coordinate + :type pix0: int + :return: coordinate of a neighbour + :rtype: ndarray(2) or None + + Is a neighbouring point of the coordinate ``seed`` that is not within the + region containing the coordinate ``seed``, ie. it is a neighbour but + outside. + """ + p = None + + for d in _dirs: + p = seed + d + try: + if im[p[1], p[0]] != pix0: + return p + except: + pass + + return None + + +# Implementation of Peter Corke's matlab homline function from: +# https://github.com/petercorke/spatialmath-matlab/blob/master/homline.m +def hom_line(p1, p2): + line = np.cross(np.r_[p1[0], p1[1], 1], np.r_[p2[0], p2[1], 1]) + + # normalize so that the result of x*l' is the pixel distance + # from the line + return line / np.linalg.norm(line[0:2]) + + +if __name__ == "__main__": + from roboticstoolbox import rtb_load_matfile + + vars = rtb_load_matfile("data/map1.mat") + map = vars["map"] + + bug2 = Bug2(occgrid=map) + # bug.plan() + path = bug2.run([20, 10], [50, 35], animate=True) +
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/mobile/CurvaturePolyPlanner.html b/_modules/roboticstoolbox/mobile/CurvaturePolyPlanner.html new file mode 100644 index 00000000..0a9d7521 --- /dev/null +++ b/_modules/roboticstoolbox/mobile/CurvaturePolyPlanner.html @@ -0,0 +1,299 @@ + + + + + + roboticstoolbox.mobile.CurvaturePolyPlanner — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.mobile.CurvaturePolyPlanner

+import math
+import scipy.integrate
+import scipy.optimize
+import numpy as np
+
+# import matplotlib.pyplot as plt
+from collections import namedtuple
+from roboticstoolbox.mobile import *
+
+
+def solvepath(x, q0=[0, 0, 0], **kwargs):
+    # x[:4] is 4 coeffs of curvature polynomial
+    # x[4] is total path length
+    # q0 is initial state of the vehicle
+    maxcurvature = 0
+
+    def dotfunc(s, q, poly):
+        # q = (x, y, θ)
+        # qdot = (cosθ, sinθ, ϰ)
+        k = poly[0] * s**3 + poly[1] * s**2 + poly[2] * s + poly[3]
+        # k = ((poly[0] * s + poly[1]) * s  + poly[2]) * s + poly[3]
+
+        # save maximum curvature for this path solution
+        nonlocal maxcurvature
+        maxcurvature = max(maxcurvature, abs(k))
+
+        theta = q[2]
+        return math.cos(theta), math.sin(theta), k
+
+    cpoly = x[:4]
+    s_f = x[4]
+    sol = scipy.integrate.solve_ivp(dotfunc, [0, s_f], q0, args=(cpoly,), **kwargs)
+    return sol.y, maxcurvature
+
+
+def xcurvature(x):
+    # inequality constraint function, must be non-negative
+    _, maxcurvature = solvepath(x, q0=(0, 0, 0))
+    return maxcurvature
+
+
+def costfunc(x, start, goal):
+    # final cost of path from start with params
+    # p[0:4] is polynomial: k0, a, b, c
+    # p[4] is s_f
+
+    # integrate the path for this curvature polynomial and length
+    # path is 3xN
+    path, _ = solvepath(x, q0=start)
+
+    # cost is configuration error at end of path
+    e = np.linalg.norm(path[:, -1] - np.r_[goal])
+
+    return e
+
+
+
[docs]class CurvaturePolyPlanner(PlannerBase): + def __init__(self, curvature=None): + """ + Continuous curvature path planner + + :param curvature: , defaults to None + :type curvature: _type_, optional + + ================== ======================== + Feature Capability + ================== ======================== + Plan :math:`\SE{2}` + Obstacle avoidance No + Curvature Continuous + Motion Forwards only + ================== ======================== + + Example: + + .. runblock:: pycon + + >>> from roboticstoolbox import DubinsPlanner + >>> from math import pi + >>> planner = CurvaturePolyPlanner() + >>> path, status = planner.query(start=(0, 0, pi/2), goal=(1, 0, pi/2)) + >>> print(path[:5,:]) + >>> print(status) + """ + super().__init__(ndims=3) + self.curvature = curvature + +
[docs] def query(self, start, goal): + r""" + Find a path betwee two configurations + + :param start: start configuration :math:`(x, y, \theta)` + :type start: array_like(3), optional + :param goal: goal configuration :math:`(x, y, \theta)` + :type goal: array_like(3), optional + :return: path and status + :rtype: ndarray(N,3), namedtuple + + The path comprises points equally spaced at a distance of ``stepsize``. + + The returned status value has elements: + + +-------------------+-----------------------------------------------+ + | Element | Description | + +-------------------+-----------------------------------------------+ + | ``length`` | total path length | + +-------------------+-----------------------------------------------+ + | ``maxcurvature`` | maximum curvature on path | + +-------------------+-----------------------------------------------+ + | ``poly`` | curvature polynomial coefficients | + +-------------------+-----------------------------------------------+ + + """ + goal = np.r_[goal] + start = np.r_[start] + self._start = start + self._goal = goal + + # initial estimate of path length is Euclidean distance + d = np.linalg.norm(goal[:2] - start[:2]) + # state vector is kappa_0, a, b, c, s_f + + if self.curvature is not None: + nlcontraints = ( + scipy.optimize.NonlinearConstraint(xcurvature, 0, self.curvature), + ) + else: + nlcontraints = () + + sol = scipy.optimize.minimize( + costfunc, + [0, 0, 0, 0, d], + constraints=nlcontraints, + bounds=[(None, None), (None, None), (None, None), (None, None), (d, None)], + args=(start, goal), + ) + print(sol) + path, maxcurvature = solvepath( + sol.x, q0=start, dense_output=True, max_step=1e-2 + ) + + status = namedtuple("CurvaturePolyStatus", ["length", "maxcurvature", "poly"])( + sol.x[4], maxcurvature, sol.x[:4] + ) + + return path.T, status
+ + +if __name__ == "__main__": + from math import pi + + # start = (1, 1, pi/4) + # goal = (-3, -3, -pi/4) + # start = (0, 0, -pi/4) + # goal = (1, 2, pi/4) + + start = (0, 0, pi / 2) + goal = (1, 0, pi / 2) + + planner = CurvaturePolyPlanner() + path, status = planner.query(start, goal) + print("start", path[0, :]) + print("goal", path[-1, :]) + + print(status) + planner.plot(path, block=True) + + ## attempt polynomial scaling, doesnt seem to work + # sf = status.s_f + # c = status.poly + # print(c) + + # print(solvepath(np.r_[c, sf], start)) + + # for i in range(4): + # c[i] /= sf ** (3-i) + + # print(solvepath(np.r_[c, 1], start)) + # print(c) +
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/mobile/DistanceTransformPlanner.html b/_modules/roboticstoolbox/mobile/DistanceTransformPlanner.html new file mode 100644 index 00000000..0971cb6b --- /dev/null +++ b/_modules/roboticstoolbox/mobile/DistanceTransformPlanner.html @@ -0,0 +1,512 @@ + + + + + + roboticstoolbox.mobile.DistanceTransformPlanner — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.mobile.DistanceTransformPlanner

+"""
+@Author: Peter Corke, original MATLAB code and Python version
+@Author: Kristian Gibson, initial MATLAB port
+"""
+from numpy import disp
+from scipy import integrate
+from spatialmath.base.transforms2d import *
+from spatialmath.base.vectors import *
+from spatialmath.pose2d import SE2
+from spatialmath import base
+from scipy.ndimage import *
+import matplotlib.pyplot as plt
+from matplotlib import cm
+from roboticstoolbox.mobile.PlannerBase import PlannerBase
+
+
+
[docs]class DistanceTransformPlanner(PlannerBase): + r""" + Distance transform path planner + + :param occgrid: occupancy grid + :type occgrid: :class:`BinaryOccGrid` or ndarray(h,w) + :param metric: distane metric, one of: "euclidean" [default], "manhattan" + :type metric: str optional + :param kwargs: common planner options, see :class:`PlannerBase` + + ================== ======================== + Feature Capability + ================== ======================== + Plan :math:`\mathbb{R}^2`, discrete + Obstacle avoidance Yes, occupancy grid + Curvature Discontinuous + Motion Omnidirectional + ================== ======================== + + Creates a planner that finds the path between two points in the 2D grid + using omnidirectional motion and avoiding occupied cells. The path + comprises a set of 4- or -way connected points in adjacent cells. Also + known as the wavefront, grassfire or brushfire planning algorithm. + + The map is described by a 2D occupancy ``occgrid`` whose elements are zero + if traversable of nonzero if untraversable, ie. an obstacle. + + The cells are assumed to be unit squares. Crossing the cell horizontally or + vertically is a travel distance of 1, and for the Euclidean distance + measure, the crossing the cell diagonally is a distance of :math:`\sqrt{2}`. + + Example: + + .. runblock:: pycon + + >>> from roboticstoolbox import DistanceTransformPlanner + >>> import numpy as np + >>> simplegrid = np.zeros((6, 6)); + >>> simplegrid[2:5, 3:5] = 1 + >>> dx = DistanceTransformPlanner(simplegrid, goal=(1, 1), distance="manhattan"); + >>> dx.plan() + >>> path = dx.query(start=(5, 4)) + >>> print(path.T) + + .. warning:: The distance planner is iterative and implemented in Python, will + be slow for very large occupancy grids. + + :author: Peter Corke + + :seealso: :meth:`plan` :meth:`query` :class:`PlannerBase` + """ + + def __init__(self, occgrid=None, metric="euclidean", **kwargs): + + super().__init__(occgrid=occgrid, ndims=2, **kwargs) + self._metric = metric + self._distancemap = None + + @property + def metric(self): + """ + Get the distance metric + + :return: Get the metric, either "euclidean" or "manhattan" + :rtype: str + """ + return self._metric + + @property + def distancemap(self): + """ + Get the distance map + + :return: distance map + :rtype: ndarray(h,w) + + The 2D array, the same size as the passed occupancy grid, has elements + equal to nan if they contain an obstacle, otherwise the minimum + obstacle-free distance to the goal using the particular distance metric. + """ + return self._distancemap + + def __str__(self): + s = super().__str__() + s += f"\n Distance metric: {self._metric}" + if self.distancemap is not None: + s += ", Distance map: computed " + else: + s += ", Distance map: empty " + + return s + +
[docs] def plan(self, goal=None, animate=False, summary=False): + r""" + Plan path using distance transform + + :param goal: goal position :math:`(x, y)`, defaults to previously set value + :type goal: array_like(2), optional + + Compute the distance transform for all non-obstacle cells, which is the + minimum obstacle-free distance to the goal using the particular distance + metric. + + :seealso: :meth:`query` + """ + # show = None + # if animate: + # show = 0.05 + # else: + # show = 0 + + if goal is not None: + self.goal = goal + + if self._goal is None: + raise ValueError("No goal specified here or in constructor") + + self._distancemap = distancexform( + self.occgrid.grid, + goal=self._goal, + metric=self._metric, + animate=animate, + summary=summary, + )
+ +
[docs] def next(self, position): + """ + Find next point on the path + + :param position: current robot position + :type position: array_like(2) + :raises RuntimeError: no plan has been computed + :return: next robot position + :rtype: ndarray(2) + + Return the robot position that is one step closer to the goal. Called + by :meth:`query` to find a path from start to goal. + + :seealso: :meth:`plan` :meth:`query` + """ + if self.distancemap is None: + Error("No distance map computed, you need to plan.") + + directions = np.array( + [ + # dy dx + [-1, -1], + [0, -1], + [1, -1], + [-1, 0], + [0, 0], + [1, 0], + [0, 1], + [1, 1], + ], + dtype=int, + ) + + x = int(position[0]) + y = int(position[1]) + + min_dist = np.inf + for d in directions: + try: + if self._distancemap[y + d[0], x + d[1]] < min_dist: + min_dir = d + min_dist = self._distancemap[y + d[0], x + d[1]] + except: + # come here if the neighbouring cell is outside the map bounds + # raise RuntimeError(f"Unexpected error finding next min dist at {d}") + pass + + if np.isinf(min_dist): + raise RuntimeError("no minimum found, shouldn't happen") + + x = x + min_dir[1] + y = y + min_dir[0] + + next = np.r_[x, y] + if all(next == self._goal): + return None + else: + return next
+ +
[docs] def plot_3d(self, path=None, ls=None): + """ + Plot path on 3D cost surface + + :param path: robot path, defaults to None + :type path: ndarray(N,2), optional + :param ls: dictionary of Matplotlib linestyle options, defaults to None + :type ls: dict, optional + :return: Matplotlib 3D axes + :rtype: Axes + + Creates a 3D plot showing distance from the goal as a cost surface. + Overlays the path if given. + + .. warning:: The visualization is poor because of Matplotlib's poor hidden + line/surface handling. + """ + fig = plt.figure() + ax = fig.add_subplot(111, projection="3d") + + distance = self._distancemap + X, Y = np.meshgrid(np.arange(distance.shape[1]), np.arange(distance.shape[0])) + surf = ax.plot_surface( + X, Y, distance, linewidth=1, antialiased=False # cmap='gray', + ) + + if path is not None: + # k = sub2ind(np.shape(self._distancemap), p[:, 1], p[:, 0]) + height = distance[path[:, 1], path[:, 0]] + ax.plot(path[:, 0], path[:, 1], height, **ls) + + plt.xlabel("x") + plt.ylabel("y") + ax.set_zlabel("z") + plt.show() + return ax
+ + +import numpy as np + + +def distancexform(occgrid, goal, metric="cityblock", animate=False, summary=False): + """ + Distance transform for path planning + + :param occgrid: Occupancy grid, 0 is free, >0 is occupied/obstacle + :type occgrid: NumPy ndarray + :param goal: Goal position (x,y) + :type goal: 2-element array-like + :param metric: distance metric, defaults to 'cityblock' + :type metric: str, optional + :param animate: animate the iterations of the algorithm + :return: Distance transform matrix + :rtype: NumPy ndarray + + Implements the grass/brush fire algorithm to compute, for every reachable + cell in the occupancy grid, its distance from the goal. + + The result is an array, the same size as the occupancy grid ``occgrid``, + where each cell contains the distance to the goal according to the chosen + ``metric``. In addition: + + - Obstacle cells will be set to ``nan`` + - Unreachable cells, ie. free cells _inside obstacles_ will be set + to ``inf`` + + The cells of the passed occupancy grid are: + - zero, cell is free or driveable + - one, cell is an obstacle, or not driveable + """ + + # build the matrix for performing distance transform: + # - obstacles are nan + # - other cells are inf + # - goal is zero + + goal = base.getvector(goal, 2, dtype=int) + + distance = occgrid.astype(np.float32) + distance[occgrid > 0] = np.nan # assign nan to obstacle cells + distance[occgrid == 0] = np.inf # assign inf to other cells + distance[goal[1], goal[0]] = 0 # assign zero to goal + + # create the appropriate distance matrix D + if metric.lower() in ("manhattan", "cityblock"): + # fmt: off + D = np.array([ + [ np.inf, 1, np.inf], + [ 1, 0, 1], + [ np.inf, 1, np.inf] + ]) + # fmt: on + elif metric.lower() == "euclidean": + r2 = np.sqrt(2) + # fmt: off + D = np.array([ + [ r2, 1, r2], + [ 1, 0, 1], + [ r2, 1, r2] + ]) + # fmt: on + + # get ready to iterate + count = 0 + ninf = np.inf # number of infinities in the map + + h = None + while True: + distance = grassfire_step(distance, D) + distance[occgrid > 0] = np.nan # reinsert nans for obstacles + + count += 1 + + if animate: + # TODO, needs work to update colorbar and be faster + display = distance.copy() + display[np.isinf(display)] = 0 + if h is None: + plt.figure() + plt.xlabel("x") + plt.ylabel("y") + ax = plt.gca() + plt.pause(0.001) + cmap = cm.get_cmap("gray") + cmap.set_bad("red") + cmap.set_over("white") + h = plt.imshow(display, cmap=cmap) + plt.colorbar(label="distance") + else: + h.remove() + h = plt.imshow(display, cmap=cmap) + plt.pause(0.001) + + ninfnow = np.isinf(distance).sum() # current number of Infs + if ninfnow == ninf: + # stop if the number of Infs left in the map had stopped reducing + # it may never get to zero if there are unreachable cells in the map + break + + ninf = ninfnow + + if summary: + print(f"{count:d} iterations, {ninf:d} unreachable cells") + return distance + + +def grassfire_step(G, D): + + # pad with inf + H = np.pad(G, max(D.shape) // 2, "constant", constant_values=np.inf) + rows, columns = G.shape + + # inspired by https://landscapearchaeology.org/2018/numpy-loops/ + minimum = np.full(G.shape, np.inf) + for y in range(3): + for x in range(3): + v = H[y : rows + y, x : columns + x] + D[y, x] + # we use fmin() because it ignores NaNs, behaves like MATLAB min() + minimum = np.fmin(minimum, v) + + return minimum + + +if __name__ == "__main__": + pass + + # # make a simple map, as per the MOOCs + # occgrid = np.zeros((10,10)) + # occgrid[3:6,2:7] = 1 + # occgrid[4,3:5] = 0 # hole in the obstacle + # # occgrid[7:8,7] = 1 # extra bit + + # print(occgrid) + # print() + + # goal=[5,7] + # np.set_printoptions(precision=1) + # dx = distancexform(occgrid, goal, metric='Euclidean') + # print(dx) + + # from roboticstoolbox import DistanceTransformPlanner, rtb_loadmat + + # house = rtb_loadmat('data/house.mat') + # floorplan = house['floorplan'] + # places = house['places'] + + # dx = DistanceTransformPlanner(floorplan) + # print(dx) + # dx.goal = [1,2] + # dx.plan(places.kitchen) + # path = dx.query(places.br3) + # dx.plot(path, block=True) +
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/mobile/DstarPlanner.html b/_modules/roboticstoolbox/mobile/DstarPlanner.html new file mode 100644 index 00000000..89768737 --- /dev/null +++ b/_modules/roboticstoolbox/mobile/DstarPlanner.html @@ -0,0 +1,710 @@ + + + + + + roboticstoolbox.mobile.DstarPlanner — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
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+ +

Source code for roboticstoolbox.mobile.DstarPlanner

+"""
+
+
+
+See Wikipedia article (https://en.wikipedia.org/wiki/D*)
+
+"""
+import math
+
+from sys import maxsize
+from collections import namedtuple
+from enum import IntEnum, auto
+import matplotlib.pyplot as plt
+import numpy as np
+import matplotlib.cm as cm
+from roboticstoolbox.mobile.PlannerBase import PlannerBase
+from roboticstoolbox.mobile.OccGrid import BinaryOccupancyGrid, OccupancyGrid
+import heapq
+import bisect
+import math
+
+show_animation = True
+
+# ======================================================================== #
+
+# The following code is modified from Python Robotics
+# https://github.com/AtsushiSakai/PythonRobotics/tree/master/PathPlanning
+# D* grid planning
+# Author: Nirnay Roy
+# Copyright (c) 2016 - 2022 Atsushi Sakai and other contributors: https://github.com/AtsushiSakai/PythonRobotics/contributors
+# Released under the MIT license: https://github.com/AtsushiSakai/PythonRobotics/blob/master/LICENSE
+
+# for performance reasons there are some important differences compared to
+# the version from Python Robotics:
+#
+# 1. replace the classic D* functions min__State(), get_kmin(), insert(), remove()
+#    with heapq.heappush() and heapq.heappop(). The open_list is now a list of
+#    tuples (k, _State) maintained by heapq, rather than a set
+#
+# 2. use enums rather than strings for cell state
+#
+# 3. lots of unnecessary inserting/deleting from the open_list due to arithmetic
+#    rounding in tests for costs h and k:
+#    - replace equality tests with math.isclose() which is faster than np.isclose()
+#    - add an offset to inequality tests, X > Y becomes X > Y + tol
+
+
+class _Tag(IntEnum):
+    NEW = auto()
+    OPEN = auto()
+    CLOSED = auto()
+
+
+class _State:
+    def __init__(self, x, y):
+        self.x = x
+        self.y = y
+        self.parent = None  # 'back pointer' to next _State
+        self.t = _Tag.NEW  # open closed new
+        self.h = 0  # cost to goal
+        self.k = 0  # estimate of shortest path cost
+
+    def __str__(self):
+        return f"({self.x}, {self.y})"  # [{self.h:.1f}, {self.k:.1f}]"
+
+    def __repr__(self):
+        return self.__str__()
+
+    def __lt__(self, other):
+        return True
+
+
+class _Map:
+    def __init__(self, row, col):
+        self.row = row
+        self.col = col
+        self._Map = self.init__Map()
+
+    def init__Map(self):
+        _Map_list = []  # list of rows
+        for i in range(self.row):
+            # for each row, make a list
+            tmp = []
+            for j in range(self.col):
+                tmp.append(_State(j, i))  # append column to the row
+            _Map_list.append(tmp)  # append row to _Map
+        return _Map_list
+
+    _neighbours = [(-1, -1), (-1, 0), (-1, 1), (0, -1), (0, 1), (1, -1), (1, 0), (1, 1)]
+
+    def get_neighbors(self, _State):
+        _State_list = []
+
+        for i, j in self._neighbours:
+            try:
+                if _State.x + i >= 0 and _State.y + j >= 0:
+                    _State_list.append(self._Map[_State.y + j][_State.x + i])
+            except IndexError:
+                pass
+        return _State_list
+
+    _root2 = np.sqrt(2)
+
+    def cost(self, _State1, _State2):
+        c = (
+            self.costmap[_State1.y, _State1.x] + self.costmap[_State2.y, _State2.x]
+        ) / 2
+
+        dx = _State1.x - _State2.x
+        dy = _State1.y - _State2.y
+        if dx == 0 or dy == 0:
+            # NSEW movement, distance of 1
+            return c
+        else:
+            # diagonal movement, distance of sqrt(2)
+            return c * self._root2
+        # return c * math.sqrt(()**2 +
+        #                  ()**2)
+
+    def show_h(self):
+        h = np.empty((self.col, self.row))
+
+        for x in range(self.col):
+            for y in range(self.row):
+                h[y, x] = self._Map[y][x].h
+        print(h)
+
+
+class _Dstar:
+    def __init__(self, _Map, tol=1e-6):
+        self._Map = _Map
+        # self.open_list = set()
+        self.open_list = []
+        self.nexpand = 0
+        self.tol = tol
+
+    def process__State(self, verbose=False):
+        if verbose:
+            print(
+                "FRONTIER ", " ".join([f"({x[1].x}, {x[1].y})" for x in self.open_list])
+            )
+
+        # get _State from frontier
+        if len(self.open_list) == 0:
+            if verbose:
+                print("  x is None ")
+            return -1
+        k_old, x = heapq.heappop(self.open_list)
+        x.t = _Tag.CLOSED
+
+        self.nexpand += 1
+
+        if verbose:
+            print(f"EXPAND {x}, {k_old:.1f}")
+
+        if x.h > k_old + self.tol:
+            # RAISE _State
+            if verbose:
+                print("  raise")
+            for y in self._Map.get_neighbors(x):
+                if (
+                    y.t is not _Tag.NEW
+                    and y.h <= k_old - self.tol
+                    and x.h > y.h + self._Map.cost(x, y) + self.tol
+                ):
+                    if verbose:
+                        print(
+                            f"  {x} cost from {x.h:.1f} to {y.h + self._Map.cost(x, y):.1f}; parent from {x.parent} to {y}"
+                        )
+                    x.parent = y
+                    x.h = y.h + self._Map.cost(x, y)
+
+        if math.isclose(x.h, k_old, rel_tol=0, abs_tol=self.tol):
+            # normal _State
+            if verbose:
+                print("  normal")
+            for y in self._Map.get_neighbors(x):
+                if (
+                    y.t is _Tag.NEW
+                    or (
+                        y.parent == x
+                        and not math.isclose(
+                            y.h, x.h + self._Map.cost(x, y), rel_tol=0, abs_tol=self.tol
+                        )
+                    )
+                    or (y.parent != x and y.h > x.h + self._Map.cost(x, y) + self.tol)
+                ):
+                    if verbose:
+                        print(f"  reparent {y} from {y.parent} to {x}")
+                    y.parent = x
+                    self.insert(y, x.h + self._Map.cost(x, y))
+        else:
+            # RAISE or LOWER _State
+            if verbose:
+                print("  raise/lower")
+            for y in self._Map.get_neighbors(x):
+                if y.t is _Tag.NEW or (
+                    y.parent == x
+                    and not math.isclose(
+                        y.h, x.h + self._Map.cost(x, y), rel_tol=0, abs_tol=self.tol
+                    )
+                ):
+                    if verbose:
+                        print(
+                            f"  {y} cost from {y.h:.1f} to {y.h + self._Map.cost(x, y):.1f}; parent from {y.parent} to {x}; add to frontier"
+                        )
+                    y.parent = x
+                    self.insert(y, x.h + self._Map.cost(x, y))
+                else:
+                    if (
+                        y.parent != x
+                        and y.h > x.h + self._Map.cost(x, y) + self.tol
+                        and x.t is _Tag.CLOSED
+                    ):
+                        self.insert(x, x.h)
+                        if verbose:
+                            print(f"  {x}, {x.h:.1f} add to frontier")
+                    else:
+                        if (
+                            y.parent != x
+                            and x.h > y.h + self._Map.cost(y, x) + self.tol
+                            and y.t is _Tag.CLOSED
+                            and y.h > k_old + self.tol
+                        ):
+                            self.insert(y, y.h)
+                        if verbose:
+                            print(f"  {y}, {y.h:.1f} add to frontier")
+        if verbose:
+            print()
+
+        if len(self.open_list) == 0:
+            return -1
+        else:
+            return self.open_list[0][0]
+
+    ninsert = 0
+    nin = 0
+
+    def insert(self, _State, h_new):
+        self.ninsert += 1
+
+        if _State.t is _Tag.NEW:
+            _State.k = h_new
+
+        elif _State.t is _Tag.OPEN:
+            k_new = min(_State.k, h_new)
+            if _State.k == k_new:
+                # k hasn't changed, and vertex already in frontier
+                # just update h and be done
+                _State.h = h_new
+                return
+            else:
+                # k has changed, we need to remove the vertex from the list
+                # and re-insert it
+                _State.k = k_new
+
+                # scan the list to find vertex, then remove it
+                # this is quite slow
+                for i, item in enumerate(self.open_list):
+                    if item[1] is _State:
+                        del self.open_list[i]
+                        break
+
+            # _State.k = min(_State.k, h_new)
+            # # remove the item from the open list
+            # # print('node already in open list, remove it first')
+            # #TODO use bisect on old _State.k to find the entry
+            # for i, item in enumerate(self.open_list):
+            #     if item[1] is _State:
+            #         del self.open_list[i]
+            #         break
+
+        elif _State.t is _Tag.CLOSED:
+            _State.k = min(_State.h, h_new)
+
+        _State.h = h_new
+        _State.t = _Tag.OPEN
+
+        # self.open_list.add(_State)
+        heapq.heappush(self.open_list, (_State.k, _State))
+
+    def modify_cost(self, x, newcost):
+        self._Map.costmap[x.y, x.x] = newcost
+        if x.t is _Tag.CLOSED:
+            self.insert(x, x.parent.h + self._Map.cost(x, x.parent))
+        if len(self.open_list) == 0:
+            return -1
+        else:
+            # lowest priority item is always at index 0 according to docs
+            return self.open_list[0][0]
+
+    def showparents(self):
+        for y in range(self._Map.row - 1, -1, -1):
+            if y == self._Map.row - 1:
+                print("   ", end="")
+                for x in range(self._Map.col):
+                    print(f"  {x}   ", end="")
+                print()
+            print(f"{y}: ", end="")
+            for x in range(self._Map.col):
+                x = self._Map._Map[y][x]
+                par = x.parent
+                if par is None:
+                    print("  G   ", end="")
+                else:
+                    print(f"({par.x},{par.y}) ", end="")
+            print()
+        print()
+
+
+# ====================== RTB wrapper ============================= #
+
+# Copyright (c) 2022 Peter Corke: https://github.com/petercorke/robotics-toolbox-python
+# Released under the MIT license: https://github.com/AtsushiSakai/PythonRobotics/blob/master/LICENSE
+
[docs]class DstarPlanner(PlannerBase): + r""" + D* path planner + + :param costmap: traversability costmap + :type costmap: OccGrid or ndarray(w,h) + :param kwargs: common planner options, see :class:`PlannerBase` + + ================== ======================== + Feature Capability + ================== ======================== + Plan :math:`\mathbb{R}^2`, discrete + Obstacle avoidance Yes, occupancy grid + Curvature Discontinuous + Motion Omnidirectional + ================== ======================== + + Creates a planner that finds the minimum-cost path between two points in the + plane using omnidirectional motion. The path comprises a set of 8-way + connected points in adjacent cells. + + The map is described by a 2D ``costmap`` whose elements indicate the cost + of traversing that cell. The cost of diagonal traverse is :math:`\sqrt{2}` + the value of the cell. An infinite cost indicates an untraversable cell + or obstacle. + + Example: + + .. runblock:: pycon + + >>> from roboticstoolbox import DstarPlanner + >>> import numpy as np + >>> costmap = np.ones((6, 6)); + >>> costmap[2:5, 3:5] = 10 + >>> ds = DstarPlanner(costmap, goal=(1, 1)); + >>> ds.plan() + >>> path, status = ds.query(start=(5, 4)) + >>> print(path.T) + >>> print(status) + + :thanks: based on D* grid planning included from `Python Robotics <https://github.com/AtsushiSakai/PythonRobotics/tree/master/PathPlanning>`_ + :seealso: :class:`PlannerBase` + """ + + def __init__(self, costmap=None, **kwargs): + super().__init__(ndims=2, **kwargs) + + if isinstance(costmap, np.ndarray): + self.costmap = costmap + elif isinstance(costmap, OccupancyGrid): + self.costmap = costmap.grid + elif self.occgrid is not None: + self.costmap = np.where(self.occgrid.grid > 0, np.inf, 1) + else: + raise ValueError("unknown type of map") + self._Map = _Map(self.costmap.shape[0], self.costmap.shape[1]) + self._Map.costmap = self.costmap + self._Dstar = _Dstar(self._Map) # , tol=0) + +
[docs] def plan(self, goal=None, animate=False, progress=True, summary=False): + r""" + Plan D* path + + :param goal: goal position :math:`(x, y)`, defaults to previously set value + :type goal: array_like(2), optional + :param animate: animate the planning algorithm iterations, defaults to False + :type animate: bool, optional + :param progress: show progress bar, defaults to True + :type progress: bool, optional + + Compute the minimum-cost obstacle-free distance to the goal from all + points in the grid. + """ + if goal is not None: + self.goal = goal + + if self._goal is None: + raise ValueError("No goal specified here or in constructor") + + self._goal = self._goal.astype(int) + + goal_State = self._Map._Map[self._goal[1]][self._goal[0]] + self.goal_State = goal_State + + # self._Dstar.open_list.add(goal_State) + self._Dstar.insert(goal_State, 0) + + while True: + ret = self._Dstar.process__State() + # print('plan', ret, len(self._Dstar.open_list)) + + if ret == -1: + break + + if summary: + print(self._Dstar.ninsert, self._Dstar.nin)
+ + @property + def nexpand(self): + """ + Number of node expansions + + :return: number of expansions + :rtype: int + + This number will increase during initial planning, and also if + replanning is invoked during the :meth:`query`. + """ + return self._Dstar.nexpand + +
[docs] def query(self, start, sensor=None, animate=False, verbose=False): + """ + Find path with replanning + + :param start: start position :math:`(x,y)` + :type start: array_like(2) + :param sensor: sensor function, defaults to None + :type sensor: callable, optional + :param animate: animate the motion of the robot, defaults to False + :type animate: bool, optional + :param verbose: display detailed diagnostic information about D* operations, defaults to False + :type verbose: bool, optional + :return: path from start to goal, one point :math:`(x, y)` per row + :rtype: ndarray(N,2) + + If ``sensor`` is None then the plan determined by the ``plan`` phase + is used unaltered. + + If ``sensor`` is not None it must be callable, and is called at each + step of the path with the current robot coordintes: + + sensor((x, y)) + + and mimics the behaviour of a simple sensor onboard the robot which can + dynamically change the costmap. The function return a list (0 or more) + of 3-tuples (x, y, newcost) which are the coordinates of cells and their + cost. If the cost has changed this will trigger D* incremental + replanning. In this case the value returned by :meth:`nexpand` will + increase, according to the severity of the replanning. + + :seealso: :meth:`plan` + """ + self.start = start + start_State = self._Map._Map[start[1]][start[0]] + s = start_State + s = s.parent + tmp = start_State + + if sensor is not None and not callable(sensor): + raise ValueError("sensor must be callable") + + cost = tmp.h + self.goal_State.h = 0 + + path = [] + while True: + path.append((tmp.x, tmp.y)) + if tmp == self.goal_State: + break + + # x, y = tmp.parent.x, tmp.parent.y + + if sensor is not None: + changes = sensor((tmp.x, tmp.y)) + if changes is not None: + # make changes to the plan + for x, y, newcost in changes: + X = self._Dstar._Map._Map[y][x] + # print(f"change cost at ({x}, {y}) to {newcost}") + val = self._Dstar.modify_cost(X, newcost) + # propagate the changes to plan + print("propagate") + # self._Dstar.showparents() + while val != -1 and val < tmp.h: + val = self._Dstar.process__State(verbose=verbose) + # print('open set', len(self._Dstar.open_list)) + # self._Dstar.showparents() + + tmp = tmp.parent + # print('move to ', tmp) + + status = namedtuple( + "_DstarStatus", + [ + "cost", + ], + ) + + return np.array(path), status(cost)
+ + # # Just feed self._h into plot function from parent as p + + # def next(self, current): + # if not self._valid_plan: + # Error("Cost _Map has changed, replan") + # # x = sub2ind(np.shape(self._costmap), current[1], current[0]) + # # x = self._b[x] + # i = np.ravel_multi_index([current[1], current[0]], self._costmap.shape) + # i = self._b[i] + + # if i == 0: + # return None # we have arrived + # else: + # x = np.unravel_index((i), self._costmap.shape) + # return x[1], x[0] + + +if __name__ == "__main__": + + og = np.zeros((10, 10)) + og[4:8, 3:6] = 1 + print(og) + + ds = DstarPlanner(occgrid=og) + print(ds.costmap) + + start = (1, 1) + goal = (7, 6) + + ds.plan(goal=goal) + ds._Map.show_h() + + # path, status = ds.query(start=start) + # print(path) + # print(status) + + # ds.plot(path=path) + + def sensorfunc(pos): + if pos == (3, 3): + changes = [] + for x in range(3, 6): + for y in range(0, 4): + changes.append((x, y, 100)) + return changes + + path2, status2 = ds.query(start=start, sensor=sensorfunc, verbose=False) + print(ds._Map.costmap) + + ds._Map.show_h() + + # ds._Dstar.replan() + + # path2, status = ds.query(start=start) + print(path2) + print(status2) + + ds.plot(path=path2) + # ds.plot(path=path2, block=True) + + # obstacle zone + # m.set_cost([30, 60, 20, 60], np.inf) + + # start = [10, 10] + # goal = [70, 70] + # if show_animation: + # m.plot() + # plt.plot(start[0], start[1], "og") + # plt.plot(goal[0], goal[1], "xb") + # plt.axis("equal") + + # start = m._Map[start[0]][start[1]] + # end = m._Map[goal[0]][goal[1]] + # _Dstar = _Dstar(m) + # rx, ry = _Dstar.run(start, end) + + # if show_animation: + # plt.plot(rx, ry, "-r") + # plt.show(block=True) + + # # costly zone + # m.set_cost([30, 40, 60, 80], 1.70, modify=_Dstar) + + # _Dstar.replan() + + plt.show(block=True) +
+ +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/mobile/DubinsPlanner.html b/_modules/roboticstoolbox/mobile/DubinsPlanner.html new file mode 100644 index 00000000..f2e15d68 --- /dev/null +++ b/_modules/roboticstoolbox/mobile/DubinsPlanner.html @@ -0,0 +1,593 @@ + + + + + + roboticstoolbox.mobile.DubinsPlanner — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
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+
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+
+ +

Source code for roboticstoolbox.mobile.DubinsPlanner

+# ======================================================================== #
+
+# The following code is modified from Python Robotics
+# https://github.com/AtsushiSakai/PythonRobotics/tree/master/PathPlanning
+# Dubins planning
+# Author: Atsushi Sakai
+# Copyright (c) 2016 - 2022 Atsushi Sakai and other contributors: https://github.com/AtsushiSakai/PythonRobotics/contributors
+# Released under the MIT license: https://github.com/AtsushiSakai/PythonRobotics/blob/master/LICENSE
+
+import math
+from collections import namedtuple
+from roboticstoolbox.mobile import PlannerBase
+
+import matplotlib.pyplot as plt
+import numpy as np
+from spatialmath import *
+from spatialmath import base
+
+
+def left_straight_left(alpha, beta, d):
+    sa = math.sin(alpha)
+    sb = math.sin(beta)
+    ca = math.cos(alpha)
+    cb = math.cos(beta)
+    c_ab = math.cos(alpha - beta)
+
+    tmp0 = d + sa - sb
+
+    mode = ["L", "S", "L"]
+    p_squared = 2 + (d * d) - (2 * c_ab) + (2 * d * (sa - sb))
+    if p_squared < 0:
+        return None, None, None, mode
+    tmp1 = math.atan2((cb - ca), tmp0)
+    t = base.wrap_0_2pi(-alpha + tmp1)
+    p = math.sqrt(p_squared)
+    q = base.wrap_0_2pi(beta - tmp1)
+
+    return t, p, q, mode
+
+
+def right_straight_right(alpha, beta, d):
+    sa = math.sin(alpha)
+    sb = math.sin(beta)
+    ca = math.cos(alpha)
+    cb = math.cos(beta)
+    c_ab = math.cos(alpha - beta)
+
+    tmp0 = d - sa + sb
+    mode = ["R", "S", "R"]
+    p_squared = 2 + (d * d) - (2 * c_ab) + (2 * d * (sb - sa))
+    if p_squared < 0:
+        return None, None, None, mode
+    tmp1 = math.atan2((ca - cb), tmp0)
+    t = base.wrap_0_2pi(alpha - tmp1)
+    p = math.sqrt(p_squared)
+    q = base.wrap_0_2pi(-beta + tmp1)
+
+    return t, p, q, mode
+
+
+def left_straight_right(alpha, beta, d):
+    sa = math.sin(alpha)
+    sb = math.sin(beta)
+    ca = math.cos(alpha)
+    cb = math.cos(beta)
+    c_ab = math.cos(alpha - beta)
+
+    p_squared = -2 + (d * d) + (2 * c_ab) + (2 * d * (sa + sb))
+    mode = ["L", "S", "R"]
+    if p_squared < 0:
+        return None, None, None, mode
+    p = math.sqrt(p_squared)
+    tmp2 = math.atan2((-ca - cb), (d + sa + sb)) - math.atan2(-2.0, p)
+    t = base.wrap_0_2pi(-alpha + tmp2)
+    q = base.wrap_0_2pi(-base.wrap_0_2pi(beta) + tmp2)
+
+    return t, p, q, mode
+
+
+def right_straight_left(alpha, beta, d):
+    sa = math.sin(alpha)
+    sb = math.sin(beta)
+    ca = math.cos(alpha)
+    cb = math.cos(beta)
+    c_ab = math.cos(alpha - beta)
+
+    p_squared = (d * d) - 2 + (2 * c_ab) - (2 * d * (sa + sb))
+    mode = ["R", "S", "L"]
+    if p_squared < 0:
+        return None, None, None, mode
+    p = math.sqrt(p_squared)
+    tmp2 = math.atan2((ca + cb), (d - sa - sb)) - math.atan2(2.0, p)
+    t = base.wrap_0_2pi(alpha - tmp2)
+    q = base.wrap_0_2pi(beta - tmp2)
+
+    return t, p, q, mode
+
+
+def right_left_right(alpha, beta, d):
+    sa = math.sin(alpha)
+    sb = math.sin(beta)
+    ca = math.cos(alpha)
+    cb = math.cos(beta)
+    c_ab = math.cos(alpha - beta)
+
+    mode = ["R", "L", "R"]
+    tmp_rlr = (6.0 - d * d + 2.0 * c_ab + 2.0 * d * (sa - sb)) / 8.0
+    if abs(tmp_rlr) > 1.0:
+        return None, None, None, mode
+
+    p = base.wrap_0_2pi(2 * math.pi - math.acos(tmp_rlr))
+    t = base.wrap_0_2pi(
+        alpha - math.atan2(ca - cb, d - sa + sb) + base.wrap_0_2pi(p / 2.0)
+    )
+    q = base.wrap_0_2pi(alpha - beta - t + base.wrap_0_2pi(p))
+    return t, p, q, mode
+
+
+def left_right_left(alpha, beta, d):
+    sa = math.sin(alpha)
+    sb = math.sin(beta)
+    ca = math.cos(alpha)
+    cb = math.cos(beta)
+    c_ab = math.cos(alpha - beta)
+
+    mode = ["L", "R", "L"]
+    tmp_lrl = (6.0 - d * d + 2.0 * c_ab + 2.0 * d * (-sa + sb)) / 8.0
+    if abs(tmp_lrl) > 1:
+        return None, None, None, mode
+    p = base.wrap_0_2pi(2 * math.pi - math.acos(tmp_lrl))
+    t = base.wrap_0_2pi(-alpha - math.atan2(ca - cb, d + sa - sb) + p / 2.0)
+    q = base.wrap_0_2pi(base.wrap_0_2pi(beta) - alpha - t + base.wrap_0_2pi(p))
+
+    return t, p, q, mode
+
+
+def dubins_path_planning_from_origin(end_x, end_y, end_yaw, curvature, step_size):
+    dx = end_x
+    dy = end_y
+    D = math.hypot(dx, dy)
+    d = D * curvature
+
+    theta = base.wrap_0_2pi(math.atan2(dy, dx))
+    alpha = base.wrap_0_2pi(-theta)
+    beta = base.wrap_0_2pi(end_yaw - theta)
+
+    planners = [
+        left_straight_left,
+        right_straight_right,
+        left_straight_right,
+        right_straight_left,
+        right_left_right,
+        left_right_left,
+    ]
+
+    best_cost = float("inf")
+    bt, bp, bq, best_mode = None, None, None, None
+
+    for planner in planners:
+        t, p, q, mode = planner(alpha, beta, d)
+        if t is None:
+            continue
+
+        cost = abs(t) + abs(p) + abs(q)
+        if best_cost > cost:
+            bt, bp, bq, best_mode = t, p, q, mode
+            best_cost = cost
+    lengths = [bt, bp, bq]
+
+    x_list, y_list, yaw_list, directions = generate_local_course(
+        sum(lengths), lengths, best_mode, curvature, step_size
+    )
+
+    return x_list, y_list, yaw_list, best_mode, best_cost, lengths
+
+
+def interpolate(
+    ind,
+    length,
+    mode,
+    max_curvature,
+    origin_x,
+    origin_y,
+    origin_yaw,
+    path_x,
+    path_y,
+    path_yaw,
+    directions,
+):
+    if mode == "S":
+        path_x[ind] = origin_x + length / max_curvature * math.cos(origin_yaw)
+        path_y[ind] = origin_y + length / max_curvature * math.sin(origin_yaw)
+        path_yaw[ind] = origin_yaw
+    else:  # curve
+        ldx = math.sin(length) / max_curvature
+        ldy = 0.0
+        if mode == "L":  # left turn
+            ldy = (1.0 - math.cos(length)) / max_curvature
+        elif mode == "R":  # right turn
+            ldy = (1.0 - math.cos(length)) / -max_curvature
+        gdx = math.cos(-origin_yaw) * ldx + math.sin(-origin_yaw) * ldy
+        gdy = -math.sin(-origin_yaw) * ldx + math.cos(-origin_yaw) * ldy
+        path_x[ind] = origin_x + gdx
+        path_y[ind] = origin_y + gdy
+
+    if mode == "L":  # left turn
+        path_yaw[ind] = origin_yaw + length
+    elif mode == "R":  # right turn
+        path_yaw[ind] = origin_yaw - length
+
+    if length > 0.0:
+        directions[ind] = 1
+    else:
+        directions[ind] = -1
+
+    return path_x, path_y, path_yaw, directions
+
+
+def generate_local_course(total_length, lengths, mode, max_curvature, step_size):
+    n_point = math.trunc(total_length / step_size) + len(lengths) + 4
+
+    path_x = [0.0 for _ in range(n_point)]
+    path_y = [0.0 for _ in range(n_point)]
+    path_yaw = [0.0 for _ in range(n_point)]
+    directions = [0.0 for _ in range(n_point)]
+    index = 1
+
+    if lengths[0] > 0.0:
+        directions[0] = 1
+    else:
+        directions[0] = -1
+
+    ll = 0.0
+
+    for (m, l, i) in zip(mode, lengths, range(len(mode))):
+        if l > 0.0:
+            d = step_size
+        else:
+            d = -step_size
+
+        # set origin state
+        origin_x, origin_y, origin_yaw = path_x[index], path_y[index], path_yaw[index]
+
+        index -= 1
+        if i >= 1 and (lengths[i - 1] * lengths[i]) > 0:
+            pd = -d - ll
+        else:
+            pd = d - ll
+
+        while abs(pd) <= abs(l):
+            index += 1
+            path_x, path_y, path_yaw, directions = interpolate(
+                index,
+                pd,
+                m,
+                max_curvature,
+                origin_x,
+                origin_y,
+                origin_yaw,
+                path_x,
+                path_y,
+                path_yaw,
+                directions,
+            )
+            pd += d
+
+        ll = l - pd - d  # calc remain length
+
+        index += 1
+        path_x, path_y, path_yaw, directions = interpolate(
+            index,
+            l,
+            m,
+            max_curvature,
+            origin_x,
+            origin_y,
+            origin_yaw,
+            path_x,
+            path_y,
+            path_yaw,
+            directions,
+        )
+
+    if len(path_x) <= 1:
+        return [], [], [], []
+
+    # remove unused data
+    while len(path_x) >= 1 and path_x[-1] == 0.0:
+        path_x.pop()
+        path_y.pop()
+        path_yaw.pop()
+        directions.pop()
+
+    return path_x, path_y, path_yaw, directions
+
+
+def path_planning(start, goal, curvature, step_size=0.1):
+    """
+    Dubins path planner
+
+    input:
+        s_x x position of start point [m]
+        s_y y position of start point [m]
+        s_yaw yaw angle of start point [rad]
+        g_x x position of end point [m]
+        g_y y position of end point [m]
+        g_yaw yaw angle of end point [rad]
+        c curvature [1/m]
+
+    """
+    s_x, s_y, s_yaw = start
+    g_x, g_y, g_yaw = goal
+
+    g_x = g_x - s_x
+    g_y = g_y - s_y
+
+    l_rot = base.rot2(s_yaw)
+    le_xy = np.stack([g_x, g_y]).T @ l_rot
+    le_yaw = g_yaw - s_yaw
+
+    lp_x, lp_y, lp_yaw, mode, length, lengths = dubins_path_planning_from_origin(
+        le_xy[0], le_xy[1], le_yaw, curvature, step_size
+    )
+
+    rot = base.rot2(-s_yaw)
+    converted_xy = np.stack([lp_x, lp_y]).T @ rot
+    x_list = converted_xy[:, 0] + s_x
+    y_list = converted_xy[:, 1] + s_y
+    yaw_list = [base.wrap_mpi_pi(i_yaw + s_yaw) for i_yaw in lp_yaw]
+
+    path = np.c_[x_list, y_list, yaw_list]
+    return path, length, mode, lengths
+
+
+# ====================== RTB wrapper ============================= #
+
+# Copyright (c) 2022 Peter Corke: https://github.com/petercorke/robotics-toolbox-python
+# Released under the MIT license: https://github.com/AtsushiSakai/PythonRobotics/blob/master/LICENSE
+
[docs]class DubinsPlanner(PlannerBase): + r""" + Dubins path planner + + :param curvature: maximum path curvature, defaults to 1.0 + :type curvature: float, optional + :param stepsize: spacing between points on the path, defaults to 0.1 + :type stepsize: float, optional + :return: Dubins path planner + :rtype: DubinsPlanner instance + + ================== ======================== + Feature Capability + ================== ======================== + Plan :math:`\SE{2}` + Obstacle avoidance No + Curvature Discontinuous + Motion Forwards only + ================== ======================== + + Creates a planner that finds the path between two configurations in the + plane using forward motion only. The path comprises upto 3 segments that are + straight lines, or arcs with curvature of :math:`\pm` ``curvature``. + + Example: + + .. runblock:: pycon + + >>> from roboticstoolbox import DubinsPlanner + >>> from math import pi + >>> dubins = DubinsPlanner(curvature=1.0) + >>> path, status = dubins.query(start=(0, 0, pi/2), goal=(1, 0, pi/2)) + >>> print(path[:5,:]) + >>> print(status) + + :reference: On Curves of Minimal Length with a Constraint on Average + Curvature, and with Prescribed Initial and Terminal Positions and + Tangents, Dubins, L.E. (July 1957), American Journal of Mathematics. + 79(3): 497–516. + + :thanks: based on Dubins path planning from `Python Robotics <https://github.com/AtsushiSakai/PythonRobotics/tree/master/PathPlanning>`_ + :seealso: :class:`ReedsSheppPlanner` :class:`PlannerBase` + """ + + def __init__(self, curvature=1, stepsize=0.1, **kwargs): + + super().__init__(ndims=3, **kwargs) + self._curvature = curvature + self._stepsize = stepsize + + def __str__(self): + s = ( + super().__str__() + + f"\n curvature={self.curvature}, stepsize={self.stepsize}" + ) + return s + + @property + def curvature(self): + return self._curvature + + @property + def stepsize(self): + return self._stepsize + +
[docs] def query(self, start, goal, **kwargs): + r""" + Find a path between two configurations + + :param start: start configuration :math:`(x, y, \theta)` + :type start: array_like(3), optional + :param goal: goal configuration :math:`(x, y, \theta)` + :type goal: array_like(3), optional + :return: path and status + :rtype: ndarray(N,3), namedtuple + + The path comprises points equally spaced at a distance of ``stepsize``. + + The returned status value has elements: + + +-------------+-----------------------------------------------------+ + | Element | Description | + +-------------+-----------------------------------------------------+ + |``segments`` | a list containing the type of each path segment as | + | | a single letter code: either "L", "R" or "S" for | + | | left turn, right turn or straight line respectively.| + +-------------+-----------------------------------------------------+ + | ``length`` | total path length | + +-------------+-----------------------------------------------------+ + |``lengths`` | the length of each path segment. The sign of the | + | | length indicates the direction of travel. | + +-------------+-----------------------------------------------------+ + + """ + super().query(start=start, goal=goal, next=False, **kwargs) + + path, length, mode, lengths = path_planning( + start=self.start, + goal=self.goal, + curvature=self._curvature, + step_size=self._stepsize, + ) + + status = namedtuple("DubinsStatus", ["segments", "length", "seglengths"]) + + return path, status(mode, sum(lengths), lengths)
+ + +if __name__ == "__main__": + from math import pi + + # start = (1, 1, pi/4) + # goal = (-3, -3, -pi/4) + + start = (0, 0, pi / 2) + goal = (1, 0, pi / 2) + + dubins = DubinsPlanner(curvature=1.0) + path, status = dubins.query(start, goal) + + print(path) + print(status) + dubins.plot(path, configspace=True) + + plt.show(block=True) + # plt.plot(path_x, path_y, label="final course " + "".join(mode)) + + # # plotting + # plot_arrow(start_x, start_y, start_yaw) + # plot_arrow(end_x, end_y, end_yaw) + + # plt.legend() + # plt.grid(True) + # plt.axis("equal") + # plt.show(block=True) +
+ +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/mobile/EKF.html b/_modules/roboticstoolbox/mobile/EKF.html new file mode 100644 index 00000000..97059acf --- /dev/null +++ b/_modules/roboticstoolbox/mobile/EKF.html @@ -0,0 +1,1739 @@ + + + + + + roboticstoolbox.mobile.EKF — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +

Source code for roboticstoolbox.mobile.EKF

+"""
+Python EKF Planner
+@Author: Peter Corke, original MATLAB code and Python version
+@Author: Kristian Gibson, initial MATLAB port
+"""
+
+from collections import namedtuple
+import numpy as np
+from math import pi
+from scipy import integrate
+from scipy.linalg import sqrtm, block_diag
+from scipy.stats.distributions import chi2
+import matplotlib.pyplot as plt
+from matplotlib import animation
+
+from spatialmath.base.animate import Animate
+from spatialmath import base, SE2
+from roboticstoolbox.mobile import VehicleBase
+from roboticstoolbox.mobile.landmarkmap import LandmarkMap
+from roboticstoolbox.mobile.sensors import SensorBase
+
+
+
[docs]class EKF: +
[docs] def __init__( + self, + robot, + sensor=None, + map=None, + P0=None, + x_est=None, + joseph=True, + animate=True, + x0=[0, 0, 0], + verbose=False, + history=True, + workspace=None, + ): + r""" + Extended Kalman filter + + :param robot: robot motion model + :type robot: 2-tuple + :param sensor: vehicle mounted sensor model, defaults to None + :type sensor: 2-tuple, optional + :param map: landmark map, defaults to None + :type map: :class:`LandmarkMap`, optional + :param P0: initial covariance matrix, defaults to None + :type P0: ndarray(n,n), optional + :param x_est: initial state estimate, defaults to None + :type x_est: array_like(n), optional + :param joseph: use Joseph update of covariance, defaults to True + :type joseph: bool, optional + :param animate: show animation of vehicle motion, defaults to True + :type animate: bool, optional + :param x0: initial EKF state, defaults to [0, 0, 0] + :type x0: array_like(n), optional + :param verbose: display extra debug information, defaults to False + :type verbose: bool, optional + :param history: retain step-by-step history, defaults to True + :type history: bool, optional + :param workspace: dimension of workspace, see :func:`~spatialmath.base.graphics.expand_dims` + :type workspace: scalar, array_like(2), array_like(4) + + This class solves several classical robotic estimation problems, which are + selected according to the arguments: + + ====================== ====== ========== ========== ======= ====== + Problem len(x) ``robot`` ``sensor`` ``map`` ``P0`` + ====================== ====== ========== ========== ======= ====== + Dead reckoning 3 (veh,V) None None P0 + Map-based localization 3 (veh,V) (smodel,W) yes P0 + Map creation 2N (veh,None) (smodel,W) None None + SLAM 3+2N (veh,V) (smodel,W) None P0 + ====================== ====== ========== ========== ======= ====== + + where: + + - ``veh`` models the robotic vehicle kinematics and odometry and is a :class:`VehicleBase` subclass + - ``V`` is the estimated odometry (process) noise covariance as an ndarray(3,3) + - ``smodel`` models the robot mounted sensor and is a :class:`SensorBase` subclass + - ``W`` is the estimated sensor (measurement) noise covariance as an ndarray(2,2) + + The state vector has different lengths depending on the particular + estimation problem, see below. + + At each iteration of the EKF: + + - invoke the step method of the ``robot`` + - obtains the next control input from the driver agent, and apply it + as the vehicle control input + - the vehicle returns a noisy odometry estimate + - the state prediction is computed + - the true pose is used to determine a noisy sensor observation + - the state is corrected, new landmarks are added to the map + + The working area of the robot is defined by ``workspace`` or inherited + from the landmark map attached to the ``sensor`` (see + :func:`~spatialmath.base.graphics.expand_dims`): + + ============== ======= ======= + ``workspace`` x-range y-range + ============== ======= ======= + A (scalar) -A:A -A:A + [A, B] A:B A:B + [A, B, C, D] A:B C:D + ============== ======= ======= + + **Dead-reckoning localization** + + The state :math:`\vec{x} = (x, y, \theta)` is the estimated vehicle + configuration. + + Create a vehicle with odometry covariance ``V``, add a driver to it, + run the Kalman filter with estimated covariances ``V`` and initial + vehicle state covariance ``P0``:: + + V = np.diag([0.02, np.radians(0.5)]) ** 2; + robot = Bicycle(covar=V) + robot.control = RandomPath(workspace=10) + + x_sdev = [0.05, 0.05, np.radians(0.5)] + P0 = np.diag(x_sdev) ** 2 + ekf = EKF(robot=(robot, V), P0=P0) + + ekf.run(T=20) # run the simulation for 20 seconds + + robot.plot_xy(color="b") # plot the true vehicle path + ekf.plot_xy(color="r") # overlay the estimated path + ekf.plot_ellipse(filled=True, facecolor="g", alpha=0.3) # overlay uncertainty ellipses + + # plot the covariance against time + t = ekf.get_t(); + pn = ekf.get_Pnorm() + plt.plot(t, pn); + + **Map-based vehicle localization** + + The state :math:`\vec{x} = (x, y, \theta)` is the estimated vehicle + configuration. + + Create a vehicle with odometry covariance ``V``, add a driver to it, + create a map with 20 point landmarks, create a sensor that uses the map + and vehicle state to estimate landmark range and bearing with covariance + ``W``, the Kalman filter with estimated covariances ``V`` and ``W`` and + initial vehicle state covariance ``P0``:: + + V = np.diag([0.02, np.radians(0.5)]) ** 2; + robot = Bicycle(covar=V) + robot.control = RandomPath(workspace=10) + + map = LandmarkMap(nlandmarks=20, workspace=10) + + W = np.diag([0.1, np.radians(1)]) ** 2 + sensor = RangeBearingSensor(robot=robot, map=map, covar=W, angle=[-np.pi/2, np.pi/2], range=4, animate=True) + + x_sdev = [0.05, 0.05, np.radians(0.5)] + P0 = np.diag(x_sdev) ** 2 + ekf = EKF(robot=(robot, V), P0=P0, map=map, sensor=(sensor, W)) + + ekf.run(T=20) # run the simulation for 20 seconds + + map.plot() # plot the map + robot.plot_xy(color="b") # plot the true vehicle path + ekf.plot_xy(color="r") # overlay the estimated path + ekf.plot_ellipse() # overlay uncertainty ellipses + + # plot the covariance against time + t = ekf.get_t(); + pn = ekf.get_Pnorm() + plt.plot(t, pn); + + **Vehicle-based map making** + + The state :math:`\vec{x} = (x_0, y_0, \dots, x_{N-1}, y_{N-1})` is the + estimated landmark positions where :math:`N` is the number of landmarks. + The state vector is initially empty, and is extended by 2 elements every + time a new landmark is observed. + + Create a vehicle with perfect odometry (no covariance), add a driver to it, + create a sensor that uses the map and vehicle state to estimate landmark range + and bearing with covariance ``W``, the Kalman filter with estimated sensor + covariance ``W``, then run the filter for N time steps:: + + robot = Bicycle() + robot.add_driver(RandomPath(20, 2)) + + map = LandmarkMap(nlandmarks=20, workspace=10, seed=0) + + W = np.diag([0.1, np.radians(1)]) ** 2 + sensor = RangeBearingSensor(robot, map, W) + + ekf = EKF(robot=(robot, None), sensor=(sensor, W)) + + ekf.run(T=20) # run the simulation for 20 seconds + + map.plot() # plot the map + robot.plot_xy(color="b") # plot the true vehicle path + + **Simultaneous localization and mapping (SLAM)** + + The state :math:`\vec{x} = (x, y, \theta, x_0, y_0, \dots, x_{N-1}, + y_{N-1})` is the estimated vehicle configuration followed by the + estimated landmark positions where :math:`N` is the number of landmarks. + The state vector is initially of length 3, and is extended by 2 elements + every time a new landmark is observed. + + Create a vehicle with odometry covariance ``V``, add a driver to it, + create a map with 20 point landmarks, create a sensor that uses the map + and vehicle state to estimate landmark range and bearing with covariance + ``W``, the Kalman filter with estimated covariances ``V`` and ``W`` and + initial state covariance ``P0``, then run the filter to estimate the + vehicle state at each time step and the map:: + + V = np.diag([0.02, np.radians(0.5)]) ** 2; + robot = Bicycle(covar=V) + robot.control = RandomPath(workspace=10) + + map = LandmarkMap(nlandmarks=20, workspace=10) + + W = np.diag([0.1, np.radians(1)]) ** 2 + sensor = RangeBearingSensor(robot=robot, map=map, covar=W, angle=[-np.pi/2, np.pi/2], range=4, animate=True) + + ekf = EKF(robot=(robot, V), P0=P0, sensor=(sensor, W)) + + ekf.run(T=20) # run the simulation for 20 seconds + + map.plot(); # plot true map + ekf.plot_map(); # plot estimated landmark position + + robot.plot_xy(); # plot true path + ekf.plot_xy(); # plot estimated robot path + ekf.plot_ellipse(); # plot estimated covariance + + # plot the covariance against time + t = ekf.get_t(); + pn = ekf.get_Pnorm() + plt.plot(t, pn); + + :seealso: :meth:`run` + """ + + if robot is not None: + if ( + not isinstance(robot, tuple) + or len(robot) != 2 + or not isinstance(robot[0], VehicleBase) + ): + raise TypeError("robot must be tuple (vehicle, V_est)") + self._robot = robot[0] # reference to the robot vehicle + self._V_est = robot[1] # estimate of vehicle state covariance V + + if sensor is not None: + if ( + not isinstance(sensor, tuple) + or len(sensor) != 2 + or not isinstance(sensor[0], SensorBase) + ): + raise TypeError("sensor must be tuple (sensor, W_est)") + self._sensor = sensor[0] # reference to the sensor + self._W_est = sensor[1] # estimate of sensor covariance W + else: + self._sensor = None + self._W_est = None + + if map is not None and not isinstance(map, LandmarkMap): + raise TypeError("map must be LandmarkMap instance") + self._ekf_map = map # prior map for localization + + if animate: + if map is not None: + self._workspace = map.workspace + self._robot._workspace = map.workspace + elif sensor is not None: + self._workspace = sensor[0].map.workspace + self._robot._workspace = sensor[0].map.workspace + elif self.robot.workspace is None: + raise ValueError("for animation robot must have a defined workspace") + self.animate = animate + + self._P0 = P0 # initial system covariance + self._x0 = x0 # initial vehicle state + + self._x_est = x_est # estimated state + self._landmarks = None # ekf_map state + + self._est_vehicle = False + self._est_ekf_map = False + if self._V_est is not None: + # estimating vehicle pose by: + # - DR if sensor is None + # - localization if sensor is not None and map is not None + self._est_vehicle = True + + # perfect vehicle case + if map is None and sensor is not None: + # estimating ekf_map + self._est_ekf_map = True + self._joseph = joseph # flag: use Joseph form to compute p + + self._verbose = verbose + + self._keep_history = history # keep history + self._htuple = namedtuple("EKFlog", "t xest odo P innov S K lm z") + + if workspace is not None: + self._dim = base.expand_dims(dim) + elif self.sensor is not None: + self._dim = self.sensor.map.workspace + else: + self._dim = self._robot.workspace + + # self.robot.init() + + # # clear the history + # self._history = [] + + if self.V_est is None: + # perfect vehicle case + + self._est_vehicle = False + self._x_est = None + self._P_est = None + else: + # noisy odometry case + if self.V_est.shape != (2, 2): + raise ValueError("vehicle state covariance V_est must be 2x2") + self._x_est = self.robot.x + self._P_est = P0 + self._est_vehicle = True + + if self.W_est is not None: + if self.W_est.shape != (2, 2): + raise ValueError("sensor covariance W_est must be 2x2") + + # if np.any(self._sensor): + # self._landmarks = None*np.zeros(2, self._sensor.ekf_map.nlandmarks) + + # # check types for passed objects + # if np.any(self._map) and not isinstance(self._map, 'LandmarkMap'): + # raise ValueError('expecting LandmarkMap object') + + # if np.any(sensor) and not isinstance(sensor, 'Sensor'): + # raise ValueError('expecting Sensor object') + + self.init() + + self.xxdata = ([], [])
+ + def __str__(self): + s = f"EKF object: {len(self._x_est)} states" + + def indent(s, n=2): + spaces = " " * n + return s.replace("\n", "\n" + spaces) + + estimating = [] + if self._est_vehicle is not None: + estimating.append("vehicle pose") + if self._est_ekf_map is not None: + estimating.append("map") + if len(estimating) > 0: + s += ", estimating: " + ", ".join(estimating) + if self.robot is not None: + s += indent("\nrobot: " + str(self.robot)) + if self.V_est is not None: + s += indent("\nV_est: " + base.array2str(self.V_est)) + + if self.sensor is not None: + s += indent("\nsensor: " + str(self.sensor)) + if self.W_est is not None: + s += indent("\nW_est: " + base.array2str(self.W_est)) + + return s + + def __repr__(self): + return str(self) + + @property + def x_est(self): + """ + Get EKF state + + :return: state vector + :rtype: ndarray(n) + + Returns the value of the estimated state vector at the end of + simulation. The dimensions depend on the problem being solved. + """ + return self._x_est + + @property + def P_est(self): + """ + Get EKF covariance + + :return: covariance matrix + :rtype: ndarray(n,n) + + Returns the value of the estimated covariance matrix at the end of + simulation. The dimensions depend on the problem being solved. + """ + return self._P_est + + @property + def P0(self): + """ + Get initial EKF covariance + + :return: covariance matrix + :rtype: ndarray(n,n) + + Returns the value of the covariance matrix passed to the constructor. + """ + return self._P0 + + @property + def V_est(self): + """ + Get estimated odometry covariance + + :return: odometry covariance + :rtype: ndarray(2,2) + + Returns the value of the estimated odometry covariance matrix passed to + the constructor + """ + return self._V_est + + @property + def W_est(self): + """ + Get estimated sensor covariance + + :return: sensor covariance + :rtype: ndarray(2,2) + + Returns the value of the estimated sensor covariance matrix passed to + the constructor + """ + return self._W_est + + @property + def robot(self): + """ + Get robot object + + :return: robot used in simulation + :rtype: :class:`VehicleBase` subclass + """ + return self._robot + + @property + def sensor(self): + """ + Get sensor object + + :return: sensor used in simulation + :rtype: :class:`SensorBase` subclass + """ + return self._sensor + + @property + def map(self): + """ + Get map object + + :return: map used in simulation + :rtype: :class:`LandmarkMap` subclass + """ + return self._map + + @property + def verbose(self): + """ + Get verbosity state + + :return: verbosity + :rtype: bool + """ + return self._verbose + + @property + def history(self): + """ + Get EKF simulation history + + :return: simulation history + :rtype: list of namedtuples + + At each simulation timestep a namedtuple of is appended to the history + list. It contains, for that time step, estimated state and covariance, + and sensor observation. + + :seealso: :meth:`get_t` :meth:`get_xyt` :meth:`get_map` :meth:`get_P` + :meth:`get_Pnorm` + """ + return self._history + + @property + def workspace(self): + """ + Size of robot workspace + + :return: workspace bounds [xmin, xmax, ymin, ymax] + :rtype: ndarray(4) + + Returns the bounds of the workspace as specified by the constructor + option ``workspace`` + """ + return self._workspace + + @property + def landmarks(self): + """ + Get landmark information + + :return: landmark information + :rtype: dict + + The dictionary is indexed by the landmark id and gives a 3-tuple: + + - order in which landmark was seen + - number of times seen + + The order in which the landmark was first seen. The first observed + landmark has order 0 and so on. + + :seealso: :meth:`landmark` + """ + return self._landmarks + +
[docs] def landmark(self, id): + """ + Landmark information + + :param id: landmark index + :type id: int + :return: order in which it was first seen, number of times seen + :rtype: int, int + + The first observed landmark has order 0 and so on. + + :seealso: :meth:`landmarks` :meth:`landmark_index` :meth:`landmark_mindex` + """ + try: + l = self._landmarks[id] + return l[0], l[1] + except KeyError: + raise ValueError(f"unknown landmark {id}") from None
+ +
[docs] def landmark_index(self, id): + """ + Landmark index in complete state vector + + :param id: landmark index + :type id: int + :return: index in the state vector + :rtype: int + + The return value ``j`` is the index of the x-coordinate of the landmark + in the EKF state vector, and ``j+1`` is the index of the y-coordinate. + + :seealso: :meth:`landmark` + """ + try: + jx = self._landmarks[id][0] * 2 + if self._est_vehicle: + jx += 3 + return jx + except KeyError: + raise ValueError(f"unknown landmark {id}") from None
+ +
[docs] def landmark_mindex(self, id): + """ + Landmark index in map state vector + + :param id: landmark index + :type id: int + :return: index in the state vector + :rtype: int + + The return value ``j`` is the index of the x-coordinate of the landmark + in the map vector, and ``j+1`` is the index of the y-coordinate. + + :seealso: :meth:`landmark` + """ + try: + return self._landmarks[id][0] * 2 + except KeyError: + raise ValueError(f"unknown landmark {id}") from None
+ +
[docs] def landmark_x(self, id): + """ + Landmark position + + :param id: landmark index + :type id: int + :return: landmark position :math:`(x,y)` + :rtype: ndarray(2) + + Returns the landmark position from the current state vector. + """ + jx = self.landmark_index(id) + return self._x_est[jx : jx + 2]
+ +
[docs] def init(self): + # EKF.init Reset the filter + # + # E.init() resets the filter state and clears landmarks and history. + self.robot.init() + if self.sensor is not None: + self.sensor.init() + + # clear the history + self._history = [] + + if self._V_est is None: + # perfect vehicle case + self._estVehicle = False + self._x_est = np.empty((0,)) + self._P_est = np.empty((0, 0)) + else: + # noisy odometry case + self._x_est = self._x0 + self._P_est = self._P0 + self._estVehicle = True + + if self.sensor is not None: + # landmark dictionary maps lm_id to list[index, nseen] + self._landmarks = {}
+ + # np.full((2, len(self.sensor.map)), -1, dtype=int) + +
[docs] def run(self, T, animate=False): + """ + Run the EKF simulation + + :param T: maximum simulation time in seconds + :type T: float + :param animate: animate motion of vehicle, defaults to False + :type animate: bool, optional + + Simulates the motion of a vehicle (under the control of a driving agent) + and the EKF estimator. The steps are: + + - initialize the filter, vehicle and vehicle driver agent, sensor + - for each time step: + + - step the vehicle and its driver agent, obtain odometry + - take a sensor reading + - execute the EKF + - save information as a namedtuple to the history list for later display + + :seealso: :meth:`history` :meth:`landmark` :meth:`landmarks` + :meth:`get_xyt` :meth:`get_t` :meth:`get_map` :meth:`get_P` :meth:`get_Pnorm` + :meth:`plot_xy` :meth:`plot_ellipse` :meth:`plot_error` :meth:`plot_map` + :meth:`run_animation` + """ + self.init() + if animate: + if self.sensor is not None: + self.sensor.map.plot() + + plt.xlabel("X") + plt.ylabel("Y") + + for k in range(round(T / self.robot.dt)): + if animate: + # self.robot.plot() + self.robot._animation.update(self.robot.x) + self.step()
+ +
[docs] def run_animation(self, T=10, x0=None, control=None, format=None, file=None): + r""" + Run the EKF simulation + + :param T: maximum simulation time in seconds + :type T: float + :param format: Output format + :type format: str, optional + :param file: File name + :type file: str, optional + :return: Matplotlib animation object + :rtype: :meth:`matplotlib.animation.FuncAnimation` + + Simulates the motion of a vehicle (under the control of a driving agent) + and the EKF estimator for ``T`` seconds and returns an animation + in various formats:: + + ``format`` ``file`` description + ============ ========= ============================ + ``"html"`` str, None return HTML5 video + ``"jshtml"`` str, None return JS+HTML video + ``"gif"`` str return animated GIF + ``"mp4"`` str return MP4/H264 video + ``None`` return a ``FuncAnimation`` object + + If ``file`` can be ``None`` then return the video as a string, otherwise it + must be a filename. + + The simulation steps are: + + - initialize the filter, vehicle and vehicle driver agent, sensor + - for each time step: + + - step the vehicle and its driver agent, obtain odometry + - take a sensor reading + - execute the EKF + - save information as a namedtuple to the history list for later display + + :seealso: :meth:`history` :meth:`landmark` :meth:`landmarks` + :meth:`get_xyt` :meth:`get_t` :meth:`get_map` :meth:`get_P` :meth:`get_Pnorm` + :meth:`plot_xy` :meth:`plot_ellipse` :meth:`plot_error` :meth:`plot_map` + :meth:`run_animation` + """ + + fig, ax = plt.subplots() + + def init(): + self.init() + if self.sensor is not None: + self.sensor.map.plot() + ax.set_xlabel("X") + ax.set_ylabel("Y") + + def animate(i): + self.robot._animation.update(self.robot.x) + self.step(pause=False) + + nframes = round(T / self.robot._dt) + anim = animation.FuncAnimation( + fig=fig, + func=animate, + init_func=init, + frames=nframes, + interval=self.robot.dt * 1000, + blit=False, + repeat=False, + ) + + ret = None + if format == "html": + ret = anim.to_html5_video() # convert to embeddable HTML5 animation + elif format == "jshtml": + ret = anim.to_jshtml() # convert to embeddable Javascript/HTML animation + elif format == "gif": + anim.save( + file, writer=animation.PillowWriter(fps=1 / self.robot.dt) + ) # convert to GIF + ret = None + elif format == "mp4": + anim.save( + file, writer=animation.FFMpegWriter(fps=1 / self.robot.dt) + ) # convert to mp4/H264 + ret = None + elif format == None: + # return the anim object + return anim + else: + raise ValueError("unknown format") + + if ret is not None and file is not None: + with open(file, "w") as f: + f.write(ret) + ret = None + plt.close(fig) + return ret
+ +
[docs] def step(self, pause=None): + """ + Execute one timestep of the simulation + """ + + # move the robot + odo = self.robot.step(pause=pause) + + # ================================================================= + # P R E D I C T I O N + # ================================================================= + if self._est_vehicle: + # split the state vector and covariance into chunks for + # vehicle and map + xv_est = self._x_est[:3] + xm_est = self._x_est[3:] + Pvv_est = self._P_est[:3, :3] + Pmm_est = self._P_est[3:, 3:] + Pvm_est = self._P_est[:3, 3:] + else: + xm_est = self._x_est + Pmm_est = self._P_est + + if self._est_vehicle: + # evaluate the state update function and the Jacobians + # if vehicle has uncertainty, predict its covariance + xv_pred = self.robot.f(xv_est, odo) + + Fx = self.robot.Fx(xv_est, odo) + Fv = self.robot.Fv(xv_est, odo) + Pvv_pred = Fx @ Pvv_est @ Fx.T + Fv @ self.V_est @ Fv.T + else: + # otherwise we just take the true robot state + xv_pred = self._robot.x + + if self._est_ekf_map: + if self._est_vehicle: + # SLAM case, compute the correlations + Pvm_pred = Fx @ Pvm_est + + Pmm_pred = Pmm_est + xm_pred = xm_est + + # put the chunks back together again + if self._est_vehicle and not self._est_ekf_map: + # vehicle only + x_pred = xv_pred + P_pred = Pvv_pred + elif not self._est_vehicle and self._est_ekf_map: + # map only + x_pred = xm_pred + P_pred = Pmm_pred + elif self._est_vehicle and self._est_ekf_map: + # vehicle and map + x_pred = np.r_[xv_pred, xm_pred] + # fmt: off + P_pred = np.block([ + [Pvv_pred, Pvm_pred], + [Pvm_pred.T, Pmm_pred] + ]) + # fmt: on + + # at this point we have: + # xv_pred the state of the vehicle to use to + # predict observations + # xm_pred the state of the map + # x_pred the full predicted state vector + # P_pred the full predicted covariance matrix + + # initialize the variables that might be computed during + # the update phase + + doUpdatePhase = False + + # disp('x_pred:') x_pred' + + # ================================================================= + # P R O C E S S O B S E R V A T I O N S + # ================================================================= + + if self.sensor is not None: + # read the sensor + z, lm_id = self.sensor.reading() + sensorReading = z is not None + else: + lm_id = None # keep history saving happy + z = None + sensorReading = False + + if sensorReading: + # here for MBL, MM, SLAM + + # compute the innovation + z_pred = self.sensor.h(xv_pred, lm_id) + innov = np.array([z[0] - z_pred[0], base.wrap_mpi_pi(z[1] - z_pred[1])]) + + if self._est_ekf_map: + # the ekf_map is estimated MM or SLAM case + if self._isseenbefore(lm_id): + # landmark is previously seen + + # get previous estimate of its state + jx = self.landmark_mindex(lm_id) + xf = xm_pred[jx : jx + 2] + + # compute Jacobian for this particular landmark + # xf = self.sensor.g(xv_pred, z) # HACK + Hx_k = self.sensor.Hp(xv_pred, xf) + + z_pred = self.sensor.h(xv_pred, xf) + innov = np.array( + [z[0] - z_pred[0], base.wrap_mpi_pi(z[1] - z_pred[1])] + ) + + # create the Jacobian for all landmarks + Hx = np.zeros((2, len(xm_pred))) + Hx[:, jx : jx + 2] = Hx_k + + Hw = self.sensor.Hw(xv_pred, xf) + + if self._est_vehicle: + # concatenate Hx for for vehicle and ekf_map + Hxv = self.sensor.Hx(xv_pred, xf) + Hx = np.block([Hxv, Hx]) + + self._landmark_increment(lm_id) # update the count + if self._verbose: + print( + f"landmark {lm_id} seen" + f" {self._landmark_count(lm_id)} times," + f" state_idx={self.landmark_index(lm_id)}" + ) + doUpdatePhase = True + + else: + # new landmark, seen for the first time + + # extend the state vector and covariance + x_pred, P_pred = self._extend_map( + P_pred, xv_pred, xm_pred, z, lm_id + ) + # if lm_id == 17: + # print(P_pred) + # # print(x_pred[-2:], self._sensor._map.landmark(17), base.norm(x_pred[-2:] - self._sensor._map.landmark(17))) + + self._landmark_add(lm_id) + if self._verbose: + print( + f"landmark {lm_id} seen for first time," + f" state_idx={self.landmark_index(lm_id)}" + ) + doUpdatePhase = False + + else: + # LBL + Hx = self.sensor.Hx(xv_pred, lm_id) + Hw = self.sensor.Hw(xv_pred, lm_id) + doUpdatePhase = True + else: + innov = None + + # doUpdatePhase flag indicates whether or not to do + # the update phase of the filter + # + # DR always false + # map-based localization if sensor reading + # map creation if sensor reading & not first + # sighting + # SLAM if sighting of a previously + # seen landmark + + if doUpdatePhase: + # disp('do update\n') + # # we have innovation, update state and covariance + # compute x_est and P_est + + # compute innovation covariance + S = Hx @ P_pred @ Hx.T + Hw @ self._W_est @ Hw.T + + # compute the Kalman gain + K = P_pred @ Hx.T @ np.linalg.inv(S) + + # update the state vector + x_est = x_pred + K @ innov + + if self._est_vehicle: + # wrap heading state for a vehicle + x_est[2] = base.wrap_mpi_pi(x_est[2]) + + # update the covariance + if self._joseph: + # we use the Joseph form + I = np.eye(P_pred.shape[0]) + P_est = (I - K @ Hx) @ P_pred @ (I - K @ Hx).T + K @ self._W_est @ K.T + else: + P_est = P_pred - K @ S @ K.T + # enforce P to be symmetric + P_est = 0.5 * (P_est + P_est.T) + else: + # no update phase, estimate is same as prediction + x_est = x_pred + P_est = P_pred + S = None + K = None + + self._x_est = x_est + self._P_est = P_est + + if self._keep_history: + hist = self._htuple( + self.robot._t, + x_est.copy(), + odo.copy(), + P_est.copy(), + innov.copy() if innov is not None else None, + S.copy() if S is not None else None, + K.copy() if K is not None else None, + lm_id if lm_id is not None else -1, + z.copy() if z is not None else None, + ) + self._history.append(hist)
+ + ## landmark management + + def _isseenbefore(self, lm_id): + + # _landmarks[0, id] is the order in which seen + # _landmarks[1, id] is the occurence count + + return lm_id in self._landmarks + + def _landmark_increment(self, lm_id): + self._landmarks[lm_id][1] += 1 # update the count + + def _landmark_count(self, lm_id): + return self._landmarks[lm_id][1] + + def _landmark_add(self, lm_id): + self._landmarks[lm_id] = [len(self._landmarks), 1] + + def _extend_map(self, P, xv, xm, z, lm_id): + # this is a new landmark, we haven't seen it before + # estimate position of landmark in the world based on + # noisy sensor reading and current vehicle pose + + # M = None + + # estimate its position based on observation and vehicle state + xf = self.sensor.g(xv, z) + + # append this estimate to the state vector + if self._est_vehicle: + x_ext = np.r_[xv, xm, xf] + else: + x_ext = np.r_[xm, xf] + + # get the Jacobian for the new landmark + Gz = self.sensor.Gz(xv, z) + + # extend the covariance matrix + n = len(self._x_est) + if self._est_vehicle: + # estimating vehicle state + Gx = self.sensor.Gx(xv, z) + # fmt: off + Yz = np.block([ + [np.eye(n), np.zeros((n, 2)) ], + [Gx, np.zeros((2, n-3)), Gz] + ]) + # fmt: on + else: + # estimating landmarks only + # P_ext = block_diag(P, Gz @ self._W_est @ Gz.T) + # fmt: off + Yz = np.block([ + [np.eye(n), np.zeros((n, 2)) ], + [np.zeros((2, n)), Gz] + ]) + # fmt: on + P_ext = Yz @ block_diag(P, self._W_est) @ Yz.T + + return x_ext, P_ext + +
[docs] def get_t(self): + """ + Get time from simulation + + :return: simulation time vector + :rtype: ndarray(n) + + Return simulation time vector, starts at zero. The timestep is an + attribute of the ``robot`` object. + + :seealso: :meth:`run` :meth:`history` + """ + return np.array([h.t for h in self._history])
+ +
[docs] def get_xyt(self): + r""" + Get estimated vehicle trajectory + + :return: vehicle trajectory where each row is configuration :math:`(x, y, \theta)` + :rtype: ndarray(n,3) + + :seealso: :meth:`plot_xy` :meth:`run` :meth:`history` + """ + if self._est_vehicle: + xyt = np.array([h.xest[:3] for h in self._history]) + else: + xyt = None + return xyt
+ +
[docs] def plot_xy(self, *args, block=None, **kwargs): + """ + Plot estimated vehicle position + + :param args: position arguments passed to :meth:`~matplotlib.axes.Axes.plot` + :param kwargs: keywords arguments passed to :meth:`~matplotlib.axes.Axes.plot` + :param block: hold plot until figure is closed, defaults to None + :type block: bool, optional + + Plot the estimated vehicle path in the xy-plane. + + :seealso: :meth:`get_xyt` :meth:`plot_error` :meth:`plot_ellipse` :meth:`plot_P` + :meth:`run` :meth:`history` + """ + if args is None and "color" not in kwargs: + kwargs["color"] = "r" + xyt = self.get_xyt() + plt.plot(xyt[:, 0], xyt[:, 1], *args, **kwargs) + if block is not None: + plt.show(block=block)
+ +
[docs] def plot_ellipse(self, confidence=0.95, N=10, block=None, **kwargs): + """ + Plot uncertainty ellipses + + :param confidence: ellipse confidence interval, defaults to 0.95 + :type confidence: float, optional + :param N: number of ellipses to plot, defaults to 10 + :type N: int, optional + :param block: hold plot until figure is closed, defaults to None + :type block: bool, optional + :param kwargs: arguments passed to :meth:`spatialmath.base.graphics.plot_ellipse` + + Plot ``N`` uncertainty ellipses spaced evenly along the trajectory. + + :seealso: :meth:`get_P` :meth:`run` :meth:`history` + """ + nhist = len(self._history) + + if "label" in kwargs: + label = kwargs["label"] + del kwargs["label"] + else: + label = f"{confidence*100:.3g}% confidence" + + for k in np.linspace(0, nhist - 1, N): + k = round(k) + h = self._history[k] + if k == 0: + base.plot_ellipse( + h.P[:2, :2], + centre=h.xest[:2], + confidence=confidence, + label=label, + inverted=True, + **kwargs, + ) + else: + base.plot_ellipse( + h.P[:2, :2], + centre=h.xest[:2], + confidence=confidence, + inverted=True, + **kwargs, + ) + if block is not None: + plt.show(block=block)
+ +
[docs] def plot_error(self, bgcolor="r", confidence=0.95, ax=None, block=None, **kwargs): + r""" + Plot error with uncertainty bounds + + :param bgcolor: background color, defaults to 'r' + :type bgcolor: str, optional + :param confidence: confidence interval, defaults to 0.95 + :type confidence: float, optional + :param block: hold plot until figure is closed, defaults to None + :type block: bool, optional + + Plot the error between actual and estimated vehicle + path :math:`(x, y, \theta)`` versus time as three stacked plots. + Heading error is wrapped into the range :math:`[-\pi,\pi)` + + + Behind each line draw a shaded polygon ``bgcolor`` showing the specified + ``confidence`` bounds based on the covariance at each time step. + Ideally the lines should be within the shaded polygon ``confidence`` + of the time. + + .. note:: Observations will decrease the uncertainty while periods of dead-reckoning increase it. + + :seealso: :meth:`get_P` :meth:`run` :meth:`history` + """ + error = [] + bounds = [] + ppf = chi2.ppf(confidence, df=2) + + x_gt = self.robot.x_hist + for k in range(len(self.history)): + hk = self.history[k] + # error is true - estimated + e = x_gt[k, :] - hk.xest + e[2] = base.wrap_mpi_pi(e[2]) + error.append(e) + + P = np.diag(hk.P) + bounds.append(np.sqrt(ppf * P[:3])) + + error = np.array(error) + bounds = np.array(bounds) + t = self.get_t() + + if ax is None: + fig, axes = plt.subplots(3) + else: + axes = ax[:3] + labels = ["x", "y", r"$\theta$"] + + for k, ax in enumerate(axes): + if confidence is not None: + edge = np.array( + [ + np.r_[t, t[::-1]], + np.r_[bounds[:, k], -bounds[::-1, k]], + ] + ) + polygon = plt.Polygon( + edge.T, closed=True, facecolor="r", edgecolor="none", alpha=0.3 + ) + ax.add_patch(polygon) + ax.plot(error[:, k], **kwargs) + ax.grid(True) + ax.set_ylabel(labels[k] + " error") + ax.set_xlim(0, t[-1]) + + if block is not None: + plt.show(block=block)
+ + # subplot(opt.nplots*100+12) + # if opt.confidence + # edge = [pxy(:,2); -pxy(end:-1:1,2)]; + # h = patch(t, edge, opt.color); + # set(h, 'EdgeColor', 'none', 'FaceAlpha', 0.3); + # end + # hold on + # plot(err(:,2), args{:}); + # hold off + # grid + # ylabel('y error') + + # subplot(opt.nplots*100+13) + # if opt.confidence + # edge = [pxy(:,3); -pxy(end:-1:1,3)]; + # h = patch(t, edge, opt.color); + # set(h, 'EdgeColor', 'none', 'FaceAlpha', 0.3); + # end + # hold on + # plot(err(:,3), args{:}); + # hold off + # grid + # xlabel('Time step') + # ylabel('\theta error') + +
[docs] def get_map(self): + """ + Get estimated map + + :return: landmark coordinates :math:`(x, y)` + :rtype: ndarray(n,2) + + Landmarks are returned in the order they were first observed. + + :seealso: :meth:`landmarks` :meth:`run` :meth:`history` + + """ + xy = [] + for lm_id, (jx, n) in self._landmarks.items(): + # jx is an index into the *landmark* part of the state + # vector, we need to offset it to account for the vehicle + # state if we are estimating vehicle as well + if self._est_vehicle: + jx += 3 + xf = self._x_est[jx : jx + 2] + xy.append(xf) + return np.array(xy)
+ +
[docs] def plot_map(self, marker=None, ellipse=None, confidence=0.95, block=None): + """ + Plot estimated landmarks + + :param marker: plot marker for landmark, arguments passed to :meth:`~matplotlib.axes.Axes.plot`, defaults to "r+" + :type marker: dict, optional + :param ellipse: arguments passed to :meth:`~spatialmath.base.graphics.plot_ellipse`, defaults to None + :type ellipse: dict, optional + :param confidence: ellipse confidence interval, defaults to 0.95 + :type confidence: float, optional + :param block: hold plot until figure is closed, defaults to None + :type block: bool, optional + + Plot a marker and covariance ellipses for each estimated landmark. + + :seealso: :meth:`get_map` :meth:`run` :meth:`history` + """ + if marker is None: + marker = { + "marker": "+", + "markersize": 10, + "markerfacecolor": "red", + "linewidth": 0, + } + + xm = self._x_est + P = self._P_est + if self._est_vehicle: + xm = xm[3:] + P = P[3:, 3:] + + # mark the estimate as a point + xm = xm.reshape((-1, 2)) # arrange as Nx2 + plt.plot(xm[:, 0], xm[:, 1], label="estimated landmark", **marker) + + # add an ellipse + if ellipse is not None: + for i in range(xm.shape[0]): + Pi = self.P_est[i : i + 2, i : i + 2] + # put ellipse in the legend only once + if i == 0: + base.plot_ellipse( + Pi, + centre=xm[i, :], + confidence=confidence, + inverted=True, + label=f"{confidence*100:.3g}% confidence", + **ellipse, + ) + else: + base.plot_ellipse( + Pi, + centre=xm[i, :], + confidence=confidence, + inverted=True, + **ellipse, + ) + # plot_ellipse( P * chi2inv_rtb(opt.confidence, 2), xf, args{:}); + if block is not None: + plt.show(block=block)
+ +
[docs] def get_P(self, k=None): + """ + Get covariance matrices from simulation + + :param k: timestep, defaults to None + :type k: int, optional + :return: covariance matrix + :rtype: ndarray(n,n) or list of ndarray(n,n) + + If ``k`` is given return covariance from simulation timestep ``k``, else + return a list of all covariance matrices. + + :seealso: :meth:`get_Pnorm` :meth:`run` :meth:`history` + """ + if k is not None: + return self._history[k].P + else: + return [h.P for h in self._history]
+ +
[docs] def get_Pnorm(self, k=None): + """ + Get covariance norm from simulation + + :param k: timestep, defaults to None + :type k: int, optional + :return: covariance matrix norm + :rtype: float or ndarray(n) + + If ``k`` is given return covariance norm from simulation timestep ``k``, else + return all covariance norms as a 1D NumPy array. + + :seealso: :meth:`get_P` :meth:`run` :meth:`history` + """ + if k is not None: + return np.sqrt(np.linalg.det(self._history[k].P)) + else: + p = [np.sqrt(np.linalg.det(h.P)) for h in self._history] + return np.array(p)
+ +
[docs] def disp_P(self, P, colorbar=False): + """ + Display covariance matrix + + :param P: covariance matrix + :type P: ndarray(n,n) + :param colorbar: add a colorbar + :type: bool or dict + + Plot the elements of the covariance matrix as an image. If ``colorbar`` + is True add a color bar, if `colorbar` is a dict add a color bar with + these options passed to colorbar. + + .. note:: A log scale is used. + + :seealso: :meth:`~matplotlib.axes.Axes.imshow` :func:`matplotlib.pyplot.colorbar` + """ + + z = np.log10(abs(P)) + mn = min(z[~np.isinf(z)]) + z[np.isinf(z)] = mn + plt.xlabel("State") + plt.ylabel("State") + + plt.imshow(z, cmap="Reds") + if colorbar is True: + plt.colorbar(label="log covariance") + elif isinstance(colorbar, dict): + plt.colorbar(**colorbar)
+ +
[docs] def get_transform(self, map): + """ + Transformation from estimated map to true map frame + + :param map: known landmark positions + :type map: :class:`LandmarkMap` + :return: transform from ``map`` to estimated map frame + :rtype: SE2 instance + + Uses a least squares technique to find the transform between the + landmark is world frame and the estimated landmarks in the SLAM + reference frame. + + :seealso: :func:`~spatialmath.base.transforms2d.points2tr2` + """ + p = [] + q = [] + + for lm_id in self._landmarks.keys(): + p.append(map[lm_id]) + q.append(self.landmark_x(lm_id)) + + p = np.array(p).T + q = np.array(q).T + + T = base.points2tr2(p, q) + return SE2(T)
+ + +if __name__ == "__main__": + + from roboticstoolbox import * + + V = np.diag([0.02, np.deg2rad(0.5)]) ** 2 + robot = Bicycle(covar=V, animation="car") + robot.control = RandomPath(workspace=10) + P0 = np.diag([1, 1, 1]) + V = np.diag([1, 1]) + ekf = EKF(robot=(robot, V), P0=P0, animate=False) + print(ekf) + ekf.run_animation(T=20, format="mp4", file="ekf.mp4") + + # from roboticstoolbox import Bicycle + + # ### RVC2: Chapter 6 + + # ## 6.1 Dead reckoning + + # ## 6.1.1 Modeling the vehicle + # V = np.diag(np.r_[0.02, 0.5*pi/180] ** 2); + + # veh = Bicycle(covar=V) + + # odo = veh.step(1, 0.3) + + # print(veh.x) + + # veh.f([0, 0, 0], odo) + + # # veh.add_driver( RandomPath(10) ) + + # # veh.run() + + # ### 6.1.2 Estimating pose + # # veh.Fx( [0,0,0], [0.5, 0.1] ) + + # P0 = np.diag(np.r_[0.005, 0.005, 0.001]**2); + + # ekf = EKF(veh, V, P0); + + # ekf.run(1000); + + # veh.plot_xy() + + # ekf.plot_xy('r') + + # P700 = ekf.history(700).P + + # sqrt(P700(1,1)) + + # ekf.plot_ellipse('g') + + # # 6.2 Map-based localization + # # randinit + # # map = LandmarkMap(20, 10) + + # # map.plot() + + # # W = diag([0.1, 1*pi/180].^2); + + # # sensor = RangeBearingSensor(veh, map, 'covar', W) + + # # [z,i] = sensor.reading() + + # # map.landmark(17) + + # # randinit + # # map = LandmarkMap(20); + # # veh = Bicycle('covar', V); + # # veh.add_driver( RandomPath(map.dim) ); + # # sensor = RangeBearingSensor(veh, map, 'covar', W, 'angle', ... + # # [-pi/2 pi/2], 'range', 4, 'animate'); + # # ekf = EKF(veh, V, P0, sensor, W, map); + + # # ekf.run(1000); + # # map.plot() + # # veh.plot_xy(); + # # ekf.plot_xy('r'); + # # ekf.plot_ellipse('k') + + # # 6.3 Creating a map + # # randinit + # # map = LandmarkMap(20); + # # veh = Bicycle(); error free vehicle + # # veh.add_driver( RandomPath(map.dim) ); + # # W = diag([0.1, 1*pi/180].^2); + # # sensor = RangeBearingSensor(veh, map, 'covar', W); + # # ekf = EKF(veh, [], [], sensor, W, []); + + # # ekf.run(1000); + + # # map.plot(); + # # ekf.plot_map('g'); + # # veh.plot_xy('b'); + + # # ekf.landmarks(:,6) + + # # ekf.x_est(19:20)' + + # # ekf.P_est(19:20,19:20) + + # # 6.4 EKF SLAM + # # randinit + # # P0 = diag([.01, .01, 0.005].^2); + # # map = LandmarkMap(20); + # # veh = Bicycle('covar', V); + # # veh.add_driver( RandomPath(map.dim) ); + # # sensor = RangeBearingSensor(veh, map, 'covar', W); + # # ekf = EKF(veh, V, P0, sensor, W, []); + + # # ekf.run(1000); + + # # map.plot(); + # # ekf.plot_map('g'); + # # ekf.plot_xy('r'); + # # veh.plot_xy('b'); + + # # 6.6 Pose-graph SLAM + # # syms x_i y_i theta_i x_j y_j theta_j x_m y_m theta_m assume real + # # xi_e = inv( SE2(x_m, y_m, theta_m) ) * inv( SE2(x_i, y_i, theta_i) ) * SE2(x_j, y_j, theta_j); + # # fk = simplify(xi_e.xyt); + + # # jacobian ( fk, [x_i y_i theta_i] ); + # # Ai = simplify (ans) + + # # pg = PoseGraph('pg1.g2o') + + # # clf + # # pg.plot() + + # # pg.optimize('animate') + + # # pg = PoseGraph('killian-small.toro'); + + # # pg.plot() + + # # pg.optimize() + + # # 6.7 Particle filter + # # randinit + # # map = LandmarkMap(20); + # # W = diag([0.1, 1*pi/180].^2); + # # veh = Bicycle('covar', V); + # # veh.add_driver( RandomPath(10) ); + + # # V = diag([0.005, 0.5*pi/180].^2); + # # sensor = RangeBearingSensor(veh, map, 'covar', W); + + # # Q = diag([0.1, 0.1, 1*pi/180]).^2; + + # # L = diag([0.1 0.1]); + + # # pf = ParticleFilter(veh, sensor, Q, L, 1000); + + # # pf.run(1000); + + # # map.plot(); + # # veh.plot_xy('b'); + + # # clf + # # pf.plot_xy('r'); + + # # clf + # # plot(pf.std(1:100,:)) + + # # clf + # # pf.plot_pdf() + + # # 6.8 Application: Scanning laser rangefinder + + # # Laser odometry + # # pg = PoseGraph('killian.g2o', 'laser'); + + # # [r, theta] = pg.scan(2580); + # # about r theta + + # # clf + # # polar(theta, r) + + # # [x,y] = pol2cart (theta, r); + # # plot (x, y, '.') + + # # p2580 = pg.scanxy(2580); + # # p2581 = pg.scanxy(2581); + # # about p2580 + + # # T = icp( p2581, p2580, 'verbose' , 'T0', transl2(0.5, 0), 'distthresh', 3) + + # # pg.time(2581)-pg.time(2580) + + # # Laser-based map building + # # map = pg.scanmap(); + # # pg.plot_occgrid(map) +
+ +
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+ +
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+
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/mobile/LatticePlanner.html b/_modules/roboticstoolbox/mobile/LatticePlanner.html new file mode 100644 index 00000000..af42df21 --- /dev/null +++ b/_modules/roboticstoolbox/mobile/LatticePlanner.html @@ -0,0 +1,530 @@ + + + + + + roboticstoolbox.mobile.LatticePlanner — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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+ +

Source code for roboticstoolbox.mobile.LatticePlanner

+from pgraph import DGraph, DVertex, Edge
+import numpy as np
+from spatialmath import SE2
+import matplotlib.pyplot as plt
+import itertools
+from roboticstoolbox.mobile.PlannerBase import PlannerBase
+from roboticstoolbox.mobile.OccGrid import BinaryOccupancyGrid
+from collections import namedtuple
+
+
+def make_arc(dir, radius=1, npoints=20):
+    points = []
+
+    if dir == "S":
+        points.append((0, 0))
+        points.append((radius, 0))
+
+    elif dir == "L":
+        for theta in np.linspace(0, np.pi / 2, npoints):
+            x = radius * np.sin(theta)
+            y = radius * (1 - np.cos(theta))
+            points.append((x, y))
+
+    elif dir == "R":
+        for theta in np.linspace(0, np.pi / 2, npoints):
+            x = radius * np.sin(theta)
+            y = radius * (-1 + np.cos(theta))
+            points.append((x, y))
+
+    return np.array(points).T
+
+
+arcs = {}
+
+
+class LatticeVertex(DVertex):
+    def __init__(self, move=None, pose=None, name=None):
+        super().__init__(name=name)
+        self.move = move
+        self.pose = pose
+        self.coord = pose.xyt()
+
+    def icoord(self):
+        xyt = self.coord
+        ix = int(round(xyt[0]))
+        iy = int(round(xyt[1]))
+        it = int(round(xyt[2] * 2 / np.pi))
+        return f"({ix:d},{iy:d},{it:d}), {self.name}"
+
+
+class LatticeEdge(Edge):
+    def __init__(self, v1, v2, cost, pose, move):
+        super().__init__(v1, v2, cost)
+        self.pose = pose
+        self.move = move
+        self.arc = arcs[move]
+
+    def plot(self, configspace=False, unwrap=False, **kwargs):
+        T = self.pose
+        xy = self.pose * self.arc
+        if configspace:
+            # 3D plot
+            theta0 = self.pose.theta()
+            if self.move == "L":
+                thetaf = theta0 + np.pi / 2
+            elif self.move == "R":
+                thetaf = theta0 - np.pi / 2
+            elif self.move == "S":
+                thetaf = theta0
+            theta = np.linspace(theta0, thetaf, self.arc.shape[1])
+            if unwrap:
+                theta = np.unwrap(theta)
+            plt.plot(xy[0, :], xy[1, :], theta, **kwargs)
+        else:
+            # 2D plot
+            plt.plot(xy[0, :], xy[1, :], **kwargs)
+
+
+
[docs]class LatticePlanner(PlannerBase): + r""" + Lattice planner + + :param costs: cost for straight, left-turn, right-turn, defaults to :math:`(1, \pi/2, \pi/2)` + :type costs: array_like(3), optional + :param root: configuration of root node, defaults to (0,0,0) + :type root: array_like(3), optional + :param kwargs: arguments passed to ``PlannerBase`` constructor + + ================== ======================== + Feature Capability + ================== ======================== + Plan :math:`\SE{2}` + Obstacle avoidance Yes, occupancy grid + Curvature Discontinuous + Motion Forwards only + ================== ======================== + + The lattice planner incrementally builds a graph from the root vertex, at + each iteration adding three edges to the graph: + + ===== ================ + code direction + ===== ================ + 'S' straight ahead + 'L' turn left + 'R' turn right + ===== ================ + + If the configuration is already in the graph, the edge connects to that + existing vertex. The vertex is named after the sequence of moves required + to reach it from the root. This means, that any configuration, ie. + :math:`(x, y, \theta)` can be reached by multiple paths and potentially have + multiple names. The first name assigned to a vertex is permanent and is not + overriden. + + If an occupancy grid exists and the configuration is an obstacle, then + the vertex is not added. + + The path through the lattice is found using A* graph search, and ``costs`` + changes the weighting for path costs at query time. + + Example: + + .. runblock:: pycon + + >>> from roboticstoolbox import LatticePlanner + >>> import numpy as np + >>> lattice = LatticePlanner(); + >>> lattice.plan(iterations=6) + >>> path, status = lattice.query(start=(0, 0, 0), goal=(1, 2, np.pi/2)) + >>> print(path.T) + >>> print(status) + + :seealso: :meth:`plan` :meth:`query` :class:`PlannerBase` + """ + + def __init__(self, costs=None, root=(0, 0, 0), **kwargs): + + global arcs + + super().__init__(ndims=3, **kwargs) + + self.poses = [SE2(1, 0, 0), SE2(1, 1, np.pi / 2), SE2(1, -1, -np.pi / 2)] + self.moves = ["S", "L", "R"] + if costs is None: + costs = [1, np.pi / 2, np.pi / 2] + self.costs = costs + self.root = root + + # create the set of possible moves + for move in self.moves: + arcs[move] = make_arc(move) + + def __str__(self): + s = ( + super().__str__() + + f"\n curvature={self.curvature}, stepsize={self.stepsize}" + ) + + def _icoord(self, xyt): + ix = int(round(xyt[0])) + iy = int(round(xyt[1])) + it = int(round(xyt[2] * 2 / np.pi)) + return f"({ix:d},{iy:d},{it:d})" + +
[docs] def plan(self, iterations=None, verbose=False, summary=False): + """ + Create a lattice plan + + :param iterations: number of iterations, defaults to None + :type iterations: int, optional + :param verbose: show frontier and added vertices/edges at each iteration, defaults to False + :type verbose: bool, optional + + If an occupancy grid exists the if ``iterations`` is None the area of the + grid will be completely filled. + + :seealso: :meth:`query` + """ + if iterations is None and self.occgrid is None: + raise ValueError( + "iterations must be finite if no occupancy grid is specified" + ) + + self.graph = DGraph(metric="SE2") + + # add root vertex to the graph, place it in the frontier + v0 = LatticeVertex(pose=SE2(self.root)) + self.graph.add_vertex(v0, name="0") + frontier = [v0] + + iteration = 0 + while True: + + newfrontier = [] + for vertex in frontier: + if verbose: + print("EXPAND:", vertex.icoord()) + + for pose, move, cost in zip(self.poses, self.moves, self.costs): + newpose = vertex.pose * pose + xyt = newpose.xyt() + # theta is guaranteed to be in range [-pi, pi) + + if verbose: + print(" MOVE", move, self._icoord(xyt)) + + if self.isoccupied(xyt[:2]): + if verbose: + print(" is occupied") + continue + vclose, d = self.graph.closest(xyt) + + if d > 0.01: + # vertex does not already exists + vnew = LatticeVertex(move, newpose, name=vertex.name + move) + + # ix = int(xyt[0]) + # iy = int(xyt[1]) + # it = int(round(xyt[2]*2/np.pi)) + # vnew = LatticeVertex(move, newpose, name=f"{ix:d},{iy:d},{it:d}") + self.graph.add_vertex(vnew) + if verbose: + print(" add to graph as", vnew.name) + + edge = LatticeEdge( + vertex, vnew, cost=cost, pose=vertex.pose, move=move + ) + + # connect it into the graph, add to frontier + vertex.connect(vnew, edge=edge) + newfrontier.append(vnew) + else: + # vertex already exists + # print('exists', vertex, move, vclose) + + # connect it into the graph, don't add to frontier + if verbose: + print(" already in graph, connect to", vclose.icoord()) + edge = LatticeEdge( + vertex, vclose, cost=cost, pose=vertex.pose, move=move + ) + vertex.connect(vclose, edge=edge) + + frontier = newfrontier + + iteration += 1 + if iterations is None: + if len(frontier) == 0: + print(f"finished after {iteration} iterations") + break + print(f"iteration {iteration}, frontier length {len(frontier)}") + elif iteration >= iterations: + break + if summary: + print(f"{self.graph.n} vertices and {self.graph.ne} edges created")
+ +
[docs] def query(self, start, goal): + r""" + Find a path through the lattice + + :param start: start configuration :math:`(x, y, \theta)` + :type start: array_like(3), optional + :param goal: goal configuration :math:`(x, y, \theta)` + :type goal: array_like(3), optional + :return: path and status + :rtype: ndarray(N,3), namedtuple + + The returned status value has elements: + + +-------------+-----------------------------------------------------+ + | Element | Description | + +-------------+-----------------------------------------------------+ + | ``cost`` | path cost | + +-------------+-----------------------------------------------------+ + |``segments`` | a list containing the type of each path segment as | + | | a single letter code: either "L", "R" or "S" for | + | | left turn, right turn or straight line respectively.| + +-------------+-----------------------------------------------------+ + |``edges`` | successive edges of the graph ``LatticeEdge`` type | + +-------------+-----------------------------------------------------+ + + :seealso: :meth:`plan` + """ + + vs, ds = self.graph.closest(start) + if ds > 0.001: + raise ValueError("start configuration is not in the lattice") + vg, dg = self.graph.closest(goal) + if dg > 0.001: + raise ValueError("goal configuration is not in the lattice") + + try: + path, cost, _ = self.graph.path_Astar(vs, vg, verbose=False) + except TypeError: + raise RuntimeError("no path found") from None + + status = namedtuple("LatticeStatus", ["cost", "segments", "edges"]) + + segments = [] + edges = [] + for p, n in zip(path[:-1], path[1:]): + e = p.edgeto(n) + edges.append(e) + segments.append(e.move) + + return np.array([p.coord for p in path]), status(cost, segments, edges)
+ +
[docs] def plot(self, path=None, **kwargs): + super().plot(**kwargs) + + if kwargs.get("configspace", False): + + # 3D plot + for k, vertex in enumerate(self.graph): + # for every node + # if k == 0: + # plt.plot(vertex.coord[0], vertex.coord[1], vertex.coord[2], 'k>', markersize=10) + # else: + plt.plot(vertex.coord[0], vertex.coord[1], vertex.coord[2], "bo") + + for edge in self.graph.edges(): + edge.plot(color="k", **kwargs) + + if path is not None: + for p, n in zip(path[:-1], path[1:]): + # turn coordinaets back into vertices + vp, _ = self.graph.closest(p) + vn, _ = self.graph.closest(n) + e = vp.edgeto(vn) + + # e.plot(color='b', linewidth=4) + + e.plot(color="k", linewidth=4) + e.plot(color="yellow", linewidth=3, dashes=(4, 4)) + else: + # 2D plot + for k, vertex in enumerate(self.graph): + # for every node + # if k == 0: + # plt.plot(vertex.coord[0], vertex.coord[1], 'k>', markersize=10) + # else: + plt.plot(vertex.coord[0], vertex.coord[1], "bo") + + for edge in self.graph.edges(): + edge.plot(color="k") + + if path is not None: + for p, n in zip(path[:-1], path[1:]): + # turn coordinaets back into vertices + vp, _ = self.graph.closest(p) + vn, _ = self.graph.closest(n) + e = vp.edgeto(vn) + + # e.plot(color='b', linewidth=4) + + e.plot(color="k", linewidth=4) + e.plot(color="yellow", linewidth=3, dashes=(4, 4))
+ + +if __name__ == "__main__": + + lattice = LatticePlanner() + lattice.plan(iterations=6) + path = lattice.query(start=(0, 0, np.pi / 2), goal=(1, 1, 0)) + print(path) + + # og = BinaryOccupancyGrid(workspace=[-5, 5, -5, 5], value=False) + # og.set([3, 3, -2, 3], True) + + # lattice = LatticePlanner(occgrid=og) + + # lattice.plan(iterations=10) + # print(lattice.graph) + + # qs = (0, 0, np.pi/2) + # qg = (1, 0, np.pi/2) + + # print('qs') + # vs, d = lattice.graph.closest(qs) + # print(vs, d, vs.coord) + # print(vs.neighbours()) + # print() + + # print('[-1, 1, -np.pi]') + # vs, d = lattice.graph.closest([-1, 1, -np.pi]) + # print(vs, d, vs.coord) + # print(vs.neighbours()) + # print() + + # print('0L') + # vs = lattice.graph['0L'] + # print(vs, vs.coord) + # print(vs.neighbours()) + # print() + + # path, status = lattice.query(qs, qg) + + # print(path) + # print(status) + + # lattice.plot(path=path) + + # plt.show(block=True) + + # ax = plt.gca() + # ax.set_aspect('equal') + # ax.grid(True) + # ax.set_xlim(-4, 4) + # ax.set_ylim(-4, 4) +
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/mobile/OccGrid.html b/_modules/roboticstoolbox/mobile/OccGrid.html new file mode 100644 index 00000000..cfaf4b84 --- /dev/null +++ b/_modules/roboticstoolbox/mobile/OccGrid.html @@ -0,0 +1,734 @@ + + + + + + roboticstoolbox.mobile.OccGrid — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.mobile.OccGrid

+import numpy as np
+import matplotlib.pyplot as plt
+from spatialmath import base
+import scipy.ndimage as sp
+from abc import ABC
+import spatialmath.base as smb
+
+from spatialmath.geom2d import Polygon2
+
+
+
[docs]class BaseMap(ABC): + def __init__(self, workspace=None, name=None, **unused): + """ + Abstract base class for maps + + :param workspace: dimensions of 2D plot area, defaults to (-10:10) x (-10:10), + see :func:`~spatialmath.base.graphics.plotvol2` + :type workspace: float, array_like(2), array_like(4) + :param name: nae of the map, defaults to None + :type name: str, optional + + The workspace can be specified in several ways: + + ============== ======= ======= + ``workspace`` x-range y-range + ============== ======= ======= + A (scalar) -A:A -A:A + [A, B] A:B A:B + [A, B, C, D] A:B C:D + ============== ======= ======= + """ + if workspace is not None: + workspace = smb.expand_dims(workspace) + self._workspace = workspace + self.dx = workspace[1] - workspace[0] + self.dy = workspace[3] - workspace[2] + self._name = name
+ + +
[docs]class BaseOccupancyGrid(BaseMap): + def __init__(self, grid=None, origin=(0, 0), value=0, cellsize=1, **kwargs): + """ + Occupancy grid (superclass) + + :param grid: occupancy grid as a NumPy array + :type grid: ndarray(N,M) + :param value: initial value of cells + :type value: any, optional + :param origin: world coordinates of the grid element [0,0], defaults to (0, 0) + :type origin: array_like(2), optional + :param cellsize: cell size, defaults to 1 + :type cellsize: float, optional + :param kwargs: options passed to :class:`~roboticstoolbox.mobile.OccGrid.BaseMap` + + This object supports a user-defined coordinate system and grid size. + World coordinates are converted to grid coordinates to lookup the + occupancy status. + + The grid can be initialized by: + + - a 2D NumPy array + - specifying ``workspace`` and ``value`` arguments + + """ + super().__init__(**kwargs) + + if grid is not None: + self._grid = grid + self._origin = smb.getvector(origin, 2) + + elif self._workspace is not None: + self._grid = np.full( + np.floor(np.r_[self.dx, self.dy] / cellsize).astype(int) + 1, value + ) + self._origin = np.r_[self._workspace[0], self._workspace[2]] + + self._cellsize = cellsize + +
[docs] def copy(self): + """ + Copy an occupancy grid (superclass) + + :return: copy of the ocupancy grid + :rtype: OccGrid + """ + return self.__class__( + self._grid.copy(), + cellsize=self._cellsize, + origin=self._origin, + name=self._name, + )
+ + def __repr__(self): + return str(self) + +
[docs] def __str__(self): + """ + Compact string description of occupancy grid (superclass) + + :return: summary of occupancy grid characteristics + :rtype: str + """ + s = self.__class__.__name__ + if self._name is not None: + s += f"[{self._name}]" + s += f": {self._grid.shape[1]} x {self._grid.shape[0]}" + s += f", cell size={self._cellsize}" + s += f", x = [{self.xmin}, {self.xmax}], y = [{self.ymin}, {self.ymax}]" + return s
+ + @property + def grid(self): + """ + Occupancy grid as a NumPy array (superclass) + + :return: binary occupancy grid + :rtype: ndarray(N,M) of bool + + If :meth:`inflate` has been called, this will return the inflated + occupancy grid. + """ + return self._grid + + @property + def xmin(self): + """ + Minimum x-coordinate of this grid (superclass) + + :return: minimum world x-coordinate + :rtype: float + """ + return self._origin[0] + + @property + def xmax(self): + """ + Maximum x-coordinate of this grid (superclass) + + :return: maximum world x-coordinate + :rtype: float + """ + return (self._grid.shape[1] - 1) * self._cellsize + self._origin[0] + + @property + def ymin(self): + """ + Minimum y-coordinate of this grid (superclass) + + :return: minimum world y-coordinate + :rtype: float + """ + return self._origin[1] + + @property + def ymax(self): + """ + Maximum y-coordinate of this grid (superclass) + + :return: maximum world y-coordinate + :rtype: float + """ + return (self._grid.shape[0] - 1) * self._cellsize + self._origin[1] + + @property + def shape(self): + """ + Shape of the occupancy grid array (superclass) + + :return: shape of the occupancy grid array + :rtype: 2-tuple + + This is the shape of the NumPy array that holds the occupancy grid. + """ + return self._grid.shape + + @property + def maxdim(self): + """ + Maximum dimension of grid in world coordinates (superclass) + + :return: maximum side length of the occupancy grid + :rtype: float + """ + return max(self.grid.shape) * self._cellsize + + @property + def workspace(self): + """ + Bounds of the occupancy grid in world coordinates (superclass) + + :return: workspace bounds [xmin, xmax, ymin, ymax] + :rtype: ndarray(4) + + Returns the bounds of the occupancy grid in world coordinates. + """ + return np.r_[self.xmin, self.xmax, self.ymin, self.ymax] + + @property + def name(self): + """ + Occupancy grid name (superclass) + + :return: name of the occupancy grid + :rtype: str + """ + return self._name + + @name.setter + def name(self, name): + """ + Set occupancy grid name (superclass) + + :param name: new name of the occupancy grid + :type name: str + """ + self._name = name + +
[docs] def set(self, region, value): + """ + Set region of map (superclass) + + :param region: The region [xmin, ymin, xmax, ymax] + :type region: array_like(4) + :param value: value to set cells to + :type value: int, bool, float + """ + bl = self.w2g([region[0], region[2]]) + tr = self.w2g([region[1], region[3]]) + self.grid[bl[1] : tr[1] + 1, bl[0] : tr[0] + 1] = value
+ +
[docs] def g2w(self, p): + """ + Convert grid coordinate to world coordinate (superclass) + + :param p: grid coordinate (column, row) + :type p: array_like(2) + :return: world coordinate (x, y) + :rtype: ndarray(2) + + The grid cell size and offset are used to convert occupancy grid + coordinate ``p`` to a world coordinate. + """ + p = smb.getvector(p, 2) + return p * self._cellsize + self._origin
+ +
[docs] def w2g(self, p): + """ + Convert world coordinate to grid coordinate (superclass) + + :param p: world coordinate (x, y) + :type p: array_like(2) + :return: grid coordinate (column, row) + :rtype: ndarray(2) + + The grid cell size and offset are used to convert ``p`` to an occupancy + grid coordinate. The grid coordinate is rounded and cast to integer + value. No check is made on the validity of the coordinate. + """ + return (np.round((p - self._origin) / self._cellsize)).astype(int)
+ +
[docs] def plot(self, map=None, ax=None, block=None, **kwargs): + """ + Plot the occupancy grid (superclass) + + :param map: array which is plotted instead of the grid, must be same + size as the occupancy grid,defaults to None + :type map: ndarray(N,M), optional + :param ax: matplotlib axes to plot into, defaults to None + :type ax: Axes2D, optional + :param block: block until plot is dismissed, defaults to None + :type block: bool, optional + :param kwargs: arguments passed to ``imshow`` + + The grid is plotted as an image but with axes in world coordinates. + + The grid is a NumPy boolean array which has values 0 (false=unoccupied) + and 1 (true=occupied). Passing a `cmap` option to imshow can be used + to control the displayed color of free space and obstacles. + + """ + + ax = smb.axes_logic(ax, 2) + + if map is None: + map = self._grid + kwargs["extent"] = self.workspace + + ax.imshow(map, origin="lower", interpolation=None, **kwargs) + ax.set_xlabel("x") + ax.set_ylabel("y") + if block is not None: + plt.show(block=block)
+ +
[docs] def line_w(self, p1, p2): + """ + Get index of cells along a line segment (superclass) + + :param p1: start + :type p1: array_like(2) + :param p2: end + :type p2: array_like(2) + :return: index into grid + :rtype: ndarray(N) + + Get the indices of cells along a line segment defined by the end + points given in world coordinates. + + The returned indices can be applied to a raveled view of the grid. + + :seealso: :meth:`ravel` :meth:`w2g` + """ + + gp1 = self.w2g(p1) + gp2 = self.w2g(p2) + + return self._line(gp1, gp2)
+ + def _line(self, p1, p2): + + x, y = smb.bresenham(p1, p2) + z = np.ravel_multi_index(np.vstack((y, x)), self.grid.shape) + return z + + @property + def ravel(self): + """ + Ravel the grid (superclass) + + :return: 1D view of the occupancy grid + :rtype: ndarray(N) + """ + return self._grid.reshape(-1)
+ + +
[docs]class BinaryOccupancyGrid(BaseOccupancyGrid): +
[docs] def __init__(self, grid=None, **kwargs): + """ + Create a binary occupancy grid instance + + :param grid: occupancy grid as a NumPy array + :type grid: ndarray(N,M) + :param size: cell size, defaults to 1 + :type size: float, optional + :param origin: world coordinates of the grid element [0,0], defaults to (0, 0) + :type origin: array_like(2), optional + :param kwargs: options passed to :class:`BaseMap` + + The array is kept internally as a bool array. Cells are set to True + (occupied) corresponding to input values > 0. + + This object supports a user-defined coordinate system and grid size. + World coordinates are converted to grid coordinates to lookup the + occupancy status. + + Example: + + .. runblock:: pycon + + >>> from roboticstoolbox import BinaryOccupancyGrid + >>> import numpy as np + >>> og = BinaryOccupancyGrid(np.zeros((5,5))) + >>> print(og) + >>> og = BinaryOccupancyGrid(workspace=[-5,5], cellsize=0.1, value=0) + >>> print(og) + + :seealso: :class:`OccupancyGrid` + """ + + if grid is not None: + if isinstance(grid, np.ndarray): + grid = grid.astype(bool) + elif isinstance(grid, BinaryOccupancyGrid): + grid = grid.grid + else: + raise ValueError("argument must be NumPy array or BinaryOccupancyGrid") + + super().__init__(grid=grid, **kwargs)
+ + def __str__(self): + s = super().__str__() + + ncells = np.prod(self._grid.shape) + nobs = self._grid.sum() + s += f", {nobs/ncells*100:.1f}% occupied" + return s + +
[docs] def isoccupied(self, p): + """ + Test if coordinate is occupied + + :param p: world coordinate (x, y) + :type p: array_like(2) + :return: occupancy status of corresponding grid cell + :rtype: bool + + The grid cell size and offset are used to convert ``p`` to an occupancy + grid coordinate. The grid coordinate is rounded and cast to integer + value. If the coordinate is outside the bounds of the occupancy grid + it is considered to be occupied. + + :seealso: :meth:`w2g` + """ + c, r = self.w2g(p) + try: + return self._grid[r, c] + except IndexError: + return True
+ +
[docs] def inflate(self, radius): + """ + Inflate obstales + + :param radius: radius of circular structuring element in world units + :type radius: float + + A circular structuring element is created and used to dilate the + stored occupancy grid. + + Successive calls to ``inflate`` will compound the inflation. + + :seealso: :func:`scipy.ndimage.binary_dilation` + """ + # Generate a circular structuring element + r = round(radius / self._cellsize) + Y, X = np.meshgrid(np.arange(-r, r + 1), np.arange(-r, r + 1)) + SE = X**2 + Y**2 <= r**2 + SE = SE.astype(int) + + # do the inflation using SciPy + self._grid = sp.binary_dilation(self._grid, SE)
+ + +
[docs]class OccupancyGrid(BaseOccupancyGrid): + """ + General occupancy grid + + + The elements of the array are floats and can represent occupancy + probability or traversal cost. + + Example: + + .. runblock:: pycon + + >>> from roboticstoolbox import OccupancyGrid + >>> import numpy as np + >>> og = OccupancyGrid(np.zeros((5,5))) + >>> print(og) + >>> og = OccupancyGrid(workspace=[-5,5], cellsize=0.1, value=0.5) + >>> print(og) + + :seealso: :class:`BinaryOccupancyGrid` + """ + + def __str__(self): + s = super().__str__() + + g = self._grid + s += f", dtype {g.dtype}" + s += f", min {g.min()}, max {g.max()}, mean {g.mean()}" + return s
+ + +
[docs]class PolygonMap(BaseMap): +
[docs] def __init__(self, workspace=None, polygons=[]): + """ + Polygonal obstacle map + + :param workspace: dimensions of 2D plot area, defaults to (-10:10) x (-10:10), + see :func:`~spatialmath.base.graphics.plotvol2` + :type workspace: float, array_like(2), array_like(4) + :param polygons: obstacle polygons, defaults to [] + :type polygons: list, optional + + The obstacle polygons are specified as instances of :class:`~spatialmath.geom2d.Polygon2` + or ndarray(2,N). + + The workspace can be specified in several ways: + + ============== ======= ======= + ``workspace`` x-range y-range + ============== ======= ======= + A (scalar) -A:A -A:A + [A, B] A:B A:B + [A, B, C, D] A:B C:D + ============== ======= ======= + + Workspace is used only to set plot bounds. + """ + super().__init__(workspace=workspace) + + self.polygons = polygons
+ +
[docs] def add(self, polygon): + """ + Add a polygon to map + + :param polygon: a polygon + :type polygon: :class:`~spatialmath.geom2d.Polygon2` or ndarray(2,N) + """ + + if isinstance(polygon, Polygon2): + self.polygons.append(polygon) # lgtm [py/modification-of-default-value] + else: + self.polygons.append( + Polygon2(polygon) + ) # lgtm [py/modification-of-default-value]
+ +
[docs] def iscollision(self, polygon): + """ + Test for collision + + :param polygon: a polygon + :type polygon: :class:`~spatialmath.geom2d.Polygon2` or ndarray(2,N) + :return: collision + :rtype: bool + + The ``polygon`` is tested against polygons in the map, and returns True + on the first collision. + + :seealso: :meth:`add` :class:`~spatialmath.geom2d.Polygon2` + """ + return polygon.intersects(self.polygons)
+ +
[docs] def plot(self, block=None): + smb.plotvol2(self.workspace) + + for polygon in self.polygons: + polygon.plot(color="r") + + if block is not None: + plt.show(block=block)
+ +
[docs] def isoccupied(self, p): + """ + Test if point lies inside an obstacle + + :param p: a 2D point + :type p: array_like(2) + :return: enclosure + :rtype: bool + + The point is tested for enclosure by polygons in the map, and returns True + on the first enclosure. + """ + for polygon in self.polygons: + if polygon.contains(p): + return True + + return False
+ + @property + def workspace(self): + """ + Bounds of the occupancy grid + + :return: workspace bounds [xmin, xmax, ymin, ymax] + :rtype: ndarray(4) + + Returns the bounds of the occupancy grid. + """ + return self._workspace
+ + +if __name__ == "__main__": + + # g = np.zeros((100, 100)) + # g[20:30, 50:80] = 1 + + # og = OccGrid(g, size=0.1, origin=(2,4),name='bob') + # print(og) + # print(og.xmin, og.xmax, og.ymin, og.ymax) + # print(og.isoccupied((8.5, 6.5))) + # print(og.isoccupied((6, 6))) + # print(og.isoccupied((500, 500))) + # og.plot(block=False) + # og2 = og.copy() + # print(og2) + # og2.inflate(0.5) + # plt.figure() + # og2.plot(block=True) + + # g = np.zeros((10,10)) + # g[2:3, 4:5] = 1 + # og = BinaryOccupancyGrid(g) + # print(og) + + # r = og.ravel + # print(r[24]) + + # og = BinaryOccupancyGrid(workspace=[2,3,4,5], cellsize=0.2) + # print(og) + + # og = BinaryOccupancyGrid(workspace=[2,3,4,5], cellsize=0.2, value=True) + # print(og) + + # og = OccupancyGrid(workspace=[2,3,4,5], cellsize=0.2, value=3) + # print(og) + + # og = OccupancyGrid(workspace=[2,3,4,5], cellsize=0.2, value=3.0) + # print(og) + + map = PolygonMap(workspace=[0, 10]) + map.add([(5, 50), (5, 6), (6, 6), (6, 50)]) + map.add([(5, 4), (5, -50), (6, -50), (6, 4)]) + map.plot() + + og = BinaryOccupancyGrid(workspace=[-5, 5, -5, 5], value=False) + # np.set_printoptions(linewidth=300) + # og = BinaryOccupancyGrid(workspace=[-10, 10, -10, 10], value=False) + # print(og) + # og.set([1,10, -10, 10], True) + # print(og.grid) + # print(og.isoccupied((0,0))) +
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/mobile/PRMPlanner.html b/_modules/roboticstoolbox/mobile/PRMPlanner.html new file mode 100644 index 00000000..1f7b2455 --- /dev/null +++ b/_modules/roboticstoolbox/mobile/PRMPlanner.html @@ -0,0 +1,456 @@ + + + + + + roboticstoolbox.mobile.PRMPlanner — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.mobile.PRMPlanner

+"""
+Python PRM
+@Author: Peter Corke, original MATLAB code and Python version
+@Author: Kristian Gibson, initial MATLAB port
+"""
+# from multiprocessing.sharedctypes import Value
+# from numpy import disp
+# from scipy import integrate
+# from spatialmath.base.animate import Animate
+from spatialmath.base.transforms2d import *
+from spatialmath.base.vectors import *
+
+# from spatialmath.pose2d import SE2
+# from spatialmath.base import animate
+from scipy.ndimage import *
+from matplotlib import cm, pyplot as plt
+from roboticstoolbox.mobile.PlannerBase import PlannerBase
+from pgraph import UGraph
+
+# from progress.bar import FillingCirclesBar
+
+
+
[docs]class PRMPlanner(PlannerBase): + r""" + Distance transform path planner + + :param occgrid: occupancy grid + :type occgrid: :class:`BinaryOccGrid` or ndarray(h,w) + :param npoints: number of random points, defaults to 100 + :type npoints: int, optional + :param dist_thresh: distance threshold, a new point is only added to the + roadmap if it is closer than this distance to an existing vertex, + defaults to None + :type dist_thresh: float, optional + :param Planner: probabilistic roadmap path planner + :param kwargs: common planner options, see :class:`PlannerBase` + + ================== ======================== + Feature Capability + ================== ======================== + Plan Cartesian space + Obstacle avoidance Yes, occupancy grid + Curvature Discontinuous + Motion Omnidirectional + ================== ======================== + + Creates a planner that finds the path between two points in the + plane using omnidirectional motion. The path comprises a set of way points. + + Example: + + .. runblock:: pycon + + >>> from roboticstoolbox import PRMPlanner + >>> import numpy as np + >>> simplegrid = np.zeros((6, 6)); + >>> simplegrid[2:5, 3:5] = 1 + >>> prm = PRMPlanner(simplegrid); + >>> prm.plan() + >>> path = prm.query(start=(5, 4), goal=(1,1)) + >>> print(path.T) + + :author: Peter Corke + :seealso: :class:`PlannerBase` + """ + + def __init__(self, occgrid=None, npoints=100, dist_thresh=None, **kwargs): + super().__init__(occgrid, ndims=2, **kwargs) + + if dist_thresh is None: + self._dist_thresh = 0.3 * self.occgrid.maxdim + + self._npoints = npoints + # self._npoints0 = npoints + self._dist_thresh0 = self.dist_thresh + self._graph = None + self._v_goal = None + self._v_start = None + self._local_goal = None + self._local_path = None + self._v_path = None + self._g_path = None + + def __str__(self): + s = super().__str__() + if self.graph is not None: + s += "\n " + str(self.graph) + return s + + @property + def npoints(self): + """ + Number of points in the roadmap + + :return: Number of points + :rtype: int + """ + return self._npoints + + @property + def dist_thresh(self): + """ + Distance threshold + + :return: distance threshold + :rtype: float + + Edges are created between points if the distance between them is less + than this value. + """ + return self._dist_thresh + + # @property + # def npoints0(self): + # return self._npoints0 + + # @property + # def dist_thresh0(self): + # return self._dist_thresh0 + + @property + def graph(self): + """ + Roadmap graph + + :return: roadmap as an undirected graph + :rtype: :class:`pgraph.UGraph` instance + """ + return self._graph + + def _create_roadmap(self, npoints, dist_thresh, animate=None): + # a = Animate(animate, fps=5) + self.progress_start(npoints) + + x = None + y = None + for j in range(npoints): + # find a random point in freespace + while True: + # pick a random unoccupied point + x = self.random.uniform(self.occgrid.xmin, self.occgrid.xmax) + y = self.random.uniform(self.occgrid.ymin, self.occgrid.ymax) + if not self.occgrid.isoccupied((x, y)): + break + + # add it as a vertex to the graph + vnew = self.graph.add_vertex([x, y]) + + # compute distance between vertices + for vertex in self.graph: + # find distance from vertex to all other vertices + distances = [] + for othervertex in self.graph: + # skip self + if vertex is othervertex: + continue + # add (distance, vertex) tuples to list + distances.append((vertex.distance(othervertex), othervertex)) + + # sort into ascending distance + distances.sort(key=lambda x: x[0]) + + # create edges to vertex if permissible + for distance, othervertex in distances: + # test if below distance threshold + if dist_thresh is not None and distance > dist_thresh: + break # sorted into ascending order, so we are done + + # test if obstacle free path connecting them + if self._test_path(vertex, othervertex): + # add an edge + self.graph.add_edge(vertex, othervertex, cost=distance) + self.progress_next() + + self.progress_end() + # if animate is not None: + # self.plot() + # if not np.empty(movie): + # a.add() + + def _test_path(self, v1, v2, npoints=None): + # vector from v1 to v2 + dir = v2.coord - v1.coord + + # figure the number of points, essentially the line length + # TODO: should delegate this test to the OccGrid object and do it + # world units + if npoints is None: + npoints = int(round(np.linalg.norm(dir))) + + # test each point along the line from v1 to v2 + for s in np.linspace(0, 1, npoints): + if self.occgrid.isoccupied(v1.coord + s * dir): + return False + return True + +
[docs] def plan(self, npoints=None, dist_thresh=None, animate=None): + """ + Plan PRM path + + :param npoints: number of random points, defaults to ``npoints`` given + to constructor + :type npoints: int, optional + :param dist_thresh: distance threshold, defaults to ``dist_thresh`` given + to constructor + :type dist_thresh: float, optional + :param animate: animate the planning algorithm iterations, defaults to False + :type animate: bool, optional + + Create a probablistic roadmap. This is a graph connecting points + randomly selected from the free space of the occupancy grid. Edges are + created between points if the distance between them is less than + ``dist_thresh``. + + The roadmap is a pgraph :obj:`~pgraph.PGraph.UGraph` + :class:`~pgraph.UGraph` + :class:`~pgraph.PGraph.UGraph` + + :seealso: :meth:`query` :meth:`graph` + """ + + self.message("create the graph") + + if npoints is None: + npoints = self.npoints + + if dist_thresh is None: + dist_thresh = self.dist_thresh + + self._graph = UGraph() + self._v_path = np.array([]) + + self.random_init() # reset the random number generator + self._create_roadmap(npoints, dist_thresh, animate)
+ +
[docs] def query(self, start, goal, **kwargs): + """ + Find a path from start to goal using planner + + :param start: start position :math:`(x, y)`, defaults to previously set value + :type start: array_like(), optional + :param goal: goal position :math:`(x, y)`, defaults to previously set value + :type goal: array_like(), optional + :param kwargs: options passed to :meth:`PlannerBase.query` + :return: path from start to goal, one point :math:`(x, y)` per row + :rtype: ndarray(N,2) + + The path is a sparse sequence of waypoints, with variable distance + between them. + + .. warning:: Waypoints 1 to N-2 are part of the roadmap, while waypoints + 0 and N-1 are the start and goal respectively. The first and last + motion segment is not guaranteed to be obstacle free. + + """ + if self.graph.n == 0: + raise RuntimeError("no plan computed") + + super().query(start=start, goal=goal, next=False, **kwargs) + + # find roadmap vertices closest to start and goal + vstart, _ = self.graph.closest(self.start) + vgoal, _ = self.graph.closest(self.goal) + + # find A* path through the roadmap + out = self.graph.path_Astar(vstart, vgoal) + if out is None: + raise RuntimeError("no path found") + path = [v.coord for v in out[0]] + + path.insert(0, start) # insert start at head of path + path.append(goal) # append goal to end of path + + return np.array(path)
+ +
[docs] def plot(self, *args, vertex={}, edge={}, **kwargs): + """ + Plot PRM path + + :param vertex: vertex style, defaults to {} + :type vertex: dict, optional + :param edge: edge style, defaults to {} + :type edge: dict, optional + + Displays: + + - the planner background (obstacles) + - the roadmap graph + - the path + + :seealso: :meth:`UGraph.plot` + """ + # plot the obstacles and path + ax = super().plot(*args, **kwargs) + print("plotted background") + vertex = {**dict(markersize=4), **vertex} + edge = {**dict(linewidth=0.5), **edge} + + # add the roadmap graph + self.graph.plot(text=False, vopt=vertex, eopt=edge, ax=ax) + print("plotted roadmap")
+ + +if __name__ == "__main__": + import matplotlib.pyplot as plt + from roboticstoolbox import * + + house = rtb_load_matfile("data/house.mat") + floorplan = house["floorplan"] + places = house["places"] + + occgrid = floorplan.copy() + + prm = PRMPlanner(occgrid=floorplan, seed=0) + prm.plan(npoints=50) + prm.plot() + + # start and goal position + # start = (10, 10) + # goal = (50, 50) + + # occgrid = np.zeros((100, 100)) + # occgrid[20:40, 15:30] = 1 + + # prm = PRMPlanner(occgrid=occgrid, verbose=True) + + # prm.plan() + + # prm.plot() + + # path = prm.query(start, goal) + # print(path) + + # prm.plot(path, path_marker=dict(zorder=8, linewidth=2, markersize=6, color='k')) + # prm.plot(ax=plt.gca(), text=False, vertex=dict(markersize=4), edge=dict(linewidth=0.5)) + plt.show(block=True) +
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/mobile/ParticleFilter.html b/_modules/roboticstoolbox/mobile/ParticleFilter.html new file mode 100644 index 00000000..22172be0 --- /dev/null +++ b/_modules/roboticstoolbox/mobile/ParticleFilter.html @@ -0,0 +1,826 @@ + + + + + + roboticstoolbox.mobile.ParticleFilter — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.mobile.ParticleFilter

+#!/usr/bin/env python3
+
+"""
+Python EKF Planner
+@Author: Peter Corke, original MATLAB code and Python version
+@Author: Kristian Gibson, initial MATLAB port
+
+Based on code by Paul Newman, Oxford University, 
+http://www.robots.ox.ac.uk/~pnewman
+"""
+
+from collections import namedtuple
+
+import numpy as np
+import scipy as sp
+import matplotlib.pyplot as plt
+from matplotlib import animation
+
+import spatialmath.base as smb
+
+"""
+Monte-carlo based localisation for estimating vehicle pose based on
+odometry and observations of known landmarks.
+"""
+
+# TODO: refactor this and EKF, RNG, history, common plots, animation, movie
+
+
+
[docs]class ParticleFilter: +
[docs] def __init__( + self, + robot, + sensor, + R, + L, + nparticles=500, + seed=0, + x0=None, + verbose=False, + animate=False, + history=True, + workspace=None, + ): + """ + Particle filter + + :param robot: robot motion model + :type robot: :class:`VehicleBase` subclass, + :param sensor: vehicle mounted sensor model + :type sensor: :class:`SensorBase` subclass + :param R: covariance of the zero-mean Gaussian noise added to the particles at each step (diffusion) + :type R: ndarray(3,3) + :param L: covariance used in the sensor likelihood model + :type L: ndarray(2,2) + :param nparticles: number of particles, defaults to 500 + :type nparticles: int, optional + :param seed: random number seed, defaults to 0 + :type seed: int, optional + :param x0: initial state, defaults to [0, 0, 0] + :type x0: array_like(3), optional + :param verbose: display extra debug information, defaults to False + :type verbose: bool, optional + :param history: retain step-by-step history, defaults to True + :type history: bool, optional + :param workspace: dimension of workspace, see :func:`~spatialmath.base.graphics.expand_dims` + :type workspace: scalar, array_like(2), array_like(4) + + This class implements a Monte-Carlo estimator or particle filter for + vehicle state, based on odometry, a landmark map, and landmark + observations. The state of each particle is a possible vehicle + configuration :math:`(x,y,\theta)`. Bootstrap particle resampling is + used. + + The working area is defined by ``workspace`` or inherited from the + landmark map attached to the ``sensor`` (see + :func:`~spatialmath.base.graphics.expand_dims`): + + ============== ======= ======= + ``workspace`` x-range y-range + ============== ======= ======= + A (scalar) -A:A -A:A + [A, B] A:B A:B + [A, B, C, D] A:B C:D + ============== ======= ======= + + Particles are initially distributed uniform randomly over this area. + + Example:: + + V = np.diag([0.02, np.radians(0.5)]) ** 2 + robot = Bicycle(covar=V, animation="car", workspace=10) + robot.control = RandomPath(workspace=robot) + + map = LandmarkMap(nlandmarks=20, workspace=robot.workspace) + + W = np.diag([0.1, np.radians(1)]) ** 2 + sensor = RangeBearingSensor(robot, map, covar=W, plot=True) + + R = np.diag([0.1, 0.1, np.radians(1)]) ** 2 + L = np.diag([0.1, 0.1]) + pf = ParticleFilter(robot, sensor, R, L, nparticles=1000) + + pf.run(T=10) + + map.plot() + robot.plot_xy() + pf.plot_xy() + + plt.plot(pf.get_std()[:100,:]) + + .. note:: Set ``seed=0`` to get different behaviour from run to run. + + :seealso: :meth:`run` + """ + self._robot = robot + self._sensor = sensor + self.R = R + self.L = L + self.nparticles = nparticles + self._animate = animate + + # self.dim = sensor.map.dim + self._history = [] + self.x = () + self.weight = () + self.w0 = 0.05 + self._x0 = x0 + + # create a private random number stream if required + self._random = np.random.default_rng(seed) + self._seed = seed + + self._keep_history = history # keep history + self._htuple = namedtuple("PFlog", "t odo xest std weights") + + if workspace is not None: + self._dim = smb.expand_dims(workspace) + else: + self._dim = sensor.map.workspace + + self._workspace = self.robot.workspace
+ # self._init() + + def __str__(self): + # ParticleFilter.char Convert to string + # + # PF.char() is a string representing the state of the ParticleFilter + # object in human-readable form. + # + # See also ParticleFilter.display. + + def indent(s, n=2): + spaces = " " * n + return s.replace("\n", "\n" + spaces) + + s = f"ParticleFilter object: {self.nparticles} particles" + s += "\nR: " + smb.array2str(self.R) + s += "\nL: " + smb.array2str(self.L) + if self.robot is not None: + s += indent("\nrobot: " + str(self.robot)) + + if self.sensor is not None: + s += indent("\nsensor: " + str(self.sensor)) + return s + + @property + def robot(self): + """ + Get robot object + + :return: robot used in simulation + :rtype: :class:`VehicleBase` subclass + """ + return self._robot + + @property + def sensor(self): + """ + Get sensor object + + :return: sensor used in simulation + :rtype: :class:`SensorBase` subclass + """ + return self._sensor + + @property + def map(self): + """ + Get map object + + :return: map used in simulation + :rtype: :class:`LandmarkMap` subclass + """ + return self._map + + @property + def verbose(self): + """ + Get verbosity state + + :return: verbosity + :rtype: bool + """ + return self._verbose + + @property + def history(self): + """ + Get EKF simulation history + + :return: simulation history + :rtype: list of namedtuples + + At each simulation timestep a namedtuple of is appended to the history + list. It contains, for that time step, estimated state and covariance, + and sensor observation. + + :seealso: :meth:`get_t` :meth:`get_xy` :meth:`get_std` + :meth:`get_Pnorm` + """ + return self._history + + @property + def workspace(self): + """ + Size of robot workspace + + :return: workspace bounds [xmin, xmax, ymin, ymax] + :rtype: ndarray(4) + + Returns the bounds of the workspace as specified by constructor + option ``workspace`` + """ + return self._workspace + + @property + def random(self): + """ + Get private random number generator + + :return: NumPy random number generator + :rtype: :class:`numpy.random.Generator` + + Has methods including: + + - ``integers(low, high, size, endpoint)`` + - ``random(size)`` + - ``uniform`` + - ``normal(mean, std, size)`` + - ``multivariate_normal(mean, covar, size)`` + + The generator is initialized with the seed provided at constructor + time every time ``init`` is called. + + """ + return self._random + + def _init(self, x0=None, animate=False, ax=None): + # ParticleFilter.init Initialize the particle filter + # + # PF.init() initializes the particle distribution and clears the + # history. + # + # Notes:: + # - If initial particle states were given to the constructor the states are + # set to this value, else a random distribution over the map is used. + # - Invoked by the run() method. + + self.robot.init() + self.sensor.init() + + # clear the history + self._history = [] + + # create a new private random number generator + if self._seed is not None: + self._random = np.random.default_rng(self._seed) + + self._t = 0 + + # initialize particles + if x0 is None: + x0 = self._x0 + if x0 is None: + # create initial particle distribution as uniformly randomly distributed + # over the map workspace and heading angles + x = self.random.uniform( + self.workspace[0], self.workspace[1], size=(self.nparticles,) + ) + y = self.random.uniform( + self.workspace[2], self.workspace[3], size=(self.nparticles,) + ) + t = self.random.uniform(-np.pi, np.pi, size=(self.nparticles,)) + self.x = np.c_[x, y, t] + + if animate: + # display the initial particles + (self.h,) = ax.plot( + self.x[:, 0], + self.x[:, 1], + "go", + zorder=0, + markersize=3, + markeredgecolor="none", + alpha=0.3, + label="particle", + ) + + self.weight = np.ones((self.nparticles,)) + +
[docs] def run(self, T=10, x0=None): + """ + Run the particle filter simulation + + :param T: maximum simulation time in seconds + :type T: float + :param x0: Initial state, defaults to value given to Vehicle constructor + :type x0: array_like(3) or array_like(2) + + Simulates the motion of a vehicle (under the control of a driving agent) + and the particle-filter estimator. The steps are: + + - initialize the filter, vehicle and vehicle driver agent, sensor + - for each time step: + + - step the vehicle and its driver agent, obtain odometry + - take a sensor reading + - execute the EKF + - save information as a namedtuple to the history list for later display + + :seealso: :meth:`history` :meth:`landmark` :meth:`landmarks` + :meth:`get_xy` :meth:`get_t` :meth:`get_std` + :meth:`plot_xy` + """ + + self._init(x0=x0) + + # anim = Animate(opt.movie) + + # display the initial particles + ax = smb.axes_logic(None, 2) + if self._animate: + (self.h,) = ax.plot( + self.x[:, 0], + self.x[:, 1], + "go", + zorder=0, + markersize=3, + markeredgecolor="none", + alpha=0.3, + label="particle", + ) + # set(self.h, 'Tag', 'particles') + + # self.robot.plot() + + # iterate over time + import time + + for i in range(round(T / self.robot.dt)): + self._step() + # time.sleep(0.2) + plt.pause(0.2)
+ # plt.draw() + # anim.add() + # anim.close() + +
[docs] def run_animation(self, T=10, x0=None, format=None, file=None): + """ + Run the particle filter simulation + + :param T: maximum simulation time in seconds + :type T: float + :param x0: Initial state, defaults to value given to Vehicle constructor + :type x0: array_like(3) or array_like(2) + :param format: Output format + :type format: str, optional + :param file: File name + :type file: str, optional + :return: Matplotlib animation object + :rtype: :meth:`matplotlib.animation.FuncAnimation` + + Simulates the motion of a vehicle (under the control of a driving agent) + and the particle-filter estimator and returns an animation + in various formats:: + + ``format`` ``file`` description + ============ ========= ============================ + ``"html"`` str, None return HTML5 video + ``"jshtml"`` str, None return JS+HTML video + ``"gif"`` str return animated GIF + ``"mp4"`` str return MP4/H264 video + ``None`` return a ``FuncAnimation`` object + + The allowables types for ``file`` are given in the second column. A str + value is the file name. If ``None`` is an option then return the video as a string. + + For the last case, a reference to the animation object must be held if used for + animation in a Jupyter cell:: + + anim = robot.run_animation(T=20) + + The steps are: + + - initialize the filter, vehicle and vehicle driver agent, sensor + - for each time step: + + - step the vehicle and its driver agent, obtain odometry + - take a sensor reading + - execute the EKF + - save information as a namedtuple to the history list for later display + + :seealso: :meth:`history` :meth:`landmark` :meth:`landmarks` + :meth:`get_xy` :meth:`get_t` :meth:`get_std` + :meth:`plot_xy` + """ + + fig, ax = plt.subplots() + + nframes = round(T / self.robot.dt) + anim = animation.FuncAnimation( + fig=fig, + # func=lambda i: self._step(animate=True, pause=False), + # init_func=lambda: self._init(animate=True), + func=lambda i: self._step(), + init_func=lambda: self._init(ax=ax, animate=True), + frames=nframes, + interval=self.robot.dt * 1000, + blit=False, + repeat=False, + ) + # anim._interval = self.dt*1000/2 + # anim._repeat = True + ret = None + if format == "html": + ret = anim.to_html5_video() # convert to embeddable HTML5 animation + elif format == "jshtml": + ret = anim.to_jshtml() # convert to embeddable Javascript/HTML animation + elif format == "gif": + anim.save( + file, writer=animation.PillowWriter(fps=1 / self.dt) + ) # convert to GIF + ret = None + elif format == "mp4": + anim.save( + file, writer=animation.FFMpegWriter(fps=1 / self.dt) + ) # convert to mp4/H264 + ret = None + elif format == None: + # return the anim object + return anim + else: + raise ValueError("unknown format") + + if ret is not None and file is not None: + with open(file, "w") as f: + f.write(ret) + ret = None + plt.close(fig) + return ret
+ # self._init(x0=x0) + + # # anim = Animate(opt.movie) + + # # display the initial particles + # ax = smb.axes_logic(None, 2) + # if self._animate: + # (self.h,) = ax.plot( + # self.x[:, 0], + # self.x[:, 1], + # "go", + # zorder=0, + # markersize=3, + # markeredgecolor="none", + # alpha=0.3, + # label="particle", + # ) + # # set(self.h, 'Tag', 'particles') + + # # self.robot.plot() + + # # iterate over time + # import time + + # for i in range(round(T / self.robot.dt)): + # self._step() + # # time.sleep(0.2) + # plt.pause(0.2) + # # plt.draw() + # # anim.add() + # # anim.close() + + def _step(self): + + # fprintf('---- step\n') + odo = self.robot.step(animate=self._animate) # move the robot + + # update the particles based on odometry + self._predict(odo) + + # get a sensor reading + z, lm_id = self.sensor.reading() + + if z is not None: + self._observe(z, lm_id) + # fprintf(' observe beacon #d\n', lm_id) + + self._select() + + # our estimate is simply the mean of the particles + x_est = self.x.mean(axis=0) + std_est = self.x.std(axis=0) + + # std is more complex for angles, need to account for 2pi wrap + std_est[2] = np.sqrt(np.sum(smb.angdiff(self.x[:, 2], x_est[2]) ** 2)) / ( + self.nparticles - 1 + ) + + # display the updated particles + # set(self.h, 'Xdata', self.x(:,1), 'Ydata', self.x(:,2), 'Zdata', self.x(:,3)) + + if self._animate: + self.h.set_xdata(self.x[:, 0]) + self.h.set_ydata(self.x[:, 1]) + + # if ~isempty(self.anim) + # self.anim.add() + + if self._keep_history: + hist = self._htuple( + self.robot._t, odo.copy(), x_est, std_est, self.weight.copy() + ) + self._history.append(hist) + +
[docs] def plot_pdf(self): + """ + Plot particle PDF + + Displays a discrete PDF of vehicle position. Creates a 3D plot where + the x- and y-axes are the estimated vehicle position and the z-axis is + the particle weight. Each particle is represented by a a vertical line + segment of height equal to particle weight. + """ + + ax = smb.plotvol3() + for (x, y, t), weight in zip(self.x, self.weight): + # ax.plot([x, x], [y, y], [0, weight], 'r') + ax.plot([x, x], [y, y], [0, weight], "skyblue", linewidth=3) + ax.plot(x, y, weight, "k.", markersize=6) + + ax.grid(True) + ax.set_xlabel("X") + ax.set_ylabel("Y") + ax.set_xlim() + ax.set_zlabel("particle weight") + ax.view_init(29, 59)
+ + def _predict(self, odo): + # step 2 + # update the particle state based on odometry and a random perturbation + + # Straightforward code: + # + # for i=1:self.nparticles + # x = self.robot.f( self.x(i,:), odo)' + sqrt(self.R)*self.randn[2,0] + # x[2] = angdiff(x[2]) + # self.x(i,:) = x + # + # Vectorized code: + + self.x = self.robot.f(self.x, odo) + self.random.multivariate_normal( + (0, 0, 0), self.R, size=self.nparticles + ) + self.x[:, 2] = smb.angdiff(self.x[:, 2]) + + def _observe(self, z, lm_id): + # step 3 + # predict observation and score the particles + + # Straightforward code: + # + # for p = 1:self.nparticles + # # what do we expect observation to be for this particle? + # # use the sensor model h(.) + # z_pred = self.sensor.h( self.x(p,:), lm_id) + # + # # how different is it + # innov[0] = z[0] - z_pred[0] + # innov[1] = angdiff(z[1], z_pred[1]) + # + # # get likelihood (new importance). Assume Gaussian but any PDF works! + # # If predicted obs is very different from actual obs this score will be low + # # ie. this particle is not very good at predicting the observation. + # # A lower score means it is less likely to be selected for the next generation... + # # The weight is never zero. + # self.weight(p) = exp(-0.5*innov'*inv(self.L)*innov) + 0.05 + # end + # + # Vectorized code: + + invL = np.linalg.inv(self.L) + z_pred = self.sensor.h(self.x, lm_id) + z_pred[:, 0] = z[0] - z_pred[:, 0] + z_pred[:, 1] = smb.angdiff(z[1], z_pred[:, 1]) + + LL = -0.5 * np.r_[invL[0, 0], invL[1, 1], 2 * invL[0, 1]] + e = ( + np.c_[z_pred[:, 0] ** 2, z_pred[:, 1] ** 2, z_pred[:, 0] * z_pred[:, 1]] + @ LL + ) + self.weight = np.exp(e) + self.w0 + + def _select(self): + # step 4 + # select particles based on their weights + # + # particles with large weights will occupy a greater percentage of the + # y axis in a cummulative plot + cdf = np.cumsum(self.weight) / self.weight.sum() + + # so randomly (uniform) choosing y values is more likely to correspond to + # better particles... + iselect = self.random.uniform(0, 1, size=(self.nparticles,)) + + # find the particle that corresponds to each y value (just a look up) + interpfun = sp.interpolate.interp1d( + cdf, + np.arange(self.nparticles), + assume_sorted=True, + kind="nearest", + fill_value="extrapolate", + ) + inextgen = interpfun(iselect).astype(int) + + # copy selected particles for next generation.. + self.x = self.x[inextgen, :] + +
[docs] def get_t(self): + """ + Get time from simulation + + :return: simulation time vector + :rtype: ndarray(n) + + Return simulation time vector, starts at zero. The timestep is an + attribute of the ``robot`` object. + """ + return np.array([h.t for h in self._history])
+ +
[docs] def get_xyt(self): + r""" + Get estimated vehicle trajectory + + :return: vehicle trajectory where each row is configuration :math:`(x, y, \theta)` + :rtype: ndarray(n,3) + + :seealso: :meth:`plot_xy` :meth:`run` :meth:`history` + """ + return np.array([h.xest[:2] for h in self._history])
+ +
[docs] def get_std(self): + r""" + Get standard deviation of particles + + :return: standard deviation of vehicle position estimate + :rtype: ndarray(n,2) + + Return the standard deviation :math:`(\sigma_x, \sigma_y)` of the + particle cloud at each time step. + + :seealso: :meth:`get_xyt` + """ + return np.array([h.std for h in self._history])
+ +
[docs] def plot_xy(self, block=None, **kwargs): + r""" + Plot estimated vehicle position + + :param args: position arguments passed to :meth:`~matplotlib.axes.Axes.plot` + :param kwargs: keywords arguments passed to :meth:`~matplotlib.axes.Axes.plot` + :param block: hold plot until figure is closed, defaults to None + :type block: bool, optional + + Plot the estimated vehicle path in the xy-plane. + + :seealso: :meth:`get_xy` + """ + xyt = self.get_xyt() + plt.plot(xyt[:, 0], xyt[:, 1], **kwargs) + if block is not None: + plt.show(block=block)
+ + +if __name__ == "__main__": + from roboticstoolbox import * + + map = LandmarkMap(20, workspace=10) + V = np.diag([0.02, np.deg2rad(0.5)]) ** 2 + robot = Bicycle(covar=V, animation="car", workspace=map) + robot.control = RandomPath(workspace=map) + W = np.diag([0.1, np.deg2rad(1)]) ** 2 + sensor = RangeBearingSensor(robot, map, covar=W, plot=True) + R = np.diag([0.1, 0.1, np.deg2rad(1)]) ** 2 + L = np.diag([0.1, 0.1]) + pf = ParticleFilter(robot, sensor=sensor, R=R, L=L, nparticles=1000, animate=True) + pf.run(T=10) +
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/mobile/PlannerBase.html b/_modules/roboticstoolbox/mobile/PlannerBase.html new file mode 100644 index 00000000..6c87e90e --- /dev/null +++ b/_modules/roboticstoolbox/mobile/PlannerBase.html @@ -0,0 +1,1125 @@ + + + + + + roboticstoolbox.mobile.PlannerBase — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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+ +

Source code for roboticstoolbox.mobile.PlannerBase

+"""
+Python Navigation Abstract Class
+@Author: Peter Corke
+@Author: Kristian Gibson
+"""
+
+from abc import ABC
+
+# from scipy import integrate
+# from scipy.ndimage import interpolation
+from spatialmath.base.transforms2d import *
+from spatialmath.base.vectors import *
+from spatialmath.base.argcheck import getvector
+from spatialmath.base.graphics import axes_logic, plotvol2, axes_get_scale
+
+# from spatialmath import SE2, SE3
+from matplotlib import cm
+from abc import ABC
+import matplotlib as mpl
+import matplotlib.pyplot as plt
+import copy
+from roboticstoolbox.mobile.OccGrid import BaseOccupancyGrid, BinaryOccupancyGrid
+from roboticstoolbox.mobile.Animations import VehiclePolygon
+from colored import fg, attr
+
+try:
+    from progress.bar import FillingCirclesBar
+
+    _progress = True
+except ImportError:
+    _progress = False
+
+
+
[docs]class PlannerBase(ABC): + r""" + Mobile robot motion planner (superclass) + + :param occgrid: occupancy grid, defaults to None + :type occgrid: :class:`OccGrid` instance of ndarray(N,M), optional + :param start: start position :math:`(x, y)` or configuration :math:`(x, y, \theta)`, defaults to None + :type start: array_like(2) or array_like(3), optional + :param goal: goal position :math:`(x, y)` or configuration :math:`(x, y, \theta)`, defaults to None + :type goal: array_like(2) or array_like(3), optional + :param inflate: obstacle inflation, defaults to 0 + :type inflate: float, optional + :param ndims: dimensionality of the planning, either 2 for :math:`\mathbb{R}^2` or + 3 for :math:`\SE{2}` + :param ndims: int, optional + :param verbose: verbosity, defaults to False + :type verbose: bool, optional + :param msgcolor: color for message channel printing + :type msgcolor: str, defaults to yellow + :param seed: seed provided to private random number generator, defaults to None + :type seed: int, optional + + Superclass for all mobile robot motion planners. Key functionality + includes: + + - encapsulates an occupancy grid and optionally inflates it + - validates ``start`` and ``goal`` if given + - encapsulates a private random number generator with specifiable seed + - encapsulates state such as start, goal, and the plan + - provides a message channel for diagnostic output + + The start and goal can be specifed in various ways: + + - at constructor time by the arguments ``start`` or ``goal`` + - by assigning the attributes ``start`` or ``goal`` + - at planning time by specifying ``goal`` to :meth:`plan` + - at query time by specifying ``start`` to :meth:`query` + + :seealso: :class:`OccGrid` + """ + + def __init__( + self, + occgrid=None, + inflate=0, + ndims=None, + start=None, + goal=None, + verbose=False, + msgcolor="yellow", + progress=True, + marker=None, + seed=None, + **unused, + ): + + self._occgrid = None + if ndims is None: + raise ValueError("ndims must be specified") + self._ndims = ndims + self._verbose = verbose + self._msgcolor = msgcolor + self._seed = seed + self._private_random = np.random.default_rng(seed=seed) + self._inflate = inflate + + self._progress = progress and _progress + + self.marker = marker + + if occgrid is not None: + if isinstance(occgrid, np.ndarray) and occgrid.ndim == 2: + # it's a NumPy array + self._occgrid = BinaryOccupancyGrid(occgrid) + elif isinstance(occgrid, BinaryOccupancyGrid): + self._occgrid = occgrid # original occgrid for reference + + if inflate > 0: + self._occgrid0 = self._occgrid.copy() + self._occgrid.inflate(inflate) + else: + self._occgrid0 = self._occgrid + + self._start = self.validate_endpoint(start) + self._goal = self.validate_endpoint(goal) + +
[docs] def __str__(self): + """ + Compact representation of the planner + + :return: pretty printed representation + :rtype: str + """ + s = f"{self.__class__.__name__}: " + if hasattr(self, "_occgrid") and self._occgrid is not None: + s += "\n " + str(self._occgrid) + if self._start is not None: + s += f"\n Start: {self.start}" + if self._goal is not None: + s += f"\n Goal: {self.goal}" + return s
+ + def __repr__(self): + return str(self) + + @property + def start(self): + r""" + Set/get start point or configuration (superclass) + + :getter: Return start point or configuration + :rtype: ndarray(2) or ndarray(3) + :setter: Set start point or configuration + :param: array_like(2) or array_like(3) + + The start is either a point :math:`(x, y)` or a configuration :math:`(x, y, \theta)`. + + :seealso: :meth:`goal` + """ + return self._start + + @start.setter + def start(self, start): + if start is not None: + if self.isoccupied(start): + raise ValueError("Start location inside obstacle") + self._start = getvector(start) + + @property + def goal(self): + r""" + Set/get goal point or configuration (superclass) + + :getter: Return goal pointor configuration + :rtype: ndarray(2) or ndarray(3) + :setter: Set goal point or configuration + :param: array_like(2) or array_like(3) + + The goal is either a point :math:`(x, y)` or a configuration :math:`(x, y, \theta)`. + + :seealso: :meth:`goal` + """ + return self._goal + + @goal.setter + def goal(self, goal): + r""" + Set goal point or configuration for planning + + :param goal: Set goal :math:`(x, y)` or configuration :math:`(x, y, \theta)` + :type goal: array_like(2) or array_like(3) + :raises ValueError: if goal point is occupied + """ + if goal is not None: + if self.isoccupied(goal): + raise ValueError("Goal location inside obstacle") + self._goal = getvector(goal) + + @property + def verbose(self): + """ + Get verbosity + + :return: verbosity + :rtype: bool + + If ``verbosity`` print more diagnostic messages to the planner's + message channel. + """ + return self._verbose + + @verbose.setter + def verbose(self, v): + """ + Set verbosity + + :param v: verbosity + :type v: bool + + If ``verbosity`` print more diagnostic messages to the planner's + message channel. + """ + self._verbose = v + + @property + def random(self): + """ + Get private random number generator + + :return: NumPy random number generator + :rtype: :class:`numpy.random.Generator` + + Has methods including: + + - :meth:`integers(low, high, size, endpoint) <numpy.random.Generator.integers>` + - :meth:`random(size) <numpy.random.Generator.random>` + - :meth:`uniform(low, high, size) <numpy.random.Generator.uniform>` + - :meth:`normal(mean, std, size) <numpy.random.Generator.normal>` + - :meth:`multivariate_normal(mean, covar, size) <numpy.random.Generator.multivariate_normal>` + + The generator is initialized with the seed provided at constructor + time. + + :seealso: :func:`numpy.random.default_rng` + """ + return self._private_random + +
[docs] def random_init(self, seed=None): + """ + Initialize private random number generator + + :param seed: random number seed, defaults to value given to constructor + :type seed: int + + The private random number generator is initialized. The seed is ``seed`` + or the value given to the constructor. If None, the generator will + be randomly seeded using a seed from the operating system. + """ + if seed is None: + seed = self._seed + + self._private_random = np.random.default_rng(seed=seed)
+ + # def randinit(self): + # if self._seed is not None: + # self._private_random = np.random.default_rng(seed=self._seed) + +
[docs] def plan(self): + r""" + Plan path (superclass) + + :param start: start position :math:`(x, y)` or configuration :math:`(x, y, \theta)`, defaults to value passed to constructor + :type start: array_like(2) or array_like(3), optional + :param goal: goal position :math:`(x, y)` or configuration :math:`(x, y, \theta)`, defaults to value passed to constructor + :type goal: array_like(2) or array_like(3), optional + + The implementation depends on the particular planner. Some may have + no planning phase. The plan may also depend on just the start or goal. + """ + pass
+ + @property + def occgrid(self): + """ + Occupancy grid + + :return: occupancy grid used for planning + :rtype: :class:`OccGrid` or subclass or None + + Returns the occupancy grid that was optionally inflated at constructor time. + + :seealso: :meth:`validate_endpoint` :meth:`isoccupied` + """ + return self._occgrid + +
[docs] def isoccupied(self, p): + """ + Test if point is occupied (superclass) + + :param p: world coordinate (x, y) + :type p: array_like(2) + :return: occupancy status of corresponding grid cell + :rtype: bool + + The world coordinate is transformed and the status of the occupancy + grid cell is returned. If the point lies outside the bounds of + the occupancy grid return True (obstacle) + + If there is no occupancy grid this function always returns False (free). + + :seealso: :meth:`occgrid` :meth:`validate_endpoint` :meth:`BinaryOccGrid.isoccupied` + """ + if self.occgrid is None: + return False + else: + return self.occgrid.isoccupied(p)
+ +
[docs] def validate_endpoint(self, p, dtype=None): + """ + Validate start or goal point + + :param p: the point + :type p: array_like(2) + :param dtype: data type for point coordinates, defaults to None + :type dtype: str, optional + :raises ValueError: point is inside obstacle + :return: point as a NumPy array of specified dtype + :rtype: ndarray(2) + + The coordinate is tested to be a free cell in the occupancy grid. + + :seealso: :meth:`isoccupied` :meth:`occgrid` + """ + if p is not None: + p = getvector(p, self._ndims, dtype=dtype) + if self.isoccupied(p): + raise ValueError("Point is inside obstacle") + return p
+ +
[docs] def progress_start(self, n): + """ + Initialize a progress bar (superclass) + + :param n: Number of iterations in the operation + :type n: int + + Create a progress bar for an operation that has ``n`` steps, for + example:: + + planner.progress_start(100) + for i in range(100): + # ... + planner.progress_next() + planner.progress_end() + + .. warning: Requires that the ``progress`` package is installed. + + :seealso: :meth:`progress_next` :meth:`progress_end` + """ + if _progress: + self._bar = FillingCirclesBar( + self.__class__.__name__, max=n, suffix="%(percent).1f%% - %(eta)ds" + )
+ +
[docs] def progress_next(self): + """ + Increment a progress bar (superclass) + + Create a progress bar for an operation that has ``n`` steps, for + example:: + + planner.progress_start(100) + for i in range(100): + # ... + planner.progress_next() + planner.progress_end() + + .. warning: Requires that the ``progress`` package is installed. + + :seealso: :meth:`progress_start` :meth:`progress_end` + """ + if _progress: + self._bar.next()
+ +
[docs] def progress_end(self): + """ + Finalize a progress bar (superclass) + + Remove/cleanip a progress bar, for + example:: + + planner.progress_start(100) + for i in range(100): + # ... + planner.progress_next() + planner.progress_end() + + .. warning: Requires that the ``progress`` package is installed. + + :seealso: :meth:`progress_start` :meth:`progress_next` + """ + if _progress: + self._bar.finish()
+ +
[docs] def query( + self, start=None, goal=None, dtype=None, next=True, animate=False, movie=None + ): + r""" + Find a path from start to goal using planner (superclass) + + :param start: start position :math:`(x, y)` or configuration :math:`(x, y, \theta)`, defaults to value specified to constructor + :type start: array_like(), optional + :param goal: goal position :math:`(x, y)` or configuration :math:`(x, y, \theta)`, defaults to value specified to constructor + :type goal: array_like(), optional + :param dtype: data type for point coordinates, defaults to None + :type dtype: str, optional + :param next: invoke :meth:`next` method of class, defaults to True + :type next: bool, optional + :param animate: show the vehicle path, defaults to False + :type animate: bool, optional + :return: path from start to goal, one point :math:`(x, y)` or configuration :math:`(x, y, \theta)` per row + :rtype: ndarray(N,2) or ndarray(N,3) + + Find a path from ``start`` to ``goal`` using a previously computed plan. + + This is a generic method that works for any planner + (:math:`\mathbb{R}^2` or :math:`\SE{2}`) that can incrementally + determine the next point on the path. The method performs the following + steps: + + - Validate start and goal + - If ``animate``, visualize the environment using :meth:`plot` + - Iterate on the class's :meth:`next` method until the ``goal`` is + achieved, and if ``animate`` plot points. + + :seealso: :meth:`next` :meth:`plan` + """ + # make sure start and goal are set and valid + self.start = self.validate_endpoint(start, dtype=dtype) + self.goal = self.validate_endpoint(goal, dtype=dtype) + + # if movie is not None: + # animate = True + + if animate: + self.plot() + + # movie = MovieWriter(movie) + + robot = self._start + path = [robot] + + while next: + if animate: + plt.plot(robot[0], robot[1], "y.", 12) + plt.pause(0.05) + + # get next point on the path + robot = self.next(robot) + + # are we are done? + if robot is None: + path.append(self._goal) + return np.array(path).astype(int) + + path.append(robot)
+ +
[docs] def plot( + self, + path=None, + line=None, + line_r=None, + configspace=False, + unwrap=True, + direction=None, + background=True, + path_marker=None, + path_marker_reverse=None, + start_marker=None, + goal_marker=None, + start_vehicle=None, + goal_vehicle=None, + start=None, + goal=None, + ax=None, + block=None, + bgargs={}, + **unused, + ): + r""" + Plot vehicle path (superclass) + + :param path: path, defaults to None + :type path: (N, 2) or ndarray(N, 3) + :param direction: travel direction associated with each point on path, is either >0 or <0, defaults to None + :type direction: array_like(N), optional + :param line: line style for forward motion, default is striped yellow on black + :type line: sequence of dict of arguments for ``plot`` + :param line_r: line style for reverse motion, default is striped red on black + :type line_r: sequence of dict of arguments for ``plot`` + + :param configspace: plot the path in 3D configuration space, input must be 3xN. + Start and goal style will be given by ``qstart_marker`` and ``qgoal_marker``, defaults to False + :type configspace: bool, optional + :param unwrap: for configuration space plot unwrap :math:`\theta` so + there are no discontinuities at :math:`\pm \pi`, defaults to True + :type unwrap: bool, optional + :param background: plot occupancy grid if present, default True + :type background: bool, optional + :param start_marker: style for marking start point + :type start_marker: dict, optional + :param goal_marker: style for marking goal point + :type goal_marker: dict, optional + :param start_vehicle: style for vehicle animation object at start configuration + :type start_vehicle: dict + :param goal_vehicle: style for vehicle animation object at goal configuration + :type goal_vehicle: dict + :param start: start position :math:`(x, y)` or configuration :math:`(x, y, \theta)`, defaults to value previously set + :type start: array_like(2) or array_like(3), optional + :param goal: goal position :math:`(x, y)` or configuration :math:`(x, y, \theta)`, defaults to value previously set + :type goal: array_like(2) or array_like(3), optional + :param bgargs: arguments passed to :meth:`plot_bg`, defaults to None + :type bgargs: dict, optional + :param ax: axes to plot into + :type ax: matplotlib axes + :param block: block after displaying the plot + :type block: bool, optional + + Plots the start and goal location/pose if they are specified by + ``start`` or ``goal`` or were set by the object constructor or + ``plan`` or ``query`` method. + + If the ``start`` and ``goal`` have length 2, planning in + :math:`\mathbb{R}^2`, then markers are drawn using styles specified by + ``start_marker`` and ``end_marker`` which are dicts using Matplotlib + keywords, for example:: + + planner.plot(path, start=dict(marker='s', color='b')) + + If the ``start`` and ``goal`` have length 3, planning in :math:`\SE{2}`, + and ``configspace`` is False, then direction-indicating markers are used + to display start and goal configuration. These are also given as dicts + but have two items: ``'shape'`` which is the shape of the polygonal + marker and is either ``'triangle'`` or ``'car'``. The second item + ``'args'`` is passed to :func:`base.plot_poly` and Matplotlib and could + have values such as ``filled=True`` or ``color``. + + If ``configspace`` is False then a 3D plot is created and the start and + goal are indicated by Matplotlib markers specified by the dicts + ``start_marker`` and ``end_marker`` + + Default values are provided for all markers: + + - the start point is a circle + - the goal point is a star + - the start vehicle style is a ``VehiclePolygon(shape='car')`` as + an unfilled outline + - the goal vehicle style is a ``VehiclePolygon(shape='car')`` as + a transparent filled shape + + If ``background`` is True then the background of the plot is either or + both of: + + - the occupancy grid + - the distance field of the planner + + Additional arguments ``bgargs`` can be passed through to :meth:`plot_bg` + + If ``path`` is specified it has one column per point and either 2 or 3 rows: + + - 2 rows describes motion in the :math:`xy`-plane and a 2D plot is created + - 3 rows describes motion in the :math:`xy\theta`-configuration space. By + default only the :math:`xy`-plane is plotted unless ``configspace`` + is True in which case motion in :math:`xy\theta`-configuration space + is shown as a 3D plot. + + If the planner supports bi-directional motion then the ``direction`` + option gives the direction for each point on the path. + + Forward motion segments are drawn using style information from ``line`` + while reverse motion segments are drawn using style information from + ``line_r``. These are each a sequence of dicts of Matplotlib plot + options which can draw an arbitrary number of lines on top of each + other. The default:: + + line = ( + {color:'black', linewidth:4}, + {color:'yellow', linewidth:3, dashes:(5,5)} + ) + + will draw a blackline of width 4 with a dashed yellow line of width 3 + plotted on top, giving a line of alternating black and yellow dashes. + + :seealso: :meth:`plot_bg` :func:`base.plot_poly` + """ + # create default markers + # passed to Matplotlib plot() + if start_marker is None: + start_marker = { + "marker": "o", + "markeredgecolor": "k", + "markerfacecolor": "y", + "markersize": 10, + "zorder": 10, + "linestyle": "none", + } + if goal_marker is None: + goal_marker = { + "marker": "*", + "markeredgecolor": "k", + "markerfacecolor": "y", + "markersize": 16, + "zorder": 10, + "linestyle": "none", + } + + # create defaut line styles + if line is None: + line = ( + {"color": "black", "linewidth": 4}, + {"color": "yellow", "linewidth": 3, "dashes": (5, 5)}, + ) + if line_r is None: + line_r = ( + {"color": "black", "linewidth": 4}, + {"color": "red", "linewidth": 3, "dashes": (5, 5)}, + ) + + # passed to VehiclePolygon + if start_vehicle is None: + start_vehicle = {"facecolor": "none", "edgecolor": "k", "linewidth": 2} + + if goal_vehicle is None: + goal_vehicle = {"alpha": 0.5} + + ndims = self._ndims + + if ndims == 3 and not configspace: + ndims = 2 + if path is not None: + path = path[:, :2] + + if configspace and ndims < 3 and path is not None: + raise ValueError(f"path should have {ndims} rows") + + ax = axes_logic(ax, ndims) + + # plot occupancy grid background + if background: + self.plot_bg(ax=ax, **bgargs) + + # mark the path + if path is not None: + if ndims == 2: + # 2D case + if direction is not None: + # bidirectional motion + direction = np.array(direction) + if direction.shape[0] != path.shape[0]: + raise ValueError( + "direction vector must have same length as path" + ) + + while len(direction) > 0: + dir = direction[0] + change = np.argwhere(dir != direction) + if len(change) == 0: + k = -1 + else: + k = change[0, 0] + + for style in line if dir > 0 else line_r: + ax.plot(path[:k, 0], path[:k, 1], zorder=9, **style) + + if len(change) == 0: + break + direction = direction[k - 1 :] + direction[0] = direction[1] + path = path[k - 1 :, :] + + else: + # forward motion only + for style in line: + ax.plot(path[:, 0], path[:, 1], zorder=9, **style) + + elif ndims == 3: + # 3D case + if direction is not None: + # bidirectional motion + + direction = np.array(direction) + if direction.shape[0] != path.shape[0]: + raise ValueError( + "direction vector must have same length as path" + ) + theta = path[:, 2] + if unwrap: + theta = np.unwrap(theta) + + while len(direction) > 0: + dir = direction[0] + change = np.argwhere(dir != direction) + if len(change) == 0: + k = -1 + else: + k = change[0, 0] + + for style in line if dir > 0 else line_r: + ax.plot(path[:k, 0], path[:k, 1], theta[:k], **style) + + if len(change) == 0: + break + direction = direction[k - 1 :] + direction[0] = direction[1] + path = path[k - 1 :, :] + theta = theta[k - 1 :] + + else: + # forward motion only + theta = path[:, 2] + if unwrap: + theta = np.unwrap(theta) + for style in line: + ax.plot(path[:, 0], path[:, 1], theta, **style) + + # mark start and goal if requested + if start is not None: + start = self.validate_endpoint(start) + else: + start = self.start + if goal is not None: + self.goal = self.validate_endpoint(goal) + else: + goal = self.goal + + if ndims == 2 and self._ndims == 2: + # proper 2d plot + if start is not None: + ax.plot(start[0], start[1], label="start", **start_marker) + if goal is not None: + ax.plot(goal[0], goal[1], label="goal", **goal_marker) + + elif ndims == 2 and self._ndims == 3: + # 2d projection of 3d plot, show start/goal configuration + scale = axes_get_scale(ax) / 10 + + if self.marker is None: + self.marker = VehiclePolygon(shape="car", scale=scale) + + if start is not None: + self.marker.plot(start, **start_vehicle) + if goal is not None: + self.marker.plot(goal, **goal_vehicle) + + elif ndims == 3: + # 3d plot + + if start is not None: + ax.plot(start[0], start[1], start[2], label="start", **start_marker) + if goal is not None: + + if path is not None and unwrap: + theta = theta[-1] + else: + theta = goal[2] + plt.plot(goal[0], goal[1], theta, label="goal", **goal_marker) + + ax.set_xlabel("x") + ax.set_ylabel("y") + if ndims == 2: + ax.set_aspect("equal") + else: + ax.set_zlabel(r"$\theta$") + + if block: + plt.show(block=block) + + return ax
+ + def _qmarker(self, shape): + h = 0.3 + t = 0.8 # start of head taper + c = 0.5 # centre x coordinate + w = 1 # width in x direction + if shape == "car": + return np.array( + [ + [-c, h], + [t - c, h], + [w - c, 0], + [t - c, -h], + [-c, -h], + ] + ).T + elif shape == "triangle": + return np.array( + [ + [-c, h], + [w, 0], + [-c, -h], + [-c, h], + ] + ).T + +
[docs] def plot_bg( + self, + distance=None, + cmap="gray", + ax=None, + inflated=True, + colorbar=True, + block=None, + **unused, + ): + """ + Plot background (superclass) + + :param distance: override distance field, defaults to None + :type distance: ndarray(N,M), optional + :param cmap: Specify a colormap for the distance field, defaults to 'gray' + :type cmap: str or Colormap, optional + + Displays the background which is either the occupancy grid or a distance + field. The distance field encodes the distance of a point from the goal, small + distance is dark, a large distance is bright. + + If the planner has an occupancy grid then that will be displayed with: + - free cells in white + - occupied cells in red + - inflated occupied cells in pink + + If distance is provided, or the planner has a distancemap attribute + the the distance field will be used as the background and obstacle cells + (actual or inflated) will be shown in red. A colorbar is added. + """ + if self._occgrid is None: + return + + if isinstance(self._occgrid, BaseOccupancyGrid): + ax = plotvol2(dim=self._occgrid.workspace, ax=ax) + else: + ax = axes_logic(ax, 2) + + # create color map for free space + obstacle: + # free space, color index = 1, white, alpha=0 to allow background and grid lines to show + # obstacle, color index = 2, red, alpha=1 + + if self._inflate > 0 and inflated: + # 0 background (white, transparent) + # 1 inflated obstacle (pink) + # 2 original obstacle (red) + colors = [(1, 1, 1, 0), (1, 0.75, 0.8, 1), (1, 0, 0, 1)] + image = self.occgrid.grid.astype(int) + self._occgrid0.grid.astype(int) + else: + # 0 background + # 1 obstacle + colors = [(1, 1, 1, 0), (1, 0, 0, 1)] + image = self.occgrid.grid + + if distance is None and hasattr(self, "distancemap"): + distance = self.distancemap + + if distance is not None: + # distance field with obstacles + + # find largest finite value + + v = distance.ravel() + vmax = max(v[np.isfinite(v)]) + + # create a copy of greyscale color map + c_map = copy.copy(mpl.cm.get_cmap(cmap)) + # c_map.set_bad(color=(1,0,0,1)) # nan and inf are red + + # change all inf to large value, so they are not 'bad' ie. red + distance[np.isinf(distance)] = 2 * vmax + c_map.set_over(color=(0, 0, 1)) # ex-infs are now blue + + # display image + norm = mpl.colors.Normalize(vmin=0, vmax=vmax, clip=False) + ax.imshow( + distance, + origin="lower", + interpolation=None, + cmap=c_map, + norm=norm, + ) + ax.grid(True, alpha=0.1, color=(1, 1, 1)) + + # add colorbar + scalar_mappable_c_map = cm.ScalarMappable(cmap=c_map, norm=norm) + if colorbar is True: + plt.colorbar( + scalar_mappable_c_map, + # shrink=0.75, + # aspect=20 * 0.75, + label="Distance", + ) + + elif isinstance(colorbar, dict): + if "label" not in colorbar: + colorbar["label"] = "Distance" + plt.colorbar(scalar_mappable_c_map, **colorbar) + # overlay obstacles + c_map = mpl.colors.ListedColormap(colors) + self.occgrid.plot(image, cmap=c_map, zorder=1) + + else: + # occupancy grid only + + # overlay obstacles + c_map = mpl.colors.ListedColormap(colors) + # self.occgrid.plot(image, cmap=c_map, zorder=1) + self.occgrid.plot(cmap=c_map, zorder=1, ax=ax) + + ax.set_facecolor((1, 1, 1)) # create white background + ax.set_xlabel("x (cells)") + ax.set_ylabel("y (cells)") + ax.grid(True, zorder=0) + + # lock axis limits to current value + # ax.set_xlim(ax.get_xlim()) + # ax.set_ylim(ax.get_ylim()) + + # plt.draw() + if block is not None: + plt.show(block=block)
+ +
[docs] def message(self, s, color=None): + """ + Print message to message channel + + :param s: message to print + :type s: str + :param color: color to print it, defaults to color specified at + constructor time. + :type color: str, optional + + """ + if self.verbose: + if color is None: + color = self._msgcolor + print(fg(color), "Planner:: " + s, attr(0))
+ + # @staticmethod + # def show_distance(d): + # d[np.isinf(d)] = None + # ax = plt.gca() + # c_map = plt.get_cmap("Greys") + # plt.clim(0, np.max(d[:])) + # plt.figimage(d) + # plt.xlabel('X') + # plt.ylabel('Y') + # plt.show() + + +class MovieWriter: + def __init__(self, filename=None, interval=0.1, fig=None): + """ + Save animation as a movie file + + :param filename: name of movie file, or tuple containing filename and + frame interval + :type filename: str or tuple(str, float) + :param interval: frame interval, defaults to 0.1 + :type interval: float, optional + :param fig: figure to record for the movie + :type fig: figure reference + + Example:: + + movie = MovieWriter(filename) + + while ... + movie.add() + + movie.done() + + To avoid extra user-logic, if ``MovieWriter`` is called with ``filename`` equal to None, + then the writer will do nothing when the ``add`` and ``done`` methods are called. + """ + # Set up formatting for the movie files + if filename is None: + self.writer = None + return + + if isinstance(filename, (tuple, list)): + filename, interval = filename + + if os.path.exists(filename): + print("overwriting movie", filename) + else: + print("creating movie", filename) + self.writer = animation.FFMpegWriter( + fps=round(1 / interval), extra_args=["-vcodec", "libx264"] + ) + if fig is None: + fig = plt.gcf() + self.writer.setup(fig, filename) + self.filename = filename + + def add(self): + """ + Add frame to the movie + """ + if self.writer is not None: + self.writer.grab_frame() + + def done(self): + if self.writer is not None: + self.writer.finish() + self.writer = None +
+ +
+
+ +
+
+
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/mobile/QuinticPolyPlanner.html b/_modules/roboticstoolbox/mobile/QuinticPolyPlanner.html new file mode 100644 index 00000000..958dc963 --- /dev/null +++ b/_modules/roboticstoolbox/mobile/QuinticPolyPlanner.html @@ -0,0 +1,473 @@ + + + + + + roboticstoolbox.mobile.QuinticPolyPlanner — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+
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+ +

Source code for roboticstoolbox.mobile.QuinticPolyPlanner

+import math
+from collections import namedtuple
+
+import matplotlib.pyplot as plt
+import numpy as np
+from roboticstoolbox.mobile.PlannerBase import PlannerBase
+
+# parameter
+
+
+show_animation = True
+
+# ======================================================================== #
+
+# The following code is modified from Python Robotics
+# https://github.com/AtsushiSakai/PythonRobotics/tree/master/PathPlanning
+# Quintic Polynomial Planner
+# Author: Atsushi Sakai
+# Copyright (c) 2016 - 2022 Atsushi Sakai and other contributors: https://github.com/AtsushiSakai/PythonRobotics/contributors
+# Released under the MIT license: https://github.com/AtsushiSakai/PythonRobotics/blob/master/LICENSE
+
+
+class _QuinticPolynomial:
+    def __init__(self, xs, vxs, axs, xe, vxe, axe, time):
+        # calc coefficient of quintic polynomial
+        # See jupyter notebook document for derivation of this equation.
+        self.a0 = xs
+        self.a1 = vxs
+        self.a2 = axs / 2.0
+
+        A = np.array(
+            [
+                [time**3, time**4, time**5],
+                [3 * time**2, 4 * time**3, 5 * time**4],
+                [6 * time, 12 * time**2, 20 * time**3],
+            ]
+        )
+        b = np.array(
+            [
+                xe - self.a0 - self.a1 * time - self.a2 * time**2,
+                vxe - self.a1 - 2 * self.a2 * time,
+                axe - 2 * self.a2,
+            ]
+        )
+        x = np.linalg.solve(A, b)
+
+        self.a3 = x[0]
+        self.a4 = x[1]
+        self.a5 = x[2]
+
+    def calc_point(self, t):
+        xt = (
+            self.a0
+            + self.a1 * t
+            + self.a2 * t**2
+            + self.a3 * t**3
+            + self.a4 * t**4
+            + self.a5 * t**5
+        )
+
+        return xt
+
+    def calc_first_derivative(self, t):
+        xt = (
+            self.a1
+            + 2 * self.a2 * t
+            + 3 * self.a3 * t**2
+            + 4 * self.a4 * t**3
+            + 5 * self.a5 * t**4
+        )
+
+        return xt
+
+    def calc_second_derivative(self, t):
+        xt = (
+            2 * self.a2
+            + 6 * self.a3 * t
+            + 12 * self.a4 * t**2
+            + 20 * self.a5 * t**3
+        )
+
+        return xt
+
+    def calc_third_derivative(self, t):
+        xt = 6 * self.a3 + 24 * self.a4 * t + 60 * self.a5 * t**2
+
+        return xt
+
+
+def quintic_polynomials_planner(
+    sx, sy, syaw, sv, sa, gx, gy, gyaw, gv, ga, max_accel, max_jerk, dt, MIN_T, MAX_T
+):
+    """
+    quintic polynomial planner
+
+    input
+        s_x: start x position [m]
+        s_y: start y position [m]
+        s_yaw: start yaw angle [rad]
+        sa: start accel [m/ss]
+        gx: goal x position [m]
+        gy: goal y position [m]
+        gyaw: goal yaw angle [rad]
+        ga: goal accel [m/ss]
+        max_accel: maximum accel [m/ss]
+        max_jerk: maximum jerk [m/sss]
+        dt: time tick [s]
+
+    return
+        time: time result
+        rx: x position result list
+        ry: y position result list
+        ryaw: yaw angle result list
+        rv: velocity result list
+        ra: accel result list
+
+    """
+
+    vxs = sv * math.cos(syaw)
+    vys = sv * math.sin(syaw)
+    vxg = gv * math.cos(gyaw)
+    vyg = gv * math.sin(gyaw)
+
+    axs = sa * math.cos(syaw)
+    ays = sa * math.sin(syaw)
+    axg = ga * math.cos(gyaw)
+    ayg = ga * math.sin(gyaw)
+
+    time, rx, ry, ryaw, rv, ra, rj = [], [], [], [], [], [], []
+
+    for T in np.arange(MIN_T, MAX_T, MIN_T):
+        xqp = _QuinticPolynomial(sx, vxs, axs, gx, vxg, axg, T)
+        yqp = _QuinticPolynomial(sy, vys, ays, gy, vyg, ayg, T)
+
+        time, rx, ry, ryaw, rv, ra, rj = [], [], [], [], [], [], []
+
+        for t in np.arange(0.0, T + dt, dt):
+            time.append(t)
+            rx.append(xqp.calc_point(t))
+            ry.append(yqp.calc_point(t))
+
+            vx = xqp.calc_first_derivative(t)
+            vy = yqp.calc_first_derivative(t)
+            v = np.hypot(vx, vy)
+            yaw = math.atan2(vy, vx)
+            rv.append(v)
+            ryaw.append(yaw)
+
+            ax = xqp.calc_second_derivative(t)
+            ay = yqp.calc_second_derivative(t)
+            a = np.hypot(ax, ay)
+            if len(rv) >= 2 and rv[-1] - rv[-2] < 0.0:
+                a *= -1
+            ra.append(a)
+
+            jx = xqp.calc_third_derivative(t)
+            jy = yqp.calc_third_derivative(t)
+            j = np.hypot(jx, jy)
+            if len(ra) >= 2 and ra[-1] - ra[-2] < 0.0:
+                j *= -1
+            rj.append(j)
+
+        if (
+            max([abs(i) for i in ra]) <= max_accel
+            and max([abs(i) for i in rj]) <= max_jerk
+        ):
+            print("find path!!")
+            break
+
+    return time, np.c_[rx, ry, ryaw], rv, ra, rj
+
+
+# ====================== RTB wrapper ============================= #
+
+# Copyright (c) 2022 Peter Corke: https://github.com/petercorke/robotics-toolbox-python
+# Released under the MIT license: https://github.com/AtsushiSakai/PythonRobotics/blob/master/LICENSE
+
[docs]class QuinticPolyPlanner(PlannerBase): + r""" + Quintic polynomial path planner + + :param dt: time step, defaults to 0.1 + :type dt: float, optional + :param start_vel: initial velocity, defaults to 0 + :type start_vel: float, optional + :param start_acc: initial acceleration, defaults to 0 + :type start_acc: float, optional + :param goal_vel: goal velocity, defaults to 0 + :type goal_vel: float, optional + :param goal_acc: goal acceleration, defaults to 0 + :type goal_acc: float, optional + :param max_acc: maximum acceleration, defaults to 1 + :type max_acc: int, optional + :param max_jerk: maximum jerk, defaults to 0.5 + :type min_t: float, optional + :param min_t: minimum path time, defaults to 5 + :type max_t: float, optional + :param max_t: maximum path time, defaults to 100 + :type max_jerk: float, optional + :return: Quintic polynomial path planner + :rtype: QuinticPolyPlanner instance + + ================== ======================== + Feature Capability + ================== ======================== + Plan :math:`\SE{2}` + Obstacle avoidance No + Curvature Continuous + Motion Forwards only + ================== ======================== + + Creates a planner that finds the path between two configurations in the + plane using forward motion only. The path is a continuous quintic polynomial + for x and y + + .. math:: + + x(t) &= a_0 + a_1 t + a_2 t^2 + a_3 t^3 + a_4 t^4 + a_5 t^5 \\ + y(t) &= b_0 + b_1 t + b_2 t^2 + b_3 t^3 + b_4 t^4 + b_5 t^5 + + that meets the given constraints. Trajectory time is given as a range. + + :reference: "Local Path Planning And Motion Control For AGV In + Positioning", Takahashi, T. Hongo, Y. Ninomiya and G. + Sugimoto; Proceedings. IEEE/RSJ International Workshop on + Intelligent Robots and Systems (IROS '89) doi: 10.1109/IROS.1989.637936 + + .. note:: The path time is searched in the interval [``min_t``, ``max_t``] in steps + of ``min_t``. + + Example: + + .. runblock:: pycon + + >>> from roboticstoolbox import QuinticPolyPlanner + >>> import numpy as np + >>> start = (10, 10, np.deg2rad(10.0)) + >>> goal = (30, -10, np.deg2rad(20.0)) + >>> quintic = QuinticPolyPlanner(start_vel=1) + >>> path, status = quintic.query(start, goal) + >>> print(path[:5,:]) + + :thanks: based on quintic polynomial planning from `Python Robotics <https://github.com/AtsushiSakai/PythonRobotics/tree/master/PathPlanning>`_ + + + :seealso: :class:`Planner` + """ + + def __init__( + self, + dt=0.1, + start_vel=0, + start_acc=0, + goal_vel=0, + goal_acc=0, + max_acc=1, + max_jerk=0.5, + min_t=5, + max_t=100, + ): + + super().__init__(ndims=3) + self.dt = dt + self.start_vel = start_vel + self.start_acc = start_acc + self.goal_vel = goal_vel + self.goal_acc = goal_acc + self.max_acc = max_acc + self.max_jerk = max_jerk + self.min_t = min_t + self.max_t = max_t + +
[docs] def query(self, start, goal): + r""" + Find a quintic polynomial path + + :param start: start configuration :math:`(x, y, \theta)` + :type start: array_like(3), optional + :param goal: goal configuration :math:`(x, y, \theta)` + :type goal: array_like(3), optional + :return: path and status + :rtype: ndarray(N,3), namedtuple + + The returned status value has elements: + + ========== =================================================== + Element Description + ========== =================================================== + ``t`` time to execute the path + ``vel`` velocity profile along the path + ``accel`` acceleration profile along the path + ``jerk`` jerk profile along the path + ========== =================================================== + + :seealso: :meth:`Planner.query` + """ + self._start = start + self._goal = goal + + time, path, v, a, j = quintic_polynomials_planner( + start[0], + start[1], + start[2], + self.start_vel, + self.start_acc, + goal[0], + goal[1], + goal[2], + self.start_vel, + self.start_acc, + self.max_acc, + self.max_jerk, + dt=self.dt, + MIN_T=self.min_t, + MAX_T=self.max_t, + ) + + status = namedtuple("QuinticPolyStatus", ["t", "vel", "acc", "jerk"])( + time, v, a, j + ) + + return path, status
+ + +if __name__ == "__main__": + from math import pi + + start = (10, 10, np.deg2rad(10.0)) + goal = (30, -10, np.deg2rad(20.0)) + + quintic = QuinticPolyPlanner(start_vel=1) + path, status = quintic.query(start, goal) + + print(status) + quintic.plot(path) + + plt.figure() + plt.subplot(311) + plt.plot(status.t, status.vel, "-r") + plt.ylabel("velocity (m/s)") + plt.grid(True) + + plt.subplot(312) + plt.plot(status.t, status.acc, "-r") + plt.ylabel("accel (m/s2)") + plt.grid(True) + + plt.subplot(313) + plt.plot(status.t, status.jerk, "-r") + plt.xlabel("Time (s)") + plt.ylabel("jerk (m/s3)") + plt.grid(True) + + plt.show(block=True) +
+ +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/mobile/RRTPlanner.html b/_modules/roboticstoolbox/mobile/RRTPlanner.html new file mode 100644 index 00000000..b1f56fff --- /dev/null +++ b/_modules/roboticstoolbox/mobile/RRTPlanner.html @@ -0,0 +1,469 @@ + + + + + + roboticstoolbox.mobile.RRTPlanner — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+
    +
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+
+
+
+
+ +

Source code for roboticstoolbox.mobile.RRTPlanner

+# ======================================================================== #
+
+# The following code is based on code from Python Robotics
+# https://github.com/AtsushiSakai/PythonRobotics/tree/master/PathPlanning
+# RRTDubins planning
+# Author: Atsushi Sakai
+# Copyright (c) 2016 - 2022 Atsushi Sakai and other contributors: https://github.com/AtsushiSakai/PythonRobotics/contributors
+# Released under the MIT license: https://github.com/AtsushiSakai/PythonRobotics/blob/master/LICENSE
+
+import math
+from collections import namedtuple
+from roboticstoolbox.mobile.OccGrid import PolygonMap
+
+# import rvcprint
+from roboticstoolbox import *
+import numpy as np
+import matplotlib.pyplot as plt
+
+from spatialmath import Polygon2, SE2, base
+from roboticstoolbox.mobile.PlannerBase import PlannerBase
+from roboticstoolbox.mobile.DubinsPlanner import DubinsPlanner
+
+# from roboticstoolbox.mobile.OccupancyGrid import OccupancyGrid
+from pgraph import DGraph
+
+
+
[docs]class RRTPlanner(PlannerBase): + r""" + Rapidly exploring tree planner + + :param map: occupancy grid + :type map: :class:`PolygonMap` + :param vehicle: vehicle kinematic model + :type vehicle: :class:`VehicleBase` subclass + :param curvature: maximum path curvature, defaults to 1.0 + :type curvature: float, optional + :param stepsize: spacing between points on the path, defaults to 0.2 + :type stepsize: float, optional + :param npoints: number of vertices in random tree, defaults to 50 + :type npoints: int, optional + + ================== ======================== + Feature Capability + ================== ======================== + Plan :math:`\SE{2}` + Obstacle avoidance Yes, polygons + Curvature Discontinuous + Motion Bidirectional + ================== ======================== + + Creates a planner that finds the obstacle-free path between two + configurations in the plane using forward and backward motion. The path + comprises multiple Dubins curves comprising straight lines, or arcs with + curvature of :math:`\pm` ``curvature``. Motion along the segments may be in + the forward or backward direction. + + Polygons are used for obstacle avoidance: + + - the environment is defined by a set of polygons represented by a :class:`PolygonMap` + - the vehicle is defined by a single polygon specified by the ``polygon`` + argument to its constructor + + Example:: + + from roboticstoolbox import RRTPlanner + from spatialmath import Polygon2 + from math import pi + + # create polygonal obstacles + map = PolygonMap(workspace=[0, 10]) + map.add([(5, 50), (5, 6), (6, 6), (6, 50)]) + map.add([(5, 4), (5, -50), (6, -50), (6, 4)]) + + # create outline polygon for vehicle + l, w = 3, 1.5 + vpolygon = Polygon2([(-l/2, w/2), (-l/2, -w/2), (l/2, -w/2), (l/2, w/2)]) + + # create vehicle model + vehicle = Bicycle(steer_max=1, L=2, polygon=vpolygon) + + # create planner + rrt = RRTPlanner(map=map, vehicle=vehicle, npoints=50, seed=0) + # start and goal configuration + qs = (2, 8, -pi/2) + qg = (8, 2, -pi/2) + + # plan path + rrt.plan(goal=qg) + path, status = rrt.query(start=qs) + print(path[:5,:]) + print(status) + + + :seealso: :class:`DubinsPlanner` :class:`Vehicle` :class:`PlannerBase` + """ + + def __init__( + self, + map, + vehicle, + curvature=1.0, + stepsize=0.2, + npoints=50, + **kwargs, + ): + super().__init__(ndims=2, **kwargs) + + self.npoints = npoints + self.map = map + + self.g = DGraph(metric="SE2") + + self.vehicle = vehicle + if curvature is None: + if vehicle is not None: + curvature = vehicle.curvature_max + else: + curvature = 1 + + print("curvature", curvature) + self.dubins = DubinsPlanner(curvature=curvature, stepsize=stepsize) + + # self.goal_yaw_th = np.deg2rad(1.0) + # self.goal_xy_th = 0.5 + +
[docs] def plan(self, goal, showsamples=True, showvalid=True, animate=False): + r""" + Plan paths to goal using RRT + + :param goal: goal pose :math:`(x, y, \theta)`, defaults to previously set value + :type goal: array_like(3), optional + :param showsamples: display position part of configurations overlaid on the map, defaults to True + :type showsamples: bool, optional + :param showvalid: display valid configurations as vehicle polygons overlaid on the map, defaults to False + :type showvalid: bool, optional + :param animate: update the display as configurations are tested, defaults to False + :type animate: bool, optional + + Compute a rapidly exploring random tree with its root at the ``goal``. + The tree will have ``npoints`` vertices spread uniformly randomly over + the workspace which is an attribute of the ``map``. + + For every new point added, a Dubins path is computed to the nearest + vertex already in the graph. Each configuration on that path, with + spacing of ``stepsize``, is tested for obstacle intersection. + + The configurations tested are displayed (translation only) if ``showsamples`` is + True. The valid configurations are displayed as vehicle polygones if ``showvalid`` + is True. If ``animate`` is True these points are displayed during the search + process, otherwise a single figure is displayed at the end. + + :seealso: :meth:`query` + """ + # TODO use validate + self.goal = np.r_[goal] + # self.goal = np.r_[goal] + self.random_init() + + v = self.g.add_vertex(coord=goal) + v.path = None + + if showsamples or showvalid: + self.map.plot() + + self.progress_start(self.npoints) + count = 0 + while count < self.npoints: + random_point = self.qrandom_free() + + if showsamples: + plt.plot(random_point[0], random_point[1], "ok", markersize=2) + if animate: + plt.pause(0.02) + + vnearest, d = self.g.closest(random_point) + + if d > 6: + continue + path, pstatus = self.dubins.query(random_point, vnearest.coord) + if path is None: + continue + + collision = False + for config in path: + if self.map.iscollision(self.vehicle.polygon(config)): + collision = True + break + if collision: + # print('collision') + continue + + if pstatus.length > 6: + # print('too long') + continue + + # we have a valid configuration to add to the graph + count += 1 + self.progress_next() + + # add new vertex to graph + vnew = self.g.add_vertex(random_point) + self.g.add_edge(vnew, vnearest, cost=pstatus.length) + vnew.path = path + + if showvalid: + self.vehicle.polygon(random_point).plot(color="b", alpha=0.1) + if animate: + plt.pause(0.02) + + if (showvalid or showsamples) and not animate: + plt.show(block=False) + + self.progress_end()
+ +
[docs] def query(self, start): + r""" + Find a path from start configuration + + :param start: start configuration :math:`(x, y, \theta)` + :type start: array_like(3), optional + :return: path and status + :rtype: ndarray(N,3), namedtuple + + The path comprises points equally spaced at a distance of ``stepsize``. + + The returned status value has elements: + + +---------------+---------------------------------------------------+ + | Element | Description | + +---------------+---------------------------------------------------+ + | ``length`` | total path length | + +-------------+-----------------------------------------------------+ + | ``initial_d`` | distance from start to first vertex in graph | + +---------------+---------------------------------------------------+ + | ``vertices`` | sequence of vertices in the graph | + +---------------+---------------------------------------------------+ + + """ + self._start = start + vstart, d = self.g.closest(start) + + vpath, cost, _ = self.g.path_UCS(vstart, self.g[0]) + + print(vpath) + # stack the Dubins path segments + path = np.empty((0, 3)) + for vertex in vpath: + if vertex.path is not None: + path = np.vstack((path, vertex.path)) + + status = namedtuple("RRTStatus", ["length", "initial_d", "vertices"])( + cost, d, vpath + ) + + return path, status
+ + # def _generate_final_course(self, goal_ind): + # path = [[self.end.x, self.end.y]] + # node = self.node_list[goal_ind] + # while node.parent is not None: + # path.append([node.x, node.y]) + # node = node.parent + # path.append([node.x, node.y]) + + # return path + + # def _calc_dist_to_goal(self, x, y): + + # dx = x - self.goal.x + # dy = y - self.end.y + # return math.hypot(dx, dy) + +
[docs] def qrandom(self): + r""" + Random configuration + + :return: random configuration :math:`(x, y, \theta)` + :rtype: ndarray(3) + + Returns a random configuration where position :math:`(x, y)` + lies within the bounds of the ``map`` associated with this planner. + + :seealso: :meth:`qrandom_free` + """ + return self.random.uniform( + low=(self.map.workspace[0], self.map.workspace[2], -np.pi), + high=(self.map.workspace[1], self.map.workspace[3], np.pi), + )
+ +
[docs] def qrandom_free(self): + r""" + Random obstacle free configuration + + :return: random configuration :math:`(x, y, \theta)` + :rtype: ndarray(3) + + Returns a random obstacle free configuration where position :math:`(x, + y)` lies within the bounds of the ``map`` associated with this planner. + Iterates on :meth:`qrandom` + + :seealso: :meth:`qrandom` :meth:`iscollision` + """ + # iterate for a random freespace configuration + while True: + q = self.qrandom() + if not self.iscollision(q): + return q
+ +
[docs] def iscollision(self, q): + r""" + Test if configuration is collision + + :param q: vehicle configuration :math:`(x, y, \theta)` + :type q: array_like(3) + :return: collision status + :rtype: bool + + Transforms the vehicle polygon and tests for intersection against + the polygonal obstacle map. + """ + return self.map.iscollision(self.vehicle.polygon(q))
+ + +if __name__ == "__main__": + from roboticstoolbox.mobile.Vehicle import Bicycle + + # start and goal configuration + qs = (2, 8, -np.pi / 2) + qg = (8, 2, -np.pi / 2) + + # obstacle map + map = PolygonMap(workspace=[0, 10]) + map.add([(5, 50), (5, 6), (6, 6), (6, 50)]) + # map.add([(5, 0), (6, 0), (6, 4), (5, 4)]) + map.add([(5, 4), (5, -50), (6, -50), (6, 4)]) + + l = 3 + w = 1.5 + v0 = Polygon2([(-l / 2, w / 2), (-l / 2, -w / 2), (l / 2, -w / 2), (l / 2, w / 2)]) + + vehicle = Bicycle(steer_max=0.4, l=2, polygon=v0) + + rrt = RRTPlanner(map=map, vehicle=vehicle, verbose=False, seed=0) + + rrt.plan(goal=qg) + path, status = rrt.query(start=qs) + rrt.plot(path) + + plt.show(block=True) +
+ +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/mobile/ReedsSheppPlanner.html b/_modules/roboticstoolbox/mobile/ReedsSheppPlanner.html new file mode 100644 index 00000000..1f14edb7 --- /dev/null +++ b/_modules/roboticstoolbox/mobile/ReedsSheppPlanner.html @@ -0,0 +1,679 @@ + + + + + + roboticstoolbox.mobile.ReedsSheppPlanner — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
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+
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+ +

Source code for roboticstoolbox.mobile.ReedsSheppPlanner

+import math
+from collections import namedtuple
+from roboticstoolbox.mobile.PlannerBase import PlannerBase
+import matplotlib.pyplot as plt
+import numpy as np
+from spatialmath import *
+from spatialmath import base
+
+# ======================================================================== #
+
+# The following code is modified from Python Robotics
+# https://github.com/AtsushiSakai/PythonRobotics/tree/master/PathPlanning
+# D* grid planning
+# Author: Atsushi Sakai
+# Copyright (c) 2016 - 2022 Atsushi Sakai and other contributors: https://github.com/AtsushiSakai/PythonRobotics/contributors
+# Released under the MIT license: https://github.com/AtsushiSakai/PythonRobotics/blob/master/LICENSE
+
+
+class _Path:
+    def __init__(self):
+        self.lengths = []
+        self.ctypes = []
+        self.L = 0.0
+        self.x = []
+        self.y = []
+        self.yaw = []
+        self.directions = []
+
+    def __repr__(self):
+        s = f"_Path: L={self.L:.2g}, "
+        s += f"[{', '.join(['{}={:.2g}'.format(c,l) for c, l in zip(self.ctypes, self.lengths)])}]"
+        if len(self.x) > 0:
+            s += f", {len(self.x)} points on path"
+        return s
+
+
+def plot_arrow(x, y, yaw, length=1.0, width=0.5, fc="r", ec="k"):
+    """
+    Plot arrow
+    """
+
+    if not isinstance(x, float):
+        for (ix, iy, iyaw) in zip(x, y, yaw):
+            plot_arrow(ix, iy, iyaw)
+    else:
+        plt.arrow(
+            x,
+            y,
+            length * math.cos(yaw),
+            length * math.sin(yaw),
+            fc=fc,
+            ec=ec,
+            head_width=width,
+            head_length=width,
+        )
+        plt.plot(x, y)
+
+
+def straight_left_straight(x, y, phi):
+    phi = base.wrap_0_2pi(phi)
+    if y > 0.0 and 0.0 < phi < math.pi * 0.99:
+        xd = -y / math.tan(phi) + x
+        t = xd - math.tan(phi / 2.0)
+        u = phi
+        v = math.sqrt((x - xd) ** 2 + y**2) - math.tan(phi / 2.0)
+        return True, t, u, v
+    elif y < 0.0 < phi < math.pi * 0.99:
+        xd = -y / math.tan(phi) + x
+        t = xd - math.tan(phi / 2.0)
+        u = phi
+        v = -math.sqrt((x - xd) ** 2 + y**2) - math.tan(phi / 2.0)
+        return True, t, u, v
+
+    return False, 0.0, 0.0, 0.0
+
+
+def set_path(paths, lengths, ctypes):
+    path = _Path()
+    path.ctypes = ctypes
+    path.lengths = lengths
+
+    # check same path exist
+    for tpath in paths:
+        typeissame = tpath.ctypes == path.ctypes
+        if typeissame:
+            if sum(np.abs(tpath.lengths)) - sum(np.abs(path.lengths)) <= 0.01:
+                return paths  # not insert path
+
+    path.L = sum([abs(i) for i in lengths])
+
+    # Base.Test.@test path.L >= 0.01
+    if path.L >= 0.01:
+        paths.append(path)
+
+    return paths
+
+
+def straight_curve_straight(x, y, phi, paths):
+    flag, t, u, v = straight_left_straight(x, y, phi)
+    if flag:
+        paths = set_path(paths, [t, u, v], ["S", "L", "S"])
+
+    flag, t, u, v = straight_left_straight(x, -y, -phi)
+    if flag:
+        paths = set_path(paths, [t, u, v], ["S", "R", "S"])
+
+    return paths
+
+
+def polar(x, y):
+    r = math.sqrt(x**2 + y**2)
+    theta = math.atan2(y, x)
+    return r, theta
+
+
+def left_straight_left(x, y, phi):
+    u, t = polar(x - math.sin(phi), y - 1.0 + math.cos(phi))
+    if t >= 0.0:
+        v = base.wrap_0_2pi(phi - t)
+        if v >= 0.0:
+            return True, t, u, v
+
+    return False, 0.0, 0.0, 0.0
+
+
+def left_right_left(x, y, phi):
+    u1, t1 = polar(x - math.sin(phi), y - 1.0 + math.cos(phi))
+
+    if u1 <= 4.0:
+        u = -2.0 * math.asin(0.25 * u1)
+        t = base.wrap_0_2pi(t1 + 0.5 * u + math.pi)
+        v = base.wrap_0_2pi(phi - t + u)
+
+        if t >= 0.0 >= u:
+            return True, t, u, v
+
+    return False, 0.0, 0.0, 0.0
+
+
+def curve_curve_curve(x, y, phi, paths):
+    flag, t, u, v = left_right_left(x, y, phi)
+    if flag:
+        paths = set_path(paths, [t, u, v], ["L", "R", "L"])
+
+    flag, t, u, v = left_right_left(-x, y, -phi)
+    if flag:
+        paths = set_path(paths, [-t, -u, -v], ["L", "R", "L"])
+
+    flag, t, u, v = left_right_left(x, -y, -phi)
+    if flag:
+        paths = set_path(paths, [t, u, v], ["R", "L", "R"])
+
+    flag, t, u, v = left_right_left(-x, -y, phi)
+    if flag:
+        paths = set_path(paths, [-t, -u, -v], ["R", "L", "R"])
+
+    # backwards
+    xb = x * math.cos(phi) + y * math.sin(phi)
+    yb = x * math.sin(phi) - y * math.cos(phi)
+
+    flag, t, u, v = left_right_left(xb, yb, phi)
+    if flag:
+        paths = set_path(paths, [v, u, t], ["L", "R", "L"])
+
+    flag, t, u, v = left_right_left(-xb, yb, -phi)
+    if flag:
+        paths = set_path(paths, [-v, -u, -t], ["L", "R", "L"])
+
+    flag, t, u, v = left_right_left(xb, -yb, -phi)
+    if flag:
+        paths = set_path(paths, [v, u, t], ["R", "L", "R"])
+
+    flag, t, u, v = left_right_left(-xb, -yb, phi)
+    if flag:
+        paths = set_path(paths, [-v, -u, -t], ["R", "L", "R"])
+
+    return paths
+
+
+def curve_straight_curve(x, y, phi, paths):
+    flag, t, u, v = left_straight_left(x, y, phi)
+    if flag:
+        paths = set_path(paths, [t, u, v], ["L", "S", "L"])
+
+    flag, t, u, v = left_straight_left(-x, y, -phi)
+    if flag:
+        paths = set_path(paths, [-t, -u, -v], ["L", "S", "L"])
+
+    flag, t, u, v = left_straight_left(x, -y, -phi)
+    if flag:
+        paths = set_path(paths, [t, u, v], ["R", "S", "R"])
+
+    flag, t, u, v = left_straight_left(-x, -y, phi)
+    if flag:
+        paths = set_path(paths, [-t, -u, -v], ["R", "S", "R"])
+
+    flag, t, u, v = left_straight_right(x, y, phi)
+    if flag:
+        paths = set_path(paths, [t, u, v], ["L", "S", "R"])
+
+    flag, t, u, v = left_straight_right(-x, y, -phi)
+    if flag:
+        paths = set_path(paths, [-t, -u, -v], ["L", "S", "R"])
+
+    flag, t, u, v = left_straight_right(x, -y, -phi)
+    if flag:
+        paths = set_path(paths, [t, u, v], ["R", "S", "L"])
+
+    flag, t, u, v = left_straight_right(-x, -y, phi)
+    if flag:
+        paths = set_path(paths, [-t, -u, -v], ["R", "S", "L"])
+
+    return paths
+
+
+def left_straight_right(x, y, phi):
+    u1, t1 = polar(x + math.sin(phi), y - 1.0 - math.cos(phi))
+    u1 = u1**2
+    if u1 >= 4.0:
+        u = math.sqrt(u1 - 4.0)
+        theta = math.atan2(2.0, u)
+        t = base.wrap_0_2pi(t1 + theta)
+        v = base.wrap_0_2pi(t - phi)
+
+        if t >= 0.0 and v >= 0.0:
+            return True, t, u, v
+
+    return False, 0.0, 0.0, 0.0
+
+
+def generate_path(q0, q1, max_curvature):
+    """
+    Return a set of feasible paths
+
+    :param q0: initial configuration
+    :type q0: array_like(3)
+    :param q1: final configuration
+    :type q1: array_like(3)
+    :param max_curvature: maximum path curvature
+    :type max_curvature: float
+    :return: set of paths
+    :rtype: list of _Path
+    """
+    dx = q1[0] - q0[0]
+    dy = q1[1] - q0[1]
+    dth = q1[2] - q0[2]
+    c = math.cos(q0[2])
+    s = math.sin(q0[2])
+    x = (c * dx + s * dy) * max_curvature
+    y = (-s * dx + c * dy) * max_curvature
+
+    # build a list of possible paths
+    paths = []
+    paths = straight_curve_straight(x, y, dth, paths)
+    paths = curve_straight_curve(x, y, dth, paths)
+    paths = curve_curve_curve(x, y, dth, paths)
+
+    return paths
+
+
+def interpolate(
+    ind,
+    length,
+    mode,
+    max_curvature,
+    origin_x,
+    origin_y,
+    origin_yaw,
+    path_x,
+    path_y,
+    path_yaw,
+    directions,
+):
+    if mode == "S":
+        path_x[ind] = origin_x + length / max_curvature * math.cos(origin_yaw)
+        path_y[ind] = origin_y + length / max_curvature * math.sin(origin_yaw)
+        path_yaw[ind] = origin_yaw
+    else:  # curve
+        ldx = math.sin(length) / max_curvature
+        ldy = 0.0
+        if mode == "L":  # left turn
+            ldy = (1.0 - math.cos(length)) / max_curvature
+        elif mode == "R":  # right turn
+            ldy = (1.0 - math.cos(length)) / -max_curvature
+        gdx = math.cos(-origin_yaw) * ldx + math.sin(-origin_yaw) * ldy
+        gdy = -math.sin(-origin_yaw) * ldx + math.cos(-origin_yaw) * ldy
+        path_x[ind] = origin_x + gdx
+        path_y[ind] = origin_y + gdy
+
+    if mode == "L":  # left turn
+        path_yaw[ind] = origin_yaw + length
+    elif mode == "R":  # right turn
+        path_yaw[ind] = origin_yaw - length
+
+    if length > 0.0:
+        directions[ind] = 1
+    else:
+        directions[ind] = -1
+
+    return path_x, path_y, path_yaw, directions
+
+
+def generate_local_course(total_length, lengths, mode, max_curvature, step_size):
+    n_point = math.trunc(total_length / step_size) + len(lengths) + 4
+
+    px = [0.0 for _ in range(n_point)]
+    py = [0.0 for _ in range(n_point)]
+    pyaw = [0.0 for _ in range(n_point)]
+    directions = [0.0 for _ in range(n_point)]
+    ind = 1
+
+    if lengths[0] > 0.0:
+        directions[0] = 1
+    else:
+        directions[0] = -1
+
+    ll = 0.0
+
+    for (m, l, i) in zip(mode, lengths, range(len(mode))):
+        if l > 0.0:
+            d = step_size
+        else:
+            d = -step_size
+
+        # set origin state
+        ox, oy, oyaw = px[ind], py[ind], pyaw[ind]
+
+        ind -= 1
+        if i >= 1 and (lengths[i - 1] * lengths[i]) > 0:
+            pd = -d - ll
+        else:
+            pd = d - ll
+
+        while abs(pd) <= abs(l):
+            ind += 1
+            px, py, pyaw, directions = interpolate(
+                ind, pd, m, max_curvature, ox, oy, oyaw, px, py, pyaw, directions
+            )
+            pd += d
+
+        ll = l - pd - d  # calc remain length
+
+        ind += 1
+        px, py, pyaw, directions = interpolate(
+            ind, l, m, max_curvature, ox, oy, oyaw, px, py, pyaw, directions
+        )
+
+    # remove unused data
+    while abs(px[-1]) < 1e-10:
+        px.pop()
+        py.pop()
+        pyaw.pop()
+        directions.pop()
+
+    return px, py, pyaw, directions
+
+
+def pi_2_pi(angle):
+    return (angle + math.pi) % (2 * math.pi) - math.pi
+
+
+def calc_paths(sx, sy, syaw, gx, gy, gyaw, maxc, step_size):
+    q0 = [sx, sy, syaw]
+    q1 = [gx, gy, gyaw]
+
+    # find set of feasible paths
+    paths = generate_path(q0, q1, maxc)
+    for path in paths:
+        # interpolate the path
+        x, y, yaw, directions = generate_local_course(
+            path.L, path.lengths, path.ctypes, maxc, step_size * maxc
+        )
+
+        # convert global coordinate
+        path.x = [
+            math.cos(-q0[2]) * ix + math.sin(-q0[2]) * iy + q0[0]
+            for (ix, iy) in zip(x, y)
+        ]
+        path.y = [
+            -math.sin(-q0[2]) * ix + math.cos(-q0[2]) * iy + q0[1]
+            for (ix, iy) in zip(x, y)
+        ]
+        path.yaw = [pi_2_pi(iyaw + q0[2]) for iyaw in yaw]
+        path.directions = directions
+        path.lengths = [length / maxc for length in path.lengths]
+        path.L = path.L / maxc
+
+    return paths
+
+
+def reeds_shepp_path_planning(start, goal, maxc, step_size):
+
+    s_x, s_y, s_yaw = start
+    g_x, g_y, g_yaw = goal
+
+    paths = calc_paths(s_x, s_y, s_yaw, g_x, g_y, g_yaw, maxc, step_size)
+
+    if not paths:
+        return None
+
+    minL = float("Inf")
+    best_path_index = -1
+    for i, _ in enumerate(paths):
+        if paths[i].L <= minL:
+            minL = paths[i].L
+            best_path_index = i
+
+    bpath = paths[best_path_index]
+
+    return bpath
+
+
+# ====================== RTB wrapper ============================= #
+
+# Copyright (c) 2022 Peter Corke: https://github.com/petercorke/robotics-toolbox-python
+# Released under the MIT license: https://github.com/AtsushiSakai/PythonRobotics/blob/master/LICENSE
+
[docs]class ReedsSheppPlanner(PlannerBase): + r""" + Reeds-Shepp path planner + + :param curvature: maximum path curvature, defaults to 1.0 + :type curvature: float, optional + :param stepsize: spacing between points on the path, defaults to 0.1 + :type stepsize: float, optional + :param Planner: Reeds-Shepp path planner + :type Planner: ReedsSheppPlanner instance + + ================== ======================== + Feature Capability + ================== ======================== + Plan :math:`\SE{2}` + Obstacle avoidance No + Curvature Discontinuous + Motion Bidirectional + ================== ======================== + + Creates a planner that finds the path between two configurations in the + plane using forward and backward motion. The path comprises upto 3 segments + that are straight lines, or arcs with curvature of :math:`\pm` + ``curvature``. + + Example: + + .. runblock:: pycon + + >>> from roboticstoolbox import ReedsSheppPlanner + >>> from math import pi + >>> reedshepp = ReedsSheppPlanner(curvature=1.0) + >>> path, status = reedshepp.query(start=(0, 0, pi/2), goal=(1, 0, pi/2)) + >>> print(path[:5,:]) + >>> print(status) + + :reference: Optimal paths for a car that goes both forwards and backwards, + Reeds, J.A. and L.A. Shepp, Pacific J. Math., 145 (1990), + pp. 367–393. + + :thanks: based on Reeds-Shepp path planning from `Python Robotics <https://github.com/AtsushiSakai/PythonRobotics/tree/master/PathPlanning>`_ + :seealso: :class:`DubinsPlanner` :class:`PlannerBase` + """ + + def __init__(self, curvature=1, stepsize=0.1, **kwargs): + super().__init__(ndims=3, **kwargs) + self._curvature = curvature + self._stepsize = stepsize + + def __str__(self): + s = ( + super().__str__() + + f"\n curvature={self.curvature}, stepsize={self.stepsize}" + ) + +
[docs] def query(self, start, goal, **kwargs): + r""" + Find a path between two configurations + + :param start: start configuration :math:`(x, y, \theta)` + :type start: array_like(3), optional + :param goal: goal configuration :math:`(x, y, \theta)` + :type goal: array_like(3), optional + :return: path and status + :rtype: ndarray(N,3), namedtuple + + The path comprises points equally spaced at a distance of ``stepsize``. + + The returned status value has elements: + + +-------------+-----------------------------------------------------+ + | Element | Description | + +-------------+-----------------------------------------------------+ + |``segments`` | a list containing the type of each path segment as | + | | a single letter code: either "L", "R" or "S" for | + | | left turn, right turn or straight line respectively.| + +-------------+-----------------------------------------------------+ + | ``length`` | total path length | + +-------------+-----------------------------------------------------+ + |``lengths`` | the length of each path segment. The sign of the | + | | length indicates the direction of travel. | + +-------------+-----------------------------------------------------+ + |``direction``| the direction of motion at each point on the path | + +-------------+-----------------------------------------------------+ + + .. note:: The direction of turning is reversed when travelling + backwards. + + """ + super().query(start=start, goal=goal, next=False, **kwargs) + + bpath = reeds_shepp_path_planning( + start=self.start, + goal=self.goal, + maxc=self._curvature, + step_size=self._stepsize, + ) + + path = np.c_[bpath.x, bpath.y, bpath.yaw] + + status = namedtuple( + "ReedsSheppStatus", ["segments", "length", "seglengths", "direction"] + ) + + return path, status( + bpath.ctypes, + sum([abs(l) for l in bpath.lengths]), + bpath.lengths, + bpath.directions, + )
+ + +if __name__ == "__main__": + from math import pi + + # start = (-1, -4, -np.radians(20)) + # goal = (5, 5, np.radians(25)) + + start = (0, 0, 0) + goal = (0, 0, pi) + + reedsshepp = ReedsSheppPlanner(curvature=1.0, stepsize=0.1) + path, status = reedsshepp.query(start, goal) + print(status) + + # px, py, pyaw, mode, clen = reeds_shepp_path_planning( + # start_x, start_y, start_yaw, end_x, end_y, end_yaw, curvature, step_size) + + # if show_animation: # pragma: no cover + # plt.cla() + # plt.plot(px, py, label="final course " + str(mode)) + + # # plotting + # plot_arrow(start_x, start_y, start_yaw) + # plot_arrow(end_x, end_y, end_yaw) + + # plt.legend() + # plt.grid(True) + # plt.axis("equal") + # plt.show(block=True) + + reedsshepp.plot(path=path, direction=status.direction, configspace=True) + plt.show(block=True) +
+ +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/mobile/Vehicle.html b/_modules/roboticstoolbox/mobile/Vehicle.html new file mode 100644 index 00000000..9d7717ba --- /dev/null +++ b/_modules/roboticstoolbox/mobile/Vehicle.html @@ -0,0 +1,1494 @@ + + + + + + roboticstoolbox.mobile.Vehicle — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
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+
    +
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+
+
+
+
+ +

Source code for roboticstoolbox.mobile.Vehicle

+"""
+Python Vehicle
+@Author: Peter Corke, original MATLAB code and Python version
+@Author: Kristian Gibson, initial MATLAB port
+"""
+from abc import ABC, abstractmethod
+import warnings
+from math import pi, sin, cos, tan
+import numpy as np
+from scipy import interpolate
+
+import matplotlib.pyplot as plt
+from matplotlib import animation
+
+# from matplotlib import patches
+# import matplotlib.transforms as mtransforms
+
+from spatialmath import SE2, base
+from roboticstoolbox.mobile.drivers import VehicleDriverBase
+from roboticstoolbox.mobile.Animations import VehiclePolygon
+
+
+
[docs]class VehicleBase(ABC): +
[docs] def __init__( + self, + covar=None, + speed_max=np.inf, + accel_max=np.inf, + x0=[0, 0, 0], + dt=0.1, + control=None, + seed=0, + animation=None, + verbose=False, + plot=False, + workspace=None, + polygon=None, + ): + r""" + Superclass for vehicle kinematic models + + :param covar: odometry covariance, defaults to zero + :type covar: ndarray(2,2), optional + :param speed_max: maximum speed, defaults to :math:`\infty` + :type speed_max: float, optional + :param accel_max: maximum acceleration, defaults to :math:`\infty` + :type accel_max: float, optional + :param x0: Initial state, defaults to (0,0,0) + :type x0: array_like(3), optional + :param dt: sample update interval, defaults to 0.1 + :type dt: float, optional + :param control: vehicle control inputs, defaults to None + :type control: array_like(2), interp1d, VehicleDriverBase + :param animation: Graphical animation of vehicle, defaults to None + :type animation: :class:`VehicleAnimationBase` subclass, optional + :param verbose: print lots of info, defaults to False + :type verbose: bool, optional + :param workspace: dimensions of 2D plot area, defaults to (-10:10) x (-10:10), + see :func:`~spatialmath.base.graphics.plotvol2` + :type workspace: float, array_like(2), array_like(4) + :param polygon: bounding polygon of vehicle + :type polygon: :class:`~spatialmath.geom2d.Polygon2` + + This is an abstract superclass that simulates the motion of a mobile + robot under the action of a controller. The controller provides + control inputs to the vehicle, and the output odometry is returned. + The true state, effectively unknowable in practice, is computed + and accessible. + + The workspace can be specified in several ways: + + ============== ======= ======= + ``workspace`` x-range y-range + ============== ======= ======= + A (scalar) -A:A -A:A + [A, B] A:B A:B + [A, B, C, D] A:B C:D + ============== ======= ======= + + :note: Set ``seed=None`` to have it randomly initialized from the + operating system. + + :seealso: :class:`Bicycle` :class:`Unicycle` :class:`VehicleAnimationBase` + """ + + self._V = covar + self._dt = dt + if x0 is None: + x0 = np.zeros((3,), dtype=float) + else: + x0 = base.getvector(x0) + if len(x0) not in (2, 3): + raise ValueError("x0 must be length 2 or 3") + self._x0 = x0 + self._x = x0.copy() + + self._random = np.random.default_rng(seed) + self._seed = seed + self._speed_max = speed_max + self._accel_max = accel_max + self._v_prev = [0] + self._polygon = polygon + + if isinstance(animation, str): + animation = VehiclePolygon(animation) + self._animation = animation + self._ax = None + + if control is not None: + self.add_driver(control) + + self._dt = dt + self._t = 0 + self._stopsim = False + + self._verbose = verbose + self._plot = False + + self._control = None + self._x_hist = [] + + if workspace: + try: + self._workspace = workspace.workspace + except AttributeError: + self._workspace = base.expand_dims(workspace) + else: + self._workspace = None
+ +
[docs] def __str__(self): + """ + String representation of vehicle (superclass) + + :return: String representation of vehicle object + :rtype: str + """ + s = f"{self.__class__.__name__}: " + s += f"x = {base.array2str(self._x)}" + return s
+ + def __repr__(self): + return str(self) + +
[docs] def f(self, x, odo, v=None): + r""" + State transition function + + :param x: vehicle state :math:`(x, y, \theta)` + :type x: array_like(3), ndarray(n,3) + :param odo: vehicle odometry :math:`(\delta_d, \delta_\theta)` + :type odo: array_like(2) + :param v: additive odometry noise, defaults to (0,0) + :type v: array_like(2), optional + :return: predicted vehicle state + :rtype: ndarray(3) + + Predict the next state based on current state and odometry + value. ``v`` is a random variable that represents additive + odometry noise for simulation purposes. + + .. math:: + + f: \vec{x}_k, \delta_d, \delta_\theta \mapsto \vec{x}_{k+1} + + For particle filters it is useful to apply the odometry to the + states of all particles and this can be achieved if ``x`` is a 2D + array with one state per row. ``v`` is ignored in this case. + + .. note:: This is the state update equation used for EKF localization. + + :seealso: :meth:`deriv` :meth:`Fx` :meth:`Fv` + """ + odo = base.getvector(odo, 2) + + if isinstance(x, np.ndarray) and x.ndim == 2: + # x is Nx3 set of vehicle states, do vectorized form + # used by particle filter + dd, dth = odo + theta = x[:, 2] + return ( + np.array(x) + + np.c_[ + dd * np.cos(theta), dd * np.sin(theta), np.full(theta.shape, dth) + ] + ) + else: + # x is a vector + x = base.getvector(x, 3) + dd, dth = odo + theta = x[2] + + if v is not None: + v = base.getvector(v, 2) + dd += v[0] + dth += v[1] + + return x + np.r_[dd * np.cos(theta), dd * np.sin(theta), dth]
+ +
[docs] def Fx(self, x, odo): + r""" + Jacobian of state transition function df/dx + + :param x: vehicle state :math:`(x, y, \theta)` + :type x: array_like(3) + :param odo: vehicle odometry :math:`(\delta_d, \delta_\theta)` + :type odo: array_like(2) + :return: Jacobian matrix + :rtype: ndarray(3,3) + + Returns the Jacobian matrix :math:`\frac{\partial \vec{f}}{\partial \vec{x}}` for + the given state and odometry. + + :seealso: :meth:`f` :meth:`deriv` :meth:`Fv` + """ + dd, dth = odo + theta = x[2] + + # fmt: off + J = np.array([ + [1, 0, -dd * sin(theta)], + [0, 1, dd * cos(theta)], + [0, 0, 1], + ]) + # fmt: on + return J
+ +
[docs] def Fv(self, x, odo): + r""" + Jacobian of state transition function df/dv + + :param x: vehicle state :math:`(x, y, \theta)` + :type x: array_like(3) + :param odo: vehicle odometry :math:`(\delta_d, \delta_\theta)` + :type odo: array_like(2) + :return: Jacobian matrix + :rtype: ndarray(3,2) + + Returns the Jacobian matrix :math:`\frac{\partial \vec{f}}{\partial \vec{v}}` for + the given state and odometry. + + :seealso: :meth:`f` :meth:`deriv` :meth:`Fx` + """ + dd, dth = odo + theta = x[2] + + # fmt: off + J = np.array([ + [cos(theta), 0], + [sin(theta), 0], + [0, 1], + ]) + # fmt: on + return J
+ + @property + def control(self): + """ + Get/set vehicle control (superclass) + + :getter: Returns vehicle's control + :setter: Sets the vehicle's control + :type: 2-tuple, callable, interp1d or VehicleDriver + + This is the input to the vehicle during a simulation performed + by :meth:`run`. The control input can be: + + * a constant tuple as the control inputs to the vehicle + * a function called as ``f(vehicle, t, x)`` that returns a tuple + * an interpolator called as ``f(t)`` that returns a tuple + * a driver agent, subclass of :class:`~roboticstoolbox.mobile.driversVehicleDriverBase` + + Example: + + .. runblock:: pycon + + >>> from roboticstoolbox import Bicycle, RandomPath + >>> bike = Bicycle() + >>> bike.control = RandomPath(10) + >>> print(bike) + + :seealso: :meth:`run` :meth:`eval_control` :obj:`scipy.interpolate.interp1d` :class:`~roboticstoolbox.mobile.drivers.VehicleDriverBase` + """ + return self._control + + @control.setter + def control(self, control): + + # * a function called as ``f(vehicle, t, x)`` that returns a tuple + # * an interpolator called as f(t) that returns a tuple, see + # SciPy interp1d + + self._control = control + if isinstance(control, VehicleDriverBase): + # if this is a driver agent, connect it to the vehicle + control.vehicle = self + +
[docs] def eval_control(self, control, x): + """ + Evaluate vehicle control input (superclass method) + + :param control: vehicle control + :type control: [type] + :param x: vehicle state :math:`(x, y, \theta)` + :type x: array_like(3) + :raises ValueError: bad control + :return: vehicle control inputs + :rtype: ndarray(2) + + Evaluates the control for this time step and state. Control is set + by the :meth:`control` property to: + + * a constant tuple as the control inputs to the vehicle + * a function called as ``f(vehicle, t, x)`` that returns a tuple + * an interpolator called as f(t) that returns a tuple + * a driver agent, subclass of :class:`VehicleDriverBase` + + Vehicle steering, speed and acceleration limits are applied to the + result before being input to the vehicle model. + + :seealso: :meth:`run` :meth:`control` :func:`scipy.interpolate.interp1d` + """ + # was called control() in the MATLAB version + + if base.isvector(control, 2): + # control is a constant + u = base.getvector(control, 2) + + elif isinstance(control, VehicleDriverBase): + # vehicle has a driver object + u = control.demand() + + elif isinstance(control, interpolate.interpolate.interp1d): + # control is an interp1d object + u = control(self._t) + + elif callable(control): + # control is a user function of time and state + u = control(self, self._t, x) + + else: + raise ValueError("bad control specified") + + # apply limits + ulim = self.u_limited(u) + return ulim
+ +
[docs] def limits_va(self, v, v_prev): + """ + Apply velocity and acceleration limits (superclass) + + :param v: desired velocity + :type v: float + :param v_prev: previous velocity, reference to list + :type v_prev: list with single element + :return: allowed velocity + :rtype: float + + Determine allowable velocity given desired velocity, speed and + acceleration limits. + + .. note:: This function requires previous velocity, ``v_prev`` to enable + acceleration limiting. This is passed as a reference to a mutable value, + a single-element list. This is reset to zero at the start of each simulation. + """ + # acceleration limit + vp = v_prev[0] + if self._accel_max is not None: + if (v - vp) / self._dt > self._accel_max: + v = vp + self._accelmax * self._dt + elif (v - vp) / self._dt < -self._accel_max: + v = vp - self._accel_max * self._dt + v_prev[0] = v + + # speed limit + if self._speed_max is not None: + v = np.clip(v, -self._speed_max, self._speed_max) + return v
+ +
[docs] def polygon(self, q): + """ + Bounding polygon at vehicle configuration + + :param q: vehicle configuration :math:`(x, y, \theta)` + :type q: array_like(3) + :return: bounding polygon of vehicle at configuration ``q`` + :rtype: :class:`~spatialmath.geom2d.Polygon2` + + The bounding polygon of the vehicle is returned for the configuration + ``q``. Can be used for collision checking using the :meth:`~spatialmath.geom2d.Polygon2.intersects` + method. + + :seealso: :class:`~spatialmath.geom2d.Polygon2` + """ + return self._polygon.transformed(SE2(q))
+ + # This function is overridden by the child class +
[docs] @abstractmethod + def deriv(self, x, u): + pass
+ +
[docs] def add_driver(self, driver): + """ + Add a driver agent (superclass) + + :param driver: a driver agent object + :type driver: VehicleDriverBase subclass + + .. warning:: Deprecated. Use ``vehicle.control = driver`` instead. + + :seealso: :class:`VehicleDriverBase` :meth:`control` + """ + + warnings.warn("add_driver is deprecated, use veh.control=driver instead") + self._control = driver + driver._veh = self
+ +
[docs] def run(self, T=10, x0=None, control=None, animate=True): + r""" + Simulate motion of vehicle (superclass) + + :param T: Simulation time in seconds, defaults to 10 + :type T: float, optional + :param x0: Initial state, defaults to value given to Vehicle constructor + :type x0: array_like(3) or array_like(2) + :param control: vehicle control inputs, defaults to None + :type control: array_like(2), callable, driving agent + :param animate: Enable graphical animation of vehicle, defaults to False + :type animate: bool, optional + :return: State trajectory, each row is :math:`(x,y,\theta)`. + :rtype: ndarray(n,3) + + Runs the vehicle simulation for ``T`` seconds and optionally plots + an animation. The method :meth:`step` performs each time step. + + The control inputs are provided by ``control`` which can be: + + * a constant tuple as the control inputs to the vehicle + * a function called as ``f(vehicle, t, x)`` that returns a tuple + * an interpolator called as ``f(t)`` that returns a tuple, see + SciPy interp1d + * a driver agent, subclass of :class:`VehicleDriverBase` + + This is evaluated by :meth:`eval_control` which applies velocity, + acceleration and steering limits. + + The simulation can be stopped prematurely by the control function + calling :meth:`stopsim`. + + .. note:: + * the simulation is fixed-time step with the step given by the ``dt`` + attribute set by the constructor. + * integration uses rectangular integration. + + :seealso: :meth:`init` :meth:`step` :meth:`control` :meth:`run_animation` + """ + + self.init(control=control, animate=animate, x0=x0) + + for i in range(round(T / self.dt)): + self.step(animate=animate) + + # check for user requested stop + if self._stopsim: + print("USER REEQUESTED STOP AT time", self._t) + break + + return self.x_hist
+ +
[docs] def run_animation(self, T=10, x0=None, control=None, format=None, file=None): + r""" + Simulate motion of vehicle (superclass) + + :param T: Simulation time in seconds, defaults to 10 + :type T: float, optional + :param x0: Initial state, defaults to value given to Vehicle constructor + :type x0: array_like(3) or array_like(2) + :param control: vehicle control inputs, defaults to None + :type control: array_like(2), callable, driving agent + :param format: Output format + :type format: str, optional + :param file: File name + :type file: str, optional + :return: Matplotlib animation object + :rtype: :meth:`matplotlib.animation.FuncAnimation` + + Runs the vehicle simulation for ``T`` seconds and returns an animation + in various formats:: + + ``format`` ``file`` description + ============ ========= ============================ + ``"html"`` str, None return HTML5 video + ``"jshtml"`` str, None return JS+HTML video + ``"gif"`` str return animated GIF + ``"mp4"`` str return MP4/H264 video + ``None`` return a ``FuncAnimation`` object + + The allowables types for ``file`` are given in the second column. A str + value is the file name. If ``None`` is an option then return the video as a string. + + For the last case, a reference to the animation object must be held if used for + animation in a Jupyter cell:: + + anim = robot.run_animation(T=20) + + The control inputs are provided by ``control`` which can be: + + * a constant tuple as the control inputs to the vehicle + * a function called as ``f(vehicle, t, x)`` that returns a tuple + * an interpolator called as ``f(t)`` that returns a tuple, see + SciPy interp1d + * a driver agent, subclass of :class:`VehicleDriverBase` + + This is evaluated by :meth:`eval_control` which applies velocity, + acceleration and steering limits. + + The simulation can be stopped prematurely by the control function + calling :meth:`stopsim`. + + .. note:: + * the simulation is fixed-time step with the step given by the ``dt`` + attribute set by the constructor. + * integration uses rectangular integration. + + :seealso: :meth:`init` :meth:`step` :meth:`control` :meth:`run_animation` + """ + + fig, ax = plt.subplots() + + nframes = round(T / self.dt) + anim = animation.FuncAnimation( + fig=fig, + func=lambda i: self.step(animate=True, pause=False), + init_func=lambda: self.init(animate=True), + frames=nframes, + interval=self.dt * 1000, + blit=False, + repeat=False, + ) + # anim._interval = self.dt*1000/2 + # anim._repeat = True + ret = None + if format == "html": + ret = anim.to_html5_video() # convert to embeddable HTML5 animation + elif format == "jshtml": + ret = anim.to_jshtml() # convert to embeddable Javascript/HTML animation + elif format == "gif": + anim.save( + file, writer=animation.PillowWriter(fps=1 / self.dt) + ) # convert to GIF + ret = None + elif format == "mp4": + anim.save( + file, writer=animation.FFMpegWriter(fps=1 / self.dt) + ) # convert to mp4/H264 + ret = None + elif format == None: + # return the anim object + return anim + else: + raise ValueError("unknown format") + + if ret is not None and file is not None: + with open(file, "w") as f: + f.write(ret) + ret = None + plt.close(fig) + return ret
+ +
[docs] def init(self, x0=None, control=None, animate=True): + """ + Initialize for simulation (superclass) + + :param x0: Initial state, defaults to value given to Vehicle constructor + :type x0: array_like(3) or array_like(2) + + + Performs the following initializations: + + #. Clears the state history + #. Set the private random number generator to initial value + #. Sets state :math:`x = x_0` + #. Sets previous velocity to zero + #. If a driver is attached, initialize it + #. If animation is enabled, initialize that + + If animation is enabled by constructor and no animation object is given, + use a default ``VehiclePolygon("car")`` + + :seealso: :class:`VehicleAnimationBase` :meth:`run` + """ + if x0 is not None: + self._x = base.getvector(x0, 3) + else: + self._x = self._x0.copy() + + self._x_hist = [] + + if self._seed is not None: + self._random = np.random.default_rng(self._seed) + + if control is not None: + # override control + self._control = control + + if isinstance(self._control, VehicleDriverBase): + self._control.init() + + self._t = 0 + self._v_prev = [0] + + # initialize the graphics + if animate and self._animation is not None: + + # setup the plot + self._ax = base.plotvol2(self.workspace) + + self._ax.set_xlabel("x") + self._ax.set_ylabel("y") + self._ax.set_aspect("equal") + try: + self._ax.figure.canvas.manager.set_window_title( + f"Robotics Toolbox for Python (Figure {self._ax.figure.number})" + ) + except AttributeError: + pass + + self._animation.add(ax=self._ax) # add vehicle animation to axis + self._timer = plt.figtext(0.85, 0.95, "") # display time counter + + # initialize the driver + if isinstance(self._control, VehicleDriverBase): + self._control.init(ax=self._ax)
+ +
[docs] def step(self, u=None, animate=True, pause=True): + r""" + Step simulator by one time step (superclass) + + :return: odometry :math:`(\delta_d, \delta_\theta)` + :rtype: ndarray(2) + + #. Obtain the vehicle control inputs + #. Integrate the vehicle state forward one timestep + #. Updates the stored state and state history + #. Update animation if enabled. + #. Returns the odometry + + - ``veh.step((vel, steer))`` for a Bicycle vehicle model + - ``veh.step((vel, vel_diff))`` for a Unicycle vehicle model + - ``veh.step()`` as above but control is taken from the ``control`` + attribute which might be a function or driver agent. + + + Example: + + .. runblock:: pycon + + >>> from roboticstoolbox import Bicycle + >>> bike = Bicycle() # default bicycle model + >>> bike.step((1, 0.2)) # one step: v=1, γ=0.2 + >>> bike.x + >>> bike.step((1, 0.2)) # another step: v=1, γ=0.2 + >>> bike.x + + .. note:: Vehicle control input limits are applied. + + :seealso: :func:`control`, :func:`update`, :func:`run` + """ + # determine vehicle control + if u is not None: + u = self.eval_control(u, self._x) + else: + u = self.eval_control(self._control, self._x) + + # update state (used to be function control() in MATLAB version) + xd = self._dt * self.deriv(self._x, u) # delta state + + # update state vector + self._x += xd + self._x_hist.append(tuple(self._x)) + + # print('VEH', u, self.x) + + # odometry comes from change in state vector + odo = np.r_[np.linalg.norm(xd[0:2]), xd[2]] + + if self._V is not None: + odo += self.random.multivariate_normal((0, 0), self._V) + + # do the graphics + if animate and self._animation: + self._animation.update(self._x) + # if self._timer is not None: + # self._timer.set_text(f"t = {self._t:.2f}") + if pause: + plt.pause(self._dt) + # plt.show(block=False) + # pass + + self._t += self._dt + + # be verbose + if self._verbose: + print( + f"{self._t:8.2f}: u=({u[0]:8.2f}, {u[1]:8.2f}), x=({self._x[0]:8.2f}," + f" {self._x[1]:8.2f}, {self._x[2]:8.2f})" + ) + + return odo
+ + @property + def workspace(self): + """ + Size of robot workspace (superclass) + + :return: workspace bounds [xmin, xmax, ymin, ymax] + :rtype: ndarray(4) + + Returns the bounds of the workspace as specified by constructor + option ``workspace`` + """ + + # get workspace specified for Vehicle or from its driver + if self._workspace is not None: + return self._workspace + if self._control is not None and hasattr(self._control, "_workspace"): + return self._control._workspace + + @property + def x(self): + r""" + Get vehicle state/configuration (superclass) + + :return: Vehicle state :math:`(x, y, \theta)` + :rtype: ndarray(3) + + :seealso: :meth:`q` + """ + return self._x + + @property + def q(self): + r""" + Get vehicle state/configuration (superclass) + + :return: Vehicle state :math:`(x, y, \theta)` + :rtype: ndarray(3) + + :seealso: :meth:`x` + """ + return self._x + + @property + def x0(self): + r""" + Get vehicle initial state/configuration (superclass) + + :return: Vehicle state :math:`(x, y, \theta)` + :rtype: ndarray(3) + + Set by :class:`VehicleBase` subclass constructor. The state is set to this value + at the beginning of each simulation run. + + :seealso: :meth:`run` + """ + return self._x0 + + @x0.setter + def x0(self, x0): + r""" + Set vehicle initial state/configuration (superclass) + + :param x0: Vehicle state :math:`(x, y, \theta)` + :type x0: array_like(3) + + The state is set to this value at the beginning of each simulation + run. Overrides the value set by :class:`VehicleBase` subclass constructor. + + :seealso: :func:`run` + """ + self._x0 = base.getvector(x0, 3) + + @property + def random(self): + """ + Get private random number generator + + :return: NumPy random number generator + :rtype: :class:`numpy.random.Generator` + + Has methods including: + + - :meth:`integers(low, high, size, endpoint) <numpy.random.Generator.integers>` + - :meth:`random(size) <numpy.random.Generator.random>` + - :meth:`uniform(low, high, size) <numpy.random.Generator.uniform>` + - :meth:`normal(mean, std, size) <numpy.random.Generator.normal>` + - :meth:`multivariate_normal(mean, covar, size) <numpy.random.Generator.multivariate_normal>` + + The generator is initialized with the seed provided at constructor + time every time :meth:`init` is called. + + :seealso: :meth:`init` :func:`numpy.random.default_rng` + """ + return self._random + + @property + def x_hist(self): + """ + Get vehicle state/configuration history (superclass) + + :return: Vehicle state history + :rtype: ndarray(n,3) + + The state :math:`(x, y, \theta)` at each time step resulting from a + simulation run, one row per timestep. + + :seealso: :meth:`run` + """ + return np.array(self._x_hist) + + @property + def speed_max(self): + """ + Get maximum speed of vehicle (superclass) + + :return: maximum speed + :rtype: float + + Set by :meth:`VehicleBase` subclass constructor. + + :seealso: :meth:`accel_max` :meth:`steer_max` :meth:`u_limited` + """ + return self._speed_max + + @property + def accel_max(self): + """ + Get maximum acceleration of vehicle (superclass) + + :return: maximum acceleration + :rtype: float + + Set by :meth:`VehicleBase` subclass constructor. + + :seealso: :meth:`speed_max` :meth:`steer_max` :meth:`u_limited` + """ + return self._accel_max + + @property + def dt(self): + """ + Get sample time (superclass) + + :return: discrete time step for simulation + :rtype: float + + Set by :meth:`VehicleBase` subclass constructor. Used by + :meth:`step` to update state. + + :seealso: :meth:`run` + """ + return self._dt + + @property + def verbose(self): + """ + Get verbosity (superclass) + + :return: verbosity level + :rtype: bool + + Set by :class:`VehicleBase` subclass constructor. + """ + return self._verbose + + @verbose.setter + def verbose(self, verbose): + """ + Set verbosity (superclass) + + :return: verbosity level + :rtype: bool + + Set by :class:`VehicleBase` subclass constructor. + """ + self._verbose = verbose + +
[docs] def stopsim(self): + """ + Stop the simulation (superclass) + + A control function can stop the simulation initated by the ``run`` + method. + + :seealso: :meth:`run` + """ + self._stopsim = True
+ +
[docs] def plot_xy(self, *args, block=None, **kwargs): + """ + Plot xy-path from history + + :param block: block until plot dismissed, defaults to None + :type block: bool, optional + :param args: positional arguments passed to :meth:`~matplotlib.axes.Axes.plot` + :param kwargs: keyword arguments passed to :meth:`~matplotlib.axes.Axes.plot` + + + The :math:`(x,y)` trajectory from the simulation history is plotted as + :math:`x` vs :math:`y. + + :seealso: :meth:`run` :meth:`plot_xyt` + """ + if args is None and "color" not in kwargs: + kwargs["color"] = "b" + xyt = self.x_hist + plt.plot(xyt[:, 0], xyt[:, 1], *args, **kwargs) + if block is not None: + plt.show(block=block)
+ +
[docs] def plot_xyt(self, block=None, **kwargs): + """ + Plot configuration vs time from history + + :param block: block until plot dismissed, defaults to False + :type block: bool, optional + :param args: positional arguments passed to :meth:`~matplotlib.axes.Axes.plot` + :param kwargs: keyword arguments passed to :meth:`~matplotlib.axes.Axes.plot` + + The :math:`(x,y, \theta)` trajectory from the simulation history is plotted + against time. + + :seealso: :meth:`run` :meth:`plot_xy` + """ + xyt = self.x_hist + t = np.arange(0, xyt.shape[0] * self._dt, self._dt) + plt.plot(xyt[:, 0], xyt[:, :], **kwargs) + plt.legend(["x", "y", "$\\theta$"]) + if block is not None: + plt.show(block=block)
+ + +# ========================================================================= # + + +
[docs]class Bicycle(VehicleBase): +
[docs] def __init__(self, L=1, steer_max=0.45 * pi, **kwargs): + r""" + Create bicycle kinematic model + + :param L: wheel base, defaults to 1 + :type L: float, optional + :param steer_max: maximum steering angle, defaults to :math:`0.45\pi` + :type steer_max: float, optional + :param kwargs: additional arguments passed to :class:`VehicleBase` + constructor + + Model the motion of a bicycle model with equations of motion given by: + + .. math:: + + \dot{x} &= v \cos \theta \\ + \dot{y} &= v \sin \theta \\ + \dot{\theta} &= \frac{v}{L} \tan \gamma + + where :math:`v` is the velocity in body frame x-direction, and + :math:`\gamma` is the angle of the steered wheel. + + :seealso: :meth:`f` :meth:`deriv` :meth:`Fx` meth:`Fv` :class:`VehicleBase` + """ + super().__init__(**kwargs) + + self._l = L + self._steer_max = steer_max
+ + def __str__(self): + + s = super().__str__() + s += ( + f"\n L={self._l}, steer_max={self._steer_max:g}," + f" speed_max={self._speed_max:g}, accel_max={self._accel_max:g}" + ) + return s + + @property + def l(self): + """ + Vehicle wheelbase + + :return: vehicle wheelbase + :rtype: float + """ + return self._l + + @property + def radius_min(self): + r""" + Vehicle turning radius + + :return: radius of minimum possible turning circle + :rtype: float + + Minimum turning radius based on maximum steered wheel angle and wheelbase. + + .. math:: + \frac{l}{\tan \gamma_{max}} + + :seealso: :meth:`curvature_max` + """ + return self.l / np.tan(self.steer_max) + + @property + def curvature_max(self): + r""" + Vehicle maximum path curvature + + :return: maximum curvature + :rtype: float + + Maximum path curvature based on maximum steered wheel angle and wheelbase. + + .. math:: + \frac{\tan \gamma_{max}}{l} + + :seealso: :meth:`radius_min` + """ + return 1.0 / self.radius_min + + @property + def steer_max(self): + """ + Vehicle maximum steered wheel angle + + :return: maximum angle + :rtype: float + """ + return self._steer_max + +
[docs] def deriv(self, x, u, limits=True): + r""" + Time derivative of state + + :param x: vehicle state :math:`(x, y, \theta)` + :type x: array_like(3) + :param u: control input :math:`(v, \gamma)` + :type u: array_like(2) + :param limits: limits are applied to input, default True + :type limits: bool + :return: state derivative :math:`(\dot{x}, \dot{y}, \dot{\theta})` + :rtype: ndarray(3) + + Returns the time derivative of state (3x1) at the state ``x`` with velocity :math:`v` + and steered wheel angle :math:`\gamma` + + .. math:: + \dot{x} &= v \cos \theta \\ + \dot{y} &= v \sin \theta \\ + \dot{\theta} &= \frac{v}{L} \tan \gamma + + If ``limits`` is True then speed, acceleration and steer-angle limits are + applied to ``u``. + + :seealso: :meth:`f` + """ + + # unpack some variables + theta = x[2] + + if limits: + u = self.u_limited(u) + v = u[0] + gamma = u[1] + + return v * np.r_[cos(theta), sin(theta), tan(gamma) / self.l]
+ +
[docs] def u_limited(self, u): + """ + Apply vehicle velocity, acceleration and steering limits + + :param u: Desired vehicle inputs :math:`(v, \gamma)` + :type u: array_like(2) + :return: Allowable vehicle inputs :math:`(v, \gamma)` + :rtype: ndarray(2) + + Velocity and acceleration limits are applied to :math:`v` and + steered wheel angle limits are applied to :math:`\gamma`. + """ + # limit speed and steer angle + ulim = np.array(u) + ulim[0] = self.limits_va(u[0], self._v_prev) + ulim[1] = np.clip(u[1], -self._steer_max, self._steer_max) + + return ulim
+ + +# ========================================================================= # + + +
[docs]class Unicycle(VehicleBase): +
[docs] def __init__(self, W=1, steer_max=np.inf, **kwargs): + r""" + Create unicycle kinematic model + + :param W: vehicle width, defaults to 1 + :type W: float, optional + :param steer_max: maximum turn rate + :type steer_max: float + :param kwargs: additional arguments passed to :class:`VehicleBase` + constructor + + Model the motion of a unicycle model with equations of motion given by: + + .. math:: + + \dot{x} &= v \cos \theta \\ + \dot{y} &= v \sin \theta \\ + \dot{\theta} &= \omega + + where :math:`v` is the velocity in body frame x-direction, and + :math:`\omega` is the turn rate. + + :seealso: :meth:`f` :meth:`deriv` :meth:`Fx` meth:`Fv` :class:`Vehicle` + """ + super().__init__(**kwargs) + self._W = W + self._steer_max = steer_max
+ + def __str__(self): + + s = super().__str__() + s += ( + f"\n W={self._W}, steer_max={self._steer_max:g}," + f" vel_max={self._speed_max}, accel_max={self._accel_max}" + ) + return s + +
[docs] def deriv(self, x, u, limits=True): + r""" + Time derivative of state + + :param x: vehicle state :math:`(x, y, \theta)` + :type x: array_like(3) + :param u: control input :math:`(v, \omega)` + :type u: array_like(2) + :param limits: limits are applied to input, default True + :type limits: bool + :return: state derivative :math:`(\dot{x}, \dot{y}, \dot{\theta})` + :rtype: ndarray(3) + + Returns the time derivative of state (3x1) at the state ``x`` with velocity :math:`v` + and turn rate :math:`\omega` + + .. math:: + \dot{x} &= v \cos \theta \\ + \dot{y} &= v \sin \theta \\ + \dot{\theta} &= \omega + + If ``limits`` is True then speed, acceleration and steer-angle limits are + applied to ``u``. + + :seealso: :meth:`f` + """ + if limits: + u = self.u_limited(u) + # unpack some variables + theta = x[2] + v = u[0] + vdiff = u[1] + + return np.r_[v * cos(theta), v * sin(theta), vdiff / self._W]
+ +
[docs] def u_limited(self, u): + """ + Apply vehicle velocity, acceleration and steering limits + + :param u: Desired vehicle inputs :math:`(v, \omega)` + :type u: array_like(2) + :return: Allowable vehicle inputs :math:`(v, \omega)` + :rtype: ndarray(2) + + Velocity and acceleration limits are applied to :math:`v` and + turn rate limits are applied to :math:`\omega`. + """ + + # limit speed and steer angle + ulim = np.array(u) + ulim[0] = self.limits_va(u[0], self._v_prev) + ulim[1] = np.maximum(-self._steer_max, np.minimum(self._steer_max, u[1])) + + return ulim
+ + +
[docs]class DiffSteer(Unicycle): +
[docs] def __init__(self, W=1, **kwargs): + r""" + Create differential steering kinematic model + + :param W: vehicle width, defaults to 1 + :type W: float, optional + :param kwargs: additional arguments passed to :class:`VehicleBase` + constructor + + Model the motion of a unicycle model with equations of motion given by: + + .. math:: + + \dot{x} &= v \cos \theta \\ + \dot{y} &= v \sin \theta \\ + \dot{\theta} &= \omega + + where :math:`v = (v_R + v_L)/2` is the velocity in body frame x-direction, and + :math:`\omega = (v_R - v_L)/W` is the turn rate. + + :seealso: :meth:`f` :meth:`deriv` :meth:`Fx` meth:`Fv` :class:`Vehicle` + """ + super().__init__(**kwargs) + self._W = W + self._v_prev_L = [0] + self._v_prev_R = [0]
+ + def __str__(self): + + s = super().__str__() + return s + +
[docs] def init(self): + super().init() + self._v_prev_L = [0] + self._v_prev_R = [0]
+ + +
[docs] def u_limited(self, u): + """ + Apply vehicle velocity and acceleration limits + + :param u: Desired vehicle inputs :math:`(v_L, v_R)` + :type u: array_like(2) + :return: Allowable vehicle inputs :math:`(v_L, v_R)` + :rtype: ndarray(2) + + Velocity and acceleration limits are applied to :math:`v` and + turn rate limits are applied to :math:`\omega`. + """ + + # limit speed and acceleration of each wheel/track + ulim = np.array(u) + ulim[0] = self.limits_va(u[0], self._v_prev_L) + ulim[1] = self.limits_va(u[1], self._v_prev_R) + + return ulim
+ +
[docs] def deriv(self, x, u, limits=True): + r""" + Time derivative of state + + :param x: vehicle state :math:`(x, y, \theta)` + :type x: array_like(3) + :param u: Desired vehicle inputs :math:`(v_L, v_R)` + :type u: array_like(2) + :param limits: limits are applied to input, default True + :type limits: bool + :return: state derivative :math:`(\dot{x}, \dot{y}, \dot{\theta})` + :rtype: ndarray(3) + + Returns the time derivative of state (3x1) at the state ``x`` with left and + right wheel speeds ``u``. + + .. math:: + + \dot{x} &= v \cos \theta \\ + \dot{y} &= v \sin \theta \\ + \dot{\theta} &= \omega + + where :math:`v = (v_R + v_L)/2` is the velocity in body frame x-direction, and + :math:`\omega = (v_R - v_L)/W` is the turn rate. + + If ``limits`` is True then speed and acceleration limits are applied to the + wheel speeds ``u``. + + :seealso: :meth:`f` + """ + if limits: + u = self.u_limited(u) + # unpack some variables + theta = x[2] + vleft = u[0] + vright = u[1] + + # convert wheel speeds to forward and differential velocity + v = (vright + vleft) / 2.0 + vdiff = vright - vleft + + return np.r_[v * cos(theta), v * sin(theta), vdiff / self._W]
+ + + +if __name__ == "__main__": + + from roboticstoolbox import RandomPath + + import roboticstoolbox as rtb + + a = rtb.VehicleMarker(marker="s", markerfacecolor="b") + veh = rtb.Bicycle(animation=a, workspace=10) + veh.control = (4, 0.5) + veh.run(20) + + # V = np.eye(2) * 0.001 + # robot = Bicycle(covar=V, animation="car") + # odo = robot.step((1, 0.3), animate=False) + + # robot.control = RandomPath(workspace=10) + + # # robot.run(T=10, animate=True) + # # plt.show(block=True) + + # anim = robot.run_animation(T=10, format="html", file="veh.html") + + # anim.save("veh.mp4", writer=animation.FFMpegWriter(fps=1/robot.dt)) # convert to mp4/H264 + # with open("veh.html", "w") as f: + # f.write(anim.to_html5_video()) + + # from math import pi + + # V = np.diag(np.r_[0.02, 0.5 * pi / 180] ** 2) + + # v = VehiclePolygon() + # # v = VehicleIcon('greycar2', scale=2, rotation=90) + + # veh = Bicycle(covar=V, animation=v, control=RandomPath(10), verbose=False) + # print(veh) + + # odo = veh.step(1, 0.3) + # print(odo) + + # print(veh.x) + + # print(veh.f([0, 0, 0], odo)) + + # def control(v, t, x): + # goal = (6,6) + # goal_heading = atan2(goal[1]-x[1], goal[0]-x[0]) + # d_heading = base.angdiff(goal_heading, x[2]) + # v.stopif(base.norm(x[0:2] - goal) < 0.1) + + # return (1, d_heading) + + # veh.control=RandomPath(10) + # p = veh.run(1000, plot=True) + # # plt.show() + # print(p) + + # veh.plot_xyt_t() + # veh.plot(p) + + # t, x = veh.path(5, u=control) + # print(t) + + # fig, ax = plt.subplots() + + # ax.set_xlim(-5, 5) + # ax.set_ylim(-5, 5) + + # v = VehicleAnimation.Polygon(shape='triangle', maxdim=0.1, color='r') + # v = VehicleAnimation.Icon('car3.png', maxdim=2, centre=[0.3, 0.5]) + # v = VehicleAnimation.Icon('/Users/corkep/Dropbox/code/robotics-toolbox-python/roboticstoolbox/data/car1.png', maxdim=2, centre=[0.3, 0.5]) + # v = VehicleAnimation.icon('car3.png', maxdim=2, centre=[0.3, 0.5]) + # v = VehicleAnimation.marker() + # v.start() + # plt.grid(True) + # # plt.axis('equal') + + # for theta in np.linspace(0, 2 * np.pi, 100): + # v.update([0, 0, theta]) + # plt.pause(0.1) +
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/mobile/animations.html b/_modules/roboticstoolbox/mobile/animations.html new file mode 100644 index 00000000..e15a81a7 --- /dev/null +++ b/_modules/roboticstoolbox/mobile/animations.html @@ -0,0 +1,582 @@ + + + + + + + + + + roboticstoolbox.mobile.animations — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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+ + + + + + + + + + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/mobile/bug2.html b/_modules/roboticstoolbox/mobile/bug2.html new file mode 100644 index 00000000..b07b96e3 --- /dev/null +++ b/_modules/roboticstoolbox/mobile/bug2.html @@ -0,0 +1,714 @@ + + + + + + + + + + roboticstoolbox.mobile.bug2 — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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+ + + + + + + + + + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/mobile/drivers.html b/_modules/roboticstoolbox/mobile/drivers.html new file mode 100644 index 00000000..b6f6c872 --- /dev/null +++ b/_modules/roboticstoolbox/mobile/drivers.html @@ -0,0 +1,420 @@ + + + + + + roboticstoolbox.mobile.drivers — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.mobile.drivers

+"""
+Python Vehicle
+@Author: Peter Corke, original MATLAB code and Python version
+@Author: Kristian Gibson, initial MATLAB port
+"""
+from abc import ABC, abstractmethod
+import warnings
+from math import pi, sin, cos, tan, atan2
+import numpy as np
+from scipy import integrate, linalg, interpolate
+
+import matplotlib.pyplot as plt
+from matplotlib import patches
+import matplotlib.transforms as mtransforms
+
+from spatialmath import SE2, base
+import roboticstoolbox as rtb
+
+
+
[docs]class VehicleDriverBase(ABC): + """ + Abstract Vehicle driver class + + Abtract class that can drive a mobile robot. + + :seealso: :class:`RandomPath` + """ + +
[docs] @abstractmethod + def demand(self): + """ + Compute speed and heading + + :return: speed and steering for :class:`VehicleBase` + + When an instance of a :class:`VehicleDriverBase` class is attached as + the control for an instance of a :class:`VehicleBase` class, this method + is called at each time step to provide the control input. + + Has access to the vehicle and its state through the :meth:`vehicle` + property. + """ + pass
+ +
[docs] @abstractmethod + def init(self): + """ + Initialize driving agent + + Called at the start of a simulation run. Used to initialize state + including random number generator state. + """ + pass
+ + @property + def vehicle(self): + """ + Set/get the vehicle under control + + :getter: return :class:`VehicleBase` instance + :setter: set :class:`VehicleBase` instance + + .. note:: The setter is invoked by ``vehicle.control = driver`` + """ + return self._veh + + @vehicle.setter + def vehicle(self, v): + + self._veh = v + + def __repr__(self): + return str(self)
+ + +# ========================================================================= # + + +
[docs]class RandomPath(VehicleDriverBase): +
[docs] def __init__( + self, + workspace, + speed=1, + dthresh=0.05, + seed=0, + headinggain=0.3, + goalmarkerstyle=None, + ): + """ + Driving agent for random path + + :param workspace: dimension of workspace, see :func:`spatialmath.base.exand_dims` + :type workspace: scalar, array_like(2), array_like(4) + :param speed: forward speed, defaults to 1 + :type speed: float, optional + :param dthresh: distance threshold, defaults to 0.05 + :type dthresh: float, optional + + :raises ValueError: bad workspace specified + + Returns a *driver* object that drives the attached vehicle to a + sequence of random waypoints. + + The driver is connected to the vehicle by:: + + Vehicle(control=driver) + + or:: + + veh = Vehicle() + veh.control = driver + + The waypoints are positioned inside a rectangular region defined by + the vehicle that is specified by (see ``plotvol2``): + + ============== ======= ======= + ``workspace`` x-range y-range + ============== ======= ======= + A (scalar) -A:A -A:A + [A, B] A:B A:B + [A, B, C, D] A:B C:D + ============== ======= ======= + + + .. note:: + - It is possible in some cases for the vehicle to move outside the desired + region, for instance if moving to a waypoint near the edge, the limited + turning circle may cause the vehicle to temporarily move outside. + - The vehicle chooses a new waypoint when it is closer than ``dthresh`` + to the current waypoint. + - Uses its own random number generator so as to not influence the performance + of other randomized algorithms such as path planning. Set ``seed=None`` to have it randomly initialized from the + operating system. + + :seealso: :class:`Bicycle` :class:`Unicycle` :func:`~spatialmath.base.graphics.plotvol2` + """ + + # TODO options to specify region, maybe accept a Map object? + + if hasattr(workspace, "workspace"): + # workspace can be defined by an object with a workspace attribute + self._workspace = base.expand_dims(workspace.workspace) + else: + self._workspace = base.expand_dims(workspace) + + self._speed = speed + self._dthresh = dthresh * np.diff(self._workspace[0:2]) + self._goal_marker = None + if goalmarkerstyle is None: + self._goal_marker_style = { + "marker": "D", + "markersize": 6, + "color": "r", + "linestyle": "None", + } + else: + self._goal_marker_style = goalmarkerstyle + self._headinggain = headinggain + + self._d_prev = np.inf + self._random = np.random.default_rng(seed) + self._seed = seed + self.verbose = True + self._goal = None + self._dthresh = dthresh * max( + self._workspace[1] - self._workspace[0], + self._workspace[3] - self._workspace[2], + ) + + self._veh = None
+ +
[docs] def __str__(self): + """%RandomPath.char Convert to string + % + % s = R.char() is a string showing driver parameters and state in in + % a compact human readable format.""" + + s = "RandomPath driver object\n" + s += ( + f" X {self._workspace[0]} : {self._workspace[1]}; Y {self._workspace[0]} :" + f" {self._workspace[1]}, dthresh={self._dthresh}\n" + ) + s += f" current goal={self._goal}" + return s
+ + @property + def workspace(self): + """ + Size of robot driving workspace + + :return: workspace bounds [xmin, xmax, ymin, ymax] + :rtype: ndarray(4) + + Returns the bounds of the workspace as specified by constructor + option ``workspace`` + """ + return self._workspace + +
[docs] def init(self, ax=None): + """ + Initialize random path driving agent + + :param ax: axes in which to draw via points, defaults to None + :type ax: Axes, optional + + Called at the start of a simulation run. Ensures that the + random number generated is reseeded to ensure that + the sequence of random waypoints is repeatable. + """ + + if self._seed is not None: + self._random = np.random.default_rng(self._seed) + + self._goal = None + # delete(driver.h_goal); % delete the goal + # driver.h_goal = []; + if ax is not None: + self._goal_marker = plt.plot( + np.nan, np.nan, **self._goal_marker_style, label="random waypoint" + )[0]
+ +
[docs] def demand(self): + """ + Compute speed and heading for random waypoint + % + % [SPEED,STEER] = R.demand() is the speed and steer angle to + % drive the vehicle toward the next waypoint. When the vehicle is + % within R.dtresh a new waypoint is chosen. + % + % See also Vehicle.""" + + if self._goal is None: + self._new_goal() + + # if nearly at goal point, choose the next one + d = np.linalg.norm(self._veh._x[0:2] - self._goal) + if d < self._dthresh or abs(d - self._d_prev) < 1e-3: + self._new_goal() + # elif d > 2 * self._d_prev: + # self.choose_goal() + self._d_prev = d + + speed = self._speed + + goal_heading = atan2( + self._goal[1] - self._veh._x[1], self._goal[0] - self._veh._x[0] + ) + delta_heading = base.angdiff(goal_heading, self._veh._x[2]) + + return np.r_[speed, self._headinggain * delta_heading]
+ + ## private method, invoked from demand() to compute a new waypoint + + def _new_goal(self): + + # choose a uniform random goal within inner 80% of driving area + while True: + r = self._random.uniform(0.1, 0.9) + gx = self._workspace[0:2] @ np.r_[r, 1 - r] + + r = self._random.uniform(0.1, 0.9) + gy = self._workspace[2:4] @ np.r_[r, 1 - r] + + self._goal = np.r_[gx, gy] + + # check not too close to last goal + if np.linalg.norm(self._goal - self._veh._x[0:2]) > 2 * self._dthresh: + break + + if self._veh.verbose: + print(f"set goal: {self._goal}") + + # update the goal marker + if self._goal_marker is not None: + self._goal_marker.set_xdata(self._goal[0]) + self._goal_marker.set_ydata(self._goal[1])
+ + +# ========================================================================= # + + +class PurePursuit(VehicleDriverBase): + def __init__(self, speed=1, radius=1): + pass + + def __str__(self): + pass + + def init(self): + pass + + def demand(self): + pass + + +# ========================================================================= # + +if __name__ == "__main__": + + import unittest +
+ +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/mobile/dx_form.html b/_modules/roboticstoolbox/mobile/dx_form.html new file mode 100644 index 00000000..4d807ae4 --- /dev/null +++ b/_modules/roboticstoolbox/mobile/dx_form.html @@ -0,0 +1,497 @@ + + + + + + + + + + roboticstoolbox.mobile.dx_form — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + + +
+ + + + + +
+ + +
+ +
+ +
+ + + + + + + + + + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/mobile/landmarkmap.html b/_modules/roboticstoolbox/mobile/landmarkmap.html new file mode 100644 index 00000000..80d1b9ae --- /dev/null +++ b/_modules/roboticstoolbox/mobile/landmarkmap.html @@ -0,0 +1,301 @@ + + + + + + roboticstoolbox.mobile.landmarkmap — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+
    +
  • + + +
  • +
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+
+
+
+
+ +

Source code for roboticstoolbox.mobile.landmarkmap

+from abc import ABC
+import numpy as np
+import scipy as sp
+from math import pi, sin, cos
+import matplotlib.pyplot as plt
+from spatialmath import base
+import roboticstoolbox as rtb
+
+
+
[docs]class LandmarkMap: + """ + Map of planar point landmarks + + :param map: map or number of landmarks + :type map: ndarray(2, N) or int + :param workspace: workspace or map bounds, defaults to 10 + :type workspace: scalar, array_like(2), array_like(4), optional + :param verbose: display debug information, defaults to True + :type verbose: bool, optional + :param seed: random number seed, defaults to 0 + :type seed: int, optional + :return: a landmark map object + :rtype: LandmarkMap + + A LandmarkMap object represents a rectangular 2D environment with a number + of point landmarks. + + The landmarks can be specified explicitly or be uniform randomly positioned + inside a region defined by the workspace. The workspace can be numeric: + + ============== ======= ======= + ``workspace`` x-range y-range + ============== ======= ======= + A (scalar) -A:A -A:A + [A, B] A:B A:B + [A, B, C, D] A:B C:D + ============== ======= ======= + + or any object that has a ``workspace`` attribute. + + Example: + + .. runblock:: pycon + + >>> from roboticstoolbox import LandmarkMap + >>> map = LandmarkMap(20) + >>> print(map) + >>> print(map[3]) # coordinates of landmark 3 + + The object is an iterator that returns consecutive landmark coordinates. + + :Reference: + + Robotics, Vision & Control, Chap 6, + Peter Corke, + Springer 2011 + + See also :class:`~roboticstoolbox.mobile.sensors.RangeBearingSensor` :class:`~roboticstoolbox.mobile.EKF` + """ + +
[docs] def __init__(self, map, workspace=10, verbose=True, seed=0): + + try: + self._workspace = workspace.workspace + except: + self._workspace = base.expand_dims(workspace) + + if base.ismatrix(map, (2, None)): + self._map = map + self._nlandmarks = map.shape[1] + elif isinstance(map, int): + self._nlandmarks = map + + random = np.random.default_rng(seed) + x = random.uniform(self._workspace[0], self._workspace[1], self._nlandmarks) + y = random.uniform(self._workspace[2], self._workspace[3], self._nlandmarks) + self._map = np.c_[x, y].T + + else: + raise ValueError("bad type for map") + + self._verbose = verbose
+ + def __str__(self): + # s = M.char() is a string showing map parameters in + # a compact human readable format. + ws = self._workspace + s = f"LandmarkMap object with {self._nlandmarks} landmarks, workspace=" + s += f"({ws[0]}: {ws[1]}, {ws[2]}: {ws[3]})" + return s + + def __repr__(self): + return str(self) + +
[docs] def __len__(self): + """ + Number of landmarks in map + + :return: number of landmarks in the map + :rtype: int + """ + return self._nlandmarks
+ + @property + def landmarks(self): + """ + xy-coordinates of all landmarks + + :return: xy-coordinates for landmark points + :rtype: ndarray(2, n) + """ + return self._map + + @property + def workspace(self): + """ + Size of map workspace + + :return: workspace bounds [xmin, xmax, ymin, ymax] + :rtype: ndarray(4) + + Returns the bounds of the workspace as specified by constructor + option ``workspace``. + """ + return self._workspace + +
[docs] def __getitem__(self, k): + """ + Get landmark coordinates from map + + :param k: landmark index + :type k: int + :return: coordinate :math:`(x,y)` of k'th landmark + :rtype: ndarray(2) + """ + return self._map[:, k]
+ +
[docs] def plot(self, labels=False, block=None, **kwargs): + """ + Plot landmark map + + :param labels: number the points on the plot, defaults to False + :type labels: bool, optional + :param block: block until figure is closed, defaults to False + :type block: bool, optional + :param kwargs: :meth:`~matplotlib.axes.Axes.plot` options + + Plot landmark points using Matplotlib options. Default style is black + crosses. + """ + + ax = base.plotvol2(self._workspace) + ax.set_aspect("equal") + + ax.set_xlabel("x") + ax.set_ylabel("y") + + if len(kwargs) == 0: + kwargs = { + "linewidth": 0, + "marker": "x", + "color": "black", + "linestyle": "None", + } + + if "label" not in kwargs: + kwargs["label"] = "landmark" + + # plt.plot(self._map[0,:], self._map[1,:], , **kwargs) + if labels: + labels = "#{}" + else: + labels = None + base.plot_point(self._map, text=labels, **kwargs) + plt.grid(True) + if block is not None: + plt.show(block=block)
+ + +if __name__ == "__main__": + import unittest +
+ +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/mobile/navigation.html b/_modules/roboticstoolbox/mobile/navigation.html new file mode 100644 index 00000000..19b0b7da --- /dev/null +++ b/_modules/roboticstoolbox/mobile/navigation.html @@ -0,0 +1,769 @@ + + + + + + + + + + roboticstoolbox.mobile.navigation — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + + +
+ + + + + +
+ + +
+ +
+ +
+ + + + + + + + + + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/mobile/sensors.html b/_modules/roboticstoolbox/mobile/sensors.html new file mode 100644 index 00000000..37d09b50 --- /dev/null +++ b/_modules/roboticstoolbox/mobile/sensors.html @@ -0,0 +1,892 @@ + + + + + + roboticstoolbox.mobile.sensors — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+
    +
  • + + +
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+
+
+
+
+ +

Source code for roboticstoolbox.mobile.sensors

+from abc import ABC
+import numpy as np
+import scipy as sp
+from math import pi, sin, cos
+import matplotlib.pyplot as plt
+from spatialmath import base
+import roboticstoolbox as rtb
+from collections.abc import Iterable
+
+"""
+Sensor Sensor superclass
+
+An abstract superclass to represent robot navigation sensorself.
+
+Methods::
+  plot        plot a line from robot to map feature
+  display     print the parameters in human readable form
+  char        convert to string
+
+Properties::
+robot   The Vehicle object on which the sensor is mounted
+map     The PointMap object representing the landmarks around the robot
+
+Reference::
+
+  Robotics, Vision & Control,
+  Peter Corke,
+  Springer 2011
+
+See also RangeBearingSensor, EKF, Vehicle, Landmarkself.
+
+"""
+
+
+class SensorBase(ABC):
+    # TODO, pose option, wrt vehicle
+
+    # robot
+    # map
+
+    # verbose
+
+    # ls
+    # animate     # animate sensor measurements
+    # interval    # measurement return subsample factor
+    # fail
+    # delay
+
+    def __init__(
+        self,
+        robot,
+        map,
+        every=1,
+        fail=[],
+        plot=False,
+        delay=0.1,
+        seed=0,
+        verbose=False,
+    ):
+        """Sensor.Sensor Sensor object constructor
+        %
+        # S = Sensor(VEHICLE, MAP, OPTIONS) is a sensor mounted on a vehicle
+        # described by the Vehicle subclass object VEHICLE and observing landmarks
+        # in a map described by the LandmarkMap class object self.
+        %
+        # Options::
+        # 'animate'    animate the action of the laser scanner
+        # 'ls',LS      laser scan lines drawn with style ls (default 'r-')
+        # 'skip', I    return a valid reading on every I'th call
+        # 'fail',T     sensor simulates failure between timesteps T=[TMIN,TMAX]
+        %
+        # Notes::
+        # - Animation shows a ray from the vehicle position to the selected
+        #   landmark.
+        """
+        self._robot = robot
+        self._map = map
+        self._every = every
+        self._fail = fail
+
+        self._verbose = verbose
+
+        self.delay = 0.1
+
+        self._animate = plot
+
+        self._seed = seed
+        self.init()
+
+    def init(self):
+        """
+        Initialize sensor (superclass)
+
+        - reseed the random number generator
+        - reset the counter for handling the ``every`` and ``fail`` options
+        """
+        self._random = np.random.default_rng(self._seed)
+        self._count = 0
+
+    def __str__(self):
+        """
+        Convert sensor parameters to a string (superclass)
+        %
+        # s = self.char() is a string showing sensor parameters in
+        # a compact human readable format.
+        """
+        s = f"{self.__class__.__name__} sensor class\n"
+        s += "  " + str(self.map)
+        return s
+
+    def __repr__(self):
+        return str(self)
+
+    @property
+    def robot(self):
+        """
+        Robot associated with sensor (superclass)
+
+        :return: robot
+        :rtype: :class:`VehicleBase` subclass
+        """
+        return self._robot
+
+    @property
+    def map(self):
+        """
+        Landmark map associated with sensor (superclass)
+
+        :return: robot
+        :rtype: :class:`LandmarkMap`
+        """
+        return self._map
+
+    @property
+    def random(self):
+        """
+        Get private random number generator (superclass)
+
+        :return: NumPy random number generator
+        :rtype: :class:`numpy.random.Generator`
+
+        Has methods including:
+
+            - :meth:`integers(low, high, size, endpoint) <numpy.random.Generator.integers>`
+            - :meth:`random(size) <numpy.random.Generator.random>`
+            - :meth:`uniform(low, high, size) <numpy.random.Generator.uniform>`
+            - :meth:`normal(mean, std, size) <numpy.random.Generator.normal>`
+            - :meth:`multivariate_normal(mean, covar, size) <numpy.random.Generator.multivariate_normal>`
+
+        The generator is initialized with the seed provided at constructor
+        time every time :meth:`init` is called.
+
+        :seealso: :meth:`init`
+        """
+        return self._random
+
+    @property
+    def verbose(self):
+        """
+        Get verbosity state
+
+        :return: verbosity
+        :rtype: bool
+        """
+        return self._verbose
+
+    def plot(self, id):
+        """
+        Plot sensor observation
+
+        :param id: landmark id
+        :type id: int
+
+        Draws a line from the robot to landmark ``id``.
+
+        .. note::
+            - The line is drawn using the ``line_style`` given at constructor time
+
+        """
+        pass
+
+        # if isempty(self.ls)
+        #     return
+        # end
+
+        # h = findobj(gca, 'tag', 'sensor')
+        # if isempty(h)
+        #     # no sensor line, create one
+        #     h = plot(0, 0, self.ls, 'tag', 'sensor')
+        # end
+
+        # # there is a sensor line animate it
+
+        # if lm_id == 0
+        #     set(h, 'Visible', 'off')
+        # else
+        #     xi = self.self.map(:,lm_id)
+        #     set(h, 'Visible', 'on', 'XData', [self.robot.x[0], xi[0]], 'YData', [self.robot.x[1], xi[1]])
+        # end
+        # pause(self.delay)
+
+        # drawnow
+
+
+# ======================================================================== #
+
+# visibility function, for one id, or return list of visible
+# covar can be 2x2 or (2,)
+# .W property
+
[docs]class RangeBearingSensor(SensorBase): +
[docs] def __init__( + self, + robot, + map, + line_style=None, + poly_style=None, + covar=None, + range=None, + angle=None, + plot=False, + seed=0, + **kwargs, + ): + + r""" + Range and bearing angle sensor + + :param robot: model of robot carrying the sensor + :type robot: :class:`VehicleBase` subclass + :param map: map of landmarks + :type map: :class:`LandmarkMap` instance + :param polygon: polygon style for sensing region, see :class:`~spatialmath.base.graphics.plot_polygon`, defaults to None + :type polygon: dict, optional + :param covar: covariance matrix for sensor readings, defaults to None + :type covar: ndarray(2,2), optional + :param range: maximum range :math:`r_{max}` or range span :math:`[r_{min}, r_{max}]`, defaults to None + :type range: float or array_like(2), optional + :param angle: angular field of view, from :math:`[-\theta, \theta]` defaults to None + :type angle: float, optional + :param plot: animate the sensor beams, defaults to False + :type plot: bool, optional + :param seed: random number seed, defaults to 0 + :type seed: int, optional + :param kwargs: arguments passed to :class:`SensorBase` + + Sensor object that returns the range and bearing angle :math:`(r, + \beta)` to a point landmark from a robot-mounted sensor. The sensor + measurements are corrupted with zero-mean Gaussian noise with covariance + ``covar``. + + The sensor can have a maximum range, or a minimum and maximum range. The + sensor can also have a restricted angular field of view. + + The sensing region can be displayed by setting the polygon parameter + which can show an outline or a filled polygon. This is updated every + time :meth:`reading` is called, based on the current configuration of + the ``robot``. + + .. runblock:: pycon + + >>> from roboticstoolbox import Bicycle, LandmarkMap, RangeBearingSensor + >>> from math import pi + >>> robot = Bicycle() + >>> map = LandmarkMap(20) + >>> sensor = RangeBearingSensor(robot, map, range=(0.5, 20), angle=pi/4) + >>> print(sensor) + + :seealso: :class:`~roboticstoolbox.mobile.LandmarkMap` :class:`~roboticstoolbox.mobile.EKF` + """ + + # TODO change plot option to animate, but RVC3 uses plot + + # call the superclass constructor + super().__init__(robot, map, **kwargs) + + self._line_style = line_style + self._poly_style = poly_style + + if covar is None: + self._W = np.zeros((2, 2)) + elif base.isvector(covar, 2): + self._W = np.diag(covar) + elif base.ismatrix(covar, (2, 2)): + self._W = covar + else: + raise ValueError("bad value for covar, must have shape (2,) or (2,2)") + + if range is None: + self._r_range = None + elif isinstance(range, Iterable): + self._r_range = base.getvector(range, 2) + else: + self._r_range = [0, range] + + if angle is None: + self._theta_range = None + elif isinstance(angle, Iterable): + self._theta_range = base.getvector(angle, 2) + else: + self._theta_range = [-angle, angle] + + self._animate = plot + self._landmarklog = [] + + self._random = np.random.default_rng(seed)
+ + def __str__(self): + s = super().__str__() + s += f"\n W = {base.array2str(self._W)}\n" + + s += f" sampled every {self._every} samples\n" + if self._r_range is not None: + s += f" range: ({self._r_range[0]}: {self._r_range[1]})\n" + if self._theta_range is not None: + s += f" angle: ({self._theta_range[0]:.3g}: {self._theta_range[1]:.3g})\n" + return s.rstrip() + +
[docs] def init(self): + """ + Initialize sensor + + - reseed the random number generator + - reset the counter for handling the ``every`` and ``fail`` options + - reset the landmark log + - initalize plots + + :seealso: :meth:`SensorBase.init` + """ + super().init() + self._landmarklog = [] + + if self._animate: + self.map.plot()
+ + @property + def W(self): + """ + Get sensor covariance + + :return: sensor covariance + :rtype: ndarray(2,2) + + Returns the value of the sensor covariance matrix passed to + the constructor. + """ + return self._covar + +
[docs] def reading(self): + r""" + Choose landmark and return observation + + :return: range and bearing angle to a landmark, and landmark id + :rtype: ndarray(2), int + + Returns an observation of a random visible landmark (range, bearing) and + the ``id`` of that landmark. The landmark is chosen randomly from the + set of all visible landmarks, those within the angular field of view and + range limit. + + If constructor argument ``every`` is set then only return a valid + reading on every ``every`` calls. + + If constructor argument ``fail`` is set then do not return a reading + during that specified time interval. + + If no valid reading is available then return (None, None) + + .. runblock:: pycon + + >>> from roboticstoolbox import Bicycle, LandmarkMap, RangeBearingSensor + >>> from math import pi + >>> robot = Bicycle() + >>> map = LandmarkMap(20) + >>> sensor = RangeBearingSensor(robot, map, range=(0.5, 20), angle=pi/4) + >>> print(sensor.reading()) + + .. note:: + + - Noise with covariance ``W`` (set by constructor) is added to the + reading + - If ``animate`` option is set then show a line from the vehicle to + the landmark + - If ``animate`` option set and the angular and distance limits + are set then display the sensor field of view as a polygon. + + :seealso: :meth:`h` + """ + + # TODO probably should return K=0 to indicated invalid + + # model a sensor that emits readings every interval samples + self._count += 1 + + # check conditions for NOT returning a value + z = [] + lm_id = -1 + # sample interval + if self._count % self._every != 0: + self._landmarklog.append(lm_id) + return (None, None) + + # simulated failure, fail is a list of 2-tuples giving (start,end) times + # for a sensor failure + if self._fail is not None: + if any([start <= self._count < end for start, end in self._fail]): + self._landmarklog.append(lm_id) + return (None, None) + + # create a polygon to indicate the active sensing area based on range+angle limits + # if self.animate && ~isempty(self.theta_range) && ~isempty(self.r_range) + # h = findobj(gca, 'tag', 'sensor-area') + # if isempty(h) + + # th=linspace(self.theta_range[0], self.theta_range[1], 20) + # x = self.r_range[1] * cos(th) + # y = self.r_range[1] * sin(th) + # if self.r_range[0] > 0 + # th = flip(th) + # x = [x self.r_range[0] * cos(th)] + # y = [y self.r_range[0] * sin(th)] + # else + # x = [x 0] + # y = [y 0] + # end + # # no sensor zone, create one + # plot_poly([x; y], 'fillcolor', 'r', 'alpha', 0.1, 'edgecolor', 'none', 'animate', 'tag', 'sensor-area') + # else + # hg = get(h, 'Parent') + # plot_poly(h, self.robot.x) + + zk = self.visible() + if len(zk) > 1: + # more than 1 visible landmark, pick a random one + i = self._random.integers(len(zk)) + z = zk[i][0] + lm_id = zk[i][1] + if self.verbose: + print(f"Sensor:: feature {lm_id}: ({z[0]}, {z[1]})") + elif len(zk) == 1: + # just 1 visible landmark + z = zk[0][0] + lm_id = zk[0][1] + if self.verbose: + print(f"Sensor:: feature {lm_id}: ({z[0]}, {z[1]})") + else: + if self.verbose: + print("Sensor:: no features\n") + self._landmarklog.append(lm_id) + return (None, None) + + # compute the range and bearing from robot to feature + # z = self.h(self.robot.x, lm_id) + + if self._animate: + self.plot(lm_id) + + # add the reading to the landmark log + self._landmarklog.append(lm_id) + + # add noise with covariance W + z += self._random.multivariate_normal((0, 0), self._W) + + return z, lm_id
+ +
[docs] def visible(self): + """ + List of all visible landmarks + + :return: list of visible landmarks + :rtype: list of int + + Return a list of the id of all landmarks that are visible, that is, it + lies with the sensing range and field of view of the sensor at the + robot's current configuration. + + :seealso: :meth:`isvisible` :meth:`h` + """ + # get range/bearing to all landmarks + z = self.h(self.robot.x) + zk = [(z, k) for k, z in enumerate(z)] + # a list of tuples, each tuple is ((range, bearing), k) + + if self._r_range is not None: + zk = filter(lambda zk: self._r_range[0] <= zk[0][0] <= self._r_range[1], zk) + + if self._theta_range is not None: + # find all within angular range as well + zk = filter( + lambda zk: self._theta_range[0] <= zk[0][1] <= self._theta_range[1], zk + ) + + return list(zk)
+ +
[docs] def isvisible(self, id): + """ + Test if landmark is visible + + :param id: landmark id + :type id: int + :return: visibility + :rtype: bool + + The landmark ``id`` is visible if it lies with the sensing range and + field of view of the sensor at the robot's current configuration. + + :seealso: :meth:`visible` :meth:`h` + """ + z = self.h(self.robot.x, id) + + return ( + (self._r_range is None) or self._r_range[0] <= z[0] <= self._r_range[1] + ) and ( + (self._theta_range is None) + or self._theta_range[0] <= z[1] <= self._theta_range[1] + )
+ +
[docs] def h(self, x, landmark=None): + r""" + Landmark observation function + + :param x: vehicle state :math:`(x, y, \theta)` + :type x: array_like(3), array_like(N,3) + :param landmark: landmark id or position, defaults to None + :type landmark: int or array_like(2), optional + :return: range and bearing angle to landmark math:`(r,\beta)` + :rtype: ndarray(2) or ndarray(N,2) + + Return the range and bearing to a landmark: + + - ``.h(x)`` is range and bearing to all landmarks, one row per landmark + - ``.h(x, id)`` is range and bearing to landmark ``id`` + - ``.h(x, p)`` is range and bearing to landmark with coordinates ``p`` + + .. runblock:: pycon + + >>> from roboticstoolbox import Bicycle, LandmarkMap, RangeBearingSensor + >>> from math import pi + >>> robot = Bicycle() + >>> map = LandmarkMap(20) + >>> sensor = RangeBearingSensor(robot, map, range=(0.5, 20), angle=pi/4) + >>> z = sensor.h((1, 2, pi/2), 3) + >>> print(z) + + .. note:: + - Noise with covariance (property ``W``) is added to each row of ``z``. + - Performs fast vectorized operation where ``x`` is an ndarray(n,3). + - The landmark is assumed to be visible, field of view and range limits are not + applied. + + :seealso: :meth:`reading` :meth:`Hx` :meth:`Hw` :meth:`Hp` + """ + # get the landmarks, one per row + + if isinstance(x, np.ndarray) and x.ndim == 2: + # x is Nx3 set of vehicle states, do vectorized form + # used by particle filter + x, y, t = x.T + else: + x, y, t = x + + if landmark is None: + # self.h(XV) all landmarks + dx = self.map.landmarks[0, :] - x + dy = self.map.landmarks[1, :] - y + elif base.isinteger(landmark): + # landmark id + # self.h(XV, JF) + xlm = self.map[landmark] + dx = xlm[0] - x + dy = xlm[1] - y + else: + # landmark position + # self.h(XV, XF) + xlm = base.getvector(landmark, 2) + dx = xlm[0] - x + dy = xlm[1] - y + + # compute range and bearing (Vectorized code) + + z = np.c_[ + np.sqrt(dx**2 + dy**2), base.angdiff(np.arctan2(dy, dx), t) + ] # range & bearing as columns + + if z.shape[0] == 1: + return z[0] + else: + return z
+ +
[docs] def Hx(self, x, landmark): + r""" + Jacobian dh/dx + + :param x: vehicle state :math:`(x, y, \theta)` + :type x: array_like(3) + :param arg: landmark id or coordinate + :type arg: int or array_like(2) + :return: Jacobian matrix + :rtype: ndarray(2,3) + + Compute the Jacobian of the observation function with respect to vehicle + configuration :math:`\partial h/\partial x` + + - ``sensor.Hx(q, id)`` is Jacobian for landmark ``id`` + - ``sensor.h(q, p)`` is Jacobian for landmark with coordinates ``p`` + + :seealso: :meth:`h` :meth:`Hp` :meth:`Hw` + """ + + if base.isinteger(landmark): + # landmark index provided + xf = self.map[landmark] + else: + # assume it is a coordinate + xf = base.getvector(landmark, 2) + + Delta = xf - x[:2] + r = base.norm(Delta) + # fmt: off + return np.array([ + [-Delta[0] / r, -Delta[1] / r, 0], + [ Delta[1] / r**2, -Delta[0] / r**2, -1], + ])
+ # fmt: on + +
[docs] def Hp(self, x, landmark): + r""" + Jacobian dh/dp + + :param x: vehicle state :math:`(x, y, \theta)` + :type x: array_like(3) + :param arg: landmark id or coordinate + :type arg: int or array_like(2) + :return: Jacobian matrix + :rtype: ndarray(2,2) + + Compute the Jacobian of the observation function with respect + to landmark position :math:`\partial h/\partial p` + + - ``sensor.Hp(x, id)`` is Jacobian for landmark ``id`` + - ``sensor.Hp(x, p)`` is Jacobian for landmark with coordinates ``p`` + + :seealso: :meth:`h` :meth:`Hx` :meth:`Hw` + """ + if base.isinteger(landmark): + xf = self.map.landmark(landmark) + else: + xf = landmark + x = base.getvector(x, 3) + + Delta = xf - x[:2] + r = base.norm(Delta) + # fmt: off + return np.array([ + [ Delta[0] / r, Delta[1] / r], + [-Delta[1] / r**2, Delta[0] / r**2], + ])
+ # fmt: on + +
[docs] def Hw(self, x, landmark): + r""" + Jacobian dh/dw + + :param x: vehicle state :math:`(x, y, \theta)` + :type x: array_like(3) + :param arg: landmark id or coordinate + :type arg: int or array_like(2) + :return: Jacobian matrix + :rtype: ndarray(2,2) + + Compute the Jacobian of the observation function with respect + to sensor noise :math:`\partial h/\partial w` + + - ``sensor.Hw(x, id)`` is Jacobian for landmark ``id`` + - ``sensor.Hw(x, p)`` is Jacobian for landmark with coordinates ``p`` + + .. note:: ``x`` and ``landmark`` are not used to compute this. + + :seealso: :meth:`h` :meth:`Hx` :meth:`Hp` + """ + return np.eye(2)
+ +
[docs] def g(self, x, z): + r""" + Landmark position from sensor observation + + :param x: vehicle state :math:`(x, y, \theta)` + :type x: array_like(3) + :param z: landmark observation :math:`(r, \beta)` + :type z: array_like(2) + :return: landmark position :math:`(x, y)` + :rtype: ndarray(2) + + Compute the world coordinate of a landmark given + the observation ``z`` from a vehicle state with ``x``. + + :seealso: :meth:`h` :meth:`Gx` :meth:`Gz` + """ + + range = z[0] + bearing = z[1] + x[2] # bearing angle in vehicle frame + + # fmt: off + return np.r_[ + x[0] + range * cos(bearing), + x[1] + range * sin(bearing) + ]
+ # fmt: on + +
[docs] def Gx(self, x, z): + r""" + Jacobian dg/dx + + :param x: vehicle state :math:`(x, y, \theta)` + :type x: array_like(3) + :param z: landmark observation :math:`(r, \beta)` + :type z: array_like(2) + :return: Jacobian matrix + :rtype: ndarray(2,3) + + Compute the Jacobian of the landmark position function with respect + to landmark position :math:`\partial g/\partial x` + + :seealso: :meth:`g` + """ + theta = x[2] + r, bearing = z + + # fmt: off + return np.array([ + [1, 0, -r*sin(theta + bearing)], + [0, 1, r*cos(theta + bearing)], + ])
+ # fmt: on + +
[docs] def Gz(self, x, z): + r""" + Jacobian dg/dz + + :param x: vehicle state :math:`(x, y, \theta)` + :type x: array_like(3) + :param z: landmark observation :math:`(r, \beta)` + :type z: array_like(2) + :return: Jacobian matrix + :rtype: ndarray(2,2) + + Compute the Jacobian of the landmark position function with respect + to sensor observation :math:`\partial g/\partial z` + + :seealso: :meth:`g` + """ + theta = x[2] + r, bearing = z + # fmt: off + return np.array([ + [cos(theta + bearing), -r * sin(theta + bearing)], + [sin(theta + bearing), r * cos(theta + bearing)], + ])
+ # fmt: on + + +if __name__ == "__main__": + + from roboticstoolbox import Bicycle, LandmarkMap, RangeBearingSensor + from math import pi + + robot = Bicycle() + map = LandmarkMap(20) + sensor = RangeBearingSensor(robot, map, range=(0.5, 20), angle=pi / 4) + print(sensor.reading()) + print(sensor.visible()) + print(sensor.isvisible(3)) + print(sensor.isvisible(4)) +
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/mobile/vehicle.html b/_modules/roboticstoolbox/mobile/vehicle.html new file mode 100644 index 00000000..cdf08a7e --- /dev/null +++ b/_modules/roboticstoolbox/mobile/vehicle.html @@ -0,0 +1,1213 @@ + + + + + + + + + + roboticstoolbox.mobile.vehicle — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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+ + + + + + + + + + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/DH/AL5D.html b/_modules/roboticstoolbox/models/DH/AL5D.html new file mode 100644 index 00000000..a57baa7b --- /dev/null +++ b/_modules/roboticstoolbox/models/DH/AL5D.html @@ -0,0 +1,242 @@ + + + + + + roboticstoolbox.models.DH.AL5D — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.DH.AL5D

+"""
+@author: Tassos Natsakis
+"""
+
+import numpy as np
+from roboticstoolbox import DHRobot, RevoluteMDH
+from spatialmath import SE3
+
+
+
[docs]class AL5D(DHRobot): + """ + Class that models a Lynxmotion AL5D manipulator + + :param symbolic: use symbolic constants + :type symbolic: bool + + ``AL5D()`` is an object which models a Lynxmotion AL5D robot and + describes its kinematic and dynamic characteristics using modified DH + conventions. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.DH.AL5D() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration + + .. note:: + - SI units are used. + + :References: + + - 'Reference of the robot <http://www.lynxmotion.com/c-130-al5d.aspx>'_ + + .. codeauthor:: Tassos Natsakis + """ # noqa + + def __init__(self, symbolic=False): + + if symbolic: + import spatialmath.base.symbolic as sym + + # zero = sym.zero() + pi = sym.pi() + else: + from math import pi + + # zero = 0.0 + + # robot length values (metres) + a = [0, 0.002, 0.14679, 0.17751] + d = [-0.06858, 0, 0, 0] + + alpha = [pi, pi / 2, pi, pi] + offset = [pi / 2, pi, -0.0427, -0.0427 - pi / 2] + + # mass data as measured + # mass = [0.187, 0.044, 0.207, 0.081] + + # center of mass as calculated through CAD model + center_of_mass = [ + [0.01724, -0.00389, 0.00468], + [0.07084, 0.00000, 0.00190], + [0.05615, -0.00251, -0.00080], + [0.04318, 0.00735, -0.00523], + ] + + # moments of inertia are practically zero + moments_of_inertia = [ + [0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0], + ] + + joint_limits = [ + [-pi / 2, pi / 2], + [-pi / 2, pi / 2], + [-pi / 2, pi / 2], + [-pi / 2, pi / 2], + ] + + links = [] + + for j in range(3): + link = RevoluteMDH( + d=d[j], + a=a[j], + alpha=alpha[j], + offset=offset[j], + r=center_of_mass[j], + I=moments_of_inertia[j], + G=1, + B=0, + Tc=[0, 0], + qlim=joint_limits[j], + ) + links.append(link) + + tool = SE3(0.07719, 0, 0) + + super().__init__( + links, + name="AL5D", + manufacturer="Lynxmotion", + keywords=("dynamics", "symbolic"), + symbolic=symbolic, + tool=tool, + ) + + # zero angles + self.addconfiguration("home", np.array([pi / 2, pi / 2, pi / 2, pi / 2]))
+ + +if __name__ == "__main__": # pragma nocover + + al5d = AL5D(symbolic=False) + print(al5d) + # print(al5d.dyntable()) +
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/DH/Ball.html b/_modules/roboticstoolbox/models/DH/Ball.html new file mode 100644 index 00000000..c9f1682e --- /dev/null +++ b/_modules/roboticstoolbox/models/DH/Ball.html @@ -0,0 +1,208 @@ + + + + + + roboticstoolbox.models.DH.Ball — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.DH.Ball

+#!/usr/bin/env python
+"""
+@author: Peter Corke
+@author: Samuel Drew
+"""
+
+from roboticstoolbox import DHRobot, RevoluteDH
+from math import pi
+import numpy as np
+
+
+
[docs]class Ball(DHRobot): + """ + Class that models a ball manipulator + + :param N: number of links, defaults to 10 + :type N: int, optional + :param symbolic: [description], defaults to False + :type symbolic: bool, optional + + The ball robot is an *abstract* robot with an arbitrary number of joints. + At zero joint angles it is straight along the x-axis, and as the joint + angles are increased (equally) it wraps up into a 3D ball shape. + + - ``Ball()`` is an object which describes the kinematic characteristics of + a ball robot using standard DH conventions. + + - ``Ball(N)`` as above, but models a robot with ``N`` joints. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.DH.Ball() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angles + - q1, ball shaped configuration + + :references: + + - "A divide and conquer articulated-body algorithm for parallel + O(log(n)) calculation of rigid body dynamics, Part 2", + Int. J. Robotics Research, 18(9), pp 876-892. + + :seealso: :func:`Hyper`, :func:`Ball` + + .. codeauthor:: Peter Corke + """ + + def __init__(self, N=10): + + links = [] + q1 = [] + + for i in range(N): + links.append(RevoluteDH(a=0.1, alpha=pi / 2)) + + # and build a serial link manipulator + super(Ball, self).__init__(links, name="ball") + + self.qr = np.array([_fract(i) for i in range(N)]) + self.qz = np.zeros(N) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+ + +def _fract(i): + # i is "i-1" as per the paper + theta1 = 1 + theta2 = -2 / 3 + + if i == 0: + return theta1 + elif i % 3 == 1: + return theta1 + elif i % 3 == 2: + return theta2 + elif i % 3 == 0: + return _fract(i / 3) + + +if __name__ == "__main__": # pragma nocover + + ball = Ball() + print(ball) +
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/DH/Baxter.html b/_modules/roboticstoolbox/models/DH/Baxter.html new file mode 100644 index 00000000..ab112bb0 --- /dev/null +++ b/_modules/roboticstoolbox/models/DH/Baxter.html @@ -0,0 +1,212 @@ + + + + + + roboticstoolbox.models.DH.Baxter — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.DH.Baxter

+import numpy as np
+from math import pi
+from roboticstoolbox import DHRobot, RevoluteDH
+from spatialmath import SE3
+
+
+
[docs]class Baxter(DHRobot): + """ + Class that models a Baxter manipulator + + :param symbolic: use symbolic constants + :type symbolic: bool + + ``Baxter()`` is an object which models the left arm of the two 7-joint + arms of a Rethink Robotics Baxter robot using standard DH conventions. + + ``Baxter(which)`` as above but models the specified arm and ``which`` is + either 'left' or 'right'. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.DH.Baxter() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration + - qr, vertical 'READY' configuration + - qs, arm is stretched out in the X direction + - qd, lower arm horizontal as per data sheet + + .. note:: SI units are used. + + .. warning:: The base transform is set to reflect the pose of the arm's + shoulder with respect to the base. Changing the base attribute of the + arm will overwrite this, IT DOES NOT CHANGE THE POSE OF BAXTER's base. + Instead do ``baxter.base = newbase * baxter.base``. + + :References: + - "Kinematics Modeling and Experimental Verification of Baxter Robot" + Z. Ju, C. Yang, H. Ma, Chinese Control Conf, 2015. + + .. codeauthor:: Peter Corke + """ # noqa + + def __init__(self, arm="left"): + + links = [ + RevoluteDH(d=0.27, a=0.069, alpha=-pi / 2), + RevoluteDH(d=0, a=0, alpha=pi / 2, offset=pi / 2), + RevoluteDH(d=0.102 + 0.262, a=0.069, alpha=-pi / 2), + RevoluteDH(d=0, a=0, alpha=pi / 2), + RevoluteDH(d=0.103 + 0.271, a=0.010, alpha=-pi / 2), + RevoluteDH(d=0, a=0, alpha=pi / 2), + RevoluteDH(d=0.28, a=0, alpha=0), + ] + + super().__init__( + links, + name=f"Baxter-{arm}", + manufacturer="Rethink Robotics", + ) + + self.qr = np.array([0, -pi / 2, -pi / 2, 0, 0, 0, 0]) + self.qz = np.zeros(7) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz) + + # straight and horizontal + self.addconfiguration_attr("qs", np.array([0, 0, -pi / 2, 0, 0, 0, 0])) + + # nominal table top picking pose + self.addconfiguration_attr("qn", np.array([0, pi / 4, pi / 2, 0, pi / 4, 0, 0])) + + if arm == "left": + self.base = SE3(0.064614, 0.25858, 0.119) * SE3.Rz(pi / 4) + else: + self.base = SE3(0.063534, -0.25966, 0.119) * SE3.Rz(-pi / 4)
+ + +if __name__ == "__main__": # pragma nocover + + baxter = Baxter("left") + print(baxter.name, baxter.base) + # baxter.plot(baxter.qz, block=False) + + # baxter = Baxter('right') + # print(baxter) + + # baxter.plot(baxter.qz) +
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/DH/Cobra600.html b/_modules/roboticstoolbox/models/DH/Cobra600.html new file mode 100644 index 00000000..5eca4e2c --- /dev/null +++ b/_modules/roboticstoolbox/models/DH/Cobra600.html @@ -0,0 +1,183 @@ + + + + + + roboticstoolbox.models.DH.Cobra600 — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.DH.Cobra600

+#!/usr/bin/env python
+"""
+@author: Peter Corke
+@author: Samuel Drew
+"""
+
+from roboticstoolbox import DHRobot, RevoluteDH, PrismaticDH
+from math import pi
+import numpy as np
+
+
+
[docs]class Cobra600(DHRobot): + """ + Class that models a Adept Cobra 600 SCARA manipulator + + ``Cobra600()`` is a class which models an Adept Cobra 600 SCARA robot and + describes its kinematic characteristics using standard DH + conventions. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.DH.Cobra600() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration, 'L' shaped configuration + + .. note:: + - SI units are used. + - Robot has only 4 DoF. + + .. codeauthor:: Peter Corke + """ + + def __init__(self): + deg = pi / 180 + + L = [ + RevoluteDH(d=0.387, a=0.325, qlim=[-50 * deg, 50 * deg]), + RevoluteDH(a=0.275, alpha=pi, qlim=[-88 * deg, 88 * deg]), + PrismaticDH(qlim=[0, 0.210]), + RevoluteDH(), + ] + + super().__init__(L, name="Cobra600", manufacturer="Omron") + + # zero angles, L shaped pose + self.addconfiguration_attr("qz", np.array([0, 0, 0, 0])) + + self.qr = np.zeros(4) + self.qz = np.zeros(4) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+ + +if __name__ == "__main__": # pragma nocover + deg = pi / 180 + a = RevoluteDH(d=0.387, a=0.325, qlim=[-50 * deg, 50 * deg]) + cobra = Cobra600() + print(cobra) +
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/DH/Coil.html b/_modules/roboticstoolbox/models/DH/Coil.html new file mode 100644 index 00000000..f93e309e --- /dev/null +++ b/_modules/roboticstoolbox/models/DH/Coil.html @@ -0,0 +1,201 @@ + + + + + + roboticstoolbox.models.DH.Coil — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.DH.Coil

+#!/usr/bin/env python
+"""
+@author: Peter Corke
+"""
+
+import numpy as np
+from roboticstoolbox import DHRobot, RevoluteDH
+
+
+
[docs]class Coil(DHRobot): + """ + Create model of a coil manipulator + + :param N: number of links, defaults to 10 + :type N: int, optional + :param symbolic: [description], defaults to False + :type symbolic: bool, optional + + The coil robot is an *abstract* robot with an arbitrary number of joints + that folds into a helix shape. At zero joint angles it is straight along + the x-axis, and as the joint angles are increased (equally) it wraps up into + a 3D helix shape. + + - ``Coil()`` is an object which describes the kinematic characteristics of + a coil robot using standard DH conventions. + + - ``Coil(N)`` as above, but models a robot with ``N`` joints. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.DH.Coil() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration + + :references: + - "A divide and conquer articulated-body algorithm for parallel O(log(n)) + calculation of rigid body dynamics, Part 2", + Int. J. Robotics Research, 18(9), pp 876-892. + + :seealso: :func:`Hyper`, :func:`Ball` + + .. codeauthor:: Peter Corke + """ # noqa + + def __init__(self, N=10, symbolic=False): + + if symbolic: + import spatialmath.base.symbolic as sym + + pi = sym.pi() + else: + from math import pi + + a = 1 / N + + links = [] + for i in range(N): + links.append(RevoluteDH(a=a, alpha=5 * pi / N)) + + super().__init__( + links, name="Coil" + str(N), keywords=("symbolic",), symbolic=symbolic + ) + + self.qr = np.array(np.ones(N) * 10 * pi / N) + self.qz = np.zeros(N) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+ + +if __name__ == "__main__": # pragma nocover + + coil = Coil(N=10, symbolic=False) + print(coil) + + # print(coil.fkine(coil.qz)) + # print(coil.fkine(coil.qf)) +
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/DH/Hyper.html b/_modules/roboticstoolbox/models/DH/Hyper.html new file mode 100644 index 00000000..17227855 --- /dev/null +++ b/_modules/roboticstoolbox/models/DH/Hyper.html @@ -0,0 +1,205 @@ + + + + + + roboticstoolbox.models.DH.Hyper — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.DH.Hyper

+#!/usr/bin/env python
+"""
+@author: Peter Corke
+"""
+
+
+# from math import pi
+import numpy as np
+from roboticstoolbox import DHRobot, RevoluteDH
+
+
+
[docs]class Hyper(DHRobot): + """ + Create model of a hyper redundant planar manipulator + + :param N: number of links, defaults to 10 + :type N: int, optional + :param a: link length, defaults total ``1/N`` giving a reach of 1 + :type a: int or symbolic, optional + :param symbolic: [description], defaults to False + :type symbolic: bool, optional + + - ``Hyper()`` is an object which describes the kinematics of a serial link + manipulator with 10 joints which moves in the xy-plane, using standard DH + conventions. At zero angles it forms a straight line along the x-axis. + + - ``Hyper(N)`` as above, but models a robot with ``N`` joints. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.DH.Hyper() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration + + :References: + + - "A divide and conquer articulated-body algorithm for parallel O(log(n)) + calculation of rigid body dynamics, Part 2", + Int. J. Robotics Research, 18(9), pp 876-892. + + :seealso: :func:`Coil`, :func:`Ball` + + .. codeauthor:: Peter Corke + """ # noqa + + def __init__(self, N=10, a=None, symbolic=False): + + if symbolic: + import spatialmath.base.symbolic as sym + + zero = sym.zero() + pi = sym.pi() + else: + from math import pi + + zero = 0.0 + + if a is None: + a = 1 / N + + links = [] + for i in range(N): + links.append(RevoluteDH(a=a, alpha=zero)) + + super().__init__( + links, name="Hyper" + str(N), keywords=("symbolic",), symbolic=symbolic + ) + + self.qr = np.array(N) + self.qz = np.zeros(N) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+ + +if __name__ == "__main__": # pragma nocover + + hyper = Hyper(N=10, symbolic=False) + print(hyper) + + # print(hyper.fkine(hyper.qz)) +
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/DH/Hyper3d.html b/_modules/roboticstoolbox/models/DH/Hyper3d.html new file mode 100644 index 00000000..d8359207 --- /dev/null +++ b/_modules/roboticstoolbox/models/DH/Hyper3d.html @@ -0,0 +1,206 @@ + + + + + + roboticstoolbox.models.DH.Hyper3d — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.DH.Hyper3d

+#!/usr/bin/env python
+"""
+@author: Peter Corke
+"""
+
+# from math import pi
+import numpy as np
+from roboticstoolbox import DHRobot, RevoluteDH
+
+
+
[docs]class Hyper3d(DHRobot): + """ + Create model of a hyper redundant manipulator + + :param N: number of links, defaults to 10 + :type N: int, optional + :param a: link length, defaults total ``1/N`` giving a reach of 1 + :type a: int or symbolic, optional + :param symbolic: [description], defaults to False + :type symbolic: bool, optional + + - ``Hyper3d()`` is an object which describes the kinematics of a serial link + manipulator with 10 joints which moves in the xy-plane, using standard DH + conventions. At zero angles it forms a straight line along the x-axis. + + - ``Hyper3d(N)`` as above, but models a robot with ``N`` joints. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.DH.Hyper3d() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration + + :References: + + - "A divide and conquer articulated-body algorithm for parallel O(log(n)) + calculation of rigid body dynamics, Part 2", + Int. J. Robotics Research, 18(9), pp 876-892. + + :seealso: :func:`Coil`, :func:`Ball` + + .. codeauthor:: Peter Corke + """ # noqa + + def __init__(self, N=10, a=None, symbolic=False): + + if symbolic: + import spatialmath.base.symbolic as sym + + # zero = sym.zero() + pi = sym.pi() + else: + from math import pi + + # zero = 0.0 + + if a is None: + a = 1 / N + + links = [] + for i in range(N): + links.append(RevoluteDH(a=a, alpha=pi / 2)) + + super().__init__( + links, name="Hyper3d" + str(N), keywords=("symbolic",), symbolic=symbolic + ) + + self.qr = np.array(N) + self.qz = np.zeros(N) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+ + +if __name__ == "__main__": # pragma nocover + + hyper = Hyper3d(N=10, symbolic=False) + print(hyper) + + # hyper.plot([2]*10, block=True) + + # print(hyper.fkine(hyper.qz)) +
+ +
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/DH/IRB140.html b/_modules/roboticstoolbox/models/DH/IRB140.html new file mode 100644 index 00000000..ceb37951 --- /dev/null +++ b/_modules/roboticstoolbox/models/DH/IRB140.html @@ -0,0 +1,274 @@ + + + + + + roboticstoolbox.models.DH.IRB140 — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.DH.IRB140

+"""
+@author: Peter Corke
+@author: Samuel Drew
+"""
+
+from roboticstoolbox import DHRobot, RevoluteDH
+from math import pi
+import numpy as np
+
+
+
[docs]class IRB140(DHRobot): + """ + Class that models an ABB IRB140 manipulator + + ``IRB140()`` is a class which models a Unimation Puma560 robot and + describes its kinematic and dynamic characteristics using standard DH + conventions. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.DH.IRB140() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration + - qr, vertical 'READY' configuration + - qd, lower arm horizontal as per data sheet + + .. note:: SI units of metres are used. + + :references: + - "IRB 140 data sheet", ABB Robotics. + - "Utilizing the Functional Work Space Evaluation Tool for Assessing a + System Design and Reconfiguration Alternatives" + A. Djuric and R. J. Urbanic + - https://github.com/4rtur1t0/ARTE/blob/master/robots/ABB/IRB140/parameters.m + + .. codeauthor:: Peter Corke + """ # noqa + + def __init__(self): + deg = pi / 180 + + # robot length values (metres) + d1 = 0.352 + a1 = 0.070 + a2 = 0.360 + d4 = 0.380 + d6 = 0.065 + + # Create Links + # Updated values form ARTE git. Old values left as comments + + L = [ + RevoluteDH( + d=d1, + a=a1, + alpha=-pi / 2, + m=34655.36e-3, + r=np.array([27.87, 43.12, -89.03]) * 1e-3, + I=np.array( + [ + 512052539.74, + 1361335.88, + 51305020.72, + 1361335.88, + 464074688.59, + 70335556.04, + 51305020.72, + 70335556.04, + 462745526.12, + ] + ) + * 1e-9, + qlim=[-180 * deg, 180 * deg], + ), + RevoluteDH( + d=0, + a=a2, + alpha=0, + m=15994.59e-3, + r=np.array([198.29, 9.73, 92.43]) * 1e03, + I=np.array( + [ + 94817914.40, + -3859712.77, + 37932017.01, + -3859712.77, + 328604163.24, + -1088970.86, + 37932017.01, + -1088970.86, + 277463004.88, + ] + ) + * 1e-9, + qlim=[-100 * deg, 100 * deg], + ), + RevoluteDH( + d=0, + a=0, + alpha=-pi / 2, # alpha=pi/2, + m=20862.05e-3, + r=np.array([-4.56, -79.96, -5.86]), + I=np.array( + [ + 500060915.95, + -1863252.17, + 934875.78, + -1863252.17, + 75152670.69, + -15204130.09, + 934875.78, + -15204130.09, + 515424754.34, + ] + ) + * 1e-9, + qlim=[-220 * deg, 60 * deg], + ), + RevoluteDH( + d=d4, a=0, alpha=pi / 2, qlim=[-200 * deg, 200 * deg] # alpha=-pi/2, + ), + RevoluteDH( + d=0, a=0, alpha=-pi / 2, qlim=[-120 * deg, 120 * deg] # alpha=pi/2, + ), + RevoluteDH(d=d6, a=0, alpha=0, qlim=[-400 * deg, 400 * deg]), # alpha=pi/2, + ] + + super().__init__( + L, + # basemesh="ABB/IRB140/link0.stl", + name="IRB 140", + manufacturer="ABB", + meshdir="meshes/ABB/IRB140", + ) + + self.qr = np.array([0, -90 * deg, 90 * deg, 0, 90 * deg, -90 * deg]) + self.qz = np.zeros(6) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz) + + self.addconfiguration_attr( + "qd", np.array([0, -90 * deg, 180 * deg, 0, 0, -90 * deg]) + )
+ + +if __name__ == "__main__": # pragma nocover + + robot = IRB140() + print(robot) +
+ +
+
+ +
+
+
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/DH/Jaco.html b/_modules/roboticstoolbox/models/DH/Jaco.html new file mode 100644 index 00000000..e8291ca6 --- /dev/null +++ b/_modules/roboticstoolbox/models/DH/Jaco.html @@ -0,0 +1,223 @@ + + + + + + roboticstoolbox.models.DH.Jaco — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.DH.Jaco

+#!/usr/bin/env python
+
+from math import sin, cos
+import numpy as np
+from roboticstoolbox import DHRobot, RevoluteDH
+# from spatialmath import SE3
+
+
+
[docs]class Jaco(DHRobot): + """ + Class that models a Kinova Jaco manipulator + + :param symbolic: use symbolic constants + :type symbolic: bool + + ``Jaco()`` is an object which models a Kinova Jaco robot and + describes its kinematic characteristics using standard DH + conventions. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.DH.Puma560() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration, 'L' shaped configuration + - qr, vertical 'READY' configuration + - qs, arm is stretched out in the x-direction + - qn, arm is at a nominal non-singular configuration + + .. note:: + - SI units are used. + + :references: + - "DH Parameters of Jaco" Version 1.0.8, July 25, 2013. + + :seealso: :func:`Mico` + """ + + def __init__(self, symbolic=False): + + if symbolic: + import spatialmath.base.symbolic as sym + + # zero = sym.zero() + pi = sym.pi() + else: + from math import pi + + # zero = 0.0 + + deg = pi / 180 + # robot length values (metres) + D1 = 0.2755 + D2 = 0.4100 + D3 = 0.2073 + D4 = 0.0743 + D5 = 0.0743 + D6 = 0.1687 + e2 = 0.0098 + + # alternate parameters + aa = 30 * deg + # ca = cos(aa) + sa = sin(aa) + # c2a = cos(2 * aa) + s2a = sin(2 * aa) + d4b = D3 + sa / s2a * D4 + d5b = sa / s2a * D4 + sa / s2a * D5 + d6b = sa / s2a * D5 + D6 + + # and build a serial link manipulator + + # offsets from the table on page 4, "Mico" angles are the passed joint + # angles. "DH Algo" are the result after adding the joint angle offset. + + super().__init__( + [ + RevoluteDH(alpha=pi / 2, a=0, d=D1, flip=True), + RevoluteDH(alpha=pi, a=D2, d=0, offset=-pi / 2), + RevoluteDH(alpha=pi / 2, a=0, d=-e2, offset=pi / 2), + RevoluteDH(alpha=2 * aa, a=0, d=-d4b), + RevoluteDH(alpha=2 * aa, a=0, d=-d5b, offset=-pi), + RevoluteDH(alpha=pi, a=0, d=-d6b, offset=100 * deg), + ], + name="Jaco", + manufacturer="Kinova", + keywords=("symbolic",), + ) + + self.qr = np.array([270, 180, 180, 0, 0, 0]) * deg + self.qz = np.zeros(6) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+ + +if __name__ == "__main__": # pragma nocover + + jaco = Jaco(symbolic=False) + print(jaco) +
+ +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/DH/KR5.html b/_modules/roboticstoolbox/models/DH/KR5.html new file mode 100644 index 00000000..fb6a8a63 --- /dev/null +++ b/_modules/roboticstoolbox/models/DH/KR5.html @@ -0,0 +1,223 @@ + + + + + + roboticstoolbox.models.DH.KR5 — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.DH.KR5

+"""
+@author: Gautam Sinha, Indian Institute of Technology, Kanpur
+  (original MATLAB version)
+@author: Peter Corke
+@author: Samuel Drew
+"""
+
+from roboticstoolbox import DHRobot, RevoluteDH
+from math import pi
+import numpy as np
+
+
+
[docs]class KR5(DHRobot): + """ + Class that models a Kuka KR5 manipulator + + ``KR5()`` is a class which models a Kuka KR5 robot and + describes its kinematic characteristics using standard DH + conventions. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.DH.KR5() + >>> print(robot) + + Defined joint configurations are: + + - qk1, nominal working position 1 + - qk2, nominal working position 2 + - qk3, nominal working position 3 + + .. note:: + - SI units of metres are used. + - Includes an 11.5cm tool in the z-direction + + :references: + - https://github.com/4rtur1t0/ARTE/blob/master/robots/KUKA/KR5_arc/parameters.m + + .. codeauthor:: Gautam Sinha, Indian Institute of Technology, Kanpur (original MATLAB version) + .. codeauthor:: Samuel Drew + .. codeauthor:: Peter Corke + """ # noqa + + def __init__(self): + deg = pi / 180 + + # Updated values form ARTE git. Old values left as comments + + L1 = RevoluteDH( + a=0.18, d=0.4, alpha=-pi / 2, qlim=[-155 * deg, 155 * deg] # alpha=pi / 2, + ) + L2 = RevoluteDH( + a=0.6, d=0, alpha=0, qlim=[-180 * deg, 65 * deg] # d=0.135, # alpha=pi, + ) + L3 = RevoluteDH( + a=0.12, + d=0, # d=0.135, + alpha=pi / 2, # alpha=-pi / 2, + qlim=[-15 * deg, 158 * deg], + ) + L4 = RevoluteDH( + a=0.0, + d=-0.62, # d=0.62, + alpha=-pi / 2, # alpha=pi / 2, + qlim=[-350 * deg, 350 * deg], + ) + L5 = RevoluteDH( + a=0.0, d=0.0, alpha=pi / 2, qlim=[-130 * deg, 130 * deg] # alpha=-pi / 2, + ) + L6 = RevoluteDH(a=0, d=-0.115, alpha=pi, qlim=[-350 * deg, 350 * deg]) + + L = [L1, L2, L3, L4, L5, L6] + + # Create SerialLink object + super().__init__( + L, + # meshdir="KUKA/KR5_arc", + name="KR5", + manufacturer="KUKA", + meshdir="meshes/KUKA/KR5_arc", + ) + + self.qr = np.array([pi / 4, pi / 3, pi / 4, pi / 6, pi / 4, pi / 6]) + self.qz = np.zeros(6) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz) + + self.addconfiguration_attr( + "qk1", np.array([pi / 4, pi / 3, pi / 4, pi / 6, pi / 4, pi / 6]) + ) + self.addconfiguration_attr( + "qk2", np.array([pi / 4, pi / 3, pi / 6, pi / 3, pi / 4, pi / 6]) + ) + self.addconfiguration_attr( + "qk3", np.array([pi / 6, pi / 3, pi / 6, pi / 3, pi / 6, pi / 3]) + )
+ + +if __name__ == "__main__": # pragma nocover + robot = KR5() + print(robot) +
+ +
+
+ +
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/DH/LWR4.html b/_modules/roboticstoolbox/models/DH/LWR4.html new file mode 100644 index 00000000..666fe97c --- /dev/null +++ b/_modules/roboticstoolbox/models/DH/LWR4.html @@ -0,0 +1,206 @@ + + + + + + roboticstoolbox.models.DH.LWR4 — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.DH.LWR4

+#!/usr/bin/env python
+
+import numpy as np
+from spatialmath.base import trotz, transl
+from roboticstoolbox import DHRobot, RevoluteDH
+
+
+
[docs]class LWR4(DHRobot): + """ + Class that models a LWR-IV manipulator + + ``LWR4()`` is a class which models a Kuka LWR-IV robot and + describes its kinematic characteristics using standard DH + conventions. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.DH.LWR4() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration, 'L' shaped configuration + - qr, vertical 'READY' configuration + - qs, arm is stretched out in the X direction + - qn, arm is at a nominal non-singular configuration + + .. note:: SI units are used. + + :references: + + - http://www.diag.uniroma1.it/~deluca/rob1_en/09_Exercise_DH_KukaLWR4.pdf + + .. codeauthor:: Peter Corke + """ # noqa + + def __init__(self): + + # deg = np.pi/180 + mm = 1e-3 + tool_offset = (103) * mm + + flange = 0 * mm + # d7 = (58.4)*mm + + # This Kuka model is defined using modified + # Denavit-Hartenberg parameters + + L = [ + RevoluteDH(a=0.0, d=0, alpha=np.pi / 2, qlim=np.array([-2.8973, 2.8973])), + RevoluteDH( + a=0.0, d=0.0, alpha=-np.pi / 2, qlim=np.array([-1.7628, 1.7628]) + ), + RevoluteDH( + a=0.0, d=0.40, alpha=-np.pi / 2, qlim=np.array([-2.8973, 2.8973]) + ), + RevoluteDH( + a=0.0, d=0.0, alpha=np.pi / 2, qlim=np.array([-3.0718, -0.0698]) + ), + RevoluteDH( + a=0.0, d=0.39, alpha=np.pi / 2, qlim=np.array([-2.8973, 2.8973]) + ), + RevoluteDH( + a=0.0, d=0.0, alpha=-np.pi / 2, qlim=np.array([-0.0175, 3.7525]) + ), + RevoluteDH(a=0.0, d=flange, alpha=0.0, qlim=np.array([-2.8973, 2.8973])), + ] + + tool = transl(0, 0, tool_offset) @ trotz(-np.pi / 4) + + super().__init__(L, name="LWR-IV", manufacturer="Kuka", tool=tool) + + # tool = xyzrpy_to_trans(0, 0, d7, 0, 0, -np.pi/4) + + self.qr = np.array(7) + self.qz = np.zeros(7) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+ + +if __name__ == "__main__": # pragma nocover + + robot = LWR4() + print(robot) +
+ +
+
+ +
+
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/DH/Mico.html b/_modules/roboticstoolbox/models/DH/Mico.html new file mode 100644 index 00000000..cc74c971 --- /dev/null +++ b/_modules/roboticstoolbox/models/DH/Mico.html @@ -0,0 +1,223 @@ + + + + + + roboticstoolbox.models.DH.Mico — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.DH.Mico

+#!/usr/bin/env python
+
+from math import pi, sin, cos
+import numpy as np
+from roboticstoolbox import DHRobot, RevoluteDH
+from spatialmath import SE3
+
+
+
[docs]class Mico(DHRobot): + """ + Class that models a Kinova Mico manipulator + + :param symbolic: use symbolic constants + :type symbolic: bool + + ``Mico()`` is an object which models a Kinova Mico robot and + describes its kinematic characteristics using standard DH + conventions. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.DH.Puma560() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration, 'L' shaped configuration + - qr, vertical 'READY' configuration + - qs, arm is stretched out in the x-direction + - qn, arm is at a nominal non-singular configuration + + .. note:: + - SI units are used. + + :references: + - "DH Parameters of Mico" Version 1.0.8, July 25, 2013. + + :seealso: :func:`Mico` + """ + + def __init__(self, symbolic=False): + + if symbolic: + import spatialmath.base.symbolic as sym + + zero = sym.zero() + pi = sym.pi() + else: + from math import pi + + zero = 0.0 + + deg = pi / 180 + # robot length values (metres) + D1 = 0.2755 + D2 = 0.2900 + D3 = 0.1233 + D4 = 0.0741 + D5 = 0.0741 + D6 = 0.1600 + e2 = 0.0070 + + # alternate parameters + aa = 30 * deg + ca = cos(aa) + sa = sin(aa) + c2a = cos(2 * aa) + s2a = sin(2 * aa) + d4b = D3 + sa / s2a * D4 + d5b = sa / s2a * D4 + sa / s2a * D5 + d6b = sa / s2a * D5 + D6 + + # and build a serial link manipulator + + # offsets from the table on page 4, "Mico" angles are the passed joint + # angles. "DH Algo" are the result after adding the joint angle offset. + + super().__init__( + [ + RevoluteDH(alpha=pi / 2, a=0, d=D1, flip=True), + RevoluteDH(alpha=pi, a=D2, d=0, offset=-pi / 2), + RevoluteDH(alpha=pi / 2, a=0, d=-e2, offset=pi / 2), + RevoluteDH(alpha=2 * aa, a=0, d=-d4b), + RevoluteDH(alpha=2 * aa, a=0, d=-d5b, offset=-pi), + RevoluteDH(alpha=pi, a=0, d=-d6b, offset=100 * deg), + ], + name="Mico", + manufacturer="Kinova", + keywords=("symbolic",), + ) + + self.qr = np.array([270, 180, 180, 0, 0, 0]) * deg + self.qz = np.zeros(6) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+ + +if __name__ == "__main__": # pragma nocover + + mico = Mico(symbolic=False) + print(mico) +
+ +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/DH/Orion5.html b/_modules/roboticstoolbox/models/DH/Orion5.html new file mode 100644 index 00000000..d07e2d5e --- /dev/null +++ b/_modules/roboticstoolbox/models/DH/Orion5.html @@ -0,0 +1,212 @@ + + + + + + roboticstoolbox.models.DH.Orion5 — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.DH.Orion5

+#!/usr/bin/env python
+# Created by: Aditya Dua
+# 5 October 2017
+
+from math import pi
+import numpy as np
+from roboticstoolbox import DHRobot, RevoluteDH
+from spatialmath import SE3
+
+
+
[docs]class Orion5(DHRobot): + """ + Class that models a RAWR robotics Orion5 manipulator + + ``Orion5()`` is a class which models a RAWR Robotics Orion5 robot and + describes its kinematic characteristics using standard DH + conventions. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.DH.Orion5() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero angles, all folded up + - qv, stretched out vertically + - qh, arm horizontal, hand down + + .. note:: + + - SI units of metres are used. + - Robot has only 4 DoF. + + :references: + - https://rawrobotics.com.au/orion5 + - https://drive.google.com/file/d/0B6_9_-ZgiRdTNkVqMEkxN2RSc2c/view + + .. codeauthor:: Aditya Dua + .. codeauthor:: Peter Corke + """ + + def __init__(self, base=None): + + mm = 1e-3 + deg = pi / 180 + + # details from + + h = 53.0 * mm + r = 30.309 * mm + l2 = 170.384 * mm + l3 = 136.307 * mm + l4 = 86.00 * mm + c = 40.0 * mm + + # tool_offset = l4 + c + + # Turret, Shoulder, Elbow, Wrist, Claw + links = [ + RevoluteDH(d=h, a=0, alpha=90 * deg), # Turret + RevoluteDH(d=0, a=l2, alpha=0, qlim=[10 * deg, 122.5 * deg]), # Shoulder + RevoluteDH(d=0, a=-l3, alpha=0, qlim=[20 * deg, 340 * deg]), # Elbow + RevoluteDH(d=0, a=l4 + c, alpha=0, qlim=[45 * deg, 315 * deg]), # Wrist + ] + + super().__init__( + links, + name="Orion 5", + manufacturer="RAWR Robotics", + base=SE3(r, 0, 0), + tool=SE3.Ry(90, "deg"), + ) + + self.qr = np.array([0, 0, 180, 90]) * deg + self.qz = np.zeros(4) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz) + + # stretched out vertically + self.addconfiguration_attr("qv", np.r_[0, 90, 180, 180] * deg) + + # arm horizontal, hand down + self.addconfiguration_attr("qh", np.r_[0, 0, 180, 90] * deg)
+ + +if __name__ == "__main__": # pragma nocover + + orion = Orion5() + print(orion) +
+ +
+
+ +
+
+
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/DH/P8.html b/_modules/roboticstoolbox/models/DH/P8.html new file mode 100644 index 00000000..175f7c57 --- /dev/null +++ b/_modules/roboticstoolbox/models/DH/P8.html @@ -0,0 +1,201 @@ + + + + + + roboticstoolbox.models.DH.P8 — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
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+ +
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+ +

Source code for roboticstoolbox.models.DH.P8

+#!/usr/bin/env python
+"""
+@author: Peter Corke
+@author: Jesse Haviland
+"""
+
+# 2/8/95  changed D3 to 150.05mm which is closer to data from Lee, AKB86 and
+# Tarn fixed errors in COG for links 2 and 3
+# 29/1/91 to agree with data from Armstrong etal.  Due to their use
+#  of modified D&H params, some of the offsets Ai, Di are
+#  offset, and for links 3-5 swap Y and Z axes.
+# 14/2/91 to use Paul's value of link twist (alpha) to be consistant
+#  with ARCL.  This is the -ve of Lee's values, which means the
+#  zero angle position is a righty for Paul, and lefty for Lee.
+
+# all parameters are in SI units: m, radians, kg, kg.m2, N.m, N.m.s etc.
+
+# from math import pi
+import numpy as np
+from roboticstoolbox import DHRobot, PrismaticDH, models
+from spatialmath import SE3
+from spatialmath import base
+from math import pi
+
+
+
[docs]class P8(DHRobot): + """ + Class that models a Puma robot on an XY base + + ``P8()`` is an object which models an 8-axis robot comprising a Puma 560 + robot on an XY base. Joints 0 and 1 are the base, joints 2-7 are the robot + arm. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.DH.Puma560() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration, 'L' shaped configuration + + + .. note:: + - SI units are used. + + .. codeauthor:: Peter Corke + + :seealso: :func:`models.DH.Puma560` + """ # noqa + + def __init__(self): + + # create the base + links = [ + PrismaticDH(alpha=-pi / 2, qlim=[-1, 1]), + PrismaticDH(theta=-pi / 2, alpha=pi / 2, qlim=[-1, 1]), + ] + + # stick the Puma on top + puma = models.DH.Puma560() + links.extend(puma.links) + + super().__init__( + links, name="P8", keywords=("mobile", "redundant"), base=SE3.Ry(pi / 2) + ) + + self.qr = np.zeros(8) + self.qz = np.zeros(8) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+ + +if __name__ == "__main__": # pragma nocover + + robot = P8() + print(robot) + + # robot.plot([.5, -0.6, 0, 0, 0, 0, 0, 0], block=True) +
+ +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/DH/Panda.html b/_modules/roboticstoolbox/models/DH/Panda.html new file mode 100644 index 00000000..625c9b01 --- /dev/null +++ b/_modules/roboticstoolbox/models/DH/Panda.html @@ -0,0 +1,299 @@ + + + + + + roboticstoolbox.models.DH.Panda — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
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+
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+ +

Source code for roboticstoolbox.models.DH.Panda

+#!/usr/bin/env python
+
+import numpy as np
+from spatialmath.base import trotz, transl
+from roboticstoolbox import DHRobot, RevoluteMDH
+
+
+
[docs]class Panda(DHRobot): + """ + A class representing the Panda robot arm. + + ``Panda()`` is a class which models a Franka-Emika Panda robot and + describes its kinematic characteristics using modified DH + conventions. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.DH.Panda() + >>> print(robot) + + .. note:: + - SI units of metres are used. + - The model includes a tool offset. + + :references: + - https://frankaemika.github.io/docs/control_parameters.html + + .. codeauthor:: Samuel Drew + .. codeauthor:: Peter Corke + """ + + def __init__(self): + + # deg = np.pi/180 + mm = 1e-3 + tool_offset = (103) * mm + + flange = (107) * mm + # d7 = (58.4)*mm + + # This Panda model is defined using modified + # Denavit-Hartenberg parameters + L = [ + RevoluteMDH( + a=0.0, + d=0.333, + alpha=0.0, + qlim=np.array([-2.8973, 2.8973]), + m=4.970684, + I=[ + 7.03370e-01, + 7.06610e-01, + 9.11700e-03, + -1.39000e-04, + 1.91690e-02, + 6.77200e-03, + ], + G=1, + ), + RevoluteMDH( + a=0.0, + d=0.0, + alpha=-np.pi / 2, + qlim=np.array([-1.7628, 1.7628]), + m=0.646926, + I=[ + 7.96200e-03, + 2.81100e-02, + 2.59950e-02, + -3.92500e-03, + 7.04000e-04, + 1.02540e-02, + ], + G=1, + ), + RevoluteMDH( + a=0.0, + d=0.316, + alpha=np.pi / 2, + qlim=np.array([-2.8973, 2.8973]), + m=3.228604, + I=[ + 3.72420e-02, + 3.61550e-02, + 1.08300e-02, + -4.76100e-03, + -1.28050e-02, + -1.13960e-02, + ], + G=1, + ), + RevoluteMDH( + a=0.0825, + d=0.0, + alpha=np.pi / 2, + qlim=np.array([-3.0718, -0.0698]), + m=3.587895, + I=[ + 2.58530e-02, + 1.95520e-02, + 2.83230e-02, + 7.79600e-03, + 8.64100e-03, + -1.33200e-03, + ], + G=1, + ), + RevoluteMDH( + a=-0.0825, + d=0.384, + alpha=-np.pi / 2, + qlim=np.array([-2.8973, 2.8973]), + m=1.225946, + I=[ + 3.55490e-02, + 2.94740e-02, + 8.62700e-03, + -2.11700e-03, + 2.29000e-04, + -4.03700e-03, + ], + G=1, + ), + RevoluteMDH( + a=0.0, + d=0.0, + alpha=np.pi / 2, + qlim=np.array([-0.0175, 3.7525]), + m=1.666555, + I=[ + 1.96400e-03, + 4.35400e-03, + 5.43300e-03, + 1.09000e-04, + 3.41000e-04, + -1.15800e-03, + ], + G=1, + ), + RevoluteMDH( + a=0.088, + d=flange, + alpha=np.pi / 2, + qlim=np.array([-2.8973, 2.8973]), + m=7.35522e-01, + I=[ + 1.25160e-02, + 1.00270e-02, + 4.81500e-03, + -4.28000e-04, + -7.41000e-04, + -1.19600e-03, + ], + G=1, + ), + ] + + tool = transl(0, 0, tool_offset) @ trotz(-np.pi / 4) + + super().__init__( + L, + name="Panda", + manufacturer="Franka Emika", + meshdir="meshes/FRANKA-EMIKA/Panda", + tool=tool, + ) + + self.qr = np.array([0, -0.3, 0, -2.2, 0, 2.0, np.pi / 4]) + self.qz = np.zeros(7) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+ + +if __name__ == "__main__": # pragma nocover + + panda = Panda() + print(panda) +
+ +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/DH/Planar2.html b/_modules/roboticstoolbox/models/DH/Planar2.html new file mode 100644 index 00000000..29ed57ac --- /dev/null +++ b/_modules/roboticstoolbox/models/DH/Planar2.html @@ -0,0 +1,190 @@ + + + + + + roboticstoolbox.models.DH.Planar2 — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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+ +

Source code for roboticstoolbox.models.DH.Planar2

+"""
+@author: Luis Fernando Lara Tobar
+@author: Peter Corke
+@author: Samuel Drew
+"""
+
+from roboticstoolbox import DHRobot, RevoluteDH
+import numpy as np
+
+
+
[docs]class Planar2(DHRobot): + """ + Class that models a planar 2-link robot + + ``Planar2()`` is a class which models a 2-link planar robot and + describes its kinematic characteristics using standard DH + conventions. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.DH.Planar2() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero angles, all folded up + - q1, links are horizontal and vertical respectively + - q2, links are vertical and horizontal respectively + + .. note:: + + - Robot has only 2 DoF. + + .. codeauthor:: Peter Corke + """ + + def __init__(self, symbolic=False): + + if symbolic: + import spatialmath.base.symbolic as sym + + zero = sym.zero() + pi = sym.pi() + a1, a2 = sym.symbol("a1 a2") # type: ignore + else: + from math import pi + + zero = 0.0 + a1 = 1 + a2 = 1 + + L = [RevoluteDH(a=a1, alpha=zero), RevoluteDH(a=a2, alpha=zero)] + + super().__init__(L, name="Planar 2 link", keywords=("planar",)) + + self.qr = np.array([0, pi / 2]) + self.qz = np.zeros(2) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz) + + self.addconfiguration_attr("q1", np.array([0, pi / 2])) + self.addconfiguration_attr("q2", np.array([pi / 2, -pi / 2]))
+ + +if __name__ == "__main__": # pragma nocover + + robot = Planar2() + print(robot) +
+ +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/DH/Planar3.html b/_modules/roboticstoolbox/models/DH/Planar3.html new file mode 100644 index 00000000..e2beff76 --- /dev/null +++ b/_modules/roboticstoolbox/models/DH/Planar3.html @@ -0,0 +1,172 @@ + + + + + + roboticstoolbox.models.DH.Planar3 — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.DH.Planar3

+"""
+@author: Luis Fernando Lara Tobar
+@author: Peter Corke
+@author: Samuel Drew
+"""
+
+from roboticstoolbox import DHRobot, RevoluteDH
+import numpy as np
+
+
+
[docs]class Planar3(DHRobot): + """ + Class that models a planar 3-link robot + + ``Planar2()`` is a class which models a 3-link planar robot and + describes its kinematic characteristics using standard DH + conventions. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.DH.Planar3() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero angles, all folded up + + .. note:: + + - Robot has only 3 DoF. + + .. codeauthor:: Peter Corke + """ + + def __init__(self): + + L = [RevoluteDH(a=1), RevoluteDH(a=1), RevoluteDH(a=1)] + + super().__init__(L, name="Planar 3 link", keywords=("planar",)) + + self.qr = np.zeros(3) + self.qz = np.zeros(3) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+ + +if __name__ == "__main__": # pragma nocover + + robot = Planar3() + print(robot) +
+ +
+
+ +
+
+
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/DH/Puma560.html b/_modules/roboticstoolbox/models/DH/Puma560.html new file mode 100644 index 00000000..162df8e4 --- /dev/null +++ b/_modules/roboticstoolbox/models/DH/Puma560.html @@ -0,0 +1,448 @@ + + + + + + roboticstoolbox.models.DH.Puma560 — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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+ +

Source code for roboticstoolbox.models.DH.Puma560

+#!/usr/bin/env python
+"""
+@author: Peter Corke
+@author: Jesse Haviland
+"""
+
+# 2/8/95  changed D3 to 150.05mm which is closer to data from Lee, AKB86 and
+# Tarn fixed errors in COG for links 2 and 3
+# 29/1/91 to agree with data from Armstrong etal.  Due to their use
+#  of modified D&H params, some of the offsets Ai, Di are
+#  offset, and for links 3-5 swap Y and Z axes.
+# 14/2/91 to use Paul's value of link twist (alpha) to be consistant
+#  with ARCL.  This is the -ve of Lee's values, which means the
+#  zero angle position is a righty for Paul, and lefty for Lee.
+
+# all parameters are in SI units: m, radians, kg, kg.m2, N.m, N.m.s etc.
+
+# from math import pi
+import numpy as np
+from roboticstoolbox import DHRobot, RevoluteDH
+from spatialmath import SE3
+from spatialmath import base
+
+
+
[docs]class Puma560(DHRobot): + """ + Class that models a Puma 560 manipulator + + :param symbolic: use symbolic constants + :type symbolic: bool + + ``Puma560()`` is an object which models a Unimation Puma560 robot and + describes its kinematic and dynamic characteristics using standard DH + conventions. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.DH.Puma560() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration, 'L' shaped configuration + - qr, vertical 'READY' configuration + - qs, arm is stretched out in the x-direction + - qn, arm is at a nominal non-singular configuration + + .. note:: + - SI units are used. + - The model includes armature inertia and gear ratios. + - The value of m1 is given as 0 here. Armstrong found no value for it + and it does not appear in the equation for tau1 after the + substituion is made to inertia about link frame rather than COG + frame. + - Gravity load torque is the motor torque necessary to keep the joint + static, and is thus -ve of the gravity caused torque. + + .. warning:: Compared to the MATLAB version of the Toolbox this model + includes the pedestal, making the z-coordinates 26 inches larger. + + :references: + - "A search for consensus among model parameters reported for the PUMA + 560 robot", P. Corke and B. Armstrong-Helouvry, + Proc. IEEE Int. Conf. Robotics and Automation, (San Diego), + pp. 1608-1613, May 1994. (for kinematic and dynamic parameters) + - "A combined optimization method for solving the inverse kinematics + problem", Wang & Chen, IEEE Trans. RA 7(4) 1991 pp 489-. + (for joint angle limits) + - https://github.com/4rtur1t0/ARTE/blob/master/robots/UNIMATE/puma560/parameters.m + + .. codeauthor:: Peter Corke + """ # noqa + + def __init__(self, symbolic=False): + + if symbolic: + import spatialmath.base.symbolic as sym + + zero = sym.zero() + pi = sym.pi() + else: + from math import pi + + zero = 0.0 + + deg = pi / 180 + inch = 0.0254 + + base = 26.45 * inch # from mounting surface to shoulder axis + + L = [ + RevoluteDH( + d=base, # link length (Dennavit-Hartenberg notation) + a=0, # link offset (Dennavit-Hartenberg notation) + alpha=pi / 2, # link twist (Dennavit-Hartenberg notation) + I=[0, 0.35, 0, 0, 0, 0], + # inertia tensor of link with respect to + # center of mass I = [L_xx, L_yy, L_zz, + # L_xy, L_yz, L_xz] + r=[0, 0, 0], + # distance of ith origin to center of mass [x,y,z] + # in link reference frame + m=0, # mass of link + Jm=200e-6, # actuator inertia + G=-62.6111, # gear ratio + B=1.48e-3, # actuator viscous friction coefficient (measured + # at the motor) + Tc=[0.395, -0.435], + # actuator Coulomb friction coefficient for + # direction [-,+] (measured at the motor) + qlim=[-160 * deg, 160 * deg], # minimum and maximum joint angle + ), + RevoluteDH( + d=0, + a=0.4318, + alpha=zero, + I=[0.13, 0.524, 0.539, 0, 0, 0], + r=[-0.3638, 0.006, 0.2275], + m=17.4, + Jm=200e-6, + G=107.815, + B=0.817e-3, + Tc=[0.126, -0.071], + qlim=[-110 * deg, 110 * deg], # qlim=[-45*deg, 225*deg] + ), + RevoluteDH( + d=0.15005, + a=0.0203, + alpha=-pi / 2, + I=[0.066, 0.086, 0.0125, 0, 0, 0], + r=[-0.0203, -0.0141, 0.070], + m=4.8, + Jm=200e-6, + G=-53.7063, + B=1.38e-3, + Tc=[0.132, -0.105], + qlim=[-135 * deg, 135 * deg], # qlim=[-225*deg, 45*deg] + ), + RevoluteDH( + d=0.4318, + a=0, + alpha=pi / 2, + I=[1.8e-3, 1.3e-3, 1.8e-3, 0, 0, 0], + r=[0, 0.019, 0], + m=0.82, + Jm=33e-6, + G=76.0364, + B=71.2e-6, + Tc=[11.2e-3, -16.9e-3], + qlim=[-266 * deg, 266 * deg], # qlim=[-110*deg, 170*deg] + ), + RevoluteDH( + d=0, + a=0, + alpha=-pi / 2, + I=[0.3e-3, 0.4e-3, 0.3e-3, 0, 0, 0], + r=[0, 0, 0], + m=0.34, + Jm=33e-6, + G=71.923, + B=82.6e-6, + Tc=[9.26e-3, -14.5e-3], + qlim=[-100 * deg, 100 * deg], + ), + RevoluteDH( + d=0, + a=0, + alpha=zero, + I=[0.15e-3, 0.15e-3, 0.04e-3, 0, 0, 0], + r=[0, 0, 0.032], + m=0.09, + Jm=33e-6, + G=76.686, + B=36.7e-6, + Tc=[3.96e-3, -10.5e-3], + qlim=[-266 * deg, 266 * deg], + ), + ] + + super().__init__( + L, + name="Puma 560", + manufacturer="Unimation", + keywords=("dynamics", "symbolic", "mesh"), + symbolic=symbolic, + meshdir="meshes/UNIMATE/puma560", + ) + + self.qr = np.array([0, pi / 2, -pi / 2, 0, 0, 0]) + self.qz = np.zeros(6) + + # nominal table top picking pose + self.qn = np.array([0, pi / 4, pi, 0, pi / 4, 0]) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz) + self.addconfiguration("qn", self.qn) + + # straight and horizontal + self.addconfiguration_attr("qs", np.array([0, 0, -pi / 2, 0, 0, 0])) + +
[docs] def ikine_a(self, T, config="lun"): + """ + Analytic inverse kinematic solution + + :param T: end-effector pose + :type T: SE3 + :param config: arm configuration, defaults to "lun" + :type config: str, optional + :return: joint angle vector in radians + :rtype: ndarray(6) + + ``robot.ikine_a(T, config)`` is the joint angle vector which achieves the + end-effector pose ``T```. The configuration string selects the specific + solution and is a sting comprising the following letters: + + ====== ============================================== + Letter Meaning + ====== ============================================== + l Choose the left-handed configuration + r Choose the right-handed configuration + u Choose the elbow up configuration + d Choose the elbow down configuration + n Choose the wrist not-flipped configuration + f Choose the wrist flipped configuration + ====== ============================================== + + + :reference: + - Inverse kinematics for a PUMA 560, + Paul and Zhang, + The International Journal of Robotics Research, + Vol. 5, No. 2, Summer 1986, p. 32-44 + + :author: based on MATLAB code by Robert Biro with Gary Von McMurray, + GTRI/ATRP/IIMB, Georgia Institute of Technology, 2/13/95 + + """ + + def ik3(robot, T, config="lun"): + + config = self.config_validate(config, ("lr", "ud", "nf")) + + # solve for the first three joints + + a2 = robot.links[1].a + a3 = robot.links[2].a + d1 = robot.links[0].d + d3 = robot.links[2].d + d4 = robot.links[3].d + + # The following parameters are extracted from the Homogeneous + # Transformation as defined in equation 1, p. 34 + + Px, Py, Pz = T.t + Pz -= d1 # offset the pedestal height + theta = np.zeros((3,)) + + # Solve for theta[0] + # r is defined in equation 38, p. 39. + # theta[0] uses equations 40 and 41, p.39, + # based on the configuration parameter n1 + + r = np.sqrt(Px**2 + Py**2) + if "r" in config: + theta[0] = np.arctan2(Py, Px) + np.arcsin(d3 / r) + elif "l" in config: + theta[0] = np.arctan2(Py, Px) + np.pi - np.arcsin(d3 / r) + else: + raise ValueError("bad configuration string") + + # Solve for theta[1] + # V114 is defined in equation 43, p.39. + # r is defined in equation 47, p.39. + # Psi is defined in equation 49, p.40. + # theta[1] uses equations 50 and 51, p.40, based on the + # configuration parameter n2 + if "u" in config: + n2 = 1 + elif "d" in config: + n2 = -1 + else: + raise ValueError("bad configuration string") + + if "l" in config: + n2 = -n2 + + V114 = Px * np.cos(theta[0]) + Py * np.sin(theta[0]) + + r = np.sqrt(V114**2 + Pz**2) + + with np.errstate(invalid="raise"): + try: + Psi = np.arccos( + (a2**2 - d4**2 - a3**2 + V114**2 + Pz**2) + / (2.0 * a2 * r) + ) + except FloatingPointError: + return "Out of reach" + + theta[1] = np.arctan2(Pz, V114) + n2 * Psi + + # Solve for theta[2] + # theta[2] uses equation 57, p. 40. + num = np.cos(theta[1]) * V114 + np.sin(theta[1]) * Pz - a2 + den = np.cos(theta[1]) * Pz - np.sin(theta[1]) * V114 + theta[2] = np.arctan2(a3, d4) - np.arctan2(num, den) + + theta = base.angdiff(theta) + + return theta + + return self.ikine_6s(T, config, ik3)
+ + +if __name__ == "__main__": # pragma nocover + + puma = Puma560(symbolic=False) + print(puma) + print(puma.dynamics()) + # T = puma.fkine(puma.qn) + # print(puma.ikine_a(T, 'lu').q) + # print(puma.ikine_a(T, 'ru').q) + # print(puma.ikine_a(T, 'ld').q) + # print(puma.ikine_a(T, 'rd').q) + + # puma.plot(puma.qz) +
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/DH/Sawyer.html b/_modules/roboticstoolbox/models/DH/Sawyer.html new file mode 100644 index 00000000..49ea51f6 --- /dev/null +++ b/_modules/roboticstoolbox/models/DH/Sawyer.html @@ -0,0 +1,206 @@ + + + + + + roboticstoolbox.models.DH.Sawyer — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.DH.Sawyer

+#!/usr/bin/env python
+"""
+@author: Peter Corke
+"""
+
+import numpy as np
+from roboticstoolbox import DHRobot, RevoluteDH
+
+
+
[docs]class Sawyer(DHRobot): + """ + Class that models a Sawyer manipulator + + :param symbolic: use symbolic constants + :type symbolic: bool + + ``Sawyer()`` is an object which models a Rethink Sawyer robot and + describes its kinematic characteristics using standard DH + conventions. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.DH.Sawyer() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration, 'L' shaped configuration + + :references: + + -`Layeghi, Daniel. “Dynamic and Kinematic Modelling of the Sawyer Arm ” Google Sites, 20 Nov. 2017 <https://sites.google.com/site/daniellayeghi/daily-work-and-writing/major-project-2>`_ + + .. note:: SI units of metres are used. + + """ + + def __init__(self, symbolic=False): + + if symbolic: + import spatialmath.base.symbolic as sym + + # zero = sym.zero() + pi = sym.pi() + else: + from math import pi + + # zero = 0.0 + + # deg = pi / 180 + mm = 1e-3 + + # kinematic parameters + a = np.r_[81, 0, 0, 0, 0, 0, 0] * mm + d = np.r_[317, 192.5, 400, 168.5, 400, 136.3, 133.75] * mm + alpha = [-pi / 2, -pi / 2, -pi / 2, -pi / 2, -pi / 2, -pi / 2, 0] + + links = [] + + for j in range(7): + link = RevoluteDH(d=d[j], a=a[j], alpha=alpha[j]) + links.append(link) + + super().__init__( + links, + name="Sawyer", + manufacturer="Rethink Robotics", + keywords=( + "redundant", + "symbolic", + ), + symbolic=symbolic, + ) + + self.qr = np.zeros(7) + self.qz = np.zeros(7) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+ + +if __name__ == "__main__": # pragma nocover + + sawyer = Sawyer(symbolic=False) + print(sawyer) +
+ +
+
+ +
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/DH/Stanford.html b/_modules/roboticstoolbox/models/DH/Stanford.html new file mode 100644 index 00000000..c35684c1 --- /dev/null +++ b/_modules/roboticstoolbox/models/DH/Stanford.html @@ -0,0 +1,268 @@ + + + + + + roboticstoolbox.models.DH.Stanford — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.DH.Stanford

+#!/usr/bin/env python
+"""
+@author: Peter Corke
+"""
+
+from roboticstoolbox import DHRobot, RevoluteDH, PrismaticDH
+from math import pi
+import numpy as np
+
+
+
[docs]class Stanford(DHRobot): + """ + Class that models a Stanford arm manipulator + + ``Stanford()`` is a class which models a Unimation Puma560 robot and + describes its kinematic and dynamic characteristics using standard DH + conventions. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.DH.Stanford() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration + + .. note:: + - SI units are used. + - Gear ratios not currently known, though reflected armature inertia + is known, so gear ratios are set to 1. + + :references: + - Kinematic data from "Modelling, Trajectory calculation and Servoing + of a computer controlled arm". Stanford AIM-177. Figure 2.3 + - Dynamic data from "Robot manipulators: mathematics, programming and + control" + Paul 1981, Tables 6.5, 6.6 + - Dobrotin & Scheinman, "Design of a computer controlled manipulator + for robot research", IJCAI, 1973. + + .. codeauthor:: Peter Corke + + """ + + def __init__(self): + + deg = pi / 180 + inch = 0.0254 + + L0 = RevoluteDH( + d=0.412, # link length (Dennavit-Hartenberg notation) + a=0, # link offset (Dennavit-Hartenberg notation) + alpha=-pi / 2, # link twist (Dennavit-Hartenberg notation) + # inertia tensor of link with respect to + # center of mass I = [L_xx, L_yy, L_zz, + # L_xy, L_yz, L_xz] + I=[0.276, 0.255, 0.071, 0, 0, 0], + # distance of ith origin to center of mass [x,y,z] + # in link reference frame + r=[0, 0.0175, -0.1105], + m=9.29, # mass of link + Jm=0.953, # actuator inertia + G=1, # gear ratio + qlim=[-170 * deg, 170 * deg], + ) # minimum and maximum joint angle + + L1 = RevoluteDH( + d=0.154, + a=0.0, + alpha=pi / 2, + I=[0.108, 0.018, 0.100, 0, 0, 0], + r=[0, -1.054, 0], + m=5.01, + Jm=2.193, + G=1, + qlim=[-170 * deg, 170 * deg], + ) + + L2 = PrismaticDH( + theta=-pi / 2, + a=0.0203, + alpha=0, + I=[2.51, 2.51, 0.006, 0, 0, 0], + r=[0, 0, -6.447], + m=4.25, + Jm=0.782, + G=1, + qlim=[12 * inch, (12 + 38) * inch], + ) + + L3 = RevoluteDH( + d=0, + a=0, + alpha=-pi / 2, + I=[0.002, 0.001, 0.001, 0, 0, 0], + r=[0, 0.092, -0.054], + m=1.08, + Jm=0.106, + G=1, + qlim=[-170 * deg, 170 * deg], + ) + + L4 = RevoluteDH( + d=0, + a=0, + alpha=pi / 2, + I=[0.003, 0.0004, 0, 0, 0, 0], + r=[0, 0.566, 0.003], + m=0.630, + Jm=0.097, + G=1, + qlim=[-90 * deg, 90 * deg], + ) + + L5 = RevoluteDH( + d=0, + a=0, + alpha=0, + I=[0.013, 0.013, 0.0003, 0, 0, 0], + r=[0, 0, 1.554], + m=0.51, + Jm=0.020, + G=1, + qlim=[-170 * deg, 170 * deg], + ) + + L = [L0, L1, L2, L3, L4, L5] + + super().__init__( + L, + name="Stanford arm", + manufacturer="Victor Scheinman", + keywords=("dynamics",), + ) + + self.qr = np.zeros(6) + self.qz = np.zeros(6) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+ + +if __name__ == "__main__": # pragma nocover + + stanford = Stanford() + print(stanford) +
+ +
+
+ +
+
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/DH/TwoLink.html b/_modules/roboticstoolbox/models/DH/TwoLink.html new file mode 100644 index 00000000..10c73cd5 --- /dev/null +++ b/_modules/roboticstoolbox/models/DH/TwoLink.html @@ -0,0 +1,229 @@ + + + + + + roboticstoolbox.models.DH.TwoLink — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.DH.TwoLink

+"""
+@author: Peter Corke
+"""
+
+from roboticstoolbox import DHRobot, RevoluteDH
+# from math import pi
+from spatialmath import SE3
+import numpy as np
+
+
+
+
+
+if __name__ == "__main__":  # pragma nocover
+
+    robot = TwoLink(symbolic=True)
+    print(robot)
+
+ +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/DH/UR10.html b/_modules/roboticstoolbox/models/DH/UR10.html new file mode 100644 index 00000000..cdd697ab --- /dev/null +++ b/_modules/roboticstoolbox/models/DH/UR10.html @@ -0,0 +1,219 @@ + + + + + + roboticstoolbox.models.DH.UR10 — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.DH.UR10

+import numpy as np
+from roboticstoolbox import DHRobot, RevoluteDH
+from spatialmath import SE3
+
+
+
[docs]class UR10(DHRobot): + """ + Class that models a Universal Robotics UR10 manipulator + + :param symbolic: use symbolic constants + :type symbolic: bool + + ``UR10()`` is an object which models a Unimation Puma560 robot and + describes its kinematic and dynamic characteristics using standard DH + conventions. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.DH.UR10() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration + - qr, arm horizontal along x-axis + + .. note:: + - SI units are used. + + :References: + + - `Parameters for calculations of kinematics and dynamics <https://www.universal-robots.com/articles/ur/parameters-for-calculations-of-kinematics-and-dynamics>`_ + + :sealso: :func:`UR3`, :func:`UR5` + + + .. codeauthor:: Peter Corke + """ # noqa + + def __init__(self, symbolic=False): + + if symbolic: + import spatialmath.base.symbolic as sym + + zero = sym.zero() + pi = sym.pi() + else: + from math import pi + + zero = 0.0 + + deg = pi / 180 + inch = 0.0254 + + # robot length values (metres) + a = [0, -0.612, -0.5723, 0, 0, 0] + d = [0.1273, 0, 0, 0.163941, 0.1157, 0.0922] + + alpha = [pi / 2, zero, zero, pi / 2, -pi / 2, zero] + + # mass data, no inertia available + mass = [7.1, 12.7, 4.27, 2.000, 2.000, 0.365] + center_of_mass = [ + [0.021, 0, 0.027], + [0.38, 0, 0.158], + [0.24, 0, 0.068], + [0.0, 0.007, 0.018], + [0.0, 0.007, 0.018], + [0, 0, -0.026], + ] + links = [] + + for j in range(6): + link = RevoluteDH( + d=d[j], a=a[j], alpha=alpha[j], m=mass[j], r=center_of_mass[j], G=1 + ) + links.append(link) + + super().__init__( + links, + name="UR10", + manufacturer="Universal Robotics", + keywords=("dynamics", "symbolic"), + symbolic=symbolic, + ) + + self.qr = np.array([180, 0, 0, 0, 90, 0]) * deg + self.qz = np.zeros(6) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+ + +if __name__ == "__main__": # pragma nocover + + ur10 = UR10(symbolic=False) + print(ur10) + # print(ur10.dyntable()) +
+ +
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/DH/UR3.html b/_modules/roboticstoolbox/models/DH/UR3.html new file mode 100644 index 00000000..dd6dcb9d --- /dev/null +++ b/_modules/roboticstoolbox/models/DH/UR3.html @@ -0,0 +1,219 @@ + + + + + + roboticstoolbox.models.DH.UR3 — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.DH.UR3

+import numpy as np
+from roboticstoolbox import DHRobot, RevoluteDH
+from spatialmath import SE3
+
+
+
[docs]class UR3(DHRobot): + """ + Class that models a Universal Robotics UR3 manipulator + + :param symbolic: use symbolic constants + :type symbolic: bool + + ``UR3()`` is an object which models a Unimation Puma560 robot and + describes its kinematic and dynamic characteristics using standard DH + conventions. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.DH.UR3() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration + - qr, arm horizontal along x-axis + + .. note:: + - SI units are used. + + :References: + + - `Parameters for calculations of kinematics and dynamics <https://www.universal-robots.com/articles/ur/parameters-for-calculations-of-kinematics-and-dynamics>`_ + + :sealso: :func:`UR5`, :func:`UR10` + + + .. codeauthor:: Peter Corke + """ # noqa + + def __init__(self, symbolic=False): + + if symbolic: + import spatialmath.base.symbolic as sym + + zero = sym.zero() + pi = sym.pi() + else: + from math import pi + + zero = 0.0 + + deg = pi / 180 + inch = 0.0254 + + # robot length values (metres) + a = [0, -0.24365, -0.21325, 0, 0, 0] + d = [0.1519, 0, 0, 0.11235, 0.08535, 0.0819] + + alpha = [pi / 2, zero, zero, pi / 2, -pi / 2, zero] + + # mass data, no inertia available + mass = [2, 3.42, 1.26, 0.8, 0.8, 0.35] + center_of_mass = [ + [0, -0.02, 0], + [0.13, 0, 0.1157], + [0.05, 0, 0.0238], + [0, 0, 0.01], + [0, 0, 0.01], + [0, 0, -0.02], + ] + links = [] + + for j in range(6): + link = RevoluteDH( + d=d[j], a=a[j], alpha=alpha[j], m=mass[j], r=center_of_mass[j], G=1 + ) + links.append(link) + + super().__init__( + links, + name="UR3", + manufacturer="Universal Robotics", + keywords=("dynamics", "symbolic"), + symbolic=symbolic, + ) + + self.qr = np.array([180, 0, 0, 0, 90, 0]) * deg + self.qz = np.zeros(6) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+ + +if __name__ == "__main__": # pragma nocover + + ur3 = UR3(symbolic=False) + print(ur3) + # print(ur3.dyntable()) +
+ +
+
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/DH/UR5.html b/_modules/roboticstoolbox/models/DH/UR5.html new file mode 100644 index 00000000..22fee061 --- /dev/null +++ b/_modules/roboticstoolbox/models/DH/UR5.html @@ -0,0 +1,219 @@ + + + + + + roboticstoolbox.models.DH.UR5 — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.DH.UR5

+import numpy as np
+from roboticstoolbox import DHRobot, RevoluteDH
+from spatialmath import SE3
+
+
+
[docs]class UR5(DHRobot): + """ + Class that models a Universal Robotics UR5 manipulator + + :param symbolic: use symbolic constants + :type symbolic: bool + + ``UR5()`` is an object which models a Unimation Puma560 robot and + describes its kinematic and dynamic characteristics using standard DH + conventions. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.DH.UR5() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration + - qr, arm horizontal along x-axis + + .. note:: + - SI units are used. + + :References: + + - `Parameters for calculations of kinematics and dynamics <https://www.universal-robots.com/articles/ur/parameters-for-calculations-of-kinematics-and-dynamics>`_ + + :sealso: :func:`UR4`, :func:`UR10` + + + .. codeauthor:: Peter Corke + """ # noqa + + def __init__(self, symbolic=False): + + if symbolic: + import spatialmath.base.symbolic as sym + + zero = sym.zero() + pi = sym.pi() + else: + from math import pi + + zero = 0.0 + + deg = pi / 180 + inch = 0.0254 + + # robot length values (metres) + a = [0, -0.42500, -0.39225, 0, 0, 0] + d = [0.089459, 0, 0, 0.10915, 0.09465, 0.0823] + + alpha = [pi / 2, zero, zero, pi / 2, -pi / 2, zero] + + # mass data, no inertia available + mass = [3.7000, 8.3930, 2.33, 1.2190, 1.2190, 0.1897] + center_of_mass = [ + [0, -0.02561, 0.00193], + [0.2125, 0, 0.11336], + [0.15, 0, 0.0265], + [0, -0.0018, 0.01634], + [0, -0.0018, 0.01634], + [0, 0, -0.001159], + ] + links = [] + + for j in range(6): + link = RevoluteDH( + d=d[j], a=a[j], alpha=alpha[j], m=mass[j], r=center_of_mass[j], G=1 + ) + links.append(link) + + super().__init__( + links, + name="UR5", + manufacturer="Universal Robotics", + keywords=("dynamics", "symbolic"), + symbolic=symbolic, + ) + + self.qr = np.array([180, 0, 0, 0, 90, 0]) * deg + self.qz = np.zeros(6) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+ + +if __name__ == "__main__": # pragma nocover + + ur5 = UR5(symbolic=False) + print(ur5) + # print(ur5.dyntable()) +
+ +
+
+ +
+
+
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/ETS/Frankie.html b/_modules/roboticstoolbox/models/ETS/Frankie.html new file mode 100644 index 00000000..0c1789d3 --- /dev/null +++ b/_modules/roboticstoolbox/models/ETS/Frankie.html @@ -0,0 +1,211 @@ + + + + + + roboticstoolbox.models.ETS.Frankie — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.ETS.Frankie

+#!/usr/bin/env python
+
+import numpy as np
+from roboticstoolbox.robot.ET import ET
+from roboticstoolbox.robot.ETS import ETS
+from roboticstoolbox.robot.Robot import Robot
+from roboticstoolbox.robot.Link import Link
+
+
+
[docs]class Frankie(Robot): + """ + A class representing the Franka Emika Panda robot arm. ETS taken from [1] + based on https://frankaemika.github.io/docs/control_parameters.html + + :param et_list: List of elementary transforms which represent the robot + kinematics + :type et_list: list of etb.robot.et + :param q_idx: List of indexes within the ets_list which correspond to + joints + :type q_idx: list of int + :param name: Name of the robot + :type name: str, optional + :param manufacturer: Manufacturer of the robot + :type manufacturer: str, optional + :param base: Location of the base is the world frame + :type base: float np.ndarray(4,4), optional + :param tool: Offset of the flange of the robot to the end-effector + :type tool: float np.ndarray(4,4), optional + :param qz: The zero joint angle configuration of the robot + :type qz: float np.ndarray(7,) + :param qr: The ready state joint angle configuration of the robot + :type qr: float np.ndarray(7,) + + References: [1] Kinematic Derivatives using the Elementary Transform + Sequence, J. Haviland and P. Corke + """ + + def __init__(self): + + deg = np.pi / 180 + mm = 1e-3 + tool_offset = (103) * mm + + b0 = Link(ETS(ET.Rz()), name="base0", parent=None) + + b1 = Link(ETS(ET.tx()), name="base1", parent=b0) + + l0 = Link(ET.tz(0.333) * ET.Rz(), name="link0", parent=b1) + + l1 = Link(ET.Rx(-90 * deg) * ET.Rz(), name="link1", parent=l0) + + l2 = Link(ET.Rx(90 * deg) * ET.tz(0.316) * ET.Rz(), name="link2", parent=l1) + + l3 = Link(ET.tx(0.0825) * ET.Rx(90 * deg) * ET.Rz(), name="link3", parent=l2) + + l4 = Link( + ET.tx(-0.0825) * ET.Rx(-90 * deg) * ET.tz(0.384) * ET.Rz(), + name="link4", + parent=l3, + ) + + l5 = Link(ET.Rx(90 * deg) * ET.Rz(), name="link5", parent=l4) + + l6 = Link( + ET.tx(0.088) * ET.Rx(90 * deg) * ET.tz(0.107) * ET.Rz(), + name="link6", + parent=l5, + ) + + ee = Link(ET.tz(tool_offset) * ET.Rz(-np.pi / 4), name="ee", parent=l6) + + elinks = [b0, b1, l0, l1, l2, l3, l4, l5, l6, ee] + + super(Frankie, self).__init__( + elinks, + name="Frankie", + manufacturer="Franka Emika, Omron", + keywords=("mobile",), + ) + + self.qr = np.array([0, 0, 0, -0.3, 0, -2.2, 0, 2.0, np.pi / 4]) + self.qz = np.zeros(9) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+ + +if __name__ == "__main__": # pragma nocover + + robot = Frankie() + print(robot) +
+ +
+
+ +
+
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/ETS/GenericSeven.html b/_modules/roboticstoolbox/models/ETS/GenericSeven.html new file mode 100644 index 00000000..6ed1e56b --- /dev/null +++ b/_modules/roboticstoolbox/models/ETS/GenericSeven.html @@ -0,0 +1,174 @@ + + + + + + roboticstoolbox.models.ETS.GenericSeven — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.ETS.GenericSeven

+#!/usr/bin/env python
+
+import numpy as np
+from roboticstoolbox.robot.ET import ET
+from roboticstoolbox.robot.ETS import ETS
+from roboticstoolbox.robot.Robot import Robot
+from roboticstoolbox.robot.Link import Link
+
+
+
[docs]class GenericSeven(Robot): + """ + Create model of a generic seven degree-of-freedom robot + + robot = GenericSeven() creates a robot object. This robot is represented + using the elementary transform sequence (ETS). + + """ + + def __init__(self): + + deg = np.pi / 180 + mm = 1e-3 + tool_offset = (103) * mm + + link_length = 0.5 + + l0 = Link(ET.tz(link_length) * ET.Rz(), name="link0", parent=None) + + l1 = Link(ETS(ET.Ry()), name="link1", parent=l0) + + l2 = Link(ET.tz(link_length) * ET.Rz(), name="link2", parent=l1) + + l3 = Link(ETS(ET.Ry()), name="link3", parent=l2) + + l4 = Link(ET.tz(link_length) * ET.Rz(), name="link4", parent=l3) + + l5 = Link(ETS(ET.Ry()), name="link5", parent=l4) + + l6 = Link(ET.tx(link_length) * ET.Rz(), name="link6", parent=l5) + + ee = Link(ETS(ET.tz(-link_length)), name="ee", parent=l6) + + elinks = [l0, l1, l2, l3, l4, l5, l6, ee] + # elinks = [l0, l1, l2, l3, l4, l5, ee] + + super(GenericSeven, self).__init__( + elinks, name="Generic Seven", manufacturer="Jesse's Imagination" + ) + + self.qr = np.array([0, -0.3, 0, -2.2, 0, 2.0, np.pi / 4]) + self.qz = np.zeros(7) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+
+ +
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/ETS/Omni.html b/_modules/roboticstoolbox/models/ETS/Omni.html new file mode 100644 index 00000000..22c60bf9 --- /dev/null +++ b/_modules/roboticstoolbox/models/ETS/Omni.html @@ -0,0 +1,184 @@ + + + + + + roboticstoolbox.models.ETS.Omni — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + +
+ + +
+ +
+ +
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/ETS/Panda.html b/_modules/roboticstoolbox/models/ETS/Panda.html new file mode 100644 index 00000000..48edbe74 --- /dev/null +++ b/_modules/roboticstoolbox/models/ETS/Panda.html @@ -0,0 +1,190 @@ + + + + + + roboticstoolbox.models.ETS.Panda — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.ETS.Panda

+#!/usr/bin/env python
+
+import numpy as np
+from roboticstoolbox.robot.ET import ET
+from roboticstoolbox.robot.Robot import Robot
+from roboticstoolbox.robot.Link import Link
+
+
+
[docs]class Panda(Robot): + """ + Create model of Franka-Emika Panda manipulator + + panda = Panda() creates a robot object representing the Franka-Emika + Panda robot arm. This robot is represented using the elementary + transform sequence (ETS). + + ETS taken from [1] based on + https://frankaemika.github.io/docs/control_parameters.html + + :references: + - Kinematic Derivatives using the Elementary Transform + Sequence, J. Haviland and P. Corke + + """ + + def __init__(self): + + deg = np.pi / 180 + mm = 1e-3 + tool_offset = (103) * mm + + l0 = Link(ET.tz(0.333) * ET.Rz(), name="link0", parent=None) + + l1 = Link(ET.Rx(-90 * deg) * ET.Rz(), name="link1", parent=l0) + + l2 = Link(ET.Rx(90 * deg) * ET.tz(0.316) * ET.Rz(), name="link2", parent=l1) + + l3 = Link(ET.tx(0.0825) * ET.Rx(90, "deg") * ET.Rz(), name="link3", parent=l2) + + l4 = Link( + ET.tx(-0.0825) * ET.Rx(-90, "deg") * ET.tz(0.384) * ET.Rz(), + name="link4", + parent=l3, + ) + + l5 = Link(ET.Rx(90, "deg") * ET.Rz(), name="link5", parent=l4) + + l6 = Link( + ET.tx(0.088) * ET.Rx(90, "deg") * ET.tz(0.107) * ET.Rz(), + name="link6", + parent=l5, + ) + + ee = Link(ET.tz(tool_offset) * ET.Rz(-np.pi / 4), name="ee", parent=l6) + + elinks = [l0, l1, l2, l3, l4, l5, l6, ee] + + super(Panda, self).__init__(elinks, name="Panda", manufacturer="Franka Emika") + + self.qr = np.array([0, -0.3, 0, -2.2, 0, 2.0, np.pi / 4]) + self.qz = np.zeros(7) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+ + +if __name__ == "__main__": # pragma nocover + + robot = Panda() + print(robot) +
+ +
+
+ +
+
+
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/ETS/Planar2.html b/_modules/roboticstoolbox/models/ETS/Planar2.html new file mode 100644 index 00000000..c7031a26 --- /dev/null +++ b/_modules/roboticstoolbox/models/ETS/Planar2.html @@ -0,0 +1,170 @@ + + + + + + roboticstoolbox.models.ETS.Planar2 — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.ETS.Planar2

+#!/usr/bin/env python
+
+import numpy as np
+from roboticstoolbox.robot.ET import ET2
+from roboticstoolbox.robot.ETS import ETS2
+from roboticstoolbox.robot.Robot import Robot2
+from roboticstoolbox.robot.Link import Link2
+
+
+
[docs]class Planar2(Robot2): + """ + Create model of a branched planar manipulator:: + + L0 -- L1 -+- L2a -- L3a -- EEa + | + +- L2b -- L3b -- EEb + + ``Planar_Y()`` creates a planar branched manipulator model. + + + :references: + - Kinematic Derivatives using the Elementary Transform + Sequence, J. Haviland and P. Corke + + """ + + def __init__(self): + + a1 = 1 + a2 = 1 + + l0 = Link2(ETS2(ET2.R()), name="link0") + l1 = Link2(ETS2(ET2.tx(a1)) * ET2.R(), name="link1", parent=l0) + l2 = Link2(ETS2(ET2.tx(a2)), name="ee", parent=l1) + + super().__init__([l0, l1, l2], name="Planar2", comment="Planar 2D manipulator") + + self.qb = np.array([0, np.pi/2]) + self.qz = np.zeros(2) + + self.addconfiguration("qz", self.qz) + self.addconfiguration("qb", self.qb)
+ + +if __name__ == "__main__": # pragma nocover + + robot = Planar2() + print(robot) + + # print(robot.fkine(robot.qz)) +
+ +
+
+ +
+
+
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/ETS/Planar_Y.html b/_modules/roboticstoolbox/models/ETS/Planar_Y.html new file mode 100644 index 00000000..5c4675e0 --- /dev/null +++ b/_modules/roboticstoolbox/models/ETS/Planar_Y.html @@ -0,0 +1,186 @@ + + + + + + roboticstoolbox.models.ETS.Planar_Y — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.ETS.Planar_Y

+#!/usr/bin/env python
+
+import numpy as np
+from roboticstoolbox.robot.ET import ET
+from roboticstoolbox.robot.ETS import ETS
+from roboticstoolbox.robot.Robot import Robot
+from roboticstoolbox.robot.Link import Link
+
+
+
[docs]class Planar_Y(Robot): + """ + Create model of a branched planar manipulator:: + + L0 -- L1 -+- L2a -- L3a -- EEa + | + +- L2b -- L3b -- EEb + + ``Planar_Y()`` creates a planar branched manipulator model. + + + :references: + - Kinematic Derivatives using the Elementary Transform + Sequence, J. Haviland and P. Corke + + """ + + def __init__(self): + + # deg = np.pi / 180 + # mm = 1e-3 + tool_offset = 1 + + # trunk of the tree + l0 = Link(ETS(ET.Rz()), name="link0", jindex=0, parent=None) + + l1 = Link(ET.tx(1) * ET.Rz(), name="link1", jindex=1, parent=l0) + + # branch 1 + l2 = Link(ET.tx(1) * ET.Rz(), name="link2a", jindex=2, parent=l1) + + l3 = Link(ET.tx(1) * ET.Rz(), name="link3a", jindex=3, parent=l2) + + eea = Link(ETS(ET.tz(tool_offset)), name="eea", parent=l3) + + # branch 2 + l4 = Link(ET.tx(1) * ET.Rz(), name="link2b", jindex=4, parent=l1) + + l5 = Link(ET.tx(1) * ET.Rz(), name="link3b", jindex=5, parent=l4) + + eeb = Link(ETS(ET.tz(tool_offset)), name="eeb", parent=l5) + + elinks = [l0, l1, l2, l3, l4, l5, eea, eeb] + + super().__init__(elinks, name="Planar-Y") + + self.qr = np.array([0, 0, np.pi / 4, 0, -np.pi / 4, 0]) + self.qz = np.zeros(6) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+ + +if __name__ == "__main__": # pragma nocover + + robot = Planar_Y() + print(robot) +
+ +
+
+ +
+
+
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/ETS/Puma560.html b/_modules/roboticstoolbox/models/ETS/Puma560.html new file mode 100644 index 00000000..878f87de --- /dev/null +++ b/_modules/roboticstoolbox/models/ETS/Puma560.html @@ -0,0 +1,190 @@ + + + + + + roboticstoolbox.models.ETS.Puma560 — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.ETS.Puma560

+from roboticstoolbox import ET as ET
+from roboticstoolbox import Robot
+import numpy as np
+
+
+
[docs]class Puma560(Robot): + """ + Create model of Franka-Emika Panda manipulator + + ``puma = Puma560()`` creates a robot object representing the classic + Unimation Puma560 robot arm. This robot is represented using the elementary + transform sequence (ETS). + + .. note:: + + - The model has different joint offset conventions compared to + ``DH.Puma560()``. For this robot: + - Zero joint angles ``qz`` is the vertical configuration, + corresponding to ``qr`` with ``DH.Puma560()`` + - ``qbent`` is the bent configuration, corresponding to + ``qz`` with ``DH.Puma560()`` + + :references: + - "A Simple and Systematic Approach to Assigning Denavit–Hartenberg + Parameters,", P. I. Corke, in IEEE Transactions on Robotics, vol. 23, + no. 3, pp. 590-594, June 2007, doi: 10.1109/TRO.2007.896765. + - https://petercorke.com/robotics/a-simple-and-systematic-approach-to-assigning-denavit-hartenberg-parameters + + """ # noqa + + def __init__(self): + # Puma dimensions (m) + l1 = 0.672 + l2 = -0.2337 + l3 = 0.4318 + l4 = 0.0203 + l5 = 0.0837 + l6 = 0.4318 + + e = ( + ET.tz(l1) + * ET.Rz() + * ET.ty(l2) + * ET.Ry() + * ET.tz(l3) + * ET.tx(l4) + * ET.ty(l5) + * ET.Ry() + * ET.tz(l6) + * ET.Rz() + * ET.Ry() + * ET.Rz() + * ET.tx(0.2) + ) + + super().__init__( + e, name="Puma560", manufacturer="Unimation", comment="ETS-based model" + ) + + self.qr = np.array([0, -np.pi / 2, np.pi / 2, 0, 0, 0]) + self.qz = np.zeros(6) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+ + +if __name__ == "__main__": # pragma nocover + + robot = Puma560() + print(robot) +
+ +
+
+ +
+
+
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/ETS/XYPanda.html b/_modules/roboticstoolbox/models/ETS/XYPanda.html new file mode 100644 index 00000000..1992d851 --- /dev/null +++ b/_modules/roboticstoolbox/models/ETS/XYPanda.html @@ -0,0 +1,205 @@ + + + + + + roboticstoolbox.models.ETS.XYPanda — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.ETS.XYPanda

+#!/usr/bin/env python
+
+import numpy as np
+from roboticstoolbox.robot.ET import ET
+from roboticstoolbox.robot.ETS import ETS
+from roboticstoolbox.robot.Robot import Robot
+from roboticstoolbox.robot.Link import Link
+
+
+
[docs]class XYPanda(Robot): + """ + Create model of Franka-Emika Panda manipulator on an XY platform + + xypanda = XYPanda() creates a robot object representing the Franka-Emika + Panda robot arm mounted on an XY platform. This robot is represented using the elementary + transform sequence (ETS). + + ETS taken from [1] based on + https://frankaemika.github.io/docs/control_parameters.html + + :references: + - Kinematic Derivatives using the Elementary Transform + Sequence, J. Haviland and P. Corke + + """ + + def __init__(self, workspace=5): + """ + Create model of Franka-Emika Panda manipulator on an XY platform. + + :param workspace: workspace limits in the x and y directions, defaults to 5 + :type workspace: float, optional + + The XY part of the robot has a range -``workspace`` to ``workspace``. + """ + + deg = np.pi / 180 + mm = 1e-3 + tool_offset = (103) * mm + + lx = Link( + ETS(ET.tx()), name="link-x", parent=None, qlim=[-workspace, workspace] + ) + + ly = Link(ETS(ET.ty()), name="link-y", parent=lx, qlim=[-workspace, workspace]) + + l0 = Link(ET.tz(0.333) * ET.Rz(), name="link0", parent=ly) + + l1 = Link(ET.Rx(-90 * deg) * ET.Rz(), name="link1", parent=l0) + + l2 = Link(ET.Rx(90 * deg) * ET.tz(0.316) * ET.Rz(), name="link2", parent=l1) + + l3 = Link(ET.tx(0.0825) * ET.Rx(90, "deg") * ET.Rz(), name="link3", parent=l2) + + l4 = Link( + ET.tx(-0.0825) * ET.Rx(-90, "deg") * ET.tz(0.384) * ET.Rz(), + name="link4", + parent=l3, + ) + + l5 = Link(ET.Rx(90, "deg") * ET.Rz(), name="link5", parent=l4) + + l6 = Link( + ET.tx(0.088) * ET.Rx(90, "deg") * ET.tz(0.107) * ET.Rz(), + name="link6", + parent=l5, + ) + + ee = Link(ET.tz(tool_offset) * ET.Rz(-np.pi / 4), name="ee", parent=l6) + + elinks = [lx, ly, l0, l1, l2, l3, l4, l5, l6, ee] + + super().__init__(elinks, name="XYPanda", manufacturer="Franka Emika") + + self.qr = np.array([0, 0, 0, -0.3, 0, -2.2, 0, 2.0, np.pi / 4]) + self.qz = np.zeros(9) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+ + +if __name__ == "__main__": # pragma nocover + + robot = XYPanda() + print(robot) +
+ +
+
+ +
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/URDF/AL5D.html b/_modules/roboticstoolbox/models/URDF/AL5D.html new file mode 100644 index 00000000..8f270a3f --- /dev/null +++ b/_modules/roboticstoolbox/models/URDF/AL5D.html @@ -0,0 +1,175 @@ + + + + + + roboticstoolbox.models.URDF.AL5D — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.URDF.AL5D

+#!/usr/bin/env python
+
+import numpy as np
+from roboticstoolbox.robot.Robot import Robot
+from math import pi
+
+
+
[docs]class AL5D(Robot): + """ + Class that imports a AL5D URDF model + + ``AL5D()`` is a class which imports a Lynxmotion AL5D robot definition + from a URDF file. The model describes its kinematic and graphical + characteristics. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.URDF.AL5D() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration, 'L' shaped configuration + - up, robot poiting upwards + + .. codeauthor:: Tassos Natsakis + """ + + def __init__(self): + + links, name, urdf_string, urdf_filepath = self.URDF_read( + "al5d_description/urdf/al5d_robot.urdf" + ) + + super().__init__( + links, + name=name, + urdf_string=urdf_string, + urdf_filepath=urdf_filepath, + ) + + self.manufacturer = "Lynxmotion" + + # zero angles, upper arm pointing up, lower arm straight ahead + self.addconfiguration("qz", np.array([0, 0, 0, 0])) + + # reference pose robot pointing upwards + self.addconfiguration("up", np.array([0.0000, 0.0000, 1.5707, 0.0000]))
+ + +if __name__ == "__main__": # pragma nocover + + robot = AL5D() + print(robot) +
+ +
+
+ +
+
+
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/URDF/Fetch.html b/_modules/roboticstoolbox/models/URDF/Fetch.html new file mode 100644 index 00000000..382fa99a --- /dev/null +++ b/_modules/roboticstoolbox/models/URDF/Fetch.html @@ -0,0 +1,191 @@ + + + + + + roboticstoolbox.models.URDF.Fetch — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.URDF.Fetch

+#!/usr/bin/env python
+
+import numpy as np
+from roboticstoolbox.robot.Robot import Robot
+
+# from spatialmath import SE3
+
+
+
[docs]class Fetch(Robot): + """ + Class that imports a Fetch URDF model + + ``Fetch()`` is a class which imports a Fetch robot definition + from a URDF file. The model describes its kinematic and graphical + characteristics. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.URDF.Fetch() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration, arm is stretched out in the x-direction + - qr, tucked arm configuration + + .. codeauthor:: Kerry He + .. sectionauthor:: Peter Corke + """ + + def __init__(self): + + links, name, urdf_string, urdf_filepath = self.URDF_read( + "fetch_description/robots/fetch.urdf" + ) + + super().__init__( + links, + name=name, + manufacturer="Fetch", + gripper_links=links[11], + urdf_string=urdf_string, + urdf_filepath=urdf_filepath, + ) + + self.qdlim = np.array([4.0, 4.0, 0.1, 1.25, 1.45, 1.57, 1.52, 1.57, 2.26, 2.26]) + + self.qz = np.array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0]) + self.qr = np.array([0, 0, 0.05, 1.32, 1.4, -0.2, 1.72, 0, 1.66, 0]) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+ + +if __name__ == "__main__": # pragma nocover + + robot = Fetch() + print(robot) + + for link in robot.links: + print(link.name) + print(link.isjoint) + print(len(link.collision)) + + print() + + for link in robot.grippers[0].links: + print(link.name) + print(link.isjoint) + print(len(link.collision)) +
+ +
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/URDF/FetchCamera.html b/_modules/roboticstoolbox/models/URDF/FetchCamera.html new file mode 100644 index 00000000..581b046a --- /dev/null +++ b/_modules/roboticstoolbox/models/URDF/FetchCamera.html @@ -0,0 +1,192 @@ + + + + + + roboticstoolbox.models.URDF.FetchCamera — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.URDF.FetchCamera

+#!/usr/bin/env python
+
+import numpy as np
+from roboticstoolbox.robot.Robot import Robot
+
+# from spatialmath import SE3
+
+
+
[docs]class FetchCamera(Robot): + """ + Class that imports a FetchCamera URDF model + + ``FetchCamera()`` is a class which imports a FetchCamera robot definition + from a URDF file. The model describes its kinematic and graphical + characteristics. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.URDF.Fetch() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration + - qr, zero joint angle configuration + + .. codeauthor:: Kerry He + .. sectionauthor:: Peter Corke + """ + + def __init__(self): + + links, name, urdf_string, urdf_filepath = self.URDF_read( + "fetch_description/robots/fetch_camera.urdf" + ) + + super().__init__( + links, + name=name, + manufacturer="Fetch", + gripper_links=links[6], + urdf_string=urdf_string, + urdf_filepath=urdf_filepath, + ) + + # self.grippers[0].tool = SE3(0, 0, 0.1034) + self.qdlim = np.array([4.0, 4.0, 0.1, 1.57, 1.57]) + + self.qz = np.array([0, 0, 0, 0, 0]) + self.qr = np.array([0, 0, 0, 0, 0]) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+ + +if __name__ == "__main__": # pragma nocover + + robot = FetchCamera() + print(robot) + + for link in robot.links: + print(link.name) + print(link.isjoint) + print(len(link.collision)) + + print() + + for link in robot.grippers[0].links: + print(link.name) + print(link.isjoint) + print(len(link.collision)) +
+ +
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+ +
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/URDF/Frankie.html b/_modules/roboticstoolbox/models/URDF/Frankie.html new file mode 100644 index 00000000..d681a229 --- /dev/null +++ b/_modules/roboticstoolbox/models/URDF/Frankie.html @@ -0,0 +1,196 @@ + + + + + + roboticstoolbox.models.URDF.Frankie — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.URDF.Frankie

+#!/usr/bin/env python
+
+import numpy as np
+from roboticstoolbox.robot.Robot import Robot
+from spatialmath import SE3
+
+
+
[docs]class Frankie(Robot): + """ + Class that imports a Frankie URDF model + + ``Frankie()`` is a class which imports a Frankie robot definition + from a URDF file. The model describes its kinematic and graphical + characteristics. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.URDF.Frankie() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration, 'L' shaped configuration + - qr, vertical 'READY' configuration + - qs, arm is stretched out in the x-direction + - qn, arm is at a nominal non-singular configuration + + .. codeauthor:: Jesse Haviland + .. sectionauthor:: Peter Corke + """ + + def __init__(self): + + links, name, urdf_string, urdf_filepath = self.URDF_read( + "franka_description/robots/frankie_arm_hand.urdf.xacro" + ) + + super().__init__( + links, + name=name, + manufacturer="Franka Emika", + gripper_links=links[12], + urdf_string=urdf_string, + urdf_filepath=urdf_filepath, + ) + + self.grippers[0].tool = SE3(0, 0, 0.1034) + + self.qdlim = np.array( + [4.0, 4.0, 2.1750, 2.1750, 2.1750, 2.1750, 2.6100, 2.6100, 2.6100, 3.0, 3.0] + ) + + self.qr = np.array([0, 0, 0, -0.3, 0, -2.2, 0, 2.0, np.pi / 4]) + self.qz = np.zeros(9) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+ + +if __name__ == "__main__": # pragma nocover + + robot = Frankie() + print(robot) + + for link in robot.links: + print(link.name) + print(link.isjoint) + print(len(link.collision)) + + print() + + for link in robot.grippers[0].links: + print(link.name) + print(link.isjoint) + print(len(link.collision)) +
+ +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/URDF/FrankieOmni.html b/_modules/roboticstoolbox/models/URDF/FrankieOmni.html new file mode 100644 index 00000000..a4086d13 --- /dev/null +++ b/_modules/roboticstoolbox/models/URDF/FrankieOmni.html @@ -0,0 +1,214 @@ + + + + + + roboticstoolbox.models.URDF.FrankieOmni — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.URDF.FrankieOmni

+#!/usr/bin/env python
+
+import numpy as np
+from roboticstoolbox.robot.Robot import Robot
+from roboticstoolbox.robot.Link import Link
+from roboticstoolbox.robot.ETS import ETS
+from roboticstoolbox.robot.ET import ET
+from spatialmath import SE3
+
+
+
[docs]class FrankieOmni(Robot): + """ + Class that imports an Omnidirectional Frankie URDF model + + ``FrankieOmni()`` is a class which imports a FrankieOmni robot definition + from a URDF file. The model describes its kinematic and graphical + characteristics. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.URDF.FrankieOmni() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration, 'L' shaped configuration + - qr, vertical 'READY' configuration + - qs, arm is stretched out in the x-direction + - qn, arm is at a nominal non-singular configuration + + .. codeauthor:: Jesse Haviland + .. sectionauthor:: Peter Corke + """ + + def __init__(self): + + links_base, _, urdf_string, urdf_filepath = self.URDF_read( + "ridgeback_description/urdf/ridgeback.urdf.xacro" + ) + + links_panda, _, _, _ = self.URDF_read( + "franka_description/robots/panda_arm_hand.urdf.xacro" + ) + + base_link = links_base[9] + base_arm = Link(ETS(ET.tz(0.28)), name="base_arm", parent=base_link) + + links_panda[0]._parent = base_arm + links_all = links_base + links_panda + links_all.append(base_arm) + + super().__init__( + links_all, + name="FrankieOmni", + manufacturer="Custom", + gripper_links=links_all[28], + urdf_string=urdf_string, + urdf_filepath=urdf_filepath, + ) + + self.grippers[0].tool = SE3(0, 0, 0.1034) + + self.qdlim = np.array( + [ + 4.0, + 4.0, + 4.0, + 2.1750, + 2.1750, + 2.1750, + 2.1750, + 2.6100, + 2.6100, + 2.6100, + 3.0, + 3.0, + ] + ) + + self.qr = np.array([0, 0, 0, 0, -0.3, 0, -2.2, 0, 2.0, np.pi / 4]) + self.qz = np.zeros(10) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+ + +if __name__ == "__main__": # pragma nocover + pass + + # r = Panda() + + # for link in r.grippers[0].links: + # print(link) +
+ +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/URDF/KinovaGen3.html b/_modules/roboticstoolbox/models/URDF/KinovaGen3.html new file mode 100644 index 00000000..2f82eb39 --- /dev/null +++ b/_modules/roboticstoolbox/models/URDF/KinovaGen3.html @@ -0,0 +1,180 @@ + + + + + + roboticstoolbox.models.URDF.KinovaGen3 — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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+ +

Source code for roboticstoolbox.models.URDF.KinovaGen3

+#!/usr/bin/env python
+
+import numpy as np
+from roboticstoolbox.robot.Robot import Robot
+
+
+
[docs]class KinovaGen3(Robot): + """ + Class that imports a KinovaGen3 URDF model + + ``KinovaGen3()`` is a class which imports a KinovaGen3 robot definition + from a URDF file. The model describes its kinematic and graphical + characteristics. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.URDF.KinovaGen3() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration, 'L' shaped configuration + - qr, vertical 'READY' configuration + - qs, arm is stretched out in the x-direction + - qn, arm is at a nominal non-singular configuration + + .. codeauthor:: Jesse Haviland + .. sectionauthor:: Peter Corke + """ + + def __init__(self): + + links, name, urdf_string, urdf_filepath = self.URDF_read( + "kortex_description/robots/gen3.xacro" + ) + + super().__init__( + links, + name=name, + manufacturer="Kinova", + urdf_string=urdf_string, + urdf_filepath=urdf_filepath, + # gripper_links=elinks[9] + ) + + # self.qdlim = np.array([ + # 2.1750, 2.1750, 2.1750, 2.1750, 2.6100, 2.6100, 2.6100, 3.0, 3.0]) + + self.qr = np.array([np.pi, -0.3, 0, -1.6, 0, -1.0, np.pi / 2]) + self.qz = np.zeros(7) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+ + +if __name__ == "__main__": # pragma nocover + + robot = KinovaGen3() + print(robot) +
+ +
+
+ +
+
+
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/URDF/LBR.html b/_modules/roboticstoolbox/models/URDF/LBR.html new file mode 100644 index 00000000..c5799525 --- /dev/null +++ b/_modules/roboticstoolbox/models/URDF/LBR.html @@ -0,0 +1,180 @@ + + + + + + roboticstoolbox.models.URDF.LBR — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
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Source code for roboticstoolbox.models.URDF.LBR

+#!/usr/bin/env python
+
+import numpy as np
+from roboticstoolbox.robot.Robot import Robot
+
+
+
[docs]class LBR(Robot): + """ + Class that imports a LBR URDF model + + ``LBR()`` is a class which imports a Franka-Emika LBR robot definition + from a URDF file. The model describes its kinematic and graphical + characteristics. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.URDF.LBR() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration, 'L' shaped configuration + - qr, vertical 'READY' configuration + - qs, arm is stretched out in the x-direction + - qn, arm is at a nominal non-singular configuration + + .. codeauthor:: Jesse Haviland + .. sectionauthor:: Peter Corke + """ + + def __init__(self): + + links, name, urdf_string, urdf_filepath = self.URDF_read( + "kuka_description/kuka_lbr_iiwa/urdf/lbr_iiwa_14_r820.xacro" + ) + + super().__init__( + links, + name=name, + manufacturer="Kuka", + urdf_string=urdf_string, + urdf_filepath=urdf_filepath, + # gripper_links=elinks[9] + ) + + # self.qdlim = np.array([ + # 2.1750, 2.1750, 2.1750, 2.1750, 2.6100, 2.6100, 2.6100, 3.0, 3.0]) + + self.qr = np.array([0, -0.3, 0, -1.9, 0, 1.5, np.pi / 4]) + self.qz = np.zeros(7) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+ + +if __name__ == "__main__": # pragma nocover + + robot = LBR() + print(robot) +
+ +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/URDF/Mico.html b/_modules/roboticstoolbox/models/URDF/Mico.html new file mode 100644 index 00000000..21c91d55 --- /dev/null +++ b/_modules/roboticstoolbox/models/URDF/Mico.html @@ -0,0 +1,189 @@ + + + + + + roboticstoolbox.models.URDF.Mico — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
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+ +

Source code for roboticstoolbox.models.URDF.Mico

+#!/usr/bin/env python
+
+import numpy as np
+from roboticstoolbox.robot.Link import Link
+from roboticstoolbox.robot.Robot import Robot
+
+
+
[docs]class Mico(Robot): + """ + Class that imports a Mico URDF model + + ``Panda()`` is a class which imports a Kinova Mico robot definition + from a URDF file. The model describes its kinematic and graphical + characteristics. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.URDF.Mico() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration, 'L' shaped configuration + - qr, vertical 'READY' configuration + + .. codeauthor:: Jesse Haviland + .. sectionauthor:: Peter Corke + """ + + def __init__(self): + + links, name, urdf_string, urdf_filepath = self.URDF_read( + "kinova_description/urdf/j2n4s300_standalone.xacro" + ) + + gripper_base = links[6] + + # Find the finger links + gripper_links = [link for link in links if link.parent == gripper_base] + + # New intermediate link + gripper = Link(name="gripper", parent=gripper_base) + links.append(gripper) + + # Set the finger link parent to be the new gripper base link + for g_link in gripper_links: + g_link._parent = gripper + + super().__init__( + links, + name=name, + manufacturer="Kinova", + gripper_links=[gripper], + urdf_string=urdf_string, + urdf_filepath=urdf_filepath, + ) + + self.qr = np.array([0, 45, 60, 0]) * np.pi / 180 + self.qz = np.zeros(4) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+ + +if __name__ == "__main__": # pragma nocover + + robot = Mico() + print(robot) +
+ +
+
+ +
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/URDF/PR2.html b/_modules/roboticstoolbox/models/URDF/PR2.html new file mode 100644 index 00000000..a75240ac --- /dev/null +++ b/_modules/roboticstoolbox/models/URDF/PR2.html @@ -0,0 +1,185 @@ + + + + + + roboticstoolbox.models.URDF.PR2 — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
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Source code for roboticstoolbox.models.URDF.PR2

+#!/usr/bin/env python
+
+from roboticstoolbox.robot.Robot import Robot
+import numpy as np
+
+
+
[docs]class PR2(Robot): + def __init__(self): + + links, name, urdf_string, urdf_filepath = self.URDF_read( + "pr2_description/robots/pr2.urdf.xacro", xacro_tld="pr2_description" + ) + + super().__init__( + links, + gripper_links=[links[53], links[73]], + name=name, + urdf_string=urdf_string, + urdf_filepath=urdf_filepath, + ) + + self.grippers[0].tool = self.link_dict["r_gripper_tool_frame"].A() + self.grippers[1].tool = self.link_dict["l_gripper_tool_frame"].A() + + self.manufacturer = "Willow Garage" + + self.qr = np.zeros(31) + self.qz = np.zeros(31) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz) + + self.qdlim = 2.0 * np.ones(31)
+ + +if __name__ == "__main__": # pragma nocover + + r = PR2() + + # i = 0 + + # for link in r.links: + # if link.isjoint: + # print(i, link.name) + + # i += 1 + + # path, n, t = r.get_path(end=r.grippers[0]) + + # print(n) + # print(t) + + # for l in path[1:]: + # if len(l.collision) > 0: + # print(l.isjoint) + # print(l.name) + # print(l.parent.name) + # print() + + # for l in r.grippers[0].links: + # if len(l.collision) > 0: + # print(l.isjoint) + # print(l.name) + # print(l.parent.name) + # print() +
+ +
+
+ +
+
+
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/URDF/Panda.html b/_modules/roboticstoolbox/models/URDF/Panda.html new file mode 100644 index 00000000..b8f98bf7 --- /dev/null +++ b/_modules/roboticstoolbox/models/URDF/Panda.html @@ -0,0 +1,188 @@ + + + + + + roboticstoolbox.models.URDF.Panda — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
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Source code for roboticstoolbox.models.URDF.Panda

+#!/usr/bin/env python
+
+import numpy as np
+from roboticstoolbox.robot.Robot import Robot
+from spatialmath import SE3
+
+
+
[docs]class Panda(Robot): + """ + Class that imports a Panda URDF model + + ``Panda()`` is a class which imports a Franka-Emika Panda robot definition + from a URDF file. The model describes its kinematic and graphical + characteristics. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.URDF.Panda() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration, 'L' shaped configuration + - qr, vertical 'READY' configuration + - qs, arm is stretched out in the x-direction + - qn, arm is at a nominal non-singular configuration + + .. codeauthor:: Jesse Haviland + .. sectionauthor:: Peter Corke + """ + + def __init__(self): + + links, name, urdf_string, urdf_filepath = self.URDF_read( + "franka_description/robots/panda_arm_hand.urdf.xacro" + ) + + super().__init__( + links, + name=name, + manufacturer="Franka Emika", + gripper_links=links[9], + urdf_string=urdf_string, + urdf_filepath=urdf_filepath, + ) + + self.grippers[0].tool = SE3(0, 0, 0.1034) + + self.qdlim = np.array( + [2.1750, 2.1750, 2.1750, 2.1750, 2.6100, 2.6100, 2.6100, 3.0, 3.0] + ) + + self.qr = np.array([0, -0.3, 0, -2.2, 0, 2.0, np.pi / 4]) + self.qz = np.zeros(7) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+ + +if __name__ == "__main__": # pragma nocover + + r = Panda() + + r.qz + + for link in r.grippers[0].links: + print(link) +
+ +
+
+ +
+
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/URDF/Puma560.html b/_modules/roboticstoolbox/models/URDF/Puma560.html new file mode 100644 index 00000000..b81a69a5 --- /dev/null +++ b/_modules/roboticstoolbox/models/URDF/Puma560.html @@ -0,0 +1,218 @@ + + + + + + roboticstoolbox.models.URDF.Puma560 — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
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Source code for roboticstoolbox.models.URDF.Puma560

+#!/usr/bin/env python
+
+import numpy as np
+from roboticstoolbox.robot.Robot import Robot
+from math import pi
+
+
+
[docs]class Puma560(Robot): + """ + Class that imports a Puma 560 URDF model + + ``Puma560()`` is a class which imports a Unimation Puma560 robot definition + from a URDF file. The model describes its kinematic and graphical + characteristics. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.URDF.Puma560() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration, 'L' shaped configuration + - qr, vertical 'READY' configuration + - qs, arm is stretched out in the x-direction + - qn, arm is at a nominal non-singular configuration + + .. warning:: This file has been modified so that the zero-angle pose is the + same as the DH model in the toolbox. ``j3`` rotation is changed from + -𝜋/2 to 𝜋/2. Dimensions are also slightly different. Both models + include the pedestal height. + + .. note:: The original file is from https://github.com/nimasarli/puma560_description/blob/master/urdf/puma560_robot.urdf.xacro + + .. codeauthor:: Jesse Haviland + .. sectionauthor:: Peter Corke + """ + + def __init__(self): + + links, name, urdf_string, urdf_filepath = self.URDF_read( + "puma560_description/urdf/puma560_robot.urdf.xacro" + ) + + super().__init__( + links, + name=name, + urdf_string=urdf_string, + urdf_filepath=urdf_filepath, + ) + + self.manufacturer = "Unimation" + # self.ee_link = self.ets[9] + + # ready pose, arm up + self.qr = np.array([0, pi / 2, -pi / 2, 0, 0, 0]) + self.qz = np.zeros(6) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz) + + # zero angles, upper arm horizontal, lower up straight up + self.addconfiguration_attr("qz", np.array([0, 0, 0, 0, 0, 0])) + + # reference pose, arm to the right, elbow up + self.addconfiguration_attr( + "ru", np.array([-0.0000, 0.7854, 3.1416, -0.0000, 0.7854, 0.0000]) + ) + + # reference pose, arm to the right, elbow up + self.addconfiguration_attr( + "rd", np.array([-0.0000, -0.8335, 0.0940, -3.1416, 0.8312, 3.1416]) + ) + + # reference pose, arm to the left, elbow up + self.addconfiguration_attr( + "lu", np.array([2.6486, -3.9270, 0.0940, 2.5326, 0.9743, 0.3734]) + ) + + # reference pose, arm to the left, elbow down + self.addconfiguration_attr( + "ld", np.array([2.6486, -2.3081, 3.1416, 0.6743, 0.8604, 2.6611]) + ) + + # straight and horizontal + self.addconfiguration_attr("qs", np.array([0, 0, -pi / 2, 0, 0, 0])) + + # nominal table top picking pose + self.addconfiguration_attr("qn", np.array([0, pi / 4, pi, 0, pi / 4, 0]))
+ + +if __name__ == "__main__": # pragma nocover + + robot = Puma560() + print(robot) + + print(robot.ee_links) +
+ +
+
+ +
+
+
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/URDF/UR10.html b/_modules/roboticstoolbox/models/URDF/UR10.html new file mode 100644 index 00000000..b2b19e82 --- /dev/null +++ b/_modules/roboticstoolbox/models/URDF/UR10.html @@ -0,0 +1,174 @@ + + + + + + roboticstoolbox.models.URDF.UR10 — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.URDF.UR10

+#!/usr/bin/env python
+
+import numpy as np
+from roboticstoolbox.robot.Robot import Robot
+
+
+
[docs]class UR10(Robot): + """ + Class that imports a UR10 URDF model + + ``UR3()`` is a class which imports a Universal Robotics UR310 robot + definition from a URDF file. The model describes its kinematic and + graphical characteristics. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.URDF.UR10() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration, 'L' shaped configuration + - qr, vertical 'READY' configuration + + .. codeauthor:: Jesse Haviland + .. sectionauthor:: Peter Corke + """ + + def __init__(self): + + links, name, urdf_string, urdf_filepath = self.URDF_read( + "ur_description/urdf/ur10_joint_limited_robot.urdf.xacro" + ) + + super().__init__( + links, + name=name.upper(), + manufacturer="Universal Robotics", + urdf_string=urdf_string, + urdf_filepath=urdf_filepath, + ) + + self.qr = np.array([np.pi, 0, 0, 0, np.pi / 2, 0]) + self.qz = np.zeros(6) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+ + +if __name__ == "__main__": # pragma nocover + + robot = UR10() + print(robot) +
+ +
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+ +
+
+
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/URDF/UR3.html b/_modules/roboticstoolbox/models/URDF/UR3.html new file mode 100644 index 00000000..02a564bb --- /dev/null +++ b/_modules/roboticstoolbox/models/URDF/UR3.html @@ -0,0 +1,174 @@ + + + + + + roboticstoolbox.models.URDF.UR3 — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.URDF.UR3

+#!/usr/bin/env python
+
+import numpy as np
+from roboticstoolbox.robot.Robot import Robot
+
+
+
[docs]class UR3(Robot): + """ + Class that imports a UR3 URDF model + + ``UR3()`` is a class which imports a Universal Robotics UR3 robot + definition from a URDF file. The model describes its kinematic and + graphical characteristics. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.URDF.UR3() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration, 'L' shaped configuration + - qr, vertical 'READY' configuration + + .. codeauthor:: Jesse Haviland + .. sectionauthor:: Peter Corke + """ + + def __init__(self): + + links, name, urdf_string, urdf_filepath = self.URDF_read( + "ur_description/urdf/ur3_joint_limited_robot.urdf.xacro" + ) + + super().__init__( + links, + name=name.upper(), + manufacturer="Universal Robotics", + urdf_string=urdf_string, + urdf_filepath=urdf_filepath, + ) + + self.qr = np.array([np.pi, 0, 0, 0, np.pi / 2, 0]) + self.qz = np.zeros(6) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+ + +if __name__ == "__main__": # pragma nocover + + robot = UR3() + print(robot) +
+ +
+
+ +
+
+
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/URDF/UR5.html b/_modules/roboticstoolbox/models/URDF/UR5.html new file mode 100644 index 00000000..f4ba2ae5 --- /dev/null +++ b/_modules/roboticstoolbox/models/URDF/UR5.html @@ -0,0 +1,195 @@ + + + + + + roboticstoolbox.models.URDF.UR5 — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.URDF.UR5

+#!/usr/bin/env python
+
+import numpy as np
+from roboticstoolbox.robot.Robot import Robot
+from math import pi
+
+
+
[docs]class UR5(Robot): + """ + Class that imports a UR5 URDF model + + ``UR3()`` is a class which imports a Universal Robotics UR5 robot + definition from a URDF file. The model describes its kinematic and + graphical characteristics. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.URDF.UR5() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration, 'L' shaped configuration + - qr, vertical 'READY' configuration + + .. codeauthor:: Jesse Haviland + .. sectionauthor:: Peter Corke + """ + + def __init__(self): + + links, name, urdf_string, urdf_filepath = self.URDF_read( + "ur_description/urdf/ur5_joint_limited_robot.urdf.xacro" + ) + # for link in links: + # print(link) + + super().__init__( + links, + name=name.upper(), + manufacturer="Universal Robotics", + gripper_links=links[7], + urdf_string=urdf_string, + urdf_filepath=urdf_filepath, + ) + + self.qr = np.array([np.pi, 0, 0, 0, np.pi / 2, 0]) + self.qz = np.zeros(6) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz) + + # sol=robot.ikine_LM(SE3(0.5, -0.2, 0.2)@SE3.OA([1,0,0],[0,0,-1])) + self.addconfiguration_attr( + "qn", + np.array( + [ + -7.052413e-01, + 3.604328e-01, + -1.494176e00, + 1.133744e00, + -7.052413e-01, + 0, + ] + ), + ) + self.addconfiguration_attr("q1", [0, -pi / 2, pi / 2, 0, pi / 2, 0])
+ + +if __name__ == "__main__": # pragma nocover + + robot = UR5() + print(robot) + print(robot.ets()) +
+ +
+
+ +
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/URDF/Valkyrie.html b/_modules/roboticstoolbox/models/URDF/Valkyrie.html new file mode 100644 index 00000000..71e40250 --- /dev/null +++ b/_modules/roboticstoolbox/models/URDF/Valkyrie.html @@ -0,0 +1,205 @@ + + + + + + roboticstoolbox.models.URDF.Valkyrie — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.URDF.Valkyrie

+#!/usr/bin/env python
+
+import numpy as np
+from roboticstoolbox.robot.Robot import Robot, Link
+
+
+
[docs]class Valkyrie(Robot): + """ + Class that imports a NASA Valkyrie URDF model + + ``Valkyrie()`` is a class which imports a NASA Valkyrie robot definition + from a URDF file. The model describes its kinematic and graphical + characteristics. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.URDF.Valkyrie() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration, 'L' shaped configuration + - qr, vertical 'READY' configuration + + :reference: + - https://github.com/gkjohnson/nasa-urdf-robots + + .. codeauthor:: Peter Corke + """ + + def __init__(self, variant="A"): + + if not variant in "ABCD": + raise ValueError("variant must be in the range A-D") + + links, name, urdf_string, urdf_filepath = self.URDF_read( + f"val_description/model/robots/valkyrie_{variant}.urdf" + ) + + # We wish to add an intermediate link between gripper_r_base and + # @gripper_r_finger_r/l + # This is because gripper_r_base contains a revolute joint which is + # a part of the core kinematic chain and not the gripper. + # So we wish for gripper_r_base to be part of the robot and + # @gripper_r_finger_r/l to be in the gripper underneath a parent Link + # gripper_r_base = links[13] + # gripper_l_base = links[33] + + # # Find the finger links + # r_gripper_links = [link for link in links if link.parent == gripper_r_base] + # l_gripper_links = [link for link in links if link.parent == gripper_l_base] + + # # New intermediate links + # r_gripper = Link(name="rightGripper", parent=gripper_r_base) + # l_gripper = Link(name="leftGripper", parent=gripper_l_base) + # links.append(r_gripper) + # links.append(l_gripper) + + # # Set the finger link parent to be the new gripper base link + # for g_link in r_gripper_links: + # g_link._parent = r_gripper + + # for g_link in l_gripper_links: + # g_link._parent = l_gripper + + super().__init__( + links, + name=name, + manufacturer="NASA", + # gripper_links=[r_gripper, l_gripper], + urdf_string=urdf_string, + urdf_filepath=urdf_filepath, + )
+ + # self.addconfiguration_attr("qz", np.array([0, 0, 0, 0, 0, 0, 0])) + # self.addconfiguration_attr("qr", np.array([0, -0.3, 0, -2.2, 0, 2.0, np.pi / 4])) + + +if __name__ == "__main__": # pragma nocover + + robot = Valkyrie("B") + print(robot) + env = robot.plot(np.zeros((robot.n,))) + env.hold() +
+ +
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+ +
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/URDF/YuMi.html b/_modules/roboticstoolbox/models/URDF/YuMi.html new file mode 100644 index 00000000..fa30ee8a --- /dev/null +++ b/_modules/roboticstoolbox/models/URDF/YuMi.html @@ -0,0 +1,230 @@ + + + + + + roboticstoolbox.models.URDF.YuMi — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.URDF.YuMi

+#!/usr/bin/env python
+
+from roboticstoolbox.robot.Link import Link
+import numpy as np
+from roboticstoolbox.robot.Robot import Robot
+import spatialmath as sm
+
+
+
[docs]class YuMi(Robot): + """ + Class that imports an ABB YuMi URDF model + + ``YuMi()`` is a class which imports an ABB YuMi (IRB14000) robot definition + from a URDF file. The model describes its kinematic and graphical + characteristics. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.URDF.YuMi() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration, 'L' shaped configuration + - qr, vertical 'READY' configuration + + :reference: + - `https://github.com/OrebroUniversity/yumi <https://github.com/OrebroUniversity/yumi>`_ + + .. codeauthor:: Jesse Haviland + .. sectionauthor:: Peter Corke + """ + + def __init__(self): + + links, name, urdf_string, urdf_filepath = self.URDF_read( + "yumi_description/urdf/yumi.urdf" + ) + + # We wish to add an intermediate link between gripper_r_base and + # @gripper_r_finger_r/l + # This is because gripper_r_base contains a revolute joint which is + # a part of the core kinematic chain and not the gripper. + # So we wish for gripper_r_base to be part of the robot and + # @gripper_r_finger_r/l to be in the gripper underneath a parent Link + + gripper_r_base = links[16] + gripper_l_base = links[19] + + # Find the finger links + r_gripper_links = [link for link in links if link.parent == gripper_r_base] + l_gripper_links = [link for link in links if link.parent == gripper_l_base] + + # New intermediate links + r_gripper = Link(name="r_gripper", parent=gripper_l_base) + l_gripper = Link(name="l_gripper", parent=gripper_r_base) + links.append(r_gripper) + links.append(l_gripper) + + # Set the finger link parent to be the new gripper base link + for g_link in r_gripper_links: + g_link._parent = r_gripper + + for g_link in l_gripper_links: + g_link._parent = l_gripper + + super().__init__( + links, + name=name, + manufacturer="ABB", + gripper_links=[r_gripper, l_gripper], + urdf_string=urdf_string, + urdf_filepath=urdf_filepath, + ) + + # Set the default tool transform for the end-effectors + self.grippers[0].tool = sm.SE3.Tz(0.13) + self.grippers[1].tool = sm.SE3.Tz(0.13) + + self.qr = np.array( + [ + 0, + -0.3, + 0, + -2.2, + 0, + 2.0, + np.pi / 4, + 0, + -0.3, + 0, + -2.2, + 0, + 2.0, + np.pi / 4, + ] + ) + self.qz = np.zeros(14) + self.q1 = np.array([0, -0.4, 0, 0, 0, 0, 0, 0, -0.4, 0, 0, 0, 0, 0]) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz) + self.addconfiguration("q1", self.q1)
+ + +if __name__ == "__main__": # pragma nocover + + robot = YuMi() + print(robot) +
+ +
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+ +
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/URDF/px100.html b/_modules/roboticstoolbox/models/URDF/px100.html new file mode 100644 index 00000000..3f6976f9 --- /dev/null +++ b/_modules/roboticstoolbox/models/URDF/px100.html @@ -0,0 +1,177 @@ + + + + + + roboticstoolbox.models.URDF.px100 — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.URDF.px100

+#!/usr/bin/env python
+
+import numpy as np
+from roboticstoolbox.robot.Robot import Robot
+
+
+
[docs]class px100(Robot): + """ + Class that imports a PX100 URDF model + + ``px100()`` is a class which imports an Interbotix px100 robot definition + from a URDF file. The model describes its kinematic and graphical + characteristics. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.URDF.px100() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration, 'L' shaped configuration + - qr, vertical 'READY' configuration + + :reference: + - https://docs.trossenrobotics.com/interbotix_xsarms_docs/specifications/px100.html + + .. codeauthor:: Jesse Haviland + .. sectionauthor:: Peter Corke + """ + + def __init__(self): + + links, name, urdf_string, urdf_filepath = self.URDF_read( + "interbotix_descriptions/urdf/px100.urdf.xacro" + ) + + super().__init__( + links, + name=name, + manufacturer="Interbotix", + urdf_string=urdf_string, + urdf_filepath=urdf_filepath, + ) + + self.qr = np.array([0, -0.3, 0, -2.2, 0, 2.0, np.pi / 4]) + self.qz = np.zeros(7) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+ + +if __name__ == "__main__": # pragma nocover + + robot = px100() + print(robot) +
+ +
+
+ +
+
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/URDF/px150.html b/_modules/roboticstoolbox/models/URDF/px150.html new file mode 100644 index 00000000..986428b7 --- /dev/null +++ b/_modules/roboticstoolbox/models/URDF/px150.html @@ -0,0 +1,177 @@ + + + + + + roboticstoolbox.models.URDF.px150 — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.URDF.px150

+#!/usr/bin/env python
+
+import numpy as np
+from roboticstoolbox.robot.Robot import Robot
+
+
+
[docs]class px150(Robot): + """ + Class that imports a PX150 URDF model + + ``px150()`` is a class which imports an Interbotix px150 robot definition + from a URDF file. The model describes its kinematic and graphical + characteristics. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.URDF.px150() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration, 'L' shaped configuration + - qr, vertical 'READY' configuration + + :reference: + - https://docs.trossenrobotics.com/interbotix_xsarms_docs/specifications/px150.html + + .. codeauthor:: Jesse Haviland + .. sectionauthor:: Peter Corke + """ + + def __init__(self): + + links, name, urdf_string, urdf_filepath = self.URDF_read( + "interbotix_descriptions/urdf/px150.urdf.xacro" + ) + + super().__init__( + links, + name=name, + manufacturer="Interbotix", + urdf_string=urdf_string, + urdf_filepath=urdf_filepath, + ) + + self.qr = np.array([0, -0.3, 0, -2.2, 0, 2.0, np.pi / 4, 0]) + self.qz = np.zeros(7) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+ + +if __name__ == "__main__": # pragma nocover + + robot = px150() + print(robot) +
+ +
+
+ +
+
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/URDF/rx150.html b/_modules/roboticstoolbox/models/URDF/rx150.html new file mode 100644 index 00000000..7bee4e1c --- /dev/null +++ b/_modules/roboticstoolbox/models/URDF/rx150.html @@ -0,0 +1,177 @@ + + + + + + roboticstoolbox.models.URDF.rx150 — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.URDF.rx150

+#!/usr/bin/env python
+
+import numpy as np
+from roboticstoolbox.robot.Robot import Robot
+
+
+
[docs]class rx150(Robot): + """ + Class that imports a RX150 URDF model + + ``rx150()`` is a class which imports an Interbotix rx150 robot definition + from a URDF file. The model describes its kinematic and graphical + characteristics. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.URDF.rx150() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration, 'L' shaped configuration + - qr, vertical 'READY' configuration + + :reference: + - https://docs.trossenrobotics.com/interbotix_xsarms_docs/specifications/rx150.html + + .. codeauthor:: Jesse Haviland + .. sectionauthor:: Peter Corke + """ + + def __init__(self): + + links, name, urdf_string, urdf_filepath = self.URDF_read( + "interbotix_descriptions/urdf/rx150.urdf.xacro" + ) + + super().__init__( + links, + name=name, + manufacturer="Interbotix", + urdf_string=urdf_string, + urdf_filepath=urdf_filepath, + ) + + self.qr = np.array([0, -0.3, 0, -2.2, 0, 2.0, np.pi / 4, 0]) + self.qz = np.zeros(7) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+ + +if __name__ == "__main__": # pragma nocover + + robot = rx150() + print(robot) +
+ +
+
+ +
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+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/URDF/rx200.html b/_modules/roboticstoolbox/models/URDF/rx200.html new file mode 100644 index 00000000..00f3c183 --- /dev/null +++ b/_modules/roboticstoolbox/models/URDF/rx200.html @@ -0,0 +1,177 @@ + + + + + + roboticstoolbox.models.URDF.rx200 — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.models.URDF.rx200

+#!/usr/bin/env python
+
+import numpy as np
+from roboticstoolbox.robot.Robot import Robot
+
+
+
[docs]class rx200(Robot): + """ + Class that imports a RX200 URDF model + + ``rx200()`` is a class which imports an Interbotix rx200 robot definition + from a URDF file. The model describes its kinematic and graphical + characteristics. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.URDF.rx200() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration, 'L' shaped configuration + - qr, vertical 'READY' configuration + + :reference: + - https://docs.trossenrobotics.com/interbotix_xsarms_docs/specifications/rx200.html + + .. codeauthor:: Jesse Haviland + .. sectionauthor:: Peter Corke + """ + + def __init__(self): + + links, name, urdf_string, urdf_filepath = self.URDF_read( + "interbotix_descriptions/urdf/rx200.urdf.xacro" + ) + + super().__init__( + links, + name=name, + manufacturer="Interbotix", + urdf_string=urdf_string, + urdf_filepath=urdf_filepath, + ) + + self.qr = np.array([0, -0.3, 0, -2.2, 0, 2.0, np.pi / 4, 0]) + self.qz = np.zeros(7) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+ + +if __name__ == "__main__": # pragma nocover + + robot = rx200() + print(robot) +
+ +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/URDF/vx300.html b/_modules/roboticstoolbox/models/URDF/vx300.html new file mode 100644 index 00000000..135b30c3 --- /dev/null +++ b/_modules/roboticstoolbox/models/URDF/vx300.html @@ -0,0 +1,177 @@ + + + + + + roboticstoolbox.models.URDF.vx300 — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
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+ +

Source code for roboticstoolbox.models.URDF.vx300

+#!/usr/bin/env python
+
+import numpy as np
+from roboticstoolbox.robot.Robot import Robot
+
+
+
[docs]class vx300(Robot): + """ + Class that imports a VX300 URDF model + + ``vx300()`` is a class which imports an Interbotix vx300 robot definition + from a URDF file. The model describes its kinematic and graphical + characteristics. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.URDF.vx300() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration, 'L' shaped configuration + - qr, vertical 'READY' configuration + + :reference: + - https://docs.trossenrobotics.com/interbotix_xsarms_docs/specifications/vx300.html + + .. codeauthor:: Jesse Haviland + .. sectionauthor:: Peter Corke + """ + + def __init__(self): + + links, name, urdf_string, urdf_filepath = self.URDF_read( + "interbotix_descriptions/urdf/vx300.urdf.xacro" + ) + + super().__init__( + links, + name=name, + manufacturer="Interbotix", + urdf_string=urdf_string, + urdf_filepath=urdf_filepath, + ) + + self.qr = np.array([0, -0.3, 0, -2.2, 0, 2.0, np.pi / 4, 0]) + self.qz = np.zeros(8) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+ + +if __name__ == "__main__": # pragma nocover + + robot = vx300() + print(robot) +
+ +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/URDF/vx300s.html b/_modules/roboticstoolbox/models/URDF/vx300s.html new file mode 100644 index 00000000..21ed62ef --- /dev/null +++ b/_modules/roboticstoolbox/models/URDF/vx300s.html @@ -0,0 +1,177 @@ + + + + + + roboticstoolbox.models.URDF.vx300s — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
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+ +

Source code for roboticstoolbox.models.URDF.vx300s

+#!/usr/bin/env python
+
+import numpy as np
+from roboticstoolbox.robot.Robot import Robot
+
+
+
[docs]class vx300s(Robot): + """ + Class that imports a VX300s URDF model + + ``vx300s()`` is a class which imports an Interbotix vx300s robot definition + from a URDF file. The model describes its kinematic and graphical + characteristics. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.URDF.vx300s() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration, 'L' shaped configuration + - qr, vertical 'READY' configuration + + :reference: + - https://docs.trossenrobotics.com/interbotix_xsarms_docs/specifications/vx300s.html + + .. codeauthor:: Jesse Haviland + .. sectionauthor:: Peter Corke + """ + + def __init__(self): + + links, name, urdf_string, urdf_filepath = self.URDF_read( + "interbotix_descriptions/urdf/vx300s.urdf.xacro" + ) + + super().__init__( + links, + name=name, + manufacturer="Interbotix", + urdf_string=urdf_string, + urdf_filepath=urdf_filepath, + ) + + self.qr = np.array([0, -0.3, 0, -2.2, 0, 2.0, np.pi / 4, 0, 0]) + self.qz = np.zeros(9) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+ + +if __name__ == "__main__": # pragma nocover + + robot = vx300s() + print(robot) +
+ +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/URDF/wx200.html b/_modules/roboticstoolbox/models/URDF/wx200.html new file mode 100644 index 00000000..ad988156 --- /dev/null +++ b/_modules/roboticstoolbox/models/URDF/wx200.html @@ -0,0 +1,177 @@ + + + + + + roboticstoolbox.models.URDF.wx200 — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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+ +

Source code for roboticstoolbox.models.URDF.wx200

+#!/usr/bin/env python
+
+import numpy as np
+from roboticstoolbox.robot.Robot import Robot
+
+
+
[docs]class wx200(Robot): + """ + Class that imports a WX200 URDF model + + ``wx200()`` is a class which imports an Interbotix wx200 robot definition + from a URDF file. The model describes its kinematic and graphical + characteristics. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.URDF.wx200() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration, 'L' shaped configuration + - qr, vertical 'READY' configuration + + :reference: + - https://docs.trossenrobotics.com/interbotix_xsarms_docs/specifications/wx200.html + + .. codeauthor:: Jesse Haviland + .. sectionauthor:: Peter Corke + """ + + def __init__(self): + + links, name, urdf_string, urdf_filepath = self.URDF_read( + "interbotix_descriptions/urdf/wx200.urdf.xacro" + ) + + super().__init__( + links, + name=name, + manufacturer="Interbotix", + urdf_string=urdf_string, + urdf_filepath=urdf_filepath, + ) + + self.qr = np.array([0, -0.3, 0, -2.2, 0, 2.0, np.pi / 4, 0]) + self.qz = np.zeros(8) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+ + +if __name__ == "__main__": # pragma nocover + + robot = wx200() + print(robot) +
+ +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/URDF/wx250.html b/_modules/roboticstoolbox/models/URDF/wx250.html new file mode 100644 index 00000000..6e569423 --- /dev/null +++ b/_modules/roboticstoolbox/models/URDF/wx250.html @@ -0,0 +1,177 @@ + + + + + + roboticstoolbox.models.URDF.wx250 — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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+ +

Source code for roboticstoolbox.models.URDF.wx250

+#!/usr/bin/env python
+
+import numpy as np
+from roboticstoolbox.robot.Robot import Robot
+
+
+
[docs]class wx250(Robot): + """ + Class that imports a WX250 URDF model + + ``wx250()`` is a class which imports an Interbotix wx250 robot definition + from a URDF file. The model describes its kinematic and graphical + characteristics. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.URDF.wx250() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration, 'L' shaped configuration + - qr, vertical 'READY' configuration + + :reference: + - https://docs.trossenrobotics.com/interbotix_xsarms_docs/specifications/wx250.html + + .. codeauthor:: Jesse Haviland + .. sectionauthor:: Peter Corke + """ + + def __init__(self): + + links, name, urdf_string, urdf_filepath = self.URDF_read( + "interbotix_descriptions/urdf/wx250.urdf.xacro" + ) + + super().__init__( + links, + name=name, + manufacturer="Interbotix", + urdf_string=urdf_string, + urdf_filepath=urdf_filepath, + ) + + self.qr = np.array([0, -0.3, 0, -2.2, 0, 2.0, np.pi / 4, 0]) + self.qz = np.zeros(8) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+ + +if __name__ == "__main__": # pragma nocover + + robot = wx250() + print(robot) +
+ +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/models/URDF/wx250s.html b/_modules/roboticstoolbox/models/URDF/wx250s.html new file mode 100644 index 00000000..820a106f --- /dev/null +++ b/_modules/roboticstoolbox/models/URDF/wx250s.html @@ -0,0 +1,177 @@ + + + + + + roboticstoolbox.models.URDF.wx250s — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
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+ +

Source code for roboticstoolbox.models.URDF.wx250s

+#!/usr/bin/env python
+
+import numpy as np
+from roboticstoolbox.robot.Robot import Robot
+
+
+
[docs]class wx250s(Robot): + """ + Class that imports a wx250s URDF model + + ``wx250s()`` is a class which imports an Interbotix wx250s robot definition + from a URDF file. The model describes its kinematic and graphical + characteristics. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.URDF.wx250s() + >>> print(robot) + + Defined joint configurations are: + + - qz, zero joint angle configuration, 'L' shaped configuration + - qr, vertical 'READY' configuration + + :reference: + - https://docs.trossenrobotics.com/interbotix_xsarms_docs/specifications/wx250s.html + + .. codeauthor:: Jesse Haviland + .. sectionauthor:: Peter Corke + """ + + def __init__(self): + + links, name, urdf_string, urdf_filepath = self.URDF_read( + "interbotix_descriptions/urdf/wx250s.urdf.xacro" + ) + + super().__init__( + links, + name=name, + manufacturer="Interbotix", + urdf_string=urdf_string, + urdf_filepath=urdf_filepath, + ) + + self.qr = np.array([0, -0.3, 0, -2.2, 0, 2.0, np.pi / 4, 0, 0]) + self.qz = np.zeros(9) + + self.addconfiguration("qr", self.qr) + self.addconfiguration("qz", self.qz)
+ + +if __name__ == "__main__": # pragma nocover + + robot = wx250s() + print(robot) +
+ +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/robot/DHDynamics.html b/_modules/roboticstoolbox/robot/DHDynamics.html new file mode 100644 index 00000000..d67ca91e --- /dev/null +++ b/_modules/roboticstoolbox/robot/DHDynamics.html @@ -0,0 +1,1788 @@ + + + + + + + + + + roboticstoolbox.robot.DHDynamics — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + + +
+ + + + + +
+ + +
+ +
+ +
+ + + + + + + + + + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/robot/DHLink.html b/_modules/roboticstoolbox/robot/DHLink.html new file mode 100644 index 00000000..e2338774 --- /dev/null +++ b/_modules/roboticstoolbox/robot/DHLink.html @@ -0,0 +1,1102 @@ + + + + + + roboticstoolbox.robot.DHLink — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +

Source code for roboticstoolbox.robot.DHLink

+#!/usr/bin/env python
+"""
+@author: Jesse Haviland
+"""
+
+# import numpy as np
+# from spatialmath import SE3
+import roboticstoolbox as rp
+from roboticstoolbox.robot.Link import Link, _listen_dyn
+from roboticstoolbox.robot.ETS import ETS
+from roboticstoolbox.robot.ET import ET
+from spatialmath import SE3
+from typing import List, Union
+from functools import wraps
+from numpy import ndarray, cos, sin, array
+from spatialgeometry import Shape
+from copy import deepcopy
+
+# _eps = np.finfo(np.float64).eps
+
+
+ArrayLike = Union[list, ndarray, tuple, set]
+
+
+def _check_rne(func):
+    """
+    @_check_rne decorator
+
+    Decorator applied to any method to calls to C RNE code.  Works in
+    conjunction with::
+
+        @_listen_dyn
+        def dyn_param_setter(self, value):
+
+    which marks the dynamic parameters as having changed using the robot's
+    ``.dynchanged()`` method.
+
+    If this is the case, then the parameters are re-serialized prior to
+    invoking inverse dynamics.
+
+    :seealso: :func:`Link._listen_dyn`
+    """
+
+    @wraps(func)
+    def wrapper_check_rne(*args, **kwargs):
+        if args[0]._rne_ob is None or args[0]._dynchanged:
+            args[0].delete_rne()
+            args[0]._init_rne()
+        args[0]._rne_changed = False
+        return func(*args, **kwargs)
+
+    return wrapper_check_rne
+
+
+# --------------------------------------------------------------#
+
+try:  # pragma: no cover
+    # print('Using SymPy')
+    import sympy as sym
+
+    def _issymbol(x):  # type: ignore
+        return isinstance(x, sym.Expr)
+
+except ImportError:
+
+    def _issymbol(x):  # pylint: disable=unused-argument
+        return False
+
+
+def _cos(theta) -> float:
+    if _issymbol(theta):
+        return sym.cos(theta)  # type: ignore
+    else:
+        return cos(theta)
+
+
+def _sin(theta) -> float:
+    if _issymbol(theta):
+        return sym.sin(theta)  # type: ignore
+    else:
+        return sin(theta)
+
+
+# --------------------------------------------------------------#
+
+
+
+
+
+# -------------------------------------------------------------------------- #
+
+
+
[docs]class RevoluteDH(DHLink): + r""" + Class for revolute links using standard DH convention + :param d: kinematic - link offset + :type d: float + :param alpha: kinematic - link twist + :type alpha: float + :param a: kinematic - link length + :type a: float + :param offset: kinematic - joint variable offset + :type offset: float + :param qlim: joint variable limits [min, max] + :type qlim: float ndarray(1,2) + :param flip: joint moves in opposite direction + :type flip: bool + :param m: dynamic - link mass + :type m: float + :param r: dynamic - position of COM with respect to link frame + :type r: float ndarray(3) + :param I: dynamic - inertia of link with respect to COM + :type I: ndarray + :param Jm: dynamic - motor inertia + :type Jm: float + :param B: dynamic - motor viscous friction: B=B⁺=B⁻, [B⁺, B⁻] + :type B: float, or ndarray(2,) + :param Tc: dynamic - motor Coulomb friction [Tc⁺, Tc⁻] + :type Tc: ndarray(2,) + :param G: dynamic - gear ratio + :type G: float + + A subclass of the :class:`DHLink` class for a revolute joint that holds all + information related to a robot link such as kinematics parameters, + rigid-body inertial parameters, motor and transmission parameters. + The link transform is + :math:`\underbrace{\mathbf{T}_{rz}(q_i)}_{\mbox{variable}} \cdot \mathbf{T}_{tz}(d_i) \cdot \mathbf{T}_{tx}(a_i) \cdot \mathbf{T}_{rx}(\alpha_i)` + where :math:`q_i` is the joint variable. + :references: + - Robotics, Vision & Control in Python, 3e, P. Corke, Springer 2023, Chap 7. + + :seealso: :func:`PrismaticDH`, :func:`DHLink`, :func:`RevoluteMDH` + """ # noqa + + def __init__( + self, d=0.0, a=0.0, alpha=0.0, offset=0.0, qlim=None, flip=False, **kwargs + ): + + theta = 0.0 + sigma = 0 + mdh = False + + super().__init__( + d=d, + alpha=alpha, + theta=theta, + a=a, + sigma=sigma, + mdh=mdh, + offset=offset, + qlim=qlim, + flip=flip, + **kwargs, + )
+ + +
[docs]class PrismaticDH(DHLink): + r""" + Class for prismatic link using standard DH convention + :param theta: kinematic: joint angle + :type theta: float + :param d: kinematic - link offset + :type d: float + :param alpha: kinematic - link twist + :type alpha: float + :param a: kinematic - link length + :type a: float + :param offset: kinematic - joint variable offset + :type offset: float + :param qlim: joint variable limits [min, max] + :type qlim: float ndarray(1,2) + :param flip: joint moves in opposite direction + :type flip: bool + :param m: dynamic - link mass + :type m: float + :param r: dynamic - position of COM with respect to link frame + :type r: float ndarray(3) + :param I: dynamic - inertia of link with respect to COM + :type I: ndarray + :param Jm: dynamic - motor inertia + :type Jm: float + :param B: dynamic - motor viscous friction: B=B⁺=B⁻, [B⁺, B⁻] + :type B: float, or ndarray(2,) + :param Tc: dynamic - motor Coulomb friction [Tc⁺, Tc⁻] + :type Tc: ndarray(2,) + :param G: dynamic - gear ratio + :type G: float + A subclass of the DHLink class for a prismatic joint that holds all + information related to a robot link such as kinematics parameters, + rigid-body inertial parameters, motor and transmission parameters. + The link transform is + :math:`\mathbf{T}_{rz}(\theta_i) \cdot \underbrace{\mathbf{T}_{tz}(q_i)}_{\mbox{variable}} \cdot \mathbf{T}_{tx}(a_i) \cdot \mathbf{T}_{rx}(\alpha_i)` + where :math:`q_i` is the joint variable. + :references: + - Robotics, Vision & Control in Python, 3e, P. Corke, Springer 2023, Chap 7. + + :seealso: :func:`RevoluteDH`, :func:`DHLink`, :func:`PrismaticMDH` + """ # noqa + + def __init__( + self, theta=0.0, a=0.0, alpha=0.0, offset=0.0, qlim=None, flip=False, **kwargs + ): + + d = 0.0 + sigma = 1 + mdh = False + + super().__init__( + theta=theta, + d=d, + a=a, + alpha=alpha, + sigma=sigma, + mdh=mdh, + offset=offset, + qlim=qlim, + flip=flip, + **kwargs, + )
+ + +
[docs]class RevoluteMDH(DHLink): + r""" + Class for revolute links using modified DH convention + + :param d: kinematic - link offset + :type d: float + :param alpha: kinematic - link twist + :type alpha: float + :param a: kinematic - link length + :type a: float + :param offset: kinematic - joint variable offset + :type offset: float + + :param qlim: joint variable limits [min, max] + :type qlim: float ndarray(1,2) + :param flip: joint moves in opposite direction + :type flip: bool + + :param m: dynamic - link mass + :type m: float + :param r: dynamic - position of COM with respect to link frame + :type r: float ndarray(3) + :param I: dynamic - inertia of link with respect to COM + :type I: ndarray + :param Jm: dynamic - motor inertia + :type Jm: float + :param B: dynamic - motor viscous friction: B=B⁺=B⁻, [B⁺, B⁻] + :type B: float, or ndarray(2,) + :param Tc: dynamic - motor Coulomb friction [Tc⁺, Tc⁻] + :type Tc: ndarray(2,) + :param G: dynamic - gear ratio + :type G: float + + A subclass of the DHLink class for a revolute joint that holds all + information related to a robot link such as kinematics parameters, + rigid-body inertial parameters, motor and transmission parameters. + + The link transform is + + :math:`\mathbf{T}_{tx}(a_{i-1}) \cdot \mathbf{T}_{rx}(\alpha_{i-1}) \cdot \underbrace{\mathbf{T}_{rz}(q_i)}_{\mbox{variable}} \cdot \mathbf{T}_{tz}(d_i)` + + where :math:`q_i` is the joint variable. + + :references: + - Robotics, Vision & Control in Python, 3e, P. Corke, Springer 2023, Chap 7. + + :seealso: :func:`PrismaticMDH`, :func:`DHLink`, :func:`RevoluteDH` + """ # noqa + + def __init__( + self, d=0.0, a=0.0, alpha=0.0, offset=0.0, qlim=None, flip=False, **kwargs + ): + + theta = 0.0 + sigma = 0 + mdh = True + + super().__init__( + d=d, + alpha=alpha, + theta=theta, + a=a, + sigma=sigma, + mdh=mdh, + offset=offset, + qlim=qlim, + flip=flip, + **kwargs, + )
+ + +
[docs]class PrismaticMDH(DHLink): + r""" + Class for prismatic link using modified DH convention + + :param theta: kinematic: joint angle + :type theta: float + :param d: kinematic - link offset + :type d: float + :param alpha: kinematic - link twist + :type alpha: float + :param a: kinematic - link length + :type a: float + :param offset: kinematic - joint variable offset + :type offset: float + + :param qlim: joint variable limits [min, max] + :type qlim: float ndarray(1,2) + :param flip: joint moves in opposite direction + :type flip: bool + + :param m: dynamic - link mass + :type m: float + :param r: dynamic - position of COM with respect to link frame + :type r: float ndarray(3) + :param I: dynamic - inertia of link with respect to COM + :type I: ndarray + :param Jm: dynamic - motor inertia + :type Jm: float + :param B: dynamic - motor viscous friction: B=B⁺=B⁻, [B⁺, B⁻] + :type B: float, or ndarray(2,) + :param Tc: dynamic - motor Coulomb friction [Tc⁺, Tc⁻] + :type Tc: ndarray(2,) + :param G: dynamic - gear ratio + :type G: float + + A subclass of the DHLink class for a prismatic joint that holds all + information related to a robot link such as kinematics parameters, + rigid-body inertial parameters, motor and transmission parameters. + + The link transform is + + :math:`\mathbf{T}_{tx}(a_{i-1}) \cdot \mathbf{T}_{rx}(\alpha_{i-1}) \cdot \mathbf{T}_{rz}(\theta_i) \cdot \underbrace{\mathbf{T}_{tz}(q_i)}_{\mbox{variable}}` + + where :math:`q_i` is the joint variable. + + :references: + - Robotics, Vision & Control in Python, 3e, P. Corke, Springer 2023, Chap 7. + + :seealso: :func:`RevoluteMDH`, :func:`DHLink`, :func:`PrismaticDH` + """ # noqa + + def __init__( + self, theta=0.0, a=0.0, alpha=0.0, offset=0.0, qlim=None, flip=False, **kwargs + ): + + d = 0.0 + sigma = 1 + mdh = True + + super().__init__( + theta=theta, + d=d, + a=a, + alpha=alpha, + sigma=sigma, + mdh=mdh, + offset=offset, + qlim=qlim, + flip=flip, + **kwargs, + )
+
+ +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/robot/DHRobot.html b/_modules/roboticstoolbox/robot/DHRobot.html new file mode 100644 index 00000000..a2eeb8d3 --- /dev/null +++ b/_modules/roboticstoolbox/robot/DHRobot.html @@ -0,0 +1,2640 @@ + + + + + + roboticstoolbox.robot.DHRobot — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+
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+
+
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+ +

Source code for roboticstoolbox.robot.DHRobot

+#!/usr/bin/env python
+"""
+@author Jesse Haviland
+"""
+
+from collections import namedtuple
+from email import message
+from roboticstoolbox.tools.data import rtb_path_to_datafile
+import warnings
+import copy
+import numpy as np
+from roboticstoolbox.robot.Robot import Robot  # DHLink
+from roboticstoolbox.robot.ETS import ETS, ET
+from roboticstoolbox.robot.DHLink import DHLink
+from roboticstoolbox import rtb_set_param
+from spatialmath.base.argcheck import getvector, isscalar, verifymatrix, getmatrix
+
+# from spatialmath import base
+from spatialmath.base import (
+    tr2jac,
+    tr2eul,
+    tr2rpy,
+    t2r,
+    trlog,
+    rotvelxform,
+)
+from spatialmath import SE3, Twist3
+import spatialmath.base.symbolic as sym
+
+# from scipy.optimize import minimize, Bounds
+from ansitable import ANSITable, Column
+from scipy.linalg import block_diag
+from roboticstoolbox.robot.DHLink import _check_rne, DHLink
+from roboticstoolbox import rtb_get_param
+from roboticstoolbox.frne import init, frne, delete
+from numpy import any
+from typing import Union, Tuple
+from roboticstoolbox.robot.IK import IKSolution
+
+ArrayLike = Union[list, np.ndarray, tuple, set]
+
+# iksol = namedtuple("IKsolution", "q, success, reason")
+
+
+
[docs]class DHRobot(Robot): + """ + Class for robots defined using Denavit-Hartenberg notation + + :param L: List of links which define the robot + :type L: list(n) + :param name: Name of the robot + :type name: str + :param manufacturer: Manufacturer of the robot + :type manufacturer: str + :param base: Location of the base + :type base: SE3 + :param tool: Location of the tool + :type tool: SE3 + :param gravity: Gravitational acceleration vector + :type gravity: ndarray(3) + + A concrete superclass for arm type robots defined using Denavit-Hartenberg + notation, that represents a serial-link arm-type robot. Each link and + joint in the chain is described by a DHLink-class object using + Denavit-Hartenberg parameters (standard or modified). + + .. note:: Link subclass elements passed in must be all standard, or all + modified, DH parameters. + + :reference: + + - Robotics, Vision & Control in Python, 3e, P. Corke, Springer 2023, Chap 7-9. + - Robot, Modeling & Control, + M.Spong, S. Hutchinson & M. Vidyasagar, Wiley 2006. + + """ + + def __init__(self, links, meshdir=None, **kwargs): + # Verify L + if not isinstance(links, list): + raise TypeError("The links must be stored in a list.") + + all_links = [] + self._n = 0 + + # If we are given a list of standard DH Links, we must convert + # them to modified DH links + # if any([isinstance(link, StandardDH) for link in links]): + # links = DHLink.StandardDH(links) + + for link in links: + if isinstance(link, DHLink): + # got a link + all_links.append(link) + link.number = self._n + 1 + link.jindex = self._n + self._n += 1 + + link.name = f"link{self._n}" + + elif isinstance(link, DHRobot): + # link is actually a robot + + # copy the links + rlinks = copy.copy(link.links) + for rlink in rlinks: + all_links.append(rlink) + rlink.number = self._n + rlink.jindex = self._n + self._n += 1 + + rlink.name = f"link{self._n}" + else: + raise TypeError("Input can be only DHLink or DHRobot") + + for i, link in enumerate(all_links): + if i > 0: + link.parent = all_links[i - 1] + else: + link.parent = None + + super().__init__(all_links, **kwargs) + + self._ee_links = [self.links[-1]] + + # Check the DH convention + self._mdh = self.links[0].mdh + if not all([link.mdh == self.mdh for link in self.links]): + raise ValueError("Robot has mixed D&H link conventions") + + # load meshes if required + if meshdir is not None: + self.meshdir = rtb_path_to_datafile(meshdir) + self.basemesh = self.meshdir / "link0.stl" + for j, link in enumerate(self._links, start=1): + link.mesh = self.meshdir / "link{:d}.stl".format(j) + self._hasgeometry = True + else: + self.basemesh = None + + # rne parameters + self._rne_ob = None + +
[docs] def __str__(self): + """ + Pretty prints the DH Model of the robot. Will output angles in degrees + + :return: Pretty print of the robot model + :rtype: str + """ + + if np.array_equal(self.base.A, np.eye(4)): + base = None + else: + base = self.base + + if np.array_equal(self.tool.A, np.eye(4)): + tool = None + else: + tool = self.tool + + unicode = rtb_get_param("unicode") + border = "thin" if unicode else "ascii" + s = f"DHRobot: {self.name}" + + if self.manufacturer is not None and len(self.manufacturer) > 0: + s += f" (by {self.manufacturer})" + s += f", {self.n} joints ({self.structure})" + + deg = 180 / np.pi + + if self._hasdynamics: + s += ", dynamics" + if any([link.mesh is not None for link in self.links]): + s += ", geometry" + + if self.mdh: + dh = "modified" + else: + dh = "standard" + s += f", {dh} DH parameters\n" + + def qstr(j, link): + j += 1 + if link.isflip: + s = f"-q{j:d}" + else: + s = f" q{j:d}" + + if L.offset != 0: + sign = "+" if L.offset > 0 else "-" + offset = abs(L.offset) + if link.isprismatic: + s += f" {sign} {offset:}" + else: + s += f" {sign} {offset * deg:.3g}\u00b0" + return s + + def angle(theta, fmt=None): + if sym.issymbol(theta): # pragma nocover + return "<<red>>" + str(theta) + else: + if fmt is not None: + return fmt.format(theta * deg) + "\u00b0" + else: + return str(theta * deg) + "\u00b0" + + def format_attr(attr) -> str: + if isinstance(attr, float): + return f"{attr:.4g}" + else: + return str(attr) + + has_qlim = any([link.qlim is not None for link in self]) + if has_qlim: + qlim_columns = [ + Column("q⁻", headalign="^"), + Column("q⁺", headalign="^"), + ] + qlim = self.qlim + + else: + qlim = np.array([]) # satisfy type checker + qlim_columns = [] + + if self.mdh: + # MDH format + table = ANSITable( + Column("aⱼ₋₁", headalign="^"), + Column("⍺ⱼ₋₁", headalign="^"), + Column("θⱼ", headalign="^"), + Column("dⱼ", headalign="^"), + *qlim_columns, + border=border, + ) + for j, L in enumerate(self): + if has_qlim: + if L.isprismatic: + ql = [qlim[0, j], qlim[1, j]] + else: + ql = [angle(qlim[k, j], "{:.1f}") for k in [0, 1]] + else: + ql = [] + if L.isprismatic: + table.row(L.a, angle(L.alpha), angle(L.theta), qstr(j, L), *ql) + else: + table.row(L.a, angle(L.alpha), qstr(j, L), L.d, *ql) + else: + # DH format + table = ANSITable( + Column("θⱼ", headalign="^", colalign="<"), + Column("dⱼ", headalign="^"), + Column("aⱼ", headalign="^"), + Column("⍺ⱼ", headalign="^"), + *qlim_columns, + border=border, + ) + + for j, L in enumerate(self): + if has_qlim: + if L.isprismatic: + ql = [qlim[0, j], qlim[1, j]] + else: + ql = [angle(qlim[k, j], "{:.1f}") for k in [0, 1]] + else: + ql = [] + if L.isprismatic: + table.row( + angle(L.theta), + qstr(j, L), + format_attr(L.a), + angle(L.alpha), + *ql, + ) + else: + table.row( + qstr(j, L), + format_attr(L.d), + format_attr(L.a), + angle(L.alpha), + *ql, + ) + + s += str(table) + + # show tool and base + if self._tool is not None or self._tool is not None: + table = ANSITable( + Column("", colalign=">"), + Column("", colalign="<"), + border=border, + header=False, + ) + if base is not None: + table.row( + "base", + base.printline(orient="rpy/xyz", fmt="{:.2g}", file=None), + ) + if tool is not None: + table.row( + "tool", + tool.strline(orient="rpy/xyz", fmt="{:.2g}"), + ) + s += "\n" + str(table) + + # show named configurations + s += self.configurations_str(border=border) + + return s
+ + def __add__(self, L): + nlinks = [] + + # TODO - Should I do a deep copy here a physically copy the DHLinks + # and not just the references? + # Copy DHLink references to new list + for i in range(self.n): + nlinks.append(self.links[i]) + + if isinstance(L, DHLink): + nlinks.append(L) + elif isinstance(L, DHRobot): + for j in range(L.n): + nlinks.append(L.links[j]) + else: + raise TypeError("Can only combine DHRobots with other DHRobots or DHLinks") + + return DHRobot( + nlinks, + name=self.name, + manufacturer=self.manufacturer, + base=self.base, + tool=self.tool, + gravity=self.gravity, + ) + + def __deepcopy__(self, memo): + links = [] + + for link in self.links: + links.append(copy.deepcopy(link)) + + name = copy.deepcopy(self.name) + manufacturer = copy.deepcopy(self.manufacturer) + comment = copy.deepcopy(self.comment) + base = copy.deepcopy(self.base) + tool = copy.deepcopy(self.tool) + gravity = copy.deepcopy(self.gravity) + keywords = copy.deepcopy(self.keywords) + symbolic = copy.deepcopy(self.symbolic) + configs = copy.deepcopy(self.configs) + + try: + if self.meshdir: + meshdir = copy.deepcopy(self.meshdir) + else: + meshdir = None + except AttributeError: + meshdir = None + + # cls = self.__class__ + result = DHRobot( + links, + meshdir=meshdir, + name=name, + manufacturer=manufacturer, + comment=comment, + base=base, + tool=tool, + gravity=gravity, + keywords=keywords, + symbolic=symbolic, + configs=configs, + ) + + # if a configuration was an attribute of original robot, make it an + # attribute of the deep copy + for config in configs: + if hasattr(self, config): + setattr(result, config, configs[config]) + + try: + setattr(result, "ikine_a", getattr(self, "ikine_a")) + except AttributeError: + pass + + memo[id(self)] = result + return result + + # def copy(self): + # """ + # Copy a robot + + # :return: A deepish copy of the robot + # :rtype: Robot subclass instance + # """ + + # L = [link.copy() for link in self] + + # new = DHRobot( + # L, + # name=self.name, + # manufacturer=self.manufacturer, + # base=self.base, + # tool=self.tool, + # gravity=self.gravity) + + # new.q = self.q + # new.qd = self.qd + # new.qdd = self.qdd + + # return new + + # --------------------------------------------------------------------- # + + def _set_link_fk(self, q): + """ + robot._set_link_fk(q) evaluates fkine for each link within a + robot and stores that pose in a private variable within the link. + + This method is not for general use. + + :param q: The joint angles/configuration of the robot + :type q: float ndarray(n) + + .. note:: + + - The robot's base transform, if present, are incorporated + into the result. + """ + + q = getvector(q, self.n) + + # t = self.base + + tall = self.fkine_all(q, old=True) + + for i, link in enumerate(self.links): + # Update the link model transforms + for col in link.collision: + col.wT = tall[i] + + for gi in link.geometry: + gi.wT = tall[i] + + # --------------------------------------------------------------------- # + + @property + def mdh(self): + """ + Modified Denavit-Hartenberg status + + :return: whether robot is defined using modified Denavit-Hartenberg + notation + :rtype: bool + + Example: + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> puma = rtb.models.DH.Puma560() + >>> puma.mdh + >>> panda = rtb.models.DH.Panda() + >>> panda.mdh + + """ + return self._mdh + + @property + def d(self): + r""" + Link offset values + + :return: List of link offset values :math:`d_j`. + :rtype: ndarray(n,) + + The following are equivalent:: + + robot.links[j].d + robot.d[j] + + Example: + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.DH.Puma560() + >>> robot.d + """ + v = [] + for i in range(self.n): + v.append(self.links[i].d) + return v + + @property + def a(self): + r""" + Link length values + + :return: List of link length values :math:`a_j`. + :rtype: ndarray(n,) + + The following are equivalent:: + + robot.links[j].a + robot.a[j] + + Example: + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.DH.Puma560() + >>> robot.a + """ + v = [] + for i in range(self.n): + v.append(self.links[i].a) + return v + + @property + def theta(self): + r""" + Joint angle values + + :return: List of joint angle values :math:`\theta_j`. + :rtype: ndarray(n,) + + The following are equivalent:: + + robot.links[j].theta + robot.theta[j] + + Example: + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.DH.Puma560() + >>> robot.theta + """ + v = [] + for i in range(self.n): + v.append(self.links[i].theta) + return v + + @property + def alpha(self): + r""" + Link twist values + + :return: List of link twist values :math:`\alpha_j`. + :rtype: ndarray(n,) + + The following are equivalent:: + + robot.links[j].alpha + robot.alpha[j] + + Example: + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.DH.Puma560() + >>> robot.alpha + """ + v = [] + for i in range(self.n): + v.append(self.links[i].alpha) + return v + + @property + def r(self): + r""" + Link centre of mass values + + :return: Array of link centre of mass values :math:`r_j`. + :rtype: ndarray(3,n) + + Example: + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.DH.Puma560() + >>> robot.r + """ + # TODO tidyup + v = np.copy(self.links[0].r) + for i in range(1, self.n): + v = np.c_[v, self.links[i].r] + return v + + @property + def offset(self): + r""" + Joint offset values + + :return: List of joint offset values :math:`\bar{q}_j`. + :rtype: ndarray(n,) + + Example: + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.DH.Puma560() + >>> robot.offset + """ + v = [] + for i in range(self.n): + v.append(self.links[i].offset) + return v + + @property + def reach(self): + r""" + Reach of the robot + + :return: Maximum reach of the robot + :rtype: float + + A conservative estimate of the reach of the robot. It is computed as + :math:`\sum_j |a_j| + |d_j|` where :math:`d_j` is taken as the maximum + joint coordinate (``qlim``) if the joint is primsmatic. + + .. note:: + + - This is the *length sum* referred to in Craig's book + - Probably an overestimate of the actual reach + - Used by numerical inverse kinematics to scale translational + error. + - For a prismatic joint, uses ``qlim`` if it is set + + .. warning:: Computed on the first access. If kinematic parameters + subsequently change this will not be reflected. + """ + if self._reach is None: + d = 0 + for link in self: + d += abs(link.a) + (link.d) + if link.isprismatic and link.qlim is not None: + d += link.qlim[1] + self._reach = d + return self._reach + + @property + def nbranches(self): + """ + Number of branches + + :return: number of branches in the robot's kinematic tree + :rtype: int + + Number of branches in this robot. + + Example: + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.DH.Panda() + >>> robot.nbranches + + :seealso: :func:`n`, :func:`nlinks` + """ + return 1 + +
[docs] def A(self, j, q=None): + """ + Link forward kinematics + + :param j: Joints to compute link transform for + :type j: int, 2-tuple + :param q: The joint configuration of the robot (Optional, + if not supplied will use the stored q values) + :type q: float ndarray(1,n) + :return T: The transform between link frames + :rtype T: SE3 + + - ``robot.A(j, q)`` transform between link frames {0} and {j}. ``q`` + is a vector (n) of joint variables. + - ``robot.A([j1, j2], q)`` as above between link frames {j1} and {j2}. + - ``robot.A(j)`` as above except uses the stored q value of the + robot object. + + .. note:: Base and tool transforms are not applied. + + """ + + if isscalar(j): + j0 = 0 + jn = int(j) + else: + j = getvector(j, 2) + j0 = int(j[0]) + jn = int(j[1]) + + jn += 1 + + if jn > self.n: + raise ValueError("The joints value out of range") + + q = getvector(q) + + T = SE3() + for i in range(j0, jn): + T *= self.links[i].A(q[i]) + + return T
+ +
[docs] def islimit(self, q=None): + """ + Joint limit test + + :param q: The joint configuration of the robot (Optional, + if not supplied will use the stored q values) + :type q: ndarray(n + :return v: is a vector of boolean values, one per joint, False if + ``q[j]`` is within the joint limits, else True + :rtype v: bool list + + - ``robot.islimit(q)`` is a list of boolean values indicating if the + joint configuration ``q`` is in violation of the joint limits. + + - ``robot.jointlimit()`` as above except uses the stored q value of the + robot object. + + Example: + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.DH.Puma560() + >>> robot.islimit([0, 0, -4, 4, 0, 0]) + + """ + if q is None: + q = self.q + + return [link.islimit(qk) for (link, qk) in zip(self, q)]
+ +
[docs] def isspherical(self): + """ + Test for spherical wrist + + :return: True if spherical wrist + :rtype: bool + + Tests if the robot has a spherical wrist, that is, the last 3 axes are + revolute and their axes intersect at a point. + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.DH.Puma560() + >>> robot.isspherical() + + """ + if self.n < 3: + return False + + L = self.links[self.n - 3 : self.n] + + alpha = [-np.pi / 2, np.pi / 2] + + return ( + L[0].a == 0 + and L[1].a == 0 + and L[1].d == 0 + and ( + (L[0].alpha == alpha[0] and L[1].alpha == alpha[1]) + or (L[0].alpha == alpha[1] and L[1].alpha == alpha[0]) + ) + and L[0].sigma == 0 + and L[1].sigma == 0 + and L[2].sigma == 0 + )
+ +
[docs] def dhunique(self): + """ + Print the unique DH parameters + + Print a table showing all the non-zero DH parameters, and their + values. This is the minimum set of kinematic parameters required + to describe the robot. + + Example: + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> puma = rtb.models.DH.Puma560() + >>> puma.dhunique() + """ + + table = ANSITable( + Column("param"), + Column("value", headalign="^", colalign="<", fmt="{:.4g}"), + border="thin", + ) + for j, link in enumerate(self): + if link.isprismatic: + if link.theta != 0: + table.row(f{j}", link.theta) + elif link.isrevolute: + if link.d != 0: + table.row(f"d{j}", link.d) + if link.a != 0: + table.row(f"a{j}", link.a) + if link.alpha != 0: + table.row(f"⍺{j}", link.alpha) + table.print()
+ +
[docs] def twists(self, q=None): + """ + Joint axes as twists + + :param q: The joint configuration of the robot, defaults to zero + :return: a vector of joint axis twists + :rtype: Twist3 instance + :return: Pose of the tool + :rtype: SE3 instance + + - ``tw, T0 = twists(q)`` calculates a vector of Twist objects (n) that + represent the axes of the joints for the robot with joint coordinates + ``q`` (n). Also returns T0 which is an SE3 object representing the + pose of the tool. + + - ``tw, T0 = twists()`` as above but the joint coordinates are taken + to be zero. + + Example: + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.DH.Puma560() + >>> tw, T0 = robot.twists() + >>> tw + >>> T0 + + """ + + if q is None: + q = np.zeros((self.n,)) + + T = self.fkine_all(q)[1:] # don't use first transform which is base + tw = Twist3.Alloc(self.n) + if self.mdh: + # MDH case + for j, link in enumerate(self): + if link.sigma == 0: + tw[j] = Twist3.UnitRevolute(T[j].a, T[j].t) + else: + tw[j] = Twist3.UnitPrismatic(T[j].a) + else: + # DH case + for j, link in enumerate(self): + if j == 0: + # first link case + if link.sigma == 0: + # revolute + tw[j] = Twist3.UnitRevolute([0, 0, 1], [0, 0, 0]) + else: + tw[j] = Twist3.UnitPrismatic([0, 0, 1]) # prismatic + else: + # subsequent links + if link.sigma == 0: + tw[j] = Twist3.UnitRevolute(T[j - 1].a, T[j - 1].t) # revolute + else: + tw[j] = Twist3.UnitPrismatic(T[j - 1].a) # prismatic + + return tw, T[-1]
+ +
[docs] def ets(self, *args, **kwargs) -> ETS: + """ + Robot kinematics as an elemenary transform sequence + + :return: elementary transform sequence + :rtype: ETS + + Example: + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> puma = rtb.models.DH.Puma560() + >>> puma.ets() + """ + + # optionally start with the base transform + if np.array_equal(self.base.A, np.eye(4)): + base = None + else: + base = self.base.A + + if np.array_equal(self.tool.A, np.eye(4)): + tool = None + else: + tool = self.tool.A + + if base is None: + ets = ETS() + else: + ets = ET.SE3(base) + + # add the links + for link in self: + ets *= link.ets + + # optionally add the base transform + if tool is not None: + ets *= ET.SE3(tool) + + return ets
+ +
[docs] def fkine(self, q, **kwargs): + """ + Forward kinematics + + :param q: The joint configuration + :type q: ndarray(n) or ndarray(m,n) + :return: Forward kinematics as an SE(3) matrix + :rtype: SE3 instance + + - ``robot.fkine(q)`` computes the forward kinematics for the robot at + joint configuration ``q``. + + If q is a 2D array, the rows are interpreted as the generalized joint + coordinates for a sequence of points along a trajectory. ``q[k,j]`` is + the j'th joint coordinate for the k'th trajectory configuration, and + the returned ``SE3`` instance contains ``n`` values. + + Example: + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> puma = rtb.models.DH.Puma560() + >>> puma.fkine([0, 0, 0, 0, 0, 0]) + + .. note:: + + - The robot's base or tool transform, if present, are incorporated + into the result. + - Joint offsets, if defined, are added to ``q`` before the forward + kinematics are computed. + """ + + if np.array_equal(self.base.A, np.eye(4)): + base = None + else: + base = self.base + + if np.array_equal(self.tool.A, np.eye(4)): + tool = None + else: + tool = self.tool + + T = SE3.Empty() + for qr in getmatrix(q, (None, self.n)): + first = True + for q, L in zip(qr, self.links): + if first: + Tr = L.A(q) + first = False + else: + Tr *= L.A(q) # type: ignore + + if base is not None: + Tr = base * Tr # type: ignore + if tool is not None: + Tr = Tr * tool # type: ignore + T.append(Tr) # type: ignore + + return T
+ +
[docs] def fkine_path(self, q, old=None): + """ + Compute the pose of every link frame + + :param q: The joint configuration + :type q: darray(n) + :return: Pose of all links + :rtype: SE3 instance + + ``T = robot.fkine_path(q)`` is an SE3 instance with ``robot.nlinks + + 1`` values: + + - ``T[0]`` is the base transform + - ``T[i+1]`` is the pose of link whose ``number`` is ``i`` + + :references: + - Kinematic Derivatives using the Elementary Transform + Sequence, J. Haviland and P. Corke + """ + T = self.base + q = getvector(q) + Tj = T + + for q, L in zip(q, self.links): + Tj *= L.A(q) + T.append(Tj) + + if self._tool is not None: + T[-1] *= self._tool + + return T
+ +
[docs] def segments(self): + segments = [None] + segments.extend(self.links) + return [segments]
+ +
[docs] def fkine_all(self, q=None, old=True): + """ + Forward kinematics for all link frames + + :param q: The joint configuration of the robot (Optional, + if not supplied will use the stored q values). + :type q: ndarray(n) or ndarray(m,n) + :param old: "old" behaviour, defaults to True + :type old: bool, optional + :return: Forward kinematics as an SE(3) matrix + :rtype: SE3 instance with ``n`` values + + - ``fkine_all(q)`` evaluates fkine for each joint within a robot and + returns a sequence of link frame poses. + + - ``fkine_all()`` as above except uses the stored q value of the + robot object. + + Example: + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> puma = rtb.models.DH.Puma560() + >>> T = puma.fkine_all([0, 0, 0, 0, 0, 0]) + >>> len(T) + + .. note:: + - Old behaviour is to return a list of ``n`` frames {1} to {n}, but + if ``old=False`` it returns ``n``+1 frames {0} to {n}, ie. it + includes the base frame. + - The robot's base or tool transform, if present, are incorporated + into the result. + - Joint offsets, if defined, are added to q before the forward + kinematics are computed. + """ + + if q is None: + q = self.q + + Tj = self.base.copy() + Tall = Tj + + for q, L in zip(q, self.links): + Tj *= L.A(q) + Tall.append(Tj) + return Tall
+ +
[docs] def jacobe(self, q, half=None, **kwargs): + r""" + Manipulator Jacobian in end-effector frame + + :param q: Joint coordinate vector + :type q: ndarray(n) + :param half: return half Jacobian: 'trans' or 'rot' + :type half: str + :return J: The manipulator Jacobian in the end-effector frame + :rtype: ndarray(6,n) + + - ``robot.jacobe(q)`` is the manipulator Jacobian matrix which maps + joint velocity to end-effector spatial velocity. + + End-effector spatial velocity :math:`\nu = (v_x, v_y, v_z, \omega_x, \omega_y, \omega_z)^T` + is related to joint velocity by :math:`{}^{E}\!\nu = \mathbf{J}_m(q) \dot{q}`. + + Example: + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> puma = rtb.models.DH.Puma560() + >>> puma.jacobe([0, 0, 0, 0, 0, 0]) + + .. warning:: This is the **geometric Jacobian** as described in texts by + Corke, Spong etal., Siciliano etal. The end-effector velocity is + described in terms of translational and angular velocity, not a + velocity twist as per the text by Lynch & Park. + """ # noqa + + q = getvector(q, self.n) + + n = self.n + L = self.links + J = np.zeros((6, self.n), dtype=q.dtype) # type: ignore + + U = self.tool.A + + for j in range(n - 1, -1, -1): + if self.mdh == 0: + # standard DH convention + U = L[j].A(q[j]).A @ U # type: ignore + + if not L[j].sigma: + # revolute axis + d = np.array( + [ + -U[0, 0] * U[1, 3] + U[1, 0] * U[0, 3], + -U[0, 1] * U[1, 3] + U[1, 1] * U[0, 3], + -U[0, 2] * U[1, 3] + U[1, 2] * U[0, 3], + ] + ) + delta = U[2, :3] # nz oz az + else: + # prismatic axis + d = U[2, :3] # nz oz az + delta = np.zeros((3,)) + + J[:, j] = np.r_[d, delta] + + if self.mdh != 0: + # modified DH convention + U = L[j].A(q[j]).A @ U # type: ignore + + # return top or bottom half if asked + if half is not None: + if half == "trans": + return J[:3, :] + elif half == "rot": + return J[3:, :] + else: + raise ValueError("bad half specified") + + return J
+ +
[docs] def jacob0(self, q=None, T=None, half=None, start=None, end=None): + r""" + Manipulator Jacobian in world frame + + :param q: Joint coordinate vector + :type q: ndarray(n) + :param T: Forward kinematics if known, SE(3 matrix) + :type T: SE3 instance + :param half: return half Jacobian: 'trans' or 'rot' + :type half: str + :return J: The manipulator Jacobian in the world frame + :rtype: ndarray(6,n) + + - ``robot.jacob0(q)`` is the manipulator geometric Jacobian matrix which maps + joint velocity to end-effector spatial velocity. + + End-effector spatial velocity :math:`\nu = (v_x, v_y, v_z, \omega_x, \omega_y, \omega_z)^T` + is related to joint velocity by :math:`{}^{0}\!\nu = \mathbf{J}_0(q) \dot{q}`. + + Example: + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> puma = rtb.models.DH.Puma560() + >>> puma.jacob0([0, 0, 0, 0, 0, 0]) + + .. warning:: This is the **geometric Jacobian** is as described in texts by + Corke, Spong etal., Siciliano etal. The end-effector velocity is + described in terms of translational and angular velocity, not a + velocity twist as per the text by Lynch & Park. + + .. note:: ``T`` can be passed in to save the cost of computing forward + kinematics which is needed to transform velocity from end-effector + frame to world frame. + + """ # noqa + q = getvector(q, self.n) + + if T is None: + T = self.fkine(q) + T = T.A + + # compute Jacobian in EE frame and transform to world frame + J0 = tr2jac(T) @ self.jacobe(q) + + # TODO optimize computation above if half matrix is returned + + # return top or bottom half if asked + if half is not None: + if half == "trans": + J0 = J0[:3, :] + elif half == "rot": + J0 = J0[3:, :] + else: + raise ValueError("bad half specified") + return J0
+ +
[docs] def jacob0_analytical(self, q, representation=None, T=None): + r""" + Manipulator Jacobian in world frame + + :param q: Joint coordinate vector + :type q: ndarray(n) + :param representation: return analytical Jacobian instead of geometric Jacobian + :type representation: str + :param T: Forward kinematics if known, SE(3 matrix) + :type T: SE3 instance + :return J: The manipulator analytical Jacobian in the world frame + :rtype: ndarray(6,n) + + Return the manipulator's analytical Jacobian matrix which maps + joint velocity to end-effector spatial velocity. + + End-effector spatial velocity :math:`\nu_a = (v_x, v_y, v_z, \dot{\Gamma}_1, \dot{\Gamma}_2, \dot{\Gamma}_3)^T` + is related to joint velocity by :math:`{}^{0}\!\nu_a = \mathbf{J}_{a,0}(q) \dot{q}`. + Where :math:`\dvec{\Gamma} = (\dot{\Gamma}_1, \dot{\Gamma}_2, \dot{\Gamma}_3)` is + orientation rate expressed as one of: + + ================== ================================== + ``representation`` Rotational representation + ================== ================================== + ``'rpy/xyz'`` RPY angular rates in XYZ order + ``'rpy/zyx'`` RPY angular rates in XYZ order + ``'eul'`` Euler angular rates in ZYZ order + ``'exp'`` exponential coordinate rates + ================== ================================== + + Example: + + .. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> puma = rtb.models.DH.Puma560() + >>> puma.jacob0_analytical([0, 0, 0, 0, 0, 0], "rpy/xyz") + + .. warning:: The **geometric Jacobian** is as described in texts by + Corke, Spong etal., Siciliano etal. The end-effector velocity is + described in terms of translational and angular velocity, not a + velocity twist as per the text by Lynch & Park. + + .. note:: ``T`` can be passed in to save the cost of computing forward + kinematics which is needed to transform velocity from end-effector + frame to world frame. + + """ # noqa + q = getvector(q, self.n) + + # compute forward kinematics if not provided + if T is None: + T = self.fkine(q) + + # compute Jacobian in world frame + J0 = self.jacob0(q, T) + + if representation is None: + return J0 + + # compute rotational transform if analytical Jacobian required + + if representation == "rpy/xyz": + gamma = tr2rpy(T.A, order="xyz") + elif representation == "rpy/zyx": + gamma = tr2rpy(T.A, order="zyx") + elif representation == "eul": + gamma = tr2eul(T.A) + elif representation == "exp": + # TODO: move to SMTB.base, Horner form with skew(v) + gamma = trlog(t2r(T.A), twist=True) + else: + raise ValueError("bad analytical value specified") + + A = rotvelxform(gamma, representation=representation, inverse=True, full=True) + return A @ J0
+ +
[docs] def hessian0(self, q=None, J0=None, start=None, end=None): + r""" + Manipulator Hessian in base frame + + :param q: joint coordinates + :type q: array_like + :param J0: Jacobian in {0} frame + :type J0: ndarray(6,n) + :return: Hessian matrix + :rtype: ndarray(6,n,n) + + This method calculcates the Hessisan in the base frame. One of ``J0`` or + ``q`` is required. If ``J0`` is already calculated for the joint + coordinates ``q`` it can be passed in to to save computation time. + + If we take the time derivative of the differential kinematic + relationship + + .. math:: + + \nu &= \mat{J}(\vec{q}) \dvec{q} \\ + \alpha &= \dmat{J} \dvec{q} + \mat{J} \ddvec{q} + + where + + .. math:: + + \dmat{J} = \mat{H} \dvec{q} + + and :math:`\mat{H} \in \mathbb{R}^{6\times n \times n}` is the + Hessian tensor. + + The elements of the Hessian are + + .. math:: + + \mat{H}_{i,j,k} = \frac{d^2 u_i}{d q_j d q_k} + + where :math:`u = \{t_x, t_y, t_z, r_x, r_y, r_z\}` are the elements + of the spatial velocity vector. + + Similarly, we can write + + .. math:: + + \mat{J}_{i,j} = \frac{d u_i}{d q_j} + + :references: + - Kinematic Derivatives using the Elementary Transform + Sequence, J. Haviland and P. Corke + + :seealso: :func:`jacob0`, :func:`jacob_dot` + """ + + return self.ets().hessian0(q, J0)
+ + def _get_limit_links(self, end=None, start=None): + # For compatibility with ERobot + + return None, None, None + + # -------------------------------------------------------------------------- # + + def _init_rne(self): + # Compress link data into a 1D array + L = np.zeros(24 * self.n) + + for i in range(self.n): + j = i * 24 + L[j] = self.links[i].alpha + L[j + 1] = self.links[i].a + L[j + 2] = self.links[i].theta + L[j + 3] = self.links[i].d + L[j + 4] = self.links[i].sigma + L[j + 5] = self.links[i].offset + L[j + 6] = self.links[i].m + L[j + 7 : j + 10] = self.links[i].r.flatten() + L[j + 10 : j + 19] = self.links[i].I.flatten() + L[j + 19] = self.links[i].Jm + L[j + 20] = self.links[i].G + L[j + 21] = self.links[i].B + L[j + 22 : j + 24] = self.links[i].Tc.flatten() + + # we negate gravity here, since the C code has the sign wrong + self._rne_ob = init(self.n, self.mdh, L, -self.gravity) + +
[docs] def delete_rne(self): + """ + Frees the memory holding the robot object in c if the robot object + has been initialised in c. + """ + if self._rne_ob is not None: + delete(self._rne_ob) + self._dynchanged = False + self._rne_ob = None
+ +
[docs] @_check_rne + def rne(self, q, qd=None, qdd=None, gravity=None, fext=None, base_wrench=False): + r""" + Inverse dynamics + + :param q: Joint coordinates + :type q: ndarray(n) + :param qd: Joint velocity + :type qd: ndarray(n) + :param qdd: The joint accelerations of the robot + :type qdd: ndarray(n) + :param gravity: Gravitational acceleration to override robot's gravity + value + :type gravity: ndarray(6) + :param fext: Specify wrench acting on the end-effector + :math:`W=[F_x F_y F_z M_x M_y M_z]` + :type fext: ndarray(6) + + ``tau = rne(q, qd, qdd, grav, fext)`` is the joint torque required for + the robot to achieve the specified joint position ``q`` (1xn), velocity + ``qd`` (1xn) and acceleration ``qdd`` (1xn), where n is the number of + robot joints. ``fext`` describes the wrench acting on the end-effector + + Trajectory operation: + If q, qd and qdd (mxn) are matrices with m cols representing a + trajectory then tau (mxn) is a matrix with cols corresponding to each + trajectory step. + + .. note:: + - The torque computed contains a contribution due to armature + inertia and joint friction. + - If a model has no dynamic parameters set the result is zero. + + :seealso: :func:`rne_python` + """ + + if base_wrench: + return self.rne_python( + q, qd, qdd, gravity=gravity, fext=fext, base_wrench=base_wrench + ) + + trajn = 1 + + try: + q = getvector(q, self.n, "row") + qd = getvector(qd, self.n, "row") + qdd = getvector(qdd, self.n, "row") + except ValueError: + trajn = q.shape[0] + verifymatrix(q, (trajn, self.n)) + verifymatrix(qd, (trajn, self.n)) + verifymatrix(qdd, (trajn, self.n)) + + if gravity is None: + gravity = self.gravity + else: + gravity = getvector(gravity, 3) + + # The c function doesn't handle base rotation, so we need to hack the + # gravity vector instead + gravity = self.base.R.T @ gravity + + if fext is None: + fext = np.zeros(6) + else: + fext = getvector(fext, 6) + + tau = np.zeros((trajn, self.n)) + + for i in range(trajn): + tau[i, :] = frne( + # we negate gravity here, since the C code has the sign wrong + self._rne_ob, + q[i, :], + qd[i, :], + qdd[i, :], + -gravity, + fext, + ) + + if trajn == 1: + return tau[0, :] + else: + return tau
+ +
[docs] def rne_python( + self, + Q, + QD=None, + QDD=None, + gravity=None, + fext=None, + debug=False, + base_wrench=False, + ): + """ + Compute inverse dynamics via recursive Newton-Euler formulation + + :param Q: Joint coordinates + :param QD: Joint velocity + :param QDD: Joint acceleration + :param gravity: gravitational acceleration, defaults to attribute + of self + :type gravity: array_like(3), optional + :param fext: end-effector wrench, defaults to None + :type fext: array-like(6), optional + :param debug: print debug information to console, defaults to False + :type debug: bool, optional + :param base_wrench: compute the base wrench, defaults to False + :type base_wrench: bool, optional + :raises ValueError: for misshaped inputs + :return: Joint force/torques + :rtype: NumPy array + + Recursive Newton-Euler for standard Denavit-Hartenberg notation. + + - ``rne_dh(q, qd, qdd)`` where the arguments have shape (n,) where n is + the number of robot joints. The result has shape (n,). + - ``rne_dh(q, qd, qdd)`` where the arguments have shape (m,n) where n + is the number of robot joints and where m is the number of steps in + the joint trajectory. The result has shape (m,n). + - ``rne_dh(p)`` where the input is a 1D array ``p`` = [q, qd, qdd] with + shape (3n,), and the result has shape (n,). + - ``rne_dh(p)`` where the input is a 2D array ``p`` = [q, qd, qdd] with + shape (m,3n) and the result has shape (m,n). + + .. note:: + - This is a pure Python implementation and slower than the .rne() + which is written in C. + - This version supports symbolic model parameters + - Verified against MATLAB code + + :seealso: :func:`rne` + """ + + if np.array_equal(self.base.A, np.eye(4)): + base = None + else: + base = self.base.A + + def removesmall(x): + return x + + n = self.n + + if self.symbolic: + dtype = "O" + else: + dtype = None + + z0 = np.array([0, 0, 1], dtype=dtype) + + if gravity is None: + gravity = self.gravity # default gravity from the object + else: + gravity = getvector(gravity, 3) + + if fext is None: + fext = np.zeros((6,), dtype=dtype) + else: + fext = getvector(fext, 6) + + if debug: + print("grav", gravity) + print("fext", fext) + + # unpack the joint coordinates and derivatives + if Q is not None and QD is None and QDD is None: + # single argument case + Q = getmatrix(Q, (None, self.n * 3)) + q = Q[:, 0:n] + qd = Q[:, n : 2 * n] + qdd = Q[:, 2 * n :] + + else: + # 3 argument case + q = getmatrix(Q, (None, self.n)) + qd = getmatrix(QD, (None, self.n)) + qdd = getmatrix(QDD, (None, self.n)) + + nk = q.shape[0] + + tau = np.zeros((nk, n), dtype=dtype) + if base_wrench: + wbase = np.zeros((nk, n), dtype=dtype) + + for k in range(nk): + # take the k'th row of data + q_k = q[k, :] + qd_k = qd[k, :] + qdd_k = qdd[k, :] + + if debug: + print("q_k", q_k) + print("qd_k", qd_k) + print("qdd_k", qdd_k) + print() + + # joint vector quantities stored columwise in matrix + # m suffix for matrix + Fm = np.zeros((3, n), dtype=dtype) + Nm = np.zeros((3, n), dtype=dtype) + # if robot.issym + # pstarm = sym([]); + # else + # pstarm = []; + pstarm = np.zeros((3, n), dtype=dtype) + Rm = [] + + # rotate base velocity and acceleration into L1 frame + # base has zero angular velocity + w = np.zeros((3,), dtype=dtype) + # base has zero angular acceleration + wd = np.zeros((3,), dtype=dtype) + vd = -gravity # type: ignore + + if base is not None: + Rb = t2r(base).T + w = Rb @ w + wd = Rb @ wd + vd = Rb @ gravity + + # ---------------- initialize some variables ----------------- # + + for j in range(n): + link = self.links[j] + + # compute the link rotation matrix + if link.sigma == 0: + # revolute axis + Tj = link.A(q_k[j]).A + d = link.d + else: + # prismatic + Tj = link.A(link.theta).A + d = q_k[j] + + # compute pstar: + # O_{j-1} to O_j in {j}, negative inverse of link xform + alpha = link.alpha + if self.mdh: + pstar = np.r_[link.a, -d * sym.sin(alpha), d * sym.cos(alpha)] + if j == 0: + if base: + Tj = base @ Tj + pstar = base @ pstar + else: + pstar = np.r_[link.a, d * sym.sin(alpha), d * sym.cos(alpha)] + + # stash them for later + Rm.append(t2r(Tj)) + pstarm[:, j] = pstar + + # ----------------- the forward recursion -------------------- # + + for j, link in enumerate(self.links): + Rt = Rm[j].T # transpose!! + pstar = pstarm[:, j] + r = link.r + + # statement order is important here + + if self.mdh: + if link.isrevolute: + # revolute axis + w_ = Rt @ w + z0 * qd_k[j] + wd_ = Rt @ wd + z0 * qdd_k[j] + _cross(Rt @ w, z0 * qd_k[j]) + vd_ = Rt @ _cross(wd, pstar) + _cross(w, _cross(w, pstar)) + vd + else: + # prismatic axis + w_ = Rt @ w + wd_ = Rt @ wd + vd_ = ( + Rt @ (_cross(wd, pstar) + _cross(w, _cross(w, pstar)) + vd) + + 2 * _cross(Rt @ w, z0 * qd_k[j]) + + z0 * qdd_k[j] + ) + # trailing underscore means new value, update here + w = w_ + wd = wd_ + vd = vd_ + else: + if link.isrevolute: + # revolute axis + wd = Rt @ (wd + z0 * qdd_k[j] + _cross(w, z0 * qd_k[j])) + w = Rt @ (w + z0 * qd_k[j]) + vd = _cross(wd, pstar) + _cross(w, _cross(w, pstar)) + Rt @ vd + else: + # prismatic axis + w = Rt @ w + wd = Rt @ wd + vd = ( + Rt @ (z0 * qdd_k[j] + vd) + + _cross(wd, pstar) + + 2 * _cross(w, Rt @ z0 * qd_k[j]) + + _cross(w, _cross(w, pstar)) + ) + + vhat = _cross(wd, r) + _cross(w, _cross(w, r)) + vd + Fm[:, j] = link.m * vhat + Nm[:, j] = link.I @ wd + _cross(w, link.I @ w) + + if debug: + print("w: ", removesmall(w)) + print("wd: ", removesmall(wd)) + print("vd: ", removesmall(vd)) + print("vdbar: ", removesmall(vhat)) + print() + + if debug: + print("Fm\n", Fm) + print("Nm\n", Nm) + + # ----------------- the backward recursion -------------------- # + + f = fext[:3] # force/moments on end of arm + nn = fext[3:] + + for j in range(n - 1, -1, -1): + link = self.links[j] + r = link.r + + # + # order of these statements is important, since both + # nn and f are functions of previous f. + # + if self.mdh: + if j == (n - 1): + R = np.eye(3, dtype=dtype) + pstar = np.zeros((3,), dtype=dtype) + else: + R = Rm[j + 1] + pstar = pstarm[:, j + 1] + + f_ = R @ f + Fm[:, j] + nn_ = ( + R @ nn + + _cross(pstar, R @ f) + + _cross(pstar, Fm[:, j]) + + Nm[:, j] + ) + f = f_ + nn = nn_ + + else: + pstar = pstarm[:, j] + if j == (n - 1): + R = np.eye(3, dtype=dtype) + else: + R = Rm[j + 1] + + nn = ( + R @ (nn + _cross(R.T @ pstar, f)) + + _cross(pstar + r, Fm[:, j]) + + Nm[:, j] + ) + f = R @ f + Fm[:, j] + + if debug: + print("f: ", removesmall(f)) + print("n: ", removesmall(nn)) + + R = Rm[j] + if self.mdh: + if link.isrevolute: + # revolute axis + t = nn @ z0 + else: + # prismatic + t = f @ z0 + else: + if link.isrevolute: + # revolute axis + t = nn @ (R.T @ z0) + else: + # prismatic + t = f @ (R.T @ z0) + + # add joint inertia and friction + # no Coulomb friction if model is symbolic + tau[k, j] = ( + t + + link.G**2 * link.Jm * qdd_k[j] + - link.friction(qd_k[j], coulomb=not self.symbolic) + ) + if debug: + print( + f"j={j:}, G={link.G:}, Jm={link.Jm:}," + f" friction={link.friction(qd_k[j], coulomb=False):}" + ) # noqa + print() + + # compute the base wrench and save it + if base_wrench: + R = Rm[0] + nn = R @ nn + f = R @ f + wbase[k, :] = np.r_[f, nn] + + # if self.symbolic: + # # simplify symbolic expressions + # print( + # 'start symbolic simplification, this might take a while...') + # # from sympy import trigsimp + + # # tau = trigsimp(tau) + # # consider using multiprocessing to spread over cores + # # https://stackoverflow.com/questions/33844085/using-multiprocessing-with-sympy # noqa + # print('done') + # if tau.shape[0] == 1: + # return tau.reshape(self.n) + # else: + # return tau + + if base_wrench: + if tau.shape[0] == 1: + return tau.flatten(), wbase.flatten() + else: + return tau, wbase + else: + if tau.shape[0] == 1: + return tau.flatten() + else: + return tau
+ + # -------------------------------------------------------------------------- # + +
[docs] def ikine_6s(self, T, config, ikfunc): + # Undo base and tool transformations, but if they are not + # set, skip the operation. Nicer for symbolics + if np.array_equal(self.base.A, np.eye(4)): + base = None + else: + base = self.base + + if np.array_equal(self.tool.A, np.eye(4)): + tool = None + else: + tool = self.tool + + if base is not None: + T = base.inv() * T + if tool is not None: + T = tool.inv() * T + + # q = np.zeros((6,)) + solutions = [] + + for k, Tk in enumerate(T): + # get model specific solution for first 3 joints + theta = ikfunc(self, Tk, config) + + if isinstance(theta, np.ndarray): + # Solve for the wrist rotation + # We need to account for some random translations between the + # first and last 3 joints (d4) and also d6,a6,alpha6 in the + # final frame. + + # Transform of first 3 joints + T13 = self.A([0, 2], theta) + + # T = T13 * Tz(d4) * R * Tz(d6) Tx(a5) + Td4 = SE3(0, 0, self.links[3].d) # Tz(d4) + + # Tz(d6) Tx(a5) Rx(alpha6) + Tt = SE3(self.links[5].a, 0, self.links[5].d) * SE3.Rx( + self.links[5].alpha + ) + + R = Td4.inv() * T13.inv() * Tk * Tt.inv() + + # The spherical wrist implements Euler angles + if "f" in config: + eul = R.eul(flip=True) + else: + eul = R.eul() + theta = np.r_[theta, eul] + if self.links[3].alpha > 0: + theta[4] = -theta[4] + + # Remove the link offset angles + theta = theta - self.offset + + # solution = iksol(theta, True, "") + solution = IKSolution(q=theta, success=True) + + else: + # ikfunc can return None or a str reason + if theta is None: + solution = IKSolution(q=None, success=False) + else: + solution = IKSolution(q=None, success=False, reason=theta) + + solutions.append(solution) + + if len(T) == 1: + return solutions[0] + else: + return IKSolution( + np.vstack([sol.q for sol in solutions]), + np.array([sol.success for sol in solutions]), + [sol.reason for sol in solutions], + )
+ +
[docs] def config_validate(self, config, allowables): + """ + Validate a configuration string + + :param config: a configuration string + :type config: str + :param allowable: [description] + :type allowable: tuple of str + :raises ValueError: bad character in configuration string + :return: configuration string + :rtype: str + + For analytic inverse kinematics the Toolbox uses a string whose + letters indicate particular solutions, eg. for the Puma 560 + + ========= =================== + Character Meaning + ========= =================== + 'l' lefty + 'r' righty + 'u' elbow up + 'd' elbow down + 'n' wrist not flipped + 'f' wrist flipped + ========= =================== + + This method checks that the configuration string is valid and adds + default values for missing characters. For example: + + config = self.config_validate(config, ('lr', 'ud', 'nf')) + + indicates the valid characters, and the first character in each + string is the default, ie. if neither 'l' or 'r' is given then + 'l' will be added to the string. + + """ + for c in config: + if not any([c in allowable for allowable in allowables]): + raise ValueError(f"bad config specifier <{c}>") + for allowable in allowables: + if all([a not in config for a in allowable]): + config += allowable[0] + return config
+ + # -------------------------------------------------------------------------- # + +
[docs] def ik_lm_chan( + self, + Tep: Union[np.ndarray, SE3], + q0: Union[np.ndarray, None] = None, + ilimit: int = 30, + slimit: int = 100, + tol: float = 1e-6, + reject_jl: bool = True, + we: Union[np.ndarray, None] = None, + λ: float = 1.0, + ) -> Tuple[np.ndarray, int, int, int, float]: + """ + Numerical inverse kinematics by Levenberg-Marquadt optimization (Chan's Method) + + :param Tep: The desired end-effector pose or pose trajectory + :param q0: initial joint configuration (default to random valid joint + configuration contrained by the joint limits of the robot) + :param ilimit: maximum number of iterations per search + :param slimit: maximum number of search attempts + :param tol: final error tolerance + :param reject_jl: constrain the solution to being within the joint limits of + the robot (reject solution with invalid joint configurations and perfrom + another search up to the slimit) + :param we: a mask vector which weights the end-effector error priority. + Corresponds to translation in X, Y and Z and rotation about X, Y and Z + respectively + :param λ: value of lambda for the damping matrix Wn + + :return: inverse kinematic solution + :rtype: tuple (q, success, iterations, searches, residual) + + ``sol = ets.ik_lm_chan(Tep)`` are the joint coordinates (n) corresponding + to the robot end-effector pose ``Tep`` which is an ``SE3`` or ``ndarray`` object. + This method can be used for robots with any number of degrees of freedom. + The return value ``sol`` is a tuple with elements: + + ============ ========== =============================================== + Element Type Description + ============ ========== =============================================== + ``q`` ndarray(n) joint coordinates in units of radians or metres + ``success`` int whether a solution was found + ``iterations`` int total number of iterations + ``searches`` int total number of searches + ``residual`` float final value of cost function + ============ ========== =============================================== + + If ``success == 0`` the ``q`` values will be valid numbers, but the + solution will be in error. The amount of error is indicated by + the ``residual``. + + **Joint Limits**: + + ``sol = robot.ikine_LM(T, slimit=100)`` which is the deafualt for this method. + The solver will initialise a solution attempt with a random valid q0 and + perform a maximum of ilimit steps within this attempt. If a solution is not + found, this process is repeated up to slimit times. + + **Global search**: + + ``sol = robot.ikine_LM(T, reject_jl=True)`` is the deafualt for this method. + By setting reject_jl to True, the solver will discard any solution which + violates the defined joint limits of the robot. The solver will then + re-initialise with a new random q0 and repeat the process up to slimit times. + Note that finding a solution with valid joint coordinates takes longer than + without. + + **Underactuated robots:** + + For the case where the manipulator has fewer than 6 DOF the + solution space has more dimensions than can be spanned by the + manipulator joint coordinates. + + In this case we specify the ``we`` option where the ``we`` vector + (6) specifies the Cartesian DOF (in the wrist coordinate frame) that + will be ignored in reaching a solution. The we vector has six + elements that correspond to translation in X, Y and Z, and rotation + about X, Y and Z respectively. The value can be 0 (for ignore) + or above to assign a priority relative to other Cartesian DoF. The number + of non-zero elements must equal the number of manipulator DOF. + + For example when using a 3 DOF manipulator tool orientation might + be unimportant, in which case use the option ``we=[1, 1, 1, 0, 0, 0]``. + + + + .. note:: + + - See `Toolbox kinematics wiki page + <https://github.com/petercorke/robotics-toolbox-python/wiki/Kinematics>`_ + - Implements a Levenberg-Marquadt variable-damping solver. + - The tolerance is computed on the norm of the error between + current and desired tool pose. This norm is computed from + distances and angles without any kind of weighting. + - The inverse kinematic solution is generally not unique, and + depends on the initial guess ``q0``. + + :references: + TODO + + :seealso: + TODO + """ + + return self.ets().ik_lm_chan(Tep, q0, ilimit, slimit, tol, reject_jl, we, λ)
+ +
[docs] def ik_lm_wampler( + self, + Tep: Union[np.ndarray, SE3], + q0: Union[np.ndarray, None] = None, + ilimit: int = 30, + slimit: int = 100, + tol: float = 1e-6, + reject_jl: bool = True, + we: Union[np.ndarray, None] = None, + λ: float = 1.0, + ) -> Tuple[np.ndarray, int, int, int, float]: + """ + Numerical inverse kinematics by Levenberg-Marquadt optimization (Wamplers's Method) + + :param Tep: The desired end-effector pose or pose trajectory + :param q0: initial joint configuration (default to random valid joint + configuration contrained by the joint limits of the robot) + :param ilimit: maximum number of iterations per search + :param slimit: maximum number of search attempts + :param tol: final error tolerance + :param reject_jl: constrain the solution to being within the joint limits of + the robot (reject solution with invalid joint configurations and perfrom + another search up to the slimit) + :param we: a mask vector which weights the end-effector error priority. + Corresponds to translation in X, Y and Z and rotation about X, Y and Z + respectively + :param λ: value of lambda for the damping matrix Wn + + :return: inverse kinematic solution + :rtype: tuple (q, success, iterations, searches, residual) + + ``sol = ets.ik_lm_chan(Tep)`` are the joint coordinates (n) corresponding + to the robot end-effector pose ``Tep`` which is an ``SE3`` or ``ndarray`` object. + This method can be used for robots with any number of degrees of freedom. + The return value ``sol`` is a tuple with elements: + + ============ ========== =============================================== + Element Type Description + ============ ========== =============================================== + ``q`` ndarray(n) joint coordinates in units of radians or metres + ``success`` int whether a solution was found + ``iterations`` int total number of iterations + ``searches`` int total number of searches + ``residual`` float final value of cost function + ============ ========== =============================================== + + If ``success == 0`` the ``q`` values will be valid numbers, but the + solution will be in error. The amount of error is indicated by + the ``residual``. + + **Joint Limits**: + + ``sol = robot.ikine_LM(T, slimit=100)`` which is the deafualt for this method. + The solver will initialise a solution attempt with a random valid q0 and + perform a maximum of ilimit steps within this attempt. If a solution is not + found, this process is repeated up to slimit times. + + **Global search**: + + ``sol = robot.ikine_LM(T, reject_jl=True)`` is the deafualt for this method. + By setting reject_jl to True, the solver will discard any solution which + violates the defined joint limits of the robot. The solver will then + re-initialise with a new random q0 and repeat the process up to slimit times. + Note that finding a solution with valid joint coordinates takes longer than + without. + + **Underactuated robots:** + + For the case where the manipulator has fewer than 6 DOF the + solution space has more dimensions than can be spanned by the + manipulator joint coordinates. + + In this case we specify the ``we`` option where the ``we`` vector + (6) specifies the Cartesian DOF (in the wrist coordinate frame) that + will be ignored in reaching a solution. The we vector has six + elements that correspond to translation in X, Y and Z, and rotation + about X, Y and Z respectively. The value can be 0 (for ignore) + or above to assign a priority relative to other Cartesian DoF. The number + of non-zero elements must equal the number of manipulator DOF. + + For example when using a 3 DOF manipulator tool orientation might + be unimportant, in which case use the option ``we=[1, 1, 1, 0, 0, 0]``. + + + + .. note:: + + - See `Toolbox kinematics wiki page + <https://github.com/petercorke/robotics-toolbox-python/wiki/Kinematics>`_ + - Implements a Levenberg-Marquadt variable-damping solver. + - The tolerance is computed on the norm of the error between + current and desired tool pose. This norm is computed from + distances and angles without any kind of weighting. + - The inverse kinematic solution is generally not unique, and + depends on the initial guess ``q0``. + + :references: + TODO + + :seealso: + TODO + """ + + return self.ets().ik_lm_wampler(Tep, q0, ilimit, slimit, tol, reject_jl, we, λ)
+ +
[docs] def ik_lm_sugihara( + self, + Tep: Union[np.ndarray, SE3], + q0: Union[np.ndarray, None] = None, + ilimit: int = 30, + slimit: int = 100, + tol: float = 1e-6, + reject_jl: bool = True, + we: Union[np.ndarray, None] = None, + λ: float = 1.0, + ) -> Tuple[np.ndarray, int, int, int, float]: + """ + Numerical inverse kinematics by Levenberg-Marquadt optimization (Sugihara's Method) + + :param Tep: The desired end-effector pose or pose trajectory + :param q0: initial joint configuration (default to random valid joint + configuration contrained by the joint limits of the robot) + :param ilimit: maximum number of iterations per search + :param slimit: maximum number of search attempts + :param tol: final error tolerance + :param reject_jl: constrain the solution to being within the joint limits of + the robot (reject solution with invalid joint configurations and perfrom + another search up to the slimit) + :param we: a mask vector which weights the end-effector error priority. + Corresponds to translation in X, Y and Z and rotation about X, Y and Z + respectively + :param λ: value of lambda for the damping matrix Wn + + :return: inverse kinematic solution + :rtype: tuple (q, success, iterations, searches, residual) + + ``sol = ets.ik_lm_chan(Tep)`` are the joint coordinates (n) corresponding + to the robot end-effector pose ``Tep`` which is an ``SE3`` or ``ndarray`` object. + This method can be used for robots with any number of degrees of freedom. + The return value ``sol`` is a tuple with elements: + + ============ ========== =============================================== + Element Type Description + ============ ========== =============================================== + ``q`` ndarray(n) joint coordinates in units of radians or metres + ``success`` int whether a solution was found + ``iterations`` int total number of iterations + ``searches`` int total number of searches + ``residual`` float final value of cost function + ============ ========== =============================================== + + If ``success == 0`` the ``q`` values will be valid numbers, but the + solution will be in error. The amount of error is indicated by + the ``residual``. + + **Joint Limits**: + + ``sol = robot.ikine_LM(T, slimit=100)`` which is the deafualt for this method. + The solver will initialise a solution attempt with a random valid q0 and + perform a maximum of ilimit steps within this attempt. If a solution is not + found, this process is repeated up to slimit times. + + **Global search**: + + ``sol = robot.ikine_LM(T, reject_jl=True)`` is the deafualt for this method. + By setting reject_jl to True, the solver will discard any solution which + violates the defined joint limits of the robot. The solver will then + re-initialise with a new random q0 and repeat the process up to slimit times. + Note that finding a solution with valid joint coordinates takes longer than + without. + + **Underactuated robots:** + + For the case where the manipulator has fewer than 6 DOF the + solution space has more dimensions than can be spanned by the + manipulator joint coordinates. + + In this case we specify the ``we`` option where the ``we`` vector + (6) specifies the Cartesian DOF (in the wrist coordinate frame) that + will be ignored in reaching a solution. The we vector has six + elements that correspond to translation in X, Y and Z, and rotation + about X, Y and Z respectively. The value can be 0 (for ignore) + or above to assign a priority relative to other Cartesian DoF. The number + of non-zero elements must equal the number of manipulator DOF. + + For example when using a 3 DOF manipulator tool orientation might + be unimportant, in which case use the option ``we=[1, 1, 1, 0, 0, 0]``. + + + + .. note:: + + - See `Toolbox kinematics wiki page + <https://github.com/petercorke/robotics-toolbox-python/wiki/Kinematics>`_ + - Implements a Levenberg-Marquadt variable-damping solver. + - The tolerance is computed on the norm of the error between + current and desired tool pose. This norm is computed from + distances and angles without any kind of weighting. + - The inverse kinematic solution is generally not unique, and + depends on the initial guess ``q0``. + + :references: + TODO + + :seealso: + TODO + """ + + return self.ets().ik_lm_sugihara(Tep, q0, ilimit, slimit, tol, reject_jl, we, λ)
+ +
[docs] def ik_nr( + self, + Tep: Union[np.ndarray, SE3], + q0: Union[np.ndarray, None] = None, + ilimit: int = 30, + slimit: int = 100, + tol: float = 1e-6, + reject_jl: bool = True, + we: Union[np.ndarray, None] = None, + use_pinv: int = True, + pinv_damping: float = 0.0, + ) -> Tuple[np.ndarray, int, int, int, float]: + """ + Numerical inverse kinematics by Levenberg-Marquadt optimization (Newton-Raphson Method) + + :param Tep: The desired end-effector pose or pose trajectory + :param q0: initial joint configuration (default to random valid joint + configuration contrained by the joint limits of the robot) + :param ilimit: maximum number of iterations per search + :param slimit: maximum number of search attempts + :param tol: final error tolerance + :param reject_jl: constrain the solution to being within the joint limits of + the robot (reject solution with invalid joint configurations and perfrom + another search up to the slimit) + :param we: a mask vector which weights the end-effector error priority. + Corresponds to translation in X, Y and Z and rotation about X, Y and Z + respectively + :param λ: value of lambda for the damping matrix Wn + + :return: inverse kinematic solution + :rtype: tuple (q, success, iterations, searches, residual) + + ``sol = ets.ik_lm_chan(Tep)`` are the joint coordinates (n) corresponding + to the robot end-effector pose ``Tep`` which is an ``SE3`` or ``ndarray`` object. + This method can be used for robots with any number of degrees of freedom. + The return value ``sol`` is a tuple with elements: + + ============ ========== =============================================== + Element Type Description + ============ ========== =============================================== + ``q`` ndarray(n) joint coordinates in units of radians or metres + ``success`` int whether a solution was found + ``iterations`` int total number of iterations + ``searches`` int total number of searches + ``residual`` float final value of cost function + ============ ========== =============================================== + + If ``success == 0`` the ``q`` values will be valid numbers, but the + solution will be in error. The amount of error is indicated by + the ``residual``. + + **Joint Limits**: + + ``sol = robot.ikine_LM(T, slimit=100)`` which is the deafualt for this method. + The solver will initialise a solution attempt with a random valid q0 and + perform a maximum of ilimit steps within this attempt. If a solution is not + found, this process is repeated up to slimit times. + + **Global search**: + + ``sol = robot.ikine_LM(T, reject_jl=True)`` is the deafualt for this method. + By setting reject_jl to True, the solver will discard any solution which + violates the defined joint limits of the robot. The solver will then + re-initialise with a new random q0 and repeat the process up to slimit times. + Note that finding a solution with valid joint coordinates takes longer than + without. + + **Underactuated robots:** + + For the case where the manipulator has fewer than 6 DOF the + solution space has more dimensions than can be spanned by the + manipulator joint coordinates. + + In this case we specify the ``we`` option where the ``we`` vector + (6) specifies the Cartesian DOF (in the wrist coordinate frame) that + will be ignored in reaching a solution. The we vector has six + elements that correspond to translation in X, Y and Z, and rotation + about X, Y and Z respectively. The value can be 0 (for ignore) + or above to assign a priority relative to other Cartesian DoF. The number + of non-zero elements must equal the number of manipulator DOF. + + For example when using a 3 DOF manipulator tool orientation might + be unimportant, in which case use the option ``we=[1, 1, 1, 0, 0, 0]``. + + + + .. note:: + + - See `Toolbox kinematics wiki page + <https://github.com/petercorke/robotics-toolbox-python/wiki/Kinematics>`_ + - Implements a Levenberg-Marquadt variable-damping solver. + - The tolerance is computed on the norm of the error between + current and desired tool pose. This norm is computed from + distances and angles without any kind of weighting. + - The inverse kinematic solution is generally not unique, and + depends on the initial guess ``q0``. + + :references: + TODO + + :seealso: + TODO + """ + + return self.ets().ik_nr( + Tep, q0, ilimit, slimit, tol, reject_jl, we, use_pinv, pinv_damping + )
+ +
[docs] def ik_gn( + self, + Tep: Union[np.ndarray, SE3], + q0: Union[np.ndarray, None] = None, + ilimit: int = 30, + slimit: int = 100, + tol: float = 1e-6, + reject_jl: bool = True, + we: Union[np.ndarray, None] = None, + use_pinv: int = True, + pinv_damping: float = 0.0, + ) -> Tuple[np.ndarray, int, int, int, float]: + """ + Numerical inverse kinematics by Levenberg-Marquadt optimization (Gauss-Newton Method) + + :param Tep: The desired end-effector pose or pose trajectory + :param q0: initial joint configuration (default to random valid joint + configuration contrained by the joint limits of the robot) + :param ilimit: maximum number of iterations per search + :param slimit: maximum number of search attempts + :param tol: final error tolerance + :param reject_jl: constrain the solution to being within the joint limits of + the robot (reject solution with invalid joint configurations and perfrom + another search up to the slimit) + :param we: a mask vector which weights the end-effector error priority. + Corresponds to translation in X, Y and Z and rotation about X, Y and Z + respectively + :param λ: value of lambda for the damping matrix Wn + + :return: inverse kinematic solution + :rtype: tuple (q, success, iterations, searches, residual) + + ``sol = ets.ik_lm_chan(Tep)`` are the joint coordinates (n) corresponding + to the robot end-effector pose ``Tep`` which is an ``SE3`` or ``ndarray`` object. + This method can be used for robots with any number of degrees of freedom. + The return value ``sol`` is a tuple with elements: + + ============ ========== =============================================== + Element Type Description + ============ ========== =============================================== + ``q`` ndarray(n) joint coordinates in units of radians or metres + ``success`` int whether a solution was found + ``iterations`` int total number of iterations + ``searches`` int total number of searches + ``residual`` float final value of cost function + ============ ========== =============================================== + + If ``success == 0`` the ``q`` values will be valid numbers, but the + solution will be in error. The amount of error is indicated by + the ``residual``. + + **Joint Limits**: + + ``sol = robot.ikine_LM(T, slimit=100)`` which is the deafualt for this method. + The solver will initialise a solution attempt with a random valid q0 and + perform a maximum of ilimit steps within this attempt. If a solution is not + found, this process is repeated up to slimit times. + + **Global search**: + + ``sol = robot.ikine_LM(T, reject_jl=True)`` is the deafualt for this method. + By setting reject_jl to True, the solver will discard any solution which + violates the defined joint limits of the robot. The solver will then + re-initialise with a new random q0 and repeat the process up to slimit times. + Note that finding a solution with valid joint coordinates takes longer than + without. + + **Underactuated robots:** + + For the case where the manipulator has fewer than 6 DOF the + solution space has more dimensions than can be spanned by the + manipulator joint coordinates. + + In this case we specify the ``we`` option where the ``we`` vector + (6) specifies the Cartesian DOF (in the wrist coordinate frame) that + will be ignored in reaching a solution. The we vector has six + elements that correspond to translation in X, Y and Z, and rotation + about X, Y and Z respectively. The value can be 0 (for ignore) + or above to assign a priority relative to other Cartesian DoF. The number + of non-zero elements must equal the number of manipulator DOF. + + For example when using a 3 DOF manipulator tool orientation might + be unimportant, in which case use the option ``we=[1, 1, 1, 0, 0, 0]``. + + + + .. note:: + + - See `Toolbox kinematics wiki page + <https://github.com/petercorke/robotics-toolbox-python/wiki/Kinematics>`_ + - Implements a Levenberg-Marquadt variable-damping solver. + - The tolerance is computed on the norm of the error between + current and desired tool pose. This norm is computed from + distances and angles without any kind of weighting. + - The inverse kinematic solution is generally not unique, and + depends on the initial guess ``q0``. + + :references: + TODO + + :seealso: + TODO + """ + + return self.ets().ik_gn( + Tep, q0, ilimit, slimit, tol, reject_jl, we, use_pinv, pinv_damping + )
+ +
[docs] def ikine_LM( + self, + Tep: Union[np.ndarray, SE3], + q0: Union[ArrayLike, None] = None, + ilimit: int = 30, + slimit: int = 100, + tol: float = 1e-6, + joint_limits: bool = False, + mask: Union[ArrayLike, None] = None, + seed: Union[int, None] = None, + ): + return self.ets().ikine_LM( + Tep=Tep, + q0=q0, + ilimit=ilimit, + slimit=slimit, + tol=tol, + joint_limits=joint_limits, + mask=mask, + seed=seed, + )
+ + +class SerialLink(DHRobot): + def __init__(self, *args, **kwargs): + warnings.warn( + "SerialLink is deprecated, use DHRobot instead", DeprecationWarning + ) + super().__init__(*args, **kwargs) + + +def _cross(a, b): + return np.r_[ + a[1] * b[2] - a[2] * b[1], a[2] * b[0] - a[0] * b[2], a[0] * b[1] - a[1] * b[0] + ] + + +if __name__ == "__main__": # pragma nocover + import roboticstoolbox as rtb + + # import spatialmath.base.symbolic as sym + + # planar = rtb.models.DH.Planar2() + # J = puma.jacob0(puma.qn) + # print(J) + # print(puma.manipulability(puma.qn)) + # print(puma.manipulability(puma.qn, 'asada')) + # tw, T0 = puma.twists(puma.qz) + # print(planar) + + puma = rtb.models.DH.Puma560() + print(puma) + # print(puma.jacob0(puma.qn, analytical="eul")) + # puma.base = None + # print('base', puma.base) + # print('tool', puma.tool) + + # print(puma.ets()) + + # puma[2].flip = True + # puma[3].offset = 1 + # puma[4].flip = True + # puma[4].offset = -1 + # print(puma) + # print(puma.ets()) + + # print(puma.dyntable()) +
+ +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/robot/Dynamics.html b/_modules/roboticstoolbox/robot/Dynamics.html new file mode 100644 index 00000000..8744d307 --- /dev/null +++ b/_modules/roboticstoolbox/robot/Dynamics.html @@ -0,0 +1,1489 @@ + + + + + + + + + + roboticstoolbox.robot.Dynamics — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + + +
+ + + + + +
+ + +
+ +
+ +
+ + + + + + + + + + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/robot/ELink.html b/_modules/roboticstoolbox/robot/ELink.html new file mode 100644 index 00000000..6833670b --- /dev/null +++ b/_modules/roboticstoolbox/robot/ELink.html @@ -0,0 +1,143 @@ + + + + + + roboticstoolbox.robot.ELink — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +

Source code for roboticstoolbox.robot.ELink

+#!/usr/bin/env python
+
+"""
+@author: Jesse Haviland
+"""
+
+from roboticstoolbox.robot.Link import Link, Link2
+
+
+
+
+
+
[docs]class ELink2(Link2): + def __init__(self, *args, **kwargs): + import warnings + + warnings.warn("ELink2 is deprecated, use Link2 instead", FutureWarning) + super().__init__(*args, **kwargs)
+
+ +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/robot/ERobot.html b/_modules/roboticstoolbox/robot/ERobot.html new file mode 100644 index 00000000..d8f8d4c6 --- /dev/null +++ b/_modules/roboticstoolbox/robot/ERobot.html @@ -0,0 +1,145 @@ + + + + + + roboticstoolbox.robot.ERobot — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +

Source code for roboticstoolbox.robot.ERobot

+#!/usr/bin/env python3
+"""
+@author: Jesse Haviland
+"""
+
+from roboticstoolbox.robot.Robot import Robot, Robot2
+
+
+
[docs]class ERobot(Robot): + def __init__(self, *args, **kwargs): + + # warn("ERobot is deprecated, use iscollided instead", FutureWarning) + + super().__init__(*args, **kwargs)
+ + +# =========================================================================== # + + +
[docs]class ERobot2(Robot2): + def __init__(self, *args, **kwargs): + + # warn("ERobot2 is deprecated, use iscollided instead", FutureWarning) + + super().__init__(*args, **kwargs)
+
+ +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/robot/ET.html b/_modules/roboticstoolbox/robot/ET.html new file mode 100644 index 00000000..c973e47d --- /dev/null +++ b/_modules/roboticstoolbox/robot/ET.html @@ -0,0 +1,488 @@ + + + + + + + + + + roboticstoolbox.robot.ET — Robotics Toolbox Python 0.3.7 + documentation + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + + +
+ + + + + +
+ +
+ + + + + + + + + + + + + + + + + +
+ +
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  • + +
  • roboticstoolbox.robot.ET
  • + + +
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+ + +
+
+
+
+ +

Source code for roboticstoolbox.robot.ET

+#!/usr/bin/env python3
+"""
+@author: Jesse Haviland
+@author: Peter Corke
+"""
+from collections import UserList, namedtuple
+import numpy as np
+from spatialmath import SE3
+
+
+
[docs]class ET(UserList): # TODO should be ETS + """ + This class implements a single elementary transform (ET) + + :param axis_func: The function which calculates the transform of the ET. + :type axis_func: static et.T__ function + :param eta: The coordinate of the ET. If not supplied the ET corresponds + to a variable ET which is a joint + :type eta: float, optional + :param joint: If this ET corresponds to a joint, this corresponds to the + joint number within the robot + :type joint: int, optional + + :references: + - Kinematic Derivatives using the Elementary Transform Sequence, + J. Haviland and P. Corke + + """ + + # TODO should probably prefix with __ + STATIC = 0 + VARIABLE = 1 + et = namedtuple('ET', 'eta axis_func axis jtype T') + + def __init__(self, axis_func=None, axis=None, eta=None): + + super().__init__() # init UserList superclass + + if axis_func is None and axis is None and eta is None: + # ET() + # create instance with no values + self.data = [] + return + + if eta is not None: + # constant value specified + jtype = self.STATIC + T = axis_func(eta) + else: + # it's a variable joint + jtype = self.VARIABLE + T = None + + # Save all the params in a named tuple + e = self.et(eta, axis_func, axis, jtype, T) + + # And make it the only value of this instance + self.data = [e] + + @property + def eta(self): + return self.data[0].eta + + @property + def axis_func(self): + return self.data[0].axis_func + + @property + def axis(self): + return self.data[0].axis + + @property + def jtype(self): + return self.data[0].jtype + +
[docs] def T(self, q=None): + """ + Calculates the transformation matrix of the ET + + :param q: Is used if this ET is variable (a joint) + :type q: float (radians), required for variable ET's + :return: The transformation matrix of the ET + :rtype: SE3 + + """ + if self.jtype is self.STATIC: + return self.data[0].T + else: + return self.axis_func(q)
+ + def __str__(self): + """ + Pretty prints the ET. Will output angles in degrees + + :return: The transformation matrix of the ET + :rtype: str + + """ + es = [] + joint = 0 + + # For et in the object, display it, data comes from properties + # which come from the named tuple + for et in self: + + if et.jtype is self.STATIC: + if et.axis[0] == 'R': + s = '%s(%g)' % (et.axis, et.eta * 180 / np.pi) + else: + s = '%s(%g)' % (et.axis, et.eta) + else: + s = '%s(q%d)' % (et.axis, joint) + joint += 1 + es.append(s) + + return " * ".join(es) + + # redefine * operator to concatenate the internal lists +
[docs] def __mul__(self, rest): + prod = ET() + prod.data = self.data + rest.data + return prod
+ +
[docs] def __rmul__(self, rest): + prod = ET() + prod.data = self.data + rest + return prod
+ + # redefine so that indexing returns an ET type +
[docs] def __getitem__(self, i): + item = ET() + data = self.data[i] # can be [2] or slice, eg. [3:5] + # ensure that data is always a list + if isinstance(data, list): + item.data = data + else: + item.data = [data] + return item
+ + def __repr__(self): + return str(self) + +
[docs] @classmethod + def rx(cls, eta=None): + """ + An elementary transform (ET). A pure rotation of eta about the x-axis. + + L = TRx(eta) will instantiate an ET object which represents a pure + rotation about the x-axis by amount eta. + + L = TRx() as above except this ET representation a variable + rotation, i.e. a joint + + :param eta: The amount of rotation about the x-axis + :type eta: float (radians) + :param joint: The joint number within the robot + :type joint: int + :return: An ET object + :rtype: ET + + """ + return cls(SE3.Rx, axis='Rx', eta=eta)
+ +
[docs] @classmethod + def ry(cls, eta=None): + """ + An elementary transform (ET). A pure rotation of eta about the y-axis. + + L = TRy(eta) will instantiate an ET object which represents a pure + rotation about the y-axis by amount eta. + + L = TRy() as above except this ET representation a variable + rotation, i.e. a joint + + :param eta: The amount of rotation about the y-axis + :type eta: float (radians) + :param joint: The joint number within the robot + :type joint: int + :return: An ET object + :rtype: ET + + """ + return cls(SE3.Ry, axis='Ry', eta=eta)
+ +
[docs] @classmethod + def rz(cls, eta=None): + """ + An elementary transform (ET). A pure rotation of eta about the z-axis. + + L = TRz(eta) will instantiate an ET object which represents a pure + rotation about the z-axis by amount eta. + + L = TRz() as above except this ET representation a variable + rotation, i.e. a joint + + :param eta: The amount of rotation about the z-axis + :type eta: float (radians) + :return: An ET object + :rtype: ET + + """ + return cls(SE3.Rz, axis='Rz', eta=eta)
+ +
[docs] @classmethod + def tx(cls, eta=None): + """ + An elementary transform (ET). A pure translation of eta along the + x-axis + + L = Ttx(eta) will instantiate an ET object which represents a pure + translation along the x-axis by amount eta. + + L = Ttx() as above except this ET representation a variable + translation, i.e. a joint + + :param eta: The amount of translation along the x-axis + :type eta: float (metres) + :return: An ET object + :rtype: ET + + """ + return cls(SE3.Tx, axis='tx', eta=eta)
+ +
[docs] @classmethod + def ty(cls, eta=None): + """ + An elementary transform (ET). A pure translation of eta along the + x-axis + + L = Tty(eta) will instantiate an ET object which represents a pure + translation along the y-axis by amount eta. + + L = Tty() as above except this ET representation a variable + translation, i.e. a joint + + :param eta: The amount of translation along the x-axis + :type eta: float (metres) + :return: An ET object + :rtype: ET + + """ + return cls(SE3.Ty, axis='ty', eta=eta)
+ +
[docs] @classmethod + def tz(cls, eta=None): + """ + An elementary transform (ET). A pure translation of eta along the + z-axis + + L = Ttz(eta) will instantiate an ET object which represents a pure + translation along the z-axis by amount eta. + + L = Ttz() as above except this ET representation a variable + translation, i.e. a joint + + :param eta: The amount of translation along the x-axis + :type eta: float (metres) + :return: An ET object + :rtype: ET + + """ + return cls(SE3.Tz, axis='tz', eta=eta)
+ + +# if __name__ == "__main__": + +# from math import pi + +# e = ET.rx(pi/2) +# print(e) + +# e = ET.rx(pi/2) * ET.tx(0.3) * ET.ty(0.4) * ET.rx(-pi/2) +# print(e) + +# e = ET.rx(pi/2) * ET.tx() * ET.ty() * ET.rx(-pi/2) +# print(e) +
+ +
+ +
+ + +
+
+ +
+ +
+ + + + + + + + + + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/robot/ETS.html b/_modules/roboticstoolbox/robot/ETS.html new file mode 100644 index 00000000..d25a5bf2 --- /dev/null +++ b/_modules/roboticstoolbox/robot/ETS.html @@ -0,0 +1,3609 @@ + + + + + + roboticstoolbox.robot.ETS — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +

Source code for roboticstoolbox.robot.ETS

+#!/usr/bin/env python3
+
+"""
+@author: Jesse Haviland
+@author: Peter Corke
+"""
+
+from collections import UserList
+import numpy as np
+from numpy.random import uniform
+from numpy.linalg import inv, det, cond, svd
+from spatialmath import SE3, SE2
+from spatialmath.base import (
+    getvector,
+    issymbol,
+    tr2jac,
+    verifymatrix,
+    tr2jac2,
+    t2r,
+    rotvelxform,
+    simplify,
+    getmatrix,
+)
+from roboticstoolbox import rtb_get_param
+from roboticstoolbox.robot.IK import IK_GN, IK_LM, IK_NR, IK_QP
+
+from roboticstoolbox.fknm import (
+    ETS_init,
+    ETS_fkine,
+    ETS_jacob0,
+    ETS_jacobe,
+    ETS_hessian0,
+    ETS_hessiane,
+    IK_NR_c,
+    IK_GN_c,
+    IK_LM_c,
+)
+from copy import deepcopy
+from roboticstoolbox.robot.ET import ET, ET2
+from typing import Union, overload, List, Set, Tuple
+from typing_extensions import Literal as L
+from sys import version_info
+from roboticstoolbox.tools.types import ArrayLike, NDArray
+
+py_ver = version_info
+
+if version_info >= (3, 9):
+    from functools import cached_property
+
+    c_property = cached_property
+else:  # pragma: nocover
+    c_property = property
+
+
+class BaseETS(UserList):
+    def __init__(self, *args):
+        super().__init__(*args)
+
+    def _update_internals(self):
+        self._m = len(self.data)
+        self._n = len([True for et in self.data if et.isjoint])
+        self._fknm = ETS_init(
+            [et.fknm for et in self.data],
+            self._n,
+            self._m,
+        )
+        # self._fknm = [et.fknm for et in self.data]
+
+    def __str__(self, q: Union[str, None] = None):
+        """
+        Pretty prints the ETS
+
+        ``q`` controls how the joint variables are displayed:
+
+        - None, format depends on number of joint variables
+            - one, display joint variable as q
+            - more, display joint variables as q0, q1, ...
+            - if a joint index was provided, use this value
+        - "", display all joint variables as empty parentheses ``()``
+        - "θ", display all joint variables as ``(θ)``
+        - format string with passed joint variables ``(j, j+1)``, so "θ{0}"
+          would display joint variables as θ0, θ1, ... while "θ{1}" would
+          display joint variables as θ1, θ2, ...  ``j`` is either the joint
+          index, if provided, otherwise a sequential value.
+
+        Parameters
+        ----------
+        q
+            control how joint variables are displayed
+
+        Returns
+        -------
+        str
+            Pretty printed ETS
+
+        Examples
+        --------
+        .. runblock:: pycon
+        >>> from roboticstoolbox import ET
+        >>> e = ET.Rz() * ET.tx(1) * ET.Rz()
+        >>> print(e[:2])
+        >>> print(e)
+        >>> print(e.__str__(""))
+        >>> print(e.__str__("θ{0}"))  # numbering from 0
+        >>> print(e.__str__("θ{1}"))  # numbering from 1
+        >>> # explicit joint indices
+        >>> e = ET.Rz(jindex=3) * ET.tx(1) * ET.Rz(jindex=4)
+        >>> print(e)
+        >>> print(e.__str__("θ{0}"))
+
+        Angular parameters are converted to degrees, except if they
+        are symbolic.
+
+        .. runblock:: pycon
+        >>> from roboticstoolbox import ET
+        >>> from spatialmath.base import symbol
+        >>> theta, d = symbol('theta, d')
+        >>> e = ET.Rx(theta) * ET.tx(2) * ET.Rx(45, 'deg') * ET.Ry(0.2) * ET.ty(d)
+        >>> str(e)
+
+        """
+
+        es = []
+        j = 0
+        c = 0
+        s = None
+        unicode = rtb_get_param("unicode")
+
+        # An empty SE3
+        if len(self.data) == 0:
+            return "SE3()"
+
+        if q is None:
+            if len(self.joints()) > 1:
+                q = "q{0}"
+            else:
+                q = "q"
+
+        # For et in the object, display it, data comes from properties
+        # which come from the named tuple
+        for et in self.data:
+            if et.isjoint:
+                if q is not None:
+                    if et.jindex is None:  # pragma: nocover  this is no longer possible
+                        _j = j
+                    else:
+                        _j = et.jindex
+                    qvar = q.format(  # lgtm [py/str-format/surplus-argument]  # noqa
+                        _j, _j + 1
+                    )
+                # else:
+                #     qvar = ""
+
+                if et.isflip:
+                    s = f"{et.axis}(-{qvar})"
+                else:
+                    s = f"{et.axis}({qvar})"
+                j += 1
+
+            elif et.isrotation:
+                if issymbol(et.eta):
+                    s = f"{et.axis}({et.eta})"
+                else:
+                    s = f"{et.axis}({et.eta * 180 / np.pi:.4g}°)"
+
+            elif et.istranslation:
+                try:
+                    s = f"{et.axis}({et.eta:.4g})"
+                except TypeError:  # pragma: nocover
+                    s = f"{et.axis}({et.eta})"
+
+            elif not et.iselementary:
+                s = str(et)
+                c += 1
+
+            es.append(s)
+
+        if unicode:
+            return " \u2295 ".join(es)
+        else:  # pragma: nocover
+            return " * ".join(es)
+
+    def _repr_pretty_(self, p, cycle):
+        """
+        Pretty string for IPython
+
+        Print stringified version when variable is displayed in IPython, ie. on
+        a line by itself.
+
+        Parameters
+        ----------
+        p
+            pretty printer handle (ignored)
+        cycle
+            pretty printer flag (ignored)
+
+        Examples
+        --------
+        In [1]: e
+        Out [1]: R(q0) ⊕ tx(1) ⊕ R(q1) ⊕ tx(1)
+
+        """
+
+        print(self.__str__())  # pragma: nocover
+
+    def joint_idx(self) -> List[int]:
+        """
+        Get index of joint transforms
+
+        Returns
+        -------
+        joint_idx
+            indices of transforms that are joints
+
+        Examples
+        --------
+        .. runblock:: pycon
+        >>> from roboticstoolbox import ET
+        >>> e = ET.Rz() * ET.tx(1) * ET.Rz() * ET.tx(1)
+        >>> e.joint_idx()
+
+        """
+
+        return np.where([e.isjoint for e in self])[0]  # type: ignore
+
+    def joints(self) -> List[ET]:
+        """
+        Get a list of the variable ETs with this ETS
+
+        Returns
+        -------
+        joints
+            list of ETs that are joints
+
+        Examples
+        --------
+        .. runblock:: pycon
+        >>> from roboticstoolbox import ET
+        >>> e = ET.Rz() * ET.tx(1) * ET.Rz() * ET.tx(1)
+        >>> e.joints()
+
+        """
+
+        return [e for e in self if e.isjoint]
+
+    def jindex_set(self) -> Set[int]:  #
+        """
+        Get set of joint indices
+
+        Returns
+        -------
+        jindex_set
+            set of unique joint indices
+
+        Examples
+        --------
+        .. runblock:: pycon
+        >>> from roboticstoolbox import ET
+        >>> e = ET.Rz(jindex=1) * ET.tx(jindex=2) * ET.Rz(jindex=1) * ET.tx(1)
+        >>> e.jointset()
+
+        """
+
+        return set([self[j].jindex for j in self.joint_idx()])  # type: ignore
+
+    @c_property
+    def jindices(self) -> NDArray:
+        """
+        Get an array of joint indices
+
+        Returns
+        -------
+        jindices
+            array of unique joint indices
+
+        Examples
+        --------
+        .. runblock:: pycon
+        >>> from roboticstoolbox import ET
+        >>> e = ET.Rz(jindex=1) * ET.tx(jindex=2) * ET.Rz(jindex=1) * ET.tx(1)
+        >>> e.jointset()
+
+        """
+
+        return np.array([j.jindex for j in self.joints()])  # type: ignore
+
+    @property
+    def qlim(self):
+        r"""
+        Get/Set Joint limits
+
+        Limits are extracted from the link objects.  If joints limits are
+        not set for:
+
+        - a revolute joint [-𝜋. 𝜋] is returned
+        - a prismatic joint an exception is raised
+
+        Parameters
+        ----------
+        new_qlim
+            An ndarray(2, n) of the new joint limits to set
+
+        Returns
+        -------
+        :return: Array of joint limit values
+        :rtype: ndarray(2,n)
+
+        Raises
+        ------
+        ValueError
+            unset limits for a prismatic joint
+
+        Examples
+        --------
+        .. runblock:: pycon
+        >>> import roboticstoolbox as rtb
+        >>> robot = rtb.models.DH.Puma560()
+        >>> robot.qlim
+
+        """
+
+        limits = np.zeros((2, self.n))
+
+        for i, et in enumerate(self.joints()):
+            if et.isrotation:
+                if et.qlim is None:
+                    v = [-np.pi, np.pi]
+                else:
+                    v = et.qlim
+            elif et.istranslation:
+                if et.qlim is None:
+                    raise ValueError("undefined prismatic joint limit")
+                else:
+                    v = et.qlim
+            else:
+                raise ValueError("Undefined Joint Type")  # pragma: nocover
+            limits[:, i] = v
+
+        return limits
+
+    @qlim.setter
+    def qlim(self, new_qlim: ArrayLike):
+        new_qlim = np.array(new_qlim)
+
+        if new_qlim.shape == (2,) and self.n == 1:
+            new_qlim = new_qlim.reshape(2, 1)
+
+        if new_qlim.shape != (2, self.n):
+            raise ValueError("new_qlim must be of shape (2, n)")
+
+        for j, i in enumerate(self.joint_idx()):
+            et = self[i]
+            et.qlim = new_qlim[:, j]
+
+            self[i] = et
+
+        self._update_internals()
+
+    @property
+    def structure(self) -> str:
+        """
+        Joint structure string
+
+        A string comprising the characters 'R' or 'P' which indicate the types
+        of joints in order from left to right.
+
+        Returns
+        -------
+        structure
+            A string indicating the joint types
+
+
+
+        Examples
+        --------
+        .. runblock:: pycon
+        >>> from roboticstoolbox import ET
+        >>> e = ET.tz() * ET.tx(1) * ET.Rz() * ET.tx(1)
+        >>> e.structure
+
+        """
+
+        return "".join(
+            ["R" if self.data[i].isrotation else "P" for i in self.joint_idx()]
+        )
+
+    @property
+    def n(self) -> int:
+        """
+        Number of joints
+
+        Counts the number of joints in the ETS.
+
+        Returns
+        -------
+        n
+            the number of joints in the ETS
+
+        Examples
+        --------
+        .. runblock:: pycon
+        >>> from roboticstoolbox import ET
+        >>> e = ET.Rx() * ET.tx(1) * ET.tz()
+        >>> e.n
+
+        See Also
+        --------
+        :func:`joints`
+
+        """
+
+        return self._n
+
+    @property
+    def m(self) -> int:
+        """
+        Number of transforms
+
+        Counts the number of transforms in the ETS.
+
+        Returns
+        -------
+        m
+            the number of transforms in the ETS
+
+        Examples
+        --------
+        .. runblock:: pycon
+        >>> from roboticstoolbox import ET
+        >>> e = ET.Rx() * ET.tx(1) * ET.tz()
+        >>> e.m
+
+        """
+
+        return self._m
+
+    @overload
+    def data(self: "ETS") -> List[ET]:
+        ...  # pragma: nocover
+
+    @overload
+    def data(self: "ETS2") -> List[ET2]:
+        ...  # pragma: nocover
+
+    @property
+    def data(self):
+        return self._data
+
+    @data.setter
+    @overload
+    def data(self: "ETS", new_data: List[ET]):
+        ...  # pragma: nocover
+
+    @data.setter
+    @overload
+    def data(self: "ETS", new_data: List[ET2]):
+        ...  # pragma: nocover
+
+    @data.setter
+    def data(self, new_data):
+        self._data = new_data
+
+    @overload
+    def pop(self: "ETS", i: int = -1) -> ET:
+        ...  # pragma: nocover
+
+    @overload
+    def pop(self: "ETS2", i: int = -1) -> ET2:
+        ...  # pragma: nocover
+
+    def pop(self, i=-1):
+        """
+        Pop value
+
+        Removes a value from the value list and returns it.  The original
+        instance is modified.
+
+        Parameters
+        ----------
+        i
+            item in the list to pop, default is last
+
+        Returns
+        -------
+        pop
+            the popped value
+
+        Raises
+        ------
+        IndexError
+            if there are no values to pop
+
+        Examples
+        --------
+        .. runblock:: pycon
+        >>> from roboticstoolbox import ET
+        >>> e = ET.Rz() * ET.tx(1) * ET.Rz() * ET.tx(1)
+        >>> tail = e.pop()
+        >>> tail
+        >>> e
+
+        """
+
+        item = super().pop(i)
+        self._update_internals()
+        return item
+
+    @overload
+    def split(self: "ETS") -> List["ETS"]:
+        ...  # pragma: nocover
+
+    @overload
+    def split(self: "ETS2") -> List["ETS2"]:
+        ...  # pragma: nocover
+
+    def split(self):
+        """
+        Split ETS into link segments
+
+        Returns
+        -------
+        split
+            a list of ETS, each one, apart from the last,
+            ends with a variable ET.
+
+        """
+
+        segments = []
+        start = 0
+
+        for j, k in enumerate(self.joint_idx()):
+            ets_j = self.data[start : k + 1]
+            start = k + 1
+            segments.append(ets_j)
+
+        tail = self.data[start:]
+
+        if isinstance(tail, list):
+            tail_len = len(tail)
+        elif tail is not None:  # pragma: nocover
+            tail_len = 1
+        else:  # pragma: nocover
+            tail_len = 0
+
+        if tail_len > 0:
+            segments.append(tail)
+
+        return segments
+
+    @overload
+    def inv(self: "ETS") -> "ETS":
+        ...  # pragma: nocover
+
+    @overload
+    def inv(self: "ETS2") -> "ETS2":
+        ...  # pragma: nocover
+
+    def inv(self):
+        r"""
+        Inverse of ETS
+
+        The inverse of a given ETS.  It is computed as the inverse of the
+        individual ETs in the reverse order.
+
+        .. math::
+
+            (\mathbf{E}_0, \mathbf{E}_1 \cdots \mathbf{E}_{n-1} )^{-1} = (\mathbf{E}_{n-1}^{-1}, \mathbf{E}_{n-2}^{-1} \cdots \mathbf{E}_0^{-1}{n-1} )
+
+        Returns
+        -------
+        inv
+            Inverse of the ETS
+
+        Examples
+        --------
+        .. runblock:: pycon
+        >>> from roboticstoolbox import ET
+        >>> e = ET.Rz(jindex=2) * ET.tx(1) * ET.Rx(jindex=3,flip=True) * ET.tx(1)
+        >>> print(e)
+        >>> print(e.inv())
+
+        Notes
+        -----
+        - It is essential to use explicit joint indices to account for
+            the reversed order of the transforms.
+
+        """  # noqa
+
+        return self.__class__([et.inv() for et in reversed(self.data)])
+
+    @overload
+    def __getitem__(self: "ETS", i: int) -> ET:
+        ...  # pragma: nocover
+
+    @overload
+    def __getitem__(self: "ETS", i: slice) -> List[ET]:
+        ...  # pragma: nocover
+
+    @overload
+    def __getitem__(self: "ETS2", i: int) -> ET2:
+        ...  # pragma: nocover
+
+    @overload
+    def __getitem__(self: "ETS2", i: slice) -> List[ET2]:
+        ...  # pragma: nocover
+
+    def __getitem__(self, i):
+        """
+        Index or slice an ETS
+
+        Parameters
+        ----------
+        i
+            the index or slince
+
+        Returns
+        -------
+        et
+            Elementary transform
+
+        Examples
+        --------
+        .. runblock:: pycon
+        >>> from roboticstoolbox import ET
+        >>> e = ET.Rz() * ET.tx(1) * ET.Rz() * ET.tx(1)
+        >>> e[0]
+        >>> e[1]
+        >>> e[1:3]
+
+        """
+        return self.data[i]  # can be [2] or slice, eg. [3:5]
+
+    def __deepcopy__(self, memo):
+        new_data = []
+
+        for data in self:
+            new_data.append(deepcopy(data))
+
+        cls = self.__class__
+        result = cls(new_data)
+        memo[id(self)] = result
+        return result
+
+    def plot(self, *args, **kwargs):
+        from roboticstoolbox.robot.Robot import Robot, Robot2
+
+        if isinstance(self, ETS):
+            robot = Robot(self)
+        else:
+            robot = Robot2(self)
+
+        robot.plot(*args, **kwargs)
+
+    def teach(self, *args, **kwargs):
+        from roboticstoolbox.robot.Robot import Robot, Robot2
+
+        if isinstance(self, ETS):
+            robot = Robot(self)
+        else:
+            robot = Robot2(self)
+
+        robot.teach(*args, **kwargs)
+
+    def random_q(self, i: int = 1) -> NDArray:
+        """
+        Generate a random valid joint configuration
+
+        Generates a random q vector within the joint limits defined by
+        `self.qlim`.
+
+        Parameters
+        ----------
+        i
+            number of configurations to generate
+
+        Examples
+        --------
+        .. runblock:: pycon
+        >>> import roboticstoolbox as rtb
+        >>> robot = rtb.models.Panda()
+        >>> ets = robot.ets()
+        >>> q = ets.random_q()
+        >>> q
+
+        """
+
+        if i == 1:
+            q = np.zeros(self.n)
+
+            for i in range(self.n):
+                q[i] = uniform(self.qlim[0, i], self.qlim[1, i])
+
+        else:
+            q = np.zeros((i, self.n))
+
+            for j in range(i):
+                for i in range(self.n):
+                    q[j, i] = uniform(self.qlim[0, i], self.qlim[1, i])
+
+        return q
+
+
+
[docs]class ETS(BaseETS): + """ + This class implements an elementary transform sequence (ETS) for 3D + + An instance can contain an elementary transform (ET) or an elementary + transform sequence (ETS). It has list-like properties by subclassing + UserList, which means we can perform indexing, slicing pop, insert, as well + as using it as an iterator over its values. + + - ``ETS()`` an empty ETS list + - ``ETS(et)`` an ETS containing a single ET + - ``ETS([et0, et1, et2])`` an ETS consisting of three ET's + + Parameters + ---------- + arg + Function to compute ET value + + Examples + -------- + .. runblock:: pycon + >>> from roboticstoolbox import ETS, ET + >>> e = ET.Rz(0.3) # a single ET, rotation about z + >>> ets1 = ETS(e) + >>> len(ets1) + >>> ets2 = ET.Rz(0.3) * ET.tx(2) # an ETS + >>> len(ets2) # of length 2 + >>> ets2[1] # an ET sliced from the ETS + + References + ---------- + - J. Haviland, and P. Corke. "Manipulator Differential Kinematics Part I: + Kinematics, Velocity, and Applications." arXiv preprint arXiv:2207.01796 (2022). + - J. Haviland, and P. Corke. "Manipulator Differential Kinematics Part II: + Acceleration and Advanced Applications." arXiv preprint arXiv:2207.01794 (2022). + + + See Also + -------- + :func:`rx` + :func:`ry` + :func:`rz` + :func:`tx` + :func:`ty` + :func:`tz` + + """ + + def __init__( + self, + arg: Union[ + List[Union["ETS", ET]], List[ET], List["ETS"], ET, "ETS", None + ] = None, + ): + super().__init__() + if isinstance(arg, list): + for item in arg: + if isinstance(item, ET): + self.data.append(deepcopy(item)) + elif isinstance(item, ETS): + for ets_item in item: + self.data.append(deepcopy(ets_item)) + else: + raise TypeError("Invalid arg") + elif isinstance(arg, ET): + self.data.append(deepcopy(arg)) + elif isinstance(arg, ETS): + for ets_item in arg: + self.data.append(deepcopy(ets_item)) + elif arg is None: + self.data = [] + else: + raise TypeError("Invalid arg") + + super()._update_internals() + + self._auto_jindex = False + + # Check if jindices are set + joints = self.joints() + + # Number of joints with a jindex + jindices = 0 + + # Number of joints with a sequential jindex (j[2] -> jindex = 2) + seq_jindex = 0 + + # Count them up + for j, joint in enumerate(joints): + if joint.jindex is not None: + jindices += 1 + if joint.jindex == j: + seq_jindex += 1 + + if ( + jindices == self.n - 1 + and seq_jindex == self.n - 1 + and joints[-1].jindex is None + ): + # ets has sequential jindicies, except for the last. + joints[-1].jindex = self.n - 1 + self._auto_jindex = True + + elif jindices > 0 and not jindices == self.n: + raise ValueError( + "You can not have some jindices set for the ET's in arg. It must be all" + " or none" + ) # pragma: nocover + elif jindices == 0 and self.n > 0: + # Set them ourself + for j, joint in enumerate(joints): + joint.jindex = j + + self._auto_jindex = True + +
[docs] def __mul__(self, other: Union["ET", "ETS"]) -> "ETS": + if isinstance(other, ET): + return ETS([*self.data, other]) + else: + return ETS([*self.data, *other.data]) # pragma: nocover
+ + def __rmul__(self, other: Union["ET", "ETS"]) -> "ETS": + return ETS([other, *self.data]) # pragma: nocover + + def __imul__(self, rest: "ETS"): + return self + rest # pragma: nocover + + def __add__(self, rest) -> "ETS": + return self.__mul__(rest) # pragma: nocover + +
[docs] def compile(self) -> "ETS": + """ + Compile an ETS + + Perform constant folding for faster evaluation. Consecutive constant + ETs are compounded, leading to a constant ET which is denoted by + ``SE3`` when displayed. + + Returns + ------- + compile + optimised ETS + + Examples + -------- + .. runblock:: pycon + >>> import roboticstoolbox as rtb + >>> robot = rtb.models.ETS.Panda() + >>> ets = robot.ets() + >>> ets + >>> ets.compile() + + See Also + -------- + :func:`isconstant` + """ + const = None + ets = ETS() + + for et in self: + if et.isjoint: + # a joint + if const is not None: + # flush the constant + if not np.array_equal(const, np.eye(4)): + ets *= ET.SE3(const) + const = None + ets *= et # emit the joint ET + else: + # not a joint + if const is None: + const = et.A() + else: + const = const @ et.A() + + if const is not None: + # flush the constant, tool transform + if not np.array_equal(const, np.eye(4)): + ets *= ET.SE3(const) + return ets
+ +
[docs] def insert( + self, + arg: Union[ET, "ETS"], + i: int = -1, + ) -> None: + """ + Insert value + + Inserts an ET or ETS into the ET sequence. The inserted value is at position + ``i``. + + Parameters + ---------- + i + insert an ET or ETS into the ETS, default is at the end + arg + the elementary transform or sequence to insert + + Examples + -------- + .. runblock:: pycon + >>> from roboticstoolbox import ET + >>> e = ET.Rz() * ET.tx(1) * ET.Rz() * ET.tx(1) + >>> f = ET.Ry() + >>> e.insert(f, 2) + >>> e + + """ + + if isinstance(arg, ET): + if i == -1: + self.data.append(arg) + else: + self.data.insert(i, arg) + elif isinstance(arg, ETS): + if i == -1: + for et in arg: + self.data.append(et) + else: + for j, et in enumerate(arg): + self.data.insert(i + j, et) + self._update_internals()
+ +
[docs] def fkine( + self, + q: ArrayLike, + base: Union[NDArray, SE3, None] = None, + tool: Union[NDArray, SE3, None] = None, + include_base: bool = True, + ) -> SE3: + """ + Forward kinematics + + ``T = ets.fkine(q)`` evaluates forward kinematics for the ets at + joint configuration ``q``. + + **Trajectory operation**: + If ``q`` has multiple rows (mxn), it is considered a trajectory and the + result is an ``SE3`` instance with ``m`` values. + + Attributes + ---------- + q + Joint coordinates + base + A base transform applied before the ETS + tool + tool transform, optional + include_base + set to True if the base transform should be considered + + Returns + ------- + The transformation matrix representing the pose of the + end-effector + + Examples + -------- + The following example makes a ``panda`` robot object, gets the ets, and + solves for the forward kinematics at the listed configuration. + + .. runblock:: pycon + >>> import roboticstoolbox as rtb + >>> panda = rtb.models.Panda().ets() + >>> panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854]) + + Notes + ----- + - A tool transform, if provided, is incorporated into the result. + - Works from the end-effector link to the base + + References + ---------- + - J. Haviland, and P. Corke. "Manipulator Differential Kinematics Part I: + Kinematics, Velocity, and Applications." arXiv preprint arXiv:2207.01796 (2022). + + """ # noqa + + ret = SE3.Empty() + fk = self.eval(q, base, tool, include_base) + + if fk.dtype == "O": + # symbolic + fk = np.array(simplify(fk)) + + if fk.ndim == 3: + for T in fk: + ret.append(SE3(T, check=False)) # type: ignore + else: + ret = SE3(fk, check=False) + + return ret
+ + def eval( + self, + q: ArrayLike, + base: Union[NDArray, SE3, None] = None, + tool: Union[NDArray, SE3, None] = None, + include_base: bool = True, + ) -> NDArray: + """ + Forward kinematics + + ``T = ets.fkine(q)`` evaluates forward kinematics for the ets at + joint configuration ``q``. + + **Trajectory operation**: + If ``q`` has multiple rows (mxn), it is considered a trajectory and the + result is an ``SE3`` instance with ``m`` values. + + Attributes + ---------- + q + Joint coordinates + base + A base transform applied before the ETS + tool + tool transform, optional + include_base + set to True if the base transform should be considered + Returns + ------- + The transformation matrix representing the pose of the + end-effector + + Examples + -------- + The following example makes a ``panda`` robot object, gets the ets, and + solves for the forward kinematics at the listed configuration. + + .. runblock:: pycon + >>> import roboticstoolbox as rtb + >>> panda = rtb.models.Panda().ets() + >>> panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854]) + + Notes + ----- + - A tool transform, if provided, is incorporated into the result. + - Works from the end-effector link to the base + + References + ---------- + - J. Haviland, and P. Corke. "Manipulator Differential Kinematics Part I: + Kinematics, Velocity, and Applications." arXiv preprint arXiv:2207.01796 (2022). + + """ # noqa + + try: + return ETS_fkine(self._fknm, q, base, tool, include_base) + except BaseException: + pass + + q = getmatrix(q, (None, None)) + l, _ = q.shape # type: ignore + end = self.data[-1] + + if isinstance(tool, SE3): + tool = np.array(tool.A) + + if isinstance(base, SE3): + base = np.array(base.A) + + if base is None: + bases = None + elif np.array_equal(base, np.eye(3)): # pragma: nocover + bases = None + else: # pragma: nocover + bases = base + + if tool is None: + tools = None + elif np.array_equal(tool, np.eye(3)): # pragma: nocover + tools = None + else: # pragma: nocover + tools = tool + + if l > 1: + T = np.zeros((l, 4, 4), dtype=object) + else: + T = np.zeros((4, 4), dtype=object) + + # Tk = None + + for k, qk in enumerate(q): # type: ignore + link = end # start with last link + + jindex = 0 if link.jindex is None and link.isjoint else link.jindex + + Tk = link.A(qk[jindex]) + + if tools is not None: + Tk = Tk @ tools + + # add remaining links, back toward the base + for i in range(self.m - 2, -1, -1): + link = self.data[i] + + jindex = 0 if link.jindex is None and link.isjoint else link.jindex + A = link.A(qk[jindex]) + + if A is not None: + Tk = A @ Tk + + # add base transform if it is set + if include_base is True and bases is not None: + Tk = bases @ Tk + + # append + if l > 1: + T[k, :, :] = Tk + else: + T = Tk + + return T + +
[docs] def jacob0( + self, + q: ArrayLike, + tool: Union[NDArray, SE3, None] = None, + ) -> NDArray: + r""" + Manipulator geometric Jacobian in the base frame + + ``robot.jacobo(q)`` is the manipulator Jacobian matrix which maps + joint velocity to end-effector spatial velocity expressed in the + base frame. + + End-effector spatial velocity :math:`\nu = (v_x, v_y, v_z, \omega_x, \omega_y, \omega_z)^T` + is related to joint velocity by :math:`{}^{E}\!\nu = \mathbf{J}_m(q) \dot{q}`. + + Parameters + ---------- + q + Joint coordinate vector + tool + a static tool transformation matrix to apply to the + end of end, defaults to None + + Returns + ------- + J0 + Manipulator Jacobian in the base frame + + Examples + -------- + The following example makes a ``Puma560`` robot object, and solves for the + base-frame Jacobian at the zero joint angle configuration + + .. runblock:: pycon + >>> import roboticstoolbox as rtb + >>> puma = rtb.models.Puma560().ets() + >>> puma.jacob0([0, 0, 0, 0, 0, 0]) + + Notes + ----- + - This is the geometric Jacobian as described in texts by + Corke, Spong etal., Siciliano etal. The end-effector velocity is + described in terms of translational and angular velocity, not a + velocity twist as per the text by Lynch & Park. + + References + ---------- + - J. Haviland, and P. Corke. "Manipulator Differential Kinematics Part I: + Kinematics, Velocity, and Applications." arXiv preprint arXiv:2207.01796 (2022). + + """ # noqa + + # Use c extension + try: + return ETS_jacob0(self._fknm, q, tool) + except TypeError: + pass + + # Otherwise use Python + if tool is None: + tools = np.eye(4) + elif isinstance(tool, SE3): + tools = np.array(tool.A) + else: # pragma: nocover + tools = np.eye(4) + + q = getvector(q, None) + + T = self.eval(q, include_base=False) @ tools + + U = np.eye(4) + j = 0 + J = np.zeros((6, self.n), dtype="object") + zero = np.array([0, 0, 0]) + end = self.data[-1] + + for link in self.data: + jindex = 0 if link.jindex is None and link.isjoint else link.jindex + + if link.isjoint: + U = U @ link.A(q[jindex]) # type: ignore + + if link == end: + U = U @ tools + + Tu = SE3(U, check=False).inv().A @ T + n = U[:3, 0] + o = U[:3, 1] + a = U[:3, 2] + x = Tu[0, 3] + y = Tu[1, 3] + z = Tu[2, 3] + + if link.axis == "Rz": + J[:3, j] = (o * x) - (n * y) + J[3:, j] = a + + elif link.axis == "Ry": + J[:3, j] = (n * z) - (a * x) + J[3:, j] = o + + elif link.axis == "Rx": + J[:3, j] = (a * y) - (o * z) + J[3:, j] = n + + elif link.axis == "tx": + J[:3, j] = n + J[3:, j] = zero + + elif link.axis == "ty": + J[:3, j] = o + J[3:, j] = zero + + elif link.axis == "tz": + J[:3, j] = a + J[3:, j] = zero + + j += 1 + else: + A = link.A() + if A is not None: + U = U @ A + + return J
+ +
[docs] def jacobe( + self, + q: ArrayLike, + tool: Union[NDArray, SE3, None] = None, + ) -> NDArray: + r""" + Manipulator geometric Jacobian in the end-effector frame + + ``robot.jacobe(q)`` is the manipulator Jacobian matrix which maps + joint velocity to end-effector spatial velocity expressed in the + ``end`` frame. + + End-effector spatial velocity :math:`\nu = (v_x, v_y, v_z, \omega_x, \omega_y, \omega_z)^T` + is related to joint velocity by :math:`{}^{E}\!\nu = \mathbf{J}_m(q) \dot{q}`. + + Parameters + ---------- + q + Joint coordinate vector + end + the particular link or gripper whose velocity the Jacobian + describes, defaults to the end-effector if only one is present + start + the link considered as the base frame, defaults to the robots's base frame + tool + a static tool transformation matrix to apply to the + end of end, defaults to None + + Returns + ------- + Je + Manipulator Jacobian in the ``end`` frame + + Examples + -------- + The following example makes a ``Puma560`` robot object, and solves for the + end-effector frame Jacobian at the zero joint angle configuration + + .. runblock:: pycon + >>> import roboticstoolbox as rtb + >>> puma = rtb.models.Puma560().ets() + >>> puma.jacobe([0, 0, 0, 0, 0, 0]) + + Notes + ----- + - This is the geometric Jacobian as described in texts by + Corke, Spong etal., Siciliano etal. The end-effector velocity is + described in terms of translational and angular velocity, not a + velocity twist as per the text by Lynch & Park. + + References + ---------- + - J. Haviland, and P. Corke. "Manipulator Differential Kinematics Part I: + Kinematics, Velocity, and Applications." arXiv preprint arXiv:2207.01796 (2022). + + """ # noqa + + # Use c extension + try: + return ETS_jacobe(self._fknm, q, tool) + except TypeError: + pass + + T = self.eval(q, tool=tool, include_base=False) + return tr2jac(T.T) @ self.jacob0(q, tool=tool)
+ +
[docs] def hessian0( + self, + q: Union[ArrayLike, None] = None, + J0: Union[NDArray, None] = None, + tool: Union[NDArray, SE3, None] = None, + ) -> NDArray: + r""" + Manipulator Hessian + + The manipulator Hessian tensor maps joint acceleration to end-effector + spatial acceleration, expressed in the base frame. This + function calulcates this based on the ETS of the robot. One of J0 or q + is required. Supply J0 if already calculated to save computation time + + Parameters + ---------- + q + The joint angles/configuration of the robot (Optional, + if not supplied will use the stored q values). + J0 + The manipulator Jacobian in the base frame + tool + a static tool transformation matrix to apply to the + end of end, defaults to None + + Returns + ------- + h0 + The manipulator Hessian in the base frame + + Synopsis + -------- + This method computes the manipulator Hessian in the base frame. If + we take the time derivative of the differential kinematic relationship + .. math:: + \nu &= \mat{J}(\vec{q}) \dvec{q} \\ + \alpha &= \dmat{J} \dvec{q} + \mat{J} \ddvec{q} + where + .. math:: + \dmat{J} = \mat{H} \dvec{q} + and :math:`\mat{H} \in \mathbb{R}^{6\times n \times n}` is the + Hessian tensor. + + The elements of the Hessian are + .. math:: + \mat{H}_{i,j,k} = \frac{d^2 u_i}{d q_j d q_k} + where :math:`u = \{t_x, t_y, t_z, r_x, r_y, r_z\}` are the elements + of the spatial velocity vector. + + Similarly, we can write + .. math:: + \mat{J}_{i,j} = \frac{d u_i}{d q_j} + + Examples + -------- + The following example makes a ``Panda`` robot object, and solves for the + base frame Hessian at the given joint angle configuration + + .. runblock:: pycon + >>> import roboticstoolbox as rtb + >>> panda = rtb.models.Panda().ets() + >>> panda.hessian0([0, -0.3, 0, -2.2, 0, 2, 0.7854]) + + References + ---------- + - J. Haviland, and P. Corke. "Manipulator Differential Kinematics Part I: + Kinematics, Velocity, and Applications." arXiv preprint arXiv:2207.01796 (2022). + - J. Haviland, and P. Corke. "Manipulator Differential Kinematics Part II: + Acceleration and Advanced Applications." arXiv preprint arXiv:2207.01794 (2022). + + """ # noqa + + # Use c extension + try: + return ETS_hessian0(self._fknm, q, J0, tool) + except TypeError: + pass + + def cross(a, b): + x = a[1] * b[2] - a[2] * b[1] + y = a[2] * b[0] - a[0] * b[2] + z = a[0] * b[1] - a[1] * b[0] + return np.array([x, y, z]) + + n = self.n + + if J0 is None: + q = getvector(q, None) + J0 = self.jacob0(q, tool=tool) + else: + verifymatrix(J0, (6, self.n)) + + H = np.zeros((n, 6, n)) + + for j in range(n): + for i in range(j, n): + H[j, :3, i] = cross(J0[3:, j], J0[:3, i]) + H[j, 3:, i] = cross(J0[3:, j], J0[3:, i]) + + if i != j: + H[i, :3, j] = H[j, :3, i] + + return H
+ +
[docs] def hessiane( + self, + q: Union[ArrayLike, None] = None, + Je: Union[NDArray, None] = None, + tool: Union[NDArray, SE3, None] = None, + ) -> NDArray: + r""" + Manipulator Hessian + + The manipulator Hessian tensor maps joint acceleration to end-effector + spatial acceleration, expressed in the end-effector coordinate frame. This + function calulcates this based on the ETS of the robot. One of J0 or q + is required. Supply J0 if already calculated to save computation time + + Parameters + ---------- + q + The joint angles/configuration of the robot (Optional, + if not supplied will use the stored q values). + J0 + The manipulator Jacobian in the end-effector frame + tool + a static tool transformation matrix to apply to the + end of end, defaults to None + + Returns + ------- + he + The manipulator Hessian in end-effector frame + + Synopsis + -------- + This method computes the manipulator Hessian in the end-effector frame. If + we take the time derivative of the differential kinematic relationship + .. math:: + \nu &= \mat{J}(\vec{q}) \dvec{q} \\ + \alpha &= \dmat{J} \dvec{q} + \mat{J} \ddvec{q} + where + .. math:: + \dmat{J} = \mat{H} \dvec{q} + and :math:`\mat{H} \in \mathbb{R}^{6\times n \times n}` is the + Hessian tensor. + + The elements of the Hessian are + .. math:: + \mat{H}_{i,j,k} = \frac{d^2 u_i}{d q_j d q_k} + where :math:`u = \{t_x, t_y, t_z, r_x, r_y, r_z\}` are the elements + of the spatial velocity vector. + + Similarly, we can write + .. math:: + \mat{J}_{i,j} = \frac{d u_i}{d q_j} + + Examples + -------- + The following example makes a ``Panda`` robot object, and solves for the + end-effector frame Hessian at the given joint angle configuration + + .. runblock:: pycon + >>> import roboticstoolbox as rtb + >>> panda = rtb.models.Panda().ets() + >>> panda.hessiane([0, -0.3, 0, -2.2, 0, 2, 0.7854]) + + References + ---------- + - J. Haviland, and P. Corke. "Manipulator Differential Kinematics Part I: + Kinematics, Velocity, and Applications." arXiv preprint arXiv:2207.01796 (2022). + - J. Haviland, and P. Corke. "Manipulator Differential Kinematics Part II: + Acceleration and Advanced Applications." arXiv preprint arXiv:2207.01794 (2022). + + """ # noqa + + # Use c extension + try: + return ETS_hessiane(self._fknm, q, Je, tool) + except TypeError: + pass + + def cross(a, b): + x = a[1] * b[2] - a[2] * b[1] + y = a[2] * b[0] - a[0] * b[2] + z = a[0] * b[1] - a[1] * b[0] + return np.array([x, y, z]) + + n = self.n + + if Je is None: + q = getvector(q, None) + Je = self.jacobe(q, tool=tool) + else: + verifymatrix(Je, (6, self.n)) + + H = np.zeros((n, 6, n)) + + for j in range(n): + for i in range(j, n): + H[j, :3, i] = cross(Je[3:, j], Je[:3, i]) + H[j, 3:, i] = cross(Je[3:, j], Je[3:, i]) + + if i != j: + H[i, :3, j] = H[j, :3, i] + + return H
+ + def jacob0_analytical( + self, + q: ArrayLike, + representation: str = "rpy/xyz", + tool: Union[NDArray, SE3, None] = None, + ): + r""" + Manipulator analytical Jacobian in the base frame + + ``robot.jacob0_analytical(q)`` is the manipulator Jacobian matrix which maps + joint velocity to end-effector spatial velocity expressed in the + base frame. + + Parameters + ---------- + q + Joint coordinate vector + representation + angular representation + tool + a static tool transformation matrix to apply to the + end of end, defaults to None + + Returns + ------- + jacob0 + Manipulator Jacobian in the base frame + + Synopsis + -------- + End-effector spatial velocity :math:`\nu = (v_x, v_y, v_z, \omega_x, \omega_y, \omega_z)^T` + is related to joint velocity by :math:`{}^{E}\!\nu = \mathbf{J}_m(q) \dot{q}`. + + |``representation`` | Rotational representation | + |---------------------|-------------------------------------| + |``'rpy/xyz'`` | RPY angular rates in XYZ order | + |``'rpy/zyx'`` | RPY angular rates in XYZ order | + |``'eul'`` | Euler angular rates in ZYZ order | + |``'exp'`` | exponential coordinate rates | + + Examples + -------- + Makes a robot object and computes the analytic Jacobian for the given + joint configuration + + .. runblock:: pycon + >>> import roboticstoolbox as rtb + >>> puma = rtb.models.ETS.Puma560().ets() + >>> puma.jacob0_analytical([0, 0, 0, 0, 0, 0]) + + """ # noqa + + T = self.eval(q, tool=tool) + J = self.jacob0(q, tool=tool) + A = rotvelxform(t2r(T), full=True, inverse=True, representation=representation) + return A @ J + + def jacobm(self, q: ArrayLike) -> NDArray: + r""" + The manipulability Jacobian + + This measure relates the rate of change of the manipulability to the + joint velocities of the robot. One of J or q is required. Supply J + and H if already calculated to save computation time + + Parameters + ---------- + q + The joint angles/configuration of the robot (Optional, + if not supplied will use the stored q values). + + Returns + ------- + jacobm + The manipulability Jacobian + + Synopsis + -------- + Yoshikawa's manipulability measure + + .. math:: + + m(\vec{q}) = \sqrt{\mat{J}(\vec{q}) \mat{J}(\vec{q})^T} + + This method returns its Jacobian with respect to configuration + + .. math:: + + \frac{\partial m(\vec{q})}{\partial \vec{q}} + + References + ---------- + - J. Haviland, and P. Corke. "Manipulator Differential Kinematics Part I: + Kinematics, Velocity, and Applications." arXiv preprint arXiv:2207.01796 (2022). + - J. Haviland, and P. Corke. "Manipulator Differential Kinematics Part II: + Acceleration and Advanced Applications." arXiv preprint arXiv:2207.01794 (2022). + + """ # noqa + + J = self.jacob0(q) + H = self.hessian0(q) + + manipulability = self.manipulability(q) + + # J = J[axes, :] + # H = H[:, axes, :] + + b = inv(J @ J.T) + Jm = np.zeros((self.n, 1)) + + for i in range(self.n): + c = J @ H[i, :, :].T + Jm[i, 0] = manipulability * (c.flatten("F")).T @ b.flatten("F") + + return Jm + + def manipulability( + self, + q, + method: L["yoshikawa", "minsingular", "invcondition"] = "yoshikawa", + axes: Union[L["all", "trans", "rot"], List[bool]] = "all", + ): + """ + Manipulability measure + + ``manipulability(q)`` is the scalar manipulability index + for the ets at the joint configuration ``q``. It indicates + dexterity, that is, how well conditioned the ets is for motion + with respect to the 6 degrees of Cartesian motion. The values is + zero if the ets is at a singularity. + + Parameters + ---------- + q + Joint coordinates, one of J or q required + method + method to use, "yoshikawa" (default), "invcondition", + "minsingular" + axes + Task space axes to consider: "all" [default], + "trans", or "rot" + + Returns + ------- + manipulability + the manipulability metric + + Synopsis + -------- + + Various measures are supported: + + | Measure | Description | + |-------------------|-------------------------------------------------| + | ``"yoshikawa"`` | Volume of the velocity ellipsoid, *distance* | + | | from singularity [Yoshikawa85]_ | + | ``"invcondition"``| Inverse condition number of Jacobian, isotropy | + | | of the velocity ellipsoid [Klein87]_ | + | ``"minsingular"`` | Minimum singular value of the Jacobian, | + | | *distance* from singularity [Klein87]_ | + + **Trajectory operation**: + + If ``q`` is a matrix (m,n) then the result (m,) is a vector of + manipulability indices for each joint configuration specified by a row + of ``q``. + + Notes + ----- + - Invokes the ``jacob0`` method of the robot if ``J`` is not passed + - The "all" option includes rotational and translational + dexterity, but this involves adding different units. It can be + more useful to look at the translational and rotational + manipulability separately. + - Examples in the RVC book (1st edition) can be replicated by + using the "all" option + - Asada's measure requires inertial a robot model with inertial + parameters. + + References + ---------- + .. [Yoshikawa85] Manipulability of Robotic Mechanisms. Yoshikawa T., + The International Journal of Robotics Research. + 1985;4(2):3-9. doi:10.1177/027836498500400201 + .. [Klein87] Dexterity Measures for the Design and Control of + Kinematically Redundant Manipulators. Klein CA, Blaho BE. + The International Journal of Robotics Research. + 1987;6(2):72-83. doi:10.1177/027836498700600206 + - Robotics, Vision & Control in Python, 3e, P. Corke, Springer 2023, Chap 7. + + + .. versionchanged:: 1.0.4 + Removed 'both' option for axes, added a custom list option. + + """ + + axes_list: List[bool] = [] + + if isinstance(axes, list): + axes_list = axes + elif axes == "all": + axes_list = [True, True, True, True, True, True] + elif axes.startswith("trans"): + axes_list = [True, True, True, False, False, False] + elif axes.startswith("rot"): + axes_list = [False, False, False, True, True, True] + else: + raise ValueError("axes must be all, trans, rot or both") + + def yoshikawa(robot, J, q, axes, **kwargs): + J = J[axes, :] + if J.shape[0] == J.shape[1]: + # simplified case for square matrix + return abs(det(J)) + else: + m2 = det(J @ J.T) + return np.sqrt(abs(m2)) + + def condition(robot, J, q, axes, **kwargs): + J = J[axes, :] + return 1 / cond(J) + + def minsingular(robot, J, q, axes, **kwargs): + J = J[axes, :] + s = svd(J, compute_uv=False) + return s[-1] # return last/smallest singular value of J + + # choose the handler function + if method == "yoshikawa": + mfunc = yoshikawa + elif method == "invcondition": + mfunc = condition + elif method == "minsingular": + mfunc = minsingular + else: + raise ValueError("Invalid method chosen") + + # Otherwise use the q vector/matrix + q = np.array(getmatrix(q, (None, self.n))) + w = np.zeros(q.shape[0]) + + for k, qk in enumerate(q): + Jk = self.jacob0(qk) + w[k] = mfunc(self, Jk, qk, axes_list) + + if len(w) == 1: + return w[0] + else: + return w + + def partial_fkine0(self, q: ArrayLike, n: int) -> NDArray: + r""" + Manipulator Forward Kinematics nth Partial Derivative + + This method computes the nth derivative of the forward kinematics where ``n`` is + greater than or equal to 3. This is an extension of the differential kinematics + where the Jacobian is the first partial derivative and the Hessian is the + second. + + Parameters + ---------- + q + The joint angles/configuration of the robot (Optional, + if not supplied will use the stored q values). + end + the final link/Gripper which the Hessian represents + start + the first link which the Hessian represents + tool + a static tool transformation matrix to apply to the + end of end, defaults to None + + Returns + ------- + A + The nth Partial Derivative of the forward kinematics + + Examples + -------- + The following example makes a ``Panda`` robot object, and solves for the + base-effector frame 4th defivative of the forward kinematics at the given + joint angle configuration + + .. runblock:: pycon + >>> import roboticstoolbox as rtb + >>> panda = rtb.models.Panda().ets() + >>> panda.partial_fkine0([0, -0.3, 0, -2.2, 0, 2, 0.7854], n=4) + + References + ---------- + - J. Haviland, and P. Corke. "Manipulator Differential Kinematics Part I: + Kinematics, Velocity, and Applications." arXiv preprint arXiv:2207.01796 (2022). + - J. Haviland, and P. Corke. "Manipulator Differential Kinematics Part II: + Acceleration and Advanced Applications." arXiv preprint arXiv:2207.01794 (2022). + + """ # noqa + + # Calculate the Jacobian and Hessian + J = self.jacob0(q) + H = self.hessian0(q) + + # A list of derivatives, starting with the jacobian and hessian + dT = [J, H] + + # The tensor dimensions of the latest derivative + # Set to the current size of the Hessian + size = [self.n, 6, self.n] + + # An array which keeps track of the index of the partial derivative + # we are calculating + # It stores the indices in the order: "j, k, l. m, n, o, ..." + # where count is extended to match oder of the partial derivative + count = np.array([0, 0]) + + # The order of derivative for which we are calculating + # The Hessian is the 2nd-order so we start with c = 2 + c = 2 + + def add_indices(indices, c): + total = len(indices * 2) + new_indices = [] + + for i in range(total): + j = i // 2 + new_indices.append([]) + new_indices[i].append(indices[j][0].copy()) + new_indices[i].append(indices[j][1].copy()) + + if i % 2 == 0: + # if even number + new_indices[i][0].append(c) + else: + # if odd number + new_indices[i][1].append(c) + + return new_indices + + def add_pdi(pdi): + total = len(pdi * 2) + new_pdi = [] + + for i in range(total): + j = i // 2 + new_pdi.append([]) + new_pdi[i].append(pdi[j][0]) + new_pdi[i].append(pdi[j][1]) + + # if even number + if i % 2 == 0: + new_pdi[i][0] += 1 + # if odd number + else: + new_pdi[i][1] += 1 + + return new_pdi + + # these are the indices used for the hessian + indices = [[[1], [0]]] + + # The partial derivative indices (pdi) + # the are the pd indices used in the cross products + pdi = [[0, 0]] + + # The length of dT correspods to the number of derivatives we have calculated + while len(dT) != n: + # Add to the start of the tensor size list + size.insert(0, self.n) + + # Add an axis to the count array + count = np.concatenate(([0], count)) + + # This variables corresponds to indices within the previous + # partial derivatives + # to be cross prodded + # The order is: "[j, k, l, m, n, o, ...]" + # Although, our partial derivatives have the order: + # pd[..., o, n, m, l, k, cartesian DoF, j] + # For example, consider the Hessian Tensor H[n, 6, n], + # the index H[k, :, j]. This corrsponds + # to the second partial derivative of the kinematics of joint j with + # respect to joint k. + indices = add_indices(indices, c) + + # This variable corresponds to the indices in Td which corresponds to the + # partial derivatives we need to use + pdi = add_pdi(pdi) + + c += 1 + + # Allocate our new partial derivative tensor + pd = np.zeros(size) + + # We need to loop n^c times + # There are n^c columns to calculate + for _ in range(self.n**c): + # Allocate the rotation and translation components + rot = np.zeros(3) + trn = np.zeros(3) + + # This loop calculates a single column ([trn, rot]) + # of the tensor for dT(x) + for j in range(len(indices)): + pdr0 = dT[pdi[j][0]] + pdr1 = dT[pdi[j][1]] + + idx0 = count[indices[j][0]] + idx1 = count[indices[j][1]] + + # This is a list of indices selecting the slices of the + # previous tensor + idx0_slices = np.flip(idx0[1:]) + idx1_slices = np.flip(idx1[1:]) + + # This index selecting the column within the 2d slice of the + # previous tensor + idx0_n = idx0[0] + idx1_n = idx1[0] + + # Use our indices to select the rotational column from pdr0 and pdr1 + col0_rot = pdr0[(*idx0_slices, slice(3, 6), idx0_n)] + col1_rot = pdr1[(*idx1_slices, slice(3, 6), idx1_n)] + + # Use our indices to select the translational column from pdr1 + col1_trn = pdr1[(*idx1_slices, slice(0, 3), idx1_n)] + + # Perform the cross product as described in the maths above + rot += np.cross(col0_rot, col1_rot) + trn += np.cross(col0_rot, col1_trn) + + pd[(*np.flip(count[1:]), slice(0, 3), count[0])] = trn + pd[(*np.flip(count[1:]), slice(3, 6), count[0])] = rot + + count[0] += 1 + for j in range(len(count)): + if count[j] == self.n: + count[j] = 0 + if j != len(count) - 1: + count[j + 1] += 1 + + dT.append(pd) + + return dT[-1] + +
[docs] def ik_LM( + self, + Tep: Union[NDArray, SE3], + q0: Union[NDArray, None] = None, + ilimit: int = 30, + slimit: int = 100, + tol: float = 1e-6, + mask: Union[NDArray, None] = None, + joint_limits: bool = True, + k: float = 1.0, + method: L["chan", "wampler", "sugihara"] = "chan", + ) -> Tuple[NDArray, int, int, int, float]: + r""" + Fast levenberg-Marquadt Numerical Inverse Kinematics Solver + + A method which provides functionality to perform numerical inverse kinematics (IK) + using the Levemberg-Marquadt method. This + is a fast solver implemented in C++. + + See the :ref:`Inverse Kinematics Docs Page <IK>` for more details and for a + **tutorial** on numerical IK, see `here <https://bit.ly/3ak5GDi>`_. + + Parameters + ---------- + Tep + The desired end-effector pose + q0 + The initial joint coordinate vector + ilimit + How many iterations are allowed within a search before a new search + is started + slimit + How many searches are allowed before being deemed unsuccessful + tol + Maximum allowed residual error E + mask + A 6 vector which assigns weights to Cartesian degrees-of-freedom + error priority + joint_limits + Reject solutions with joint limit violations + seed + A seed for the private RNG used to generate random joint coordinate + vectors + k + Sets the gain value for the damping matrix Wn in the next iteration. See + synopsis + method + One of "chan", "sugihara" or "wampler". Defines which method is used + to calculate the damping matrix Wn in the ``step`` method + + Synopsis + -------- + The operation is defined by the choice of the ``method`` kwarg. + + The step is deined as + + .. math:: + + \vec{q}_{k+1} + &= + \vec{q}_k + + \left( + \mat{A}_k + \right)^{-1} + \bf{g}_k \\ + % + \mat{A}_k + &= + {\mat{J}(\vec{q}_k)}^\top + \mat{W}_e \ + {\mat{J}(\vec{q}_k)} + + + \mat{W}_n + + where :math:`\mat{W}_n = \text{diag}(\vec{w_n})(\vec{w_n} \in \mathbb{R}^n_{>0})` is a + diagonal damping matrix. The damping matrix ensures that :math:`\mat{A}_k` is + non-singular and positive definite. The performance of the LM method largely depends + on the choice of :math:`\mat{W}_n`. + + *Chan's Method* + + Chan proposed + + .. math:: + + \mat{W}_n + = + λ E_k \mat{1}_n + + where λ is a constant which reportedly does not have much influence on performance. + Use the kwarg `k` to adjust the weighting term λ. + + *Sugihara's Method* + + Sugihara proposed + + .. math:: + + \mat{W}_n + = + E_k \mat{1}_n + \text{diag}(\hat{\vec{w}}_n) + + where :math:`\hat{\vec{w}}_n \in \mathbb{R}^n`, :math:`\hat{w}_{n_i} = l^2 \sim 0.01 l^2`, + and :math:`l` is the length of a typical link within the manipulator. We provide the + variable `k` as a kwarg to adjust the value of :math:`w_n`. + + *Wampler's Method* + + Wampler proposed :math:`\vec{w_n}` to be a constant. This is set through the `k` kwarg. + + Examples + -------- + The following example gets the ``ets`` of a ``panda`` robot object, makes a goal + pose ``Tep``, and then solves for the joint coordinates which result in the pose + ``Tep`` using the `ikine_LM` method. + + .. runblock:: pycon + >>> import roboticstoolbox as rtb + >>> panda = rtb.models.Panda().ets() + >>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854]) + >>> panda.ikine_LM(Tep) + + Notes + ----- + The value for the ``k`` kwarg will depend on the ``method`` chosen and the arm you are + using. Use the following as a rough guide ``chan, k = 1.0 - 0.01``, + ``wampler, k = 0.01 - 0.0001``, and ``sugihara, k = 0.1 - 0.0001`` + + When using the this method, the initial joint coordinates :math:`q_0`, should correspond + to a non-singular manipulator pose, since it uses the manipulator Jacobian. + + This class supports null-space motion to assist with maximising manipulability and + avoiding joint limits. These are enabled by setting kq and km to non-zero values. + + References + ---------- + - J. Haviland, and P. Corke. "Manipulator Differential Kinematics Part I: + Kinematics, Velocity, and Applications." arXiv preprint arXiv:2207.01796 (2022). + - J. Haviland, and P. Corke. "Manipulator Differential Kinematics Part II: + Acceleration and Advanced Applications." arXiv preprint arXiv:2207.01794 (2022). + + See Also + -------- + ik_NR + A fast numerical inverse kinematics solver using Newton-Raphson optimisation + ik_GN + A fast numerical inverse kinematics solver using Gauss-Newton optimisation + + + .. versionchanged:: 1.0.4 + Merged the Levemberg-Marquadt IK solvers into the ik_LM method + + """ # noqa + + return IK_LM_c( + self._fknm, Tep, q0, ilimit, slimit, tol, joint_limits, mask, k, method + )
+ +
[docs] def ik_NR( + self, + Tep: Union[NDArray, SE3], + q0: Union[NDArray, None] = None, + ilimit: int = 30, + slimit: int = 100, + tol: float = 1e-6, + mask: Union[NDArray, None] = None, + joint_limits: bool = True, + pinv: int = True, + pinv_damping: float = 0.0, + ) -> Tuple[NDArray, int, int, int, float]: + r""" + Fast numerical inverse kinematics using Newton-Raphson optimization + + ``sol = ets.ik_NR(Tep)`` are the joint coordinates (n) corresponding + to the robot end-effector pose ``Tep`` which is an ``SE3`` or ``ndarray`` object. + This method can be used for robots with any number of degrees of freedom. This + is a fast solver implemented in C++. + + See the :ref:`Inverse Kinematics Docs Page <IK>` for more details and for a + **tutorial** on numerical IK, see `here <https://bit.ly/3ak5GDi>`_. + + Note + ---- + When using this method with redundant robots (>6 DoF), ``pinv`` must be set to ``True`` + + Parameters + ---------- + Tep + The desired end-effector pose or pose trajectory + q0 + initial joint configuration (default to random valid joint + configuration contrained by the joint limits of the robot) + ilimit + maximum number of iterations per search + slimit + maximum number of search attempts + tol + final error tolerance + mask + a mask vector which weights the end-effector error priority. + Corresponds to translation in X, Y and Z and rotation about X, Y and Z + respectively + joint_limits + constrain the solution to being within the joint limits of + the robot (reject solution with invalid joint configurations and perfrom + another search up to the slimit) + pinv + Use the psuedo-inverse instad of the normal matrix inverse + pinv_damping + Damping factor for the psuedo-inverse + + Returns + ------- + sol + tuple (q, success, iterations, searches, residual) + + The return value ``sol`` is a tuple with elements: + + ============ ========== =============================================== + Element Type Description + ============ ========== =============================================== + ``q`` ndarray(n) joint coordinates in units of radians or metres + ``success`` int whether a solution was found + ``iterations`` int total number of iterations + ``searches`` int total number of searches + ``residual`` float final value of cost function + ============ ========== =============================================== + + If ``success == 0`` the ``q`` values will be valid numbers, but the + solution will be in error. The amount of error is indicated by + the ``residual``. + + Synopsis + -------- + Each iteration uses the Newton-Raphson optimisation method + + .. math:: + + \vec{q}_{k+1} = \vec{q}_k + {^0\mat{J}(\vec{q}_k)}^{-1} \vec{e}_k + + Examples + -------- + The following example gets the ``ets`` of a ``panda`` robot object, makes a goal + pose ``Tep``, and then solves for the joint coordinates which result in the pose + ``Tep`` using the `ikine_GN` method. + + .. runblock:: pycon + >>> import roboticstoolbox as rtb + >>> panda = rtb.models.Panda().ets() + >>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854]) + >>> panda.ik_NR(Tep) + + Notes + ----- + When using the this method, the initial joint coordinates :math:`q_0`, should correspond + to a non-singular manipulator pose, since it uses the manipulator Jacobian. + + References + ---------- + - J. Haviland, and P. Corke. "Manipulator Differential Kinematics Part I: + Kinematics, Velocity, and Applications." arXiv preprint arXiv:2207.01796 (2022). + - J. Haviland, and P. Corke. "Manipulator Differential Kinematics Part II: + Acceleration and Advanced Applications." arXiv preprint arXiv:2207.01794 (2022). + + See Also + -------- + ik_LM + A fast numerical inverse kinematics solver using Levenberg-Marquadt optimisation + ik_GN + A fast numerical inverse kinematics solver using Gauss-Newton optimisation + + """ # noqa + + return IK_NR_c( + self._fknm, + Tep, + q0, + ilimit, + slimit, + tol, + joint_limits, + mask, + pinv, + pinv_damping, + )
+ +
[docs] def ik_GN( + self, + Tep: Union[NDArray, SE3], + q0: Union[NDArray, None] = None, + ilimit: int = 30, + slimit: int = 100, + tol: float = 1e-6, + mask: Union[NDArray, None] = None, + joint_limits: bool = True, + pinv: int = True, + pinv_damping: float = 0.0, + ) -> Tuple[NDArray, int, int, int, float]: + r""" + Fast numerical inverse kinematics by Gauss-Newton optimization + + ``sol = ets.ik_GN(Tep)`` are the joint coordinates (n) corresponding + to the robot end-effector pose ``Tep`` which is an ``SE3`` or ``ndarray`` object. + This method can be used for robots with any number of degrees of freedom. This + is a fast solver implemented in C++. + + See the :ref:`Inverse Kinematics Docs Page <IK>` for more details and for a + **tutorial** on numerical IK, see `here <https://bit.ly/3ak5GDi>`_. + + Note + ---- + When using this method with redundant robots (>6 DoF), ``pinv`` must be set to ``True`` + + Parameters + ---------- + Tep + The desired end-effector pose or pose trajectory + q0 + initial joint configuration (default to random valid joint + configuration contrained by the joint limits of the robot) + ilimit + maximum number of iterations per search + slimit + maximum number of search attempts + tol + final error tolerance + mask + a mask vector which weights the end-effector error priority. + Corresponds to translation in X, Y and Z and rotation about X, Y and Z + respectively + joint_limits + constrain the solution to being within the joint limits of + the robot (reject solution with invalid joint configurations and perfrom + another search up to the slimit) + pinv + Use the psuedo-inverse instad of the normal matrix inverse + pinv_damping + Damping factor for the psuedo-inverse + + Returns + ------- + sol + tuple (q, success, iterations, searches, residual) + + The return value ``sol`` is a tuple with elements: + + ============ ========== =============================================== + Element Type Description + ============ ========== =============================================== + ``q`` ndarray(n) joint coordinates in units of radians or metres + ``success`` int whether a solution was found + ``iterations`` int total number of iterations + ``searches`` int total number of searches + ``residual`` float final value of cost function + ============ ========== =============================================== + + If ``success == 0`` the ``q`` values will be valid numbers, but the + solution will be in error. The amount of error is indicated by + the ``residual``. + + Synopsis + -------- + Each iteration uses the Gauss-Newton optimisation method + + .. math:: + + \vec{q}_{k+1} &= \vec{q}_k + + \left( + {\mat{J}(\vec{q}_k)}^\top + \mat{W}_e \ + {\mat{J}(\vec{q}_k)} + \right)^{-1} + \bf{g}_k \\ + \bf{g}_k &= + {\mat{J}(\vec{q}_k)}^\top + \mat{W}_e + \vec{e}_k + + where :math:`\mat{J} = {^0\mat{J}}` is the base-frame manipulator Jacobian. If + :math:`\mat{J}(\vec{q}_k)` is non-singular, and :math:`\mat{W}_e = \mat{1}_n`, then + the above provides the pseudoinverse solution. However, if :math:`\mat{J}(\vec{q}_k)` + is singular, the above can not be computed and the GN solution is infeasible. + + Examples + -------- + The following example gets the ``ets`` of a ``panda`` robot object, makes a goal + pose ``Tep``, and then solves for the joint coordinates which result in the pose + ``Tep`` using the `ikine_GN` method. + + .. runblock:: pycon + >>> import roboticstoolbox as rtb + >>> panda = rtb.models.Panda().ets() + >>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854]) + >>> panda.ik_GN(Tep) + + Notes + ----- + When using the this method, the initial joint coordinates :math:`q_0`, should correspond + to a non-singular manipulator pose, since it uses the manipulator Jacobian. + + References + ---------- + - J. Haviland, and P. Corke. "Manipulator Differential Kinematics Part I: + Kinematics, Velocity, and Applications." arXiv preprint arXiv:2207.01796 (2022). + - J. Haviland, and P. Corke. "Manipulator Differential Kinematics Part II: + Acceleration and Advanced Applications." arXiv preprint arXiv:2207.01794 (2022). + + See Also + -------- + ik_NR + A fast numerical inverse kinematics solver using Newton-Raphson optimisation + ik_GN + A fast numerical inverse kinematics solver using Gauss-Newton optimisation + + """ # noqa + + return IK_GN_c( + self._fknm, + Tep, + q0, + ilimit, + slimit, + tol, + joint_limits, + mask, + pinv, + pinv_damping, + )
+ +
[docs] def ikine_LM( + self, + Tep: Union[NDArray, SE3], + q0: Union[ArrayLike, None] = None, + ilimit: int = 30, + slimit: int = 100, + tol: float = 1e-6, + mask: Union[ArrayLike, None] = None, + joint_limits: bool = True, + seed: Union[int, None] = None, + k: float = 1.0, + method: L["chan", "wampler", "sugihara"] = "chan", + kq: float = 0.0, + km: float = 0.0, + ps: float = 0.0, + pi: Union[NDArray, float] = 0.3, + **kwargs, + ): + r""" + Levemberg-Marquadt Numerical Inverse Kinematics Solver + + A method which provides functionality to perform numerical inverse kinematics (IK) + using the Levemberg-Marquadt method. + + See the :ref:`Inverse Kinematics Docs Page <IK>` for more details and for a + **tutorial** on numerical IK, see `here <https://bit.ly/3ak5GDi>`_. + + Parameters + ---------- + Tep + The desired end-effector pose + q0 + The initial joint coordinate vector + ilimit + How many iterations are allowed within a search before a new search + is started + slimit + How many searches are allowed before being deemed unsuccessful + tol + Maximum allowed residual error E + mask + A 6 vector which assigns weights to Cartesian degrees-of-freedom + error priority + joint_limits + Reject solutions with joint limit violations + seed + A seed for the private RNG used to generate random joint coordinate + vectors + k + Sets the gain value for the damping matrix Wn in the next iteration. See + synopsis + method + One of "chan", "sugihara" or "wampler". Defines which method is used + to calculate the damping matrix Wn in the ``step`` method + kq + The gain for joint limit avoidance. Setting to 0.0 will remove this + completely from the solution + km + The gain for maximisation. Setting to 0.0 will remove this completely + from the solution + ps + The minimum angle/distance (in radians or metres) in which the joint is + allowed to approach to its limit + pi + The influence angle/distance (in radians or metres) in null space motion + becomes active + + Synopsis + -------- + The operation is defined by the choice of the ``method`` kwarg. + + The step is deined as + + .. math:: + + \vec{q}_{k+1} + &= + \vec{q}_k + + \left( + \mat{A}_k + \right)^{-1} + \bf{g}_k \\ + % + \mat{A}_k + &= + {\mat{J}(\vec{q}_k)}^\top + \mat{W}_e \ + {\mat{J}(\vec{q}_k)} + + + \mat{W}_n + + where :math:`\mat{W}_n = \text{diag}(\vec{w_n})(\vec{w_n} \in \mathbb{R}^n_{>0})` is a + diagonal damping matrix. The damping matrix ensures that :math:`\mat{A}_k` is + non-singular and positive definite. The performance of the LM method largely depends + on the choice of :math:`\mat{W}_n`. + + *Chan's Method* + + Chan proposed + + .. math:: + + \mat{W}_n + = + λ E_k \mat{1}_n + + where λ is a constant which reportedly does not have much influence on performance. + Use the kwarg `k` to adjust the weighting term λ. + + *Sugihara's Method* + + Sugihara proposed + + .. math:: + + \mat{W}_n + = + E_k \mat{1}_n + \text{diag}(\hat{\vec{w}}_n) + + where :math:`\hat{\vec{w}}_n \in \mathbb{R}^n`, :math:`\hat{w}_{n_i} = l^2 \sim 0.01 l^2`, + and :math:`l` is the length of a typical link within the manipulator. We provide the + variable `k` as a kwarg to adjust the value of :math:`w_n`. + + *Wampler's Method* + + Wampler proposed :math:`\vec{w_n}` to be a constant. This is set through the `k` kwarg. + + Examples + -------- + The following example gets the ``ets`` of a ``panda`` robot object, makes a goal + pose ``Tep``, and then solves for the joint coordinates which result in the pose + ``Tep`` using the `ikine_LM` method. + + .. runblock:: pycon + >>> import roboticstoolbox as rtb + >>> panda = rtb.models.Panda().ets() + >>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854]) + >>> panda.ikine_LM(Tep) + + Notes + ----- + The value for the ``k`` kwarg will depend on the ``method`` chosen and the arm you are + using. Use the following as a rough guide ``chan, k = 1.0 - 0.01``, + ``wampler, k = 0.01 - 0.0001``, and ``sugihara, k = 0.1 - 0.0001`` + + When using the this method, the initial joint coordinates :math:`q_0`, should correspond + to a non-singular manipulator pose, since it uses the manipulator Jacobian. + + This class supports null-space motion to assist with maximising manipulability and + avoiding joint limits. These are enabled by setting kq and km to non-zero values. + + References + ---------- + - J. Haviland, and P. Corke. "Manipulator Differential Kinematics Part I: + Kinematics, Velocity, and Applications." arXiv preprint arXiv:2207.01796 (2022). + - J. Haviland, and P. Corke. "Manipulator Differential Kinematics Part II: + Acceleration and Advanced Applications." arXiv preprint arXiv:2207.01794 (2022). + + See Also + -------- + :py:class:`~roboticstoolbox.robot.IK.IK_LM` + An IK Solver class which implements the Levemberg Marquadt optimisation technique + ikine_NR + Implements the :py:class:`~roboticstoolbox.robot.IK.IK_NR` class as a method within the :py:class:`ETS` class + ikine_GN + Implements the :py:class:`~roboticstoolbox.robot.IK.IK_GN` class as a method within the :py:class:`ETS` class + ikine_QP + Implements the :py:class:`~roboticstoolbox.robot.IK.IK_QP` class as a method within the :py:class:`ETS` class + + + .. versionchanged:: 1.0.4 + Added the Levemberg-Marquadt IK solver method on the `ETS` class + + """ # noqa + + solver = IK_LM( + ilimit=ilimit, + slimit=slimit, + tol=tol, + joint_limits=joint_limits, + mask=mask, + seed=seed, + k=k, + method=method, + kq=kq, + km=km, + ps=ps, + pi=pi, + **kwargs, + ) + + # if isinstance(Tep, SE3): + # Tep = Tep.A + + return solver.solve(ets=self, Tep=Tep, q0=q0)
+ +
[docs] def ikine_NR( + self, + Tep: Union[NDArray, SE3], + q0: Union[ArrayLike, None] = None, + ilimit: int = 30, + slimit: int = 100, + tol: float = 1e-6, + mask: Union[ArrayLike, None] = None, + joint_limits: bool = True, + seed: Union[int, None] = None, + pinv: bool = False, + kq: float = 0.0, + km: float = 0.0, + ps: float = 0.0, + pi: Union[NDArray, float] = 0.3, + **kwargs, + ): + r""" + Newton-Raphson Numerical Inverse Kinematics Solver + + A method which provides functionality to perform numerical inverse kinematics (IK) + using the Newton-Raphson method. + + See the :ref:`Inverse Kinematics Docs Page <IK>` for more details and for a + **tutorial** on numerical IK, see `here <https://bit.ly/3ak5GDi>`_. + + Note + ---- + When using this method with redundant robots (>6 DoF), ``pinv`` must be set to ``True`` + + Parameters + ---------- + Tep + The desired end-effector pose + q0 + The initial joint coordinate vector + ilimit + How many iterations are allowed within a search before a new search + is started + slimit + How many searches are allowed before being deemed unsuccessful + tol + Maximum allowed residual error E + mask + A 6 vector which assigns weights to Cartesian degrees-of-freedom + error priority + joint_limits + Reject solutions with joint limit violations + seed + A seed for the private RNG used to generate random joint coordinate + vectors + pinv + If True, will use the psuedoinverse in the `step` method instead of + the normal inverse + kq + The gain for joint limit avoidance. Setting to 0.0 will remove this + completely from the solution + km + The gain for maximisation. Setting to 0.0 will remove this completely + from the solution + ps + The minimum angle/distance (in radians or metres) in which the joint is + allowed to approach to its limit + pi + The influence angle/distance (in radians or metres) in null space motion + becomes active + + Synopsis + -------- + Each iteration uses the Newton-Raphson optimisation method + + .. math:: + + \vec{q}_{k+1} = \vec{q}_k + {^0\mat{J}(\vec{q}_k)}^{-1} \vec{e}_k + + Examples + -------- + The following example gets the ``ets`` of a ``panda`` robot object, makes a goal + pose ``Tep``, and then solves for the joint coordinates which result in the pose + ``Tep`` using the `ikine_NR` method. + + .. runblock:: pycon + >>> import roboticstoolbox as rtb + >>> panda = rtb.models.Panda().ets() + >>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854]) + >>> panda.ikine_NR(Tep) + + Notes + ----- + When using the this method, the initial joint coordinates :math:`q_0`, should correspond + to a non-singular manipulator pose, since it uses the manipulator Jacobian. + + This class supports null-space motion to assist with maximising manipulability and + avoiding joint limits. These are enabled by setting kq and km to non-zero values. + + References + ---------- + - J. Haviland, and P. Corke. "Manipulator Differential Kinematics Part I: + Kinematics, Velocity, and Applications." arXiv preprint arXiv:2207.01796 (2022). + - J. Haviland, and P. Corke. "Manipulator Differential Kinematics Part II: + Acceleration and Advanced Applications." arXiv preprint arXiv:2207.01794 (2022). + + See Also + -------- + :py:class:`~roboticstoolbox.robot.IK.IK_NR` + An IK Solver class which implements the Newton-Raphson optimisation technique + ikine_LM + Implements the :py:class:`~roboticstoolbox.robot.IK.IK_LM` class as a method within the :py:class:`ETS` class + ikine_GN + Implements the :py:class:`~roboticstoolbox.robot.IK.IK_GN` class as a method within the :py:class:`ETS` class + ikine_QP + Implements the :py:class:`~roboticstoolbox.robot.IK.IK_QP` class as a method within the :py:class:`ETS` class + + + .. versionchanged:: 1.0.4 + Added the Newton-Raphson IK solver method on the `ETS` class + + """ # noqa + + solver = IK_NR( + ilimit=ilimit, + slimit=slimit, + tol=tol, + joint_limits=joint_limits, + mask=mask, + seed=seed, + pinv=pinv, + kq=kq, + km=km, + ps=ps, + pi=pi, + **kwargs, + ) + + # if isinstance(Tep, SE3): + # Tep = Tep.A + + return solver.solve(ets=self, Tep=Tep, q0=q0)
+ +
[docs] def ikine_GN( + self, + Tep: Union[NDArray, SE3], + q0: Union[ArrayLike, None] = None, + ilimit: int = 30, + slimit: int = 100, + tol: float = 1e-6, + mask: Union[ArrayLike, None] = None, + joint_limits: bool = True, + seed: Union[int, None] = None, + pinv: bool = False, + kq: float = 0.0, + km: float = 0.0, + ps: float = 0.0, + pi: Union[NDArray, float] = 0.3, + **kwargs, + ): + r""" + Gauss-Newton Numerical Inverse Kinematics Solver + + A method which provides functionality to perform numerical inverse kinematics (IK) + using the Gauss-Newton method. + + See the :ref:`Inverse Kinematics Docs Page <IK>` for more details and for a + **tutorial** on numerical IK, see `here <https://bit.ly/3ak5GDi>`_. + + Note + ---- + When using this method with redundant robots (>6 DoF), ``pinv`` must be set to ``True`` + + Parameters + ---------- + Tep + The desired end-effector pose + q0 + The initial joint coordinate vector + ilimit + How many iterations are allowed within a search before a new search + is started + slimit + How many searches are allowed before being deemed unsuccessful + tol + Maximum allowed residual error E + mask + A 6 vector which assigns weights to Cartesian degrees-of-freedom + error priority + joint_limits + Reject solutions with joint limit violations + seed + A seed for the private RNG used to generate random joint coordinate + vectors + pinv + If True, will use the psuedoinverse in the `step` method instead of + the normal inverse + kq + The gain for joint limit avoidance. Setting to 0.0 will remove this + completely from the solution + km + The gain for maximisation. Setting to 0.0 will remove this completely + from the solution + ps + The minimum angle/distance (in radians or metres) in which the joint is + allowed to approach to its limit + pi + The influence angle/distance (in radians or metres) in null space motion + becomes active + + Synopsis + -------- + Each iteration uses the Gauss-Newton optimisation method + + .. math:: + + \vec{q}_{k+1} &= \vec{q}_k + + \left( + {\mat{J}(\vec{q}_k)}^\top + \mat{W}_e \ + {\mat{J}(\vec{q}_k)} + \right)^{-1} + \bf{g}_k \\ + \bf{g}_k &= + {\mat{J}(\vec{q}_k)}^\top + \mat{W}_e + \vec{e}_k + + where :math:`\mat{J} = {^0\mat{J}}` is the base-frame manipulator Jacobian. If + :math:`\mat{J}(\vec{q}_k)` is non-singular, and :math:`\mat{W}_e = \mat{1}_n`, then + the above provides the pseudoinverse solution. However, if :math:`\mat{J}(\vec{q}_k)` + is singular, the above can not be computed and the GN solution is infeasible. + + Examples + -------- + The following example gets the ``ets`` of a ``panda`` robot object, makes a goal + pose ``Tep``, and then solves for the joint coordinates which result in the pose + ``Tep`` using the `ikine_GN` method. + + .. runblock:: pycon + >>> import roboticstoolbox as rtb + >>> panda = rtb.models.Panda().ets() + >>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854]) + >>> panda.ikine_GN(Tep) + + Notes + ----- + When using the this method, the initial joint coordinates :math:`q_0`, should correspond + to a non-singular manipulator pose, since it uses the manipulator Jacobian. + + This class supports null-space motion to assist with maximising manipulability and + avoiding joint limits. These are enabled by setting kq and km to non-zero values. + + References + ---------- + - J. Haviland, and P. Corke. "Manipulator Differential Kinematics Part I: + Kinematics, Velocity, and Applications." arXiv preprint arXiv:2207.01796 (2022). + - J. Haviland, and P. Corke. "Manipulator Differential Kinematics Part II: + Acceleration and Advanced Applications." arXiv preprint arXiv:2207.01794 (2022). + + See Also + -------- + :py:class:`~roboticstoolbox.robot.IK.IK_NR` + An IK Solver class which implements the Newton-Raphson optimisation technique + ikine_LM + Implements the :py:class:`~roboticstoolbox.robot.IK.IK_LM` class as a method within the :py:class:`ETS` class + ikine_NR + Implements the :py:class:`~roboticstoolbox.robot.IK.IK_NR` class as a method within the :py:class:`ETS` class + ikine_QP + Implements the :py:class:`~roboticstoolbox.robot.IK.IK_QP` class as a method within the :py:class:`ETS` class + + + .. versionchanged:: 1.0.4 + Added the Gauss-Newton IK solver method on the `ETS` class + + """ # noqa + + solver = IK_GN( + ilimit=ilimit, + slimit=slimit, + tol=tol, + joint_limits=joint_limits, + mask=mask, + seed=seed, + pinv=pinv, + kq=kq, + km=km, + ps=ps, + pi=pi, + **kwargs, + ) + + # if isinstance(Tep, SE3): + # Tep = Tep.A + + return solver.solve(ets=self, Tep=Tep, q0=q0)
+ +
[docs] def ikine_QP( + self, + Tep: Union[NDArray, SE3], + q0: Union[ArrayLike, None] = None, + ilimit: int = 30, + slimit: int = 100, + tol: float = 1e-6, + mask: Union[ArrayLike, None] = None, + joint_limits: bool = True, + seed: Union[int, None] = None, + kj=1.0, + ks=1.0, + kq: float = 0.0, + km: float = 0.0, + ps: float = 0.0, + pi: Union[NDArray, float] = 0.3, + **kwargs, + ): + r""" + Quadratic Programming Numerical Inverse Kinematics Solver + + A method that provides functionality to perform numerical inverse kinematics + (IK) using a quadratic progamming approach. + + See the :ref:`Inverse Kinematics Docs Page <IK>` for more details and for a + **tutorial** on numerical IK, see `here <https://bit.ly/3ak5GDi>`_. + + Parameters + ---------- + Tep + The desired end-effector pose + q0 + The initial joint coordinate vector + ilimit + How many iterations are allowed within a search before a new search + is started + slimit + How many searches are allowed before being deemed unsuccessful + tol + Maximum allowed residual error E + mask + A 6 vector which assigns weights to Cartesian degrees-of-freedom + error priority + joint_limits + Reject solutions with joint limit violations + seed + A seed for the private RNG used to generate random joint coordinate + vectors + kj + A gain for joint velocity norm minimisation + ks + A gain which adjusts the cost of slack (intentional error) + kq + The gain for joint limit avoidance. Setting to 0.0 will remove this + completely from the solution + km + The gain for maximisation. Setting to 0.0 will remove this completely + from the solution + ps + The minimum angle/distance (in radians or metres) in which the joint is + allowed to approach to its limit + pi + The influence angle/distance (in radians or metres) in null space motion + becomes active + + Raises + ------ + ImportError + If the package ``qpsolvers`` is not installed + + Synopsis + -------- + Each iteration uses the following approach + + .. math:: + + \vec{q}_{k+1} = \vec{q}_{k} + \dot{\vec{q}}. + + where the QP is defined as + + .. math:: + + \min_x \quad f_o(\vec{x}) &= \frac{1}{2} \vec{x}^\top \mathcal{Q} \vec{x}+ \mathcal{C}^\top \vec{x}, \\ + \text{subject to} \quad \mathcal{J} \vec{x} &= \vec{\nu}, \\ + \mathcal{A} \vec{x} &\leq \mathcal{B}, \\ + \vec{x}^- &\leq \vec{x} \leq \vec{x}^+ + + with + + .. math:: + + \vec{x} &= + \begin{pmatrix} + \dvec{q} \\ \vec{\delta} + \end{pmatrix} \in \mathbb{R}^{(n+6)} \\ + \mathcal{Q} &= + \begin{pmatrix} + \lambda_q \mat{1}_{n} & \mathbf{0}_{6 \times 6} \\ \mathbf{0}_{n \times n} & \lambda_\delta \mat{1}_{6} + \end{pmatrix} \in \mathbb{R}^{(n+6) \times (n+6)} \\ + \mathcal{J} &= + \begin{pmatrix} + \mat{J}(\vec{q}) & \mat{1}_{6} + \end{pmatrix} \in \mathbb{R}^{6 \times (n+6)} \\ + \mathcal{C} &= + \begin{pmatrix} + \mat{J}_m \\ \bf{0}_{6 \times 1} + \end{pmatrix} \in \mathbb{R}^{(n + 6)} \\ + \mathcal{A} &= + \begin{pmatrix} + \mat{1}_{n \times n + 6} \\ + \end{pmatrix} \in \mathbb{R}^{(l + n) \times (n + 6)} \\ + \mathcal{B} &= + \eta + \begin{pmatrix} + \frac{\rho_0 - \rho_s} + {\rho_i - \rho_s} \\ + \vdots \\ + \frac{\rho_n - \rho_s} + {\rho_i - \rho_s} + \end{pmatrix} \in \mathbb{R}^{n} \\ + \vec{x}^{-, +} &= + \begin{pmatrix} + \dvec{q}^{-, +} \\ + \vec{\delta}^{-, +} + \end{pmatrix} \in \mathbb{R}^{(n+6)}, + + where :math:`\vec{\delta} \in \mathbb{R}^6` is the slack vector, + :math:`\lambda_\delta \in \mathbb{R}^+` is a gain term which adjusts the + cost of the norm of the slack vector in the optimiser, + :math:`\dvec{q}^{-,+}` are the minimum and maximum joint velocities, and + :math:`\dvec{\delta}^{-,+}` are the minimum and maximum slack velocities. + + Examples + -------- + The following example gets the ``ets`` of a ``panda`` robot object, makes a goal + pose ``Tep``, and then solves for the joint coordinates which result in the pose + ``Tep`` using the `ikine_QP` method. + + .. runblock:: pycon + >>> import roboticstoolbox as rtb + >>> panda = rtb.models.Panda().ets() + >>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854]) + >>> panda.ikine_QP(Tep) + + Notes + ----- + When using the this method, the initial joint coordinates :math:`q_0`, should correspond + to a non-singular manipulator pose, since it uses the manipulator Jacobian. + + This class supports null-space motion to assist with maximising manipulability and + avoiding joint limits. These are enabled by setting kq and km to non-zero values. + + References + ---------- + - J. Haviland, and P. Corke. "Manipulator Differential Kinematics Part I: + Kinematics, Velocity, and Applications." arXiv preprint arXiv:2207.01796 (2022). + - J. Haviland, and P. Corke. "Manipulator Differential Kinematics Part II: + Acceleration and Advanced Applications." arXiv preprint arXiv:2207.01794 (2022). + + See Also + -------- + :py:class:`~roboticstoolbox.robot.IK.IK_NR` + An IK Solver class which implements the Newton-Raphson optimisation technique + ikine_LM + Implements the :py:class:`~roboticstoolbox.robot.IK.IK_LM` class as a method within the :py:class:`ETS` class + ikine_GN + Implements the :py:class:`~roboticstoolbox.robot.IK.IK_GN` class as a method within the :py:class:`ETS` class + ikine_NR + Implements the :py:class:`~roboticstoolbox.robot.IK.IK_NR` class as a method within the :py:class:`ETS` class + + + .. versionchanged:: 1.0.4 + Added the Quadratic Programming IK solver method on the `ETS` class + + """ # noqa: E501 + + solver = IK_QP( + ilimit=ilimit, + slimit=slimit, + tol=tol, + joint_limits=joint_limits, + mask=mask, + seed=seed, + kj=kj, + ks=ks, + kq=kq, + km=km, + ps=ps, + pi=pi, + **kwargs, + ) + + # if isinstance(Tep, SE3): + # Tep = Tep.A + + return solver.solve(ets=self, Tep=Tep, q0=q0)
+ + +
[docs]class ETS2(BaseETS): + """ + This class implements an elementary transform sequence (ETS) for 2D + + :param arg: Function to compute ET value + + An instance can contain an elementary transform (ET) or an elementary + transform sequence (ETS). It has list-like properties by subclassing + UserList, which means we can perform indexing, slicing pop, insert, as well + as using it as an iterator over its values. + + - ``ETS()`` an empty ETS list + - ``ET2.XY(η)`` is a constant elementary transform + - ``ET2.XY(η, 'deg')`` as above but the angle is expressed in degrees + - ``ET2.XY()`` is a joint variable, the value is left free until evaluation + time + - ``ET2.XY(j=J)`` as above but the joint index is explicitly given, this + might correspond to the joint number of a multi-joint robot. + - ``ET2.XY(flip=True)`` as above but the joint moves in the opposite sense + + where ``XY`` is one of ``R``, ``tx``, ``ty``. + + Example: + + .. runblock:: pycon + + >>> from roboticstoolbox import ETS2 as ET2 + >>> e = ET2.R(0.3) # a single ET, rotation about z + >>> len(e) + >>> e = ET2.R(0.3) * ET2.tx(2) # an ETS + >>> len(e) # of length 2 + >>> e[1] # an ET sliced from the ETS + + :references: + - Kinematic Derivatives using the Elementary Transform Sequence, + J. Haviland and P. Corke + + :seealso: :func:`r`, :func:`tx`, :func:`ty` + """ + + def __init__( + self, + arg: Union[ + List[Union["ETS2", ET2]], List[ET2], List["ETS2"], ET2, "ETS2", None + ] = None, + ): + super().__init__() + if isinstance(arg, list): + for item in arg: + if isinstance(item, ET2): + self.data.append(deepcopy(item)) + elif isinstance(item, ETS2): + for ets_item in item: + self.data.append(deepcopy(ets_item)) + else: + raise TypeError("bad arg") + elif isinstance(arg, ET2): + self.data.append(deepcopy(arg)) + elif isinstance(arg, ETS2): + for ets_item in arg: + self.data.append(deepcopy(ets_item)) + elif arg is None: + self.data = [] + else: + raise TypeError("Invalid arg") + + self._update_internals() + self._ndims = 2 + self._auto_jindex = False + + # Check if jindices are set + joints = self.joints() + + # Number of joints with a jindex + jindices = 0 + + # Number of joints with a sequential jindex (j[2] -> jindex = 2) + seq_jindex = 0 + + # Count them up + for j, joint in enumerate(joints): + if joint.jindex is not None: + jindices += 1 + if joint.jindex == j: + seq_jindex += 1 + + if ( + jindices == self.n - 1 + and seq_jindex == self.n - 1 + and joints[-1].jindex is None + ): + # ets has sequential jindicies, except for the last. + joints[-1].jindex = self.n - 1 + self._auto_jindex = True + elif jindices > 0 and not jindices == self.n: + raise ValueError( + "You can not have some jindices set for the ET's in arg. It must be all" + " or none" + ) # pragma: nocover + elif jindices == 0 and self.n > 0: + # Set them ourself + for j, joint in enumerate(joints): + joint.jindex = j + self._auto_jindex = True + +
[docs] def __mul__(self, other: Union[ET2, "ETS2"]) -> "ETS2": + if isinstance(other, ET2): + return ETS2([*self.data, other]) + else: + return ETS2([*self.data, *other.data]) # pragma: nocover
+ + def __rmul__(self, other: Union[ET2, "ETS2"]) -> "ETS2": + return ETS2([other, self.data]) # pragma: nocover + + def __imul__(self, rest: "ETS2"): + return self + rest # pragma: nocover + + def __add__(self, rest) -> "ETS2": + return self.__mul__(rest) # pragma: nocover + +
[docs] def compile(self) -> "ETS2": + """ + Compile an ETS2 + + :return: optimised ETS2 + + Perform constant folding for faster evaluation. Consecutive constant + ETs are compounded, leading to a constant ET which is denoted by + ``SE3`` when displayed. + + :seealso: :func:`isconstant` + """ + const = None + ets = ETS2() + + for et in self: + if et.isjoint: + # a joint + if const is not None: + # flush the constant + if not np.array_equal(const, np.eye(3)): + ets *= ET2.SE2(const) + const = None + ets *= et # emit the joint ET + else: + # not a joint + if const is None: + const = et.A() + else: + const = const @ et.A() + + if const is not None: + # flush the constant, tool transform + if not np.array_equal(const, np.eye(3)): + ets *= ET2.SE2(const) + return ets
+ +
[docs] def insert( + self, + arg: Union[ET2, "ETS2"], + i: int = -1, + ) -> None: + """ + Insert value + + :param i: insert an ET or ETS into the ETS, default is at the end + :param arg: the elementary transform or sequence to insert + + Inserts an ET or ETS into the ET sequence. The inserted value is at position + ``i``. + + Example: + + .. runblock:: pycon + + >>> from roboticstoolbox import ET2 + >>> e = ET2.R() * ET2.tx(1) * ET2.R() * ET2.tx(1) + >>> f = ET2.R() + >>> e.insert(f, 2) + >>> e + """ + + if isinstance(arg, ET2): + if i == -1: + self.data.append(arg) + else: + self.data.insert(i, arg) + elif isinstance(arg, ETS2): + if i == -1: + for et in arg: + self.data.append(et) + else: + for j, et in enumerate(arg): + self.data.insert(i + j, et) + self._update_internals()
+ +
[docs] def fkine( + self, + q: ArrayLike, + base: Union[NDArray, SE2, None] = None, + tool: Union[NDArray, SE2, None] = None, + include_base: bool = True, + ) -> SE2: + """ + Forward kinematics + :param q: Joint coordinates + :type q: ArrayLike + :param base: base transform, optional + :param tool: tool transform, optional + + :return: The transformation matrix representing the pose of the + end-effector + + - ``T = ets.fkine(q)`` evaluates forward kinematics for the robot at + joint configuration ``q``. + **Trajectory operation**: + If ``q`` has multiple rows (mxn), it is considered a trajectory and the + result is an ``SE2`` instance with ``m`` values. + .. note:: + - The robot's base tool transform, if set, is incorporated + into the result. + - A tool transform, if provided, is incorporated into the result. + - Works from the end-effector link to the base + :references: + - Kinematic Derivatives using the Elementary Transform + Sequence, J. Haviland and P. Corke + """ + + ret = SE2.Empty() + fk = self.eval(q, base, tool, include_base) + + if fk.dtype == "O": + # symbolic + fk = np.array(simplify(fk)) + + if fk.ndim == 3: + for T in fk: + ret.append(SE2(T, check=False)) # type: ignore + else: + ret = SE2(fk, check=False) + + return ret
+ + def eval( + self, + q: ArrayLike, + base: Union[NDArray, SE2, None] = None, + tool: Union[NDArray, SE2, None] = None, + include_base: bool = True, + ) -> NDArray: + """ + Forward kinematics + :param q: Joint coordinates + :type q: ArrayLike + :param base: base transform, optional + :param tool: tool transform, optional + + :return: The transformation matrix representing the pose of the + end-effector + + - ``T = ets.fkine(q)`` evaluates forward kinematics for the robot at + joint configuration ``q``. + **Trajectory operation**: + If ``q`` has multiple rows (mxn), it is considered a trajectory and the + result is an ``SE2`` instance with ``m`` values. + .. note:: + - The robot's base tool transform, if set, is incorporated + into the result. + - A tool transform, if provided, is incorporated into the result. + - Works from the end-effector link to the base + :references: + - Kinematic Derivatives using the Elementary Transform + Sequence, J. Haviland and P. Corke + """ + + q = getmatrix(q, (None, None)) + l, _ = q.shape # type: ignore + end = self[-1] + + if base is None: + bases = None + elif isinstance(base, SE2): + bases = np.array(base.A) + elif np.array_equal(base, np.eye(3)): # pragma: nocover + bases = None + else: # pragma: nocover + bases = base + + if tool is None: + tools = None + elif isinstance(tool, SE2): + tools = np.array(tool.A) + elif np.array_equal(tool, np.eye(3)): # pragma: nocover + tools = None + else: # pragma: nocover + tools = tool + + if l > 1: + T = np.zeros((l, 3, 3), dtype=object) + else: + T = np.zeros((3, 3), dtype=object) + + for k, qk in enumerate(q): # type: ignore + link = end # start with last link + + jindex = 0 if link.jindex is None and link.isjoint else link.jindex + Tk = link.A(qk[jindex]) + + if tools is not None: + Tk = Tk @ tools + + # add remaining links, back toward the base + for i in range(self.m - 2, -1, -1): + link = self.data[i] + + jindex = 0 if link.jindex is None and link.isjoint else link.jindex + A = link.A(qk[jindex]) + + if A is not None: + Tk = A @ Tk + + # add base transform if it is set + if include_base is True and bases is not None: + Tk = bases @ Tk + + # append + if l > 1: + T[k, :, :] = Tk + # ret.append(SE2(Tk, check=False)) # type: ignore + else: + T = Tk + # ret = SE2(Tk, check=False) + + return T + +
[docs] def jacob0( + self, + q: ArrayLike, + ) -> NDArray: + # very inefficient implementation, just put a 1 in last row + # if its a rotation joint + q = getvector(q) + + j = 0 + J = np.zeros((3, self.n)) + etjoints = self.joint_idx() + + if not np.all(np.array([self[i].jindex for i in etjoints])): + # not all joints have a jindex it is required, set them + for j in range(self.n): + i = etjoints[j] + self[i].jindex = j + + for j in range(self.n): + i = etjoints[j] + + if self[i].jindex is not None: + jindex = self[i].jindex + else: + jindex = 0 # pragma: nocover + + # jindex = 0 if self[i].jindex is None else self[i].jindex + + axis = self[i].axis + if axis == "R": + dTdq = np.array([[0, -1, 0], [1, 0, 0], [0, 0, 0]]) @ self[i].A( + q[jindex] # type: ignore + ) + elif axis == "tx": + dTdq = np.array([[0, 0, 1], [0, 0, 0], [0, 0, 0]]) + elif axis == "ty": + dTdq = np.array([[0, 0, 0], [0, 0, 1], [0, 0, 0]]) + else: # pragma: nocover + raise TypeError("Invalid axes") + + E0 = ETS2(self[:i]) + if len(E0) > 0: + dTdq = E0.fkine(q).A @ dTdq + + Ef = ETS2(self[i + 1 :]) + if len(Ef) > 0: + dTdq = dTdq @ Ef.fkine(q).A + + T = self.fkine(q).A + dRdt = dTdq[:2, :2] @ T[:2, :2].T + + J[:2, j] = dTdq[:2, 2] + J[2, j] = dRdt[1, 0] + + return J
+ +
[docs] def jacobe( + self, + q: ArrayLike, + ): + r""" + Jacobian in base frame + + :param q: joint coordinates + :type q: ArrayLike + :return: Jacobian matrix + + ``jacobe(q)`` is the manipulator Jacobian matrix which maps joint + velocity to end-effector spatial velocity. + + End-effector spatial velocity :math:`\nu = (v_x, v_y, v_z, \omega_x, \omega_y, \omega_z)^T` + is related to joint velocity by :math:`{}^{e}\nu = {}^{e}\mathbf{J}_0(q) \dot{q}`. + + :seealso: :func:`jacob`, :func:`hessian0` + """ # noqa + + T = self.fkine(q, include_base=False).A + return tr2jac2(T.T) @ self.jacob0(q)
+
+ +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/robot/IK.html b/_modules/roboticstoolbox/robot/IK.html new file mode 100644 index 00000000..e6cfa1b9 --- /dev/null +++ b/_modules/roboticstoolbox/robot/IK.html @@ -0,0 +1,1641 @@ + + + + + + roboticstoolbox.robot.IK — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +

Source code for roboticstoolbox.robot.IK

+#!/usr/bin/env python
+
+"""
+@author Jesse Haviland
+"""
+
+import numpy as np
+from abc import ABC, abstractmethod
+from typing import Tuple, Union
+import roboticstoolbox as rtb
+from dataclasses import dataclass
+from spatialmath import SE3
+from roboticstoolbox.tools.types import ArrayLike
+
+try:
+    import qpsolvers as qp
+
+    _qp = True
+except ImportError:  # pragma nocover
+    _qp = False
+
+
+
[docs]@dataclass +class IKSolution: + """ + A dataclass for representing an IK solution + + Attributes + ---------- + q + The joint coordinates of the solution (ndarray). Note that these + will not be valid if failed to find a solution + success + True if a valid solution was found + iterations + How many iterations were performed + searches + How many searches were performed + residual + The final error value from the cost function + reason + The reason the IK problem failed if applicable + + + .. versionchanged:: 1.0.3 + Added IKSolution dataclass to replace the IKsolution named tuple + + """ + + q: np.ndarray + success: bool + iterations: int = 0 + searches: int = 0 + residual: float = 0.0 + reason: str = "" + + def __iter__(self): + return iter( + ( + self.q, + self.success, + self.iterations, + self.searches, + self.residual, + self.reason, + ) + ) + + def __str__(self): + if self.q is not None: + q_str = np.array2string( + self.q, + separator=", ", + formatter={ + "float": lambda x: "{:.4g}".format(0 if abs(x) < 1e-6 else x) + }, + ) # np.round(self.q, 4) + else: + q_str = None + + if self.iterations == 0 and self.searches == 0: + # Check for analytic + if self.success: + return f"IKSolution: q={q_str}, success=True" + else: + return f"IKSolution: q={q_str}, success=False, reason={self.reason}" + else: + # Otherwise it is a numeric solution + if self.success: + return ( + f"IKSolution: q={q_str}, success=True," + f" iterations={self.iterations}, searches={self.searches}," + f" residual={self.residual:.3g}" + ) + else: + return ( + f"IKSolution: q={q_str}, success=False, reason={self.reason}," + f" iterations={self.iterations}, searches={self.searches}," + f" residual={np.round(self.residual, 4):.3g}" + )
+ + +
[docs]class IKSolver(ABC): + """ + An abstract super class for numerical inverse kinematics (IK) + + This class provides basic functionality to perform numerical IK. Superclasses + can inherit this class and implement the `solve` method and redefine any other + methods necessary. + + Parameters + ---------- + name + The name of the IK algorithm + ilimit + How many iterations are allowed within a search before a new search + is started + slimit + How many searches are allowed before being deemed unsuccessful + tol + Maximum allowed residual error E + mask + A 6 vector which assigns weights to Cartesian degrees-of-freedom + error priority + joint_limits + Reject solutions with joint limit violations + seed + A seed for the private RNG used to generate random joint coordinate + vectors + + See Also + -------- + IK_NR + Implements this class using the Newton-Raphson method + IK_GN + Implements this class using the Gauss-Newton method + IK_LM + Implements this class using the Levemberg-Marquadt method + IK_QP + Implements this class using a quadratic programming approach + + + .. versionchanged:: 1.0.3 + Added the abstract super class IKSolver + + """ + +
[docs] def __init__( + self, + name: str = "IK Solver", + ilimit: int = 30, + slimit: int = 100, + tol: float = 1e-6, + mask: Union[ArrayLike, None] = None, + joint_limits: bool = True, + seed: Union[int, None] = None, + ): + # Solver parameters + self.name = name + self.slimit = slimit + self.ilimit = ilimit + self.tol = tol + + # Random number generator + self._private_random = np.random.default_rng(seed=seed) + + if mask is None: + mask = np.ones(6) + + self.We = np.diag(mask) # type: ignore + self.joint_limits = joint_limits
+ +
[docs] def solve( + self, + ets: "rtb.ETS", + Tep: Union[SE3, np.ndarray], + q0: Union[ArrayLike, None] = None, + ) -> IKSolution: + """ + Solves the IK problem + + This method will attempt to solve the IK problem and obtain joint coordinates + which result the the end-effector pose `Tep`. + + Parameters + ---------- + ets + The ETS representing the manipulators kinematics + Tep + The desired end-effector pose + q0 + The initial joint coordinate vector + + Returns + ------- + q + The joint coordinates of the solution (ndarray). Note that these + will not be valid if failed to find a solution + success + True if a valid solution was found + iterations + How many iterations were performed + searches + How many searches were performed + residual + The final error value from the cost function + jl_valid + True if q is inbounds of the robots joint limits + reason + The reason the IK problem failed if applicable + + """ + # Get the largest jindex in the ETS. If this is greater than ETS.n + # then we need to pad the q vector with zeros + max_jindex: int = 0 + + for j in ets.joints(): + if j.jindex > max_jindex: # type: ignore + max_jindex = j.jindex # type: ignore + + q0_method = np.zeros((self.slimit, max_jindex + 1)) + + if q0 is None: + q0_method[:, ets.jindices] = self._random_q(ets, self.slimit) + + elif not isinstance(q0, np.ndarray): + q0 = np.array(q0) + + if q0 is not None and q0.ndim == 1: + q0_method[:, ets.jindices] = self._random_q(ets, self.slimit) + + q0_method[0, ets.jindices] = q0 + + if q0 is not None and q0.ndim == 2: + q0_method[:, ets.jindices] = self._random_q(ets, self.slimit) + + q0_method[: q0.shape[0], ets.jindices] = q0 + + q0 = q0_method + + traj = False + + methTep: np.ndarray + + if isinstance(Tep, SE3): + if len(Tep) > 1: + traj = True + methTep = np.empty((len(Tep), 4, 4)) + + for i, T in enumerate(Tep): + methTep[i] = T.A + else: + methTep = Tep.A + elif Tep.ndim == 3: + traj = True + methTep = Tep + elif Tep.shape != (4, 4): + raise ValueError("Tep must be a 4x4 SE3 matrix") + else: + methTep = Tep + + if traj: + q = np.empty((methTep.shape[0], ets.n)) + success = True + interations = 0 + searches = 0 + residual = np.inf + reason = "" + + for i, T in enumerate(methTep): + sol = self._solve(ets, T, q0) + q[i] = sol.q + if not sol.success: + success = False + reason = sol.reason + interations += sol.iterations + searches += sol.searches + + if sol.residual < residual: + residual = sol.residual + + return IKSolution( + q=q, + success=success, + iterations=interations, + searches=searches, + residual=residual, + reason=reason, + ) + + else: + sol = self._solve(ets, methTep, q0) + + return sol
+ + def _solve(self, ets: "rtb.ETS", Tep: np.ndarray, q0: np.ndarray) -> IKSolution: + # Iteration count + i = 0 + total_i = 0 + + # Error flags + found_with_limits = False + linalg_error = 0 + + # Initialise variables + E = 0.0 + q = q0[0] + + for search in range(self.slimit): + q = q0[search].copy() + i = 0 + + while i < self.ilimit: + i += 1 + + # Attempt a step + try: + E, q[ets.jindices] = self.step(ets, Tep, q) + + except np.linalg.LinAlgError: + # Abandon search and try again + linalg_error += 1 + break + + # Check if we have arrived + if E < self.tol: + # Wrap q to be within +- 180 deg + # If your robot has larger than 180 deg range on a joint + # this line should be modified in incorporate the extra range + q = (q + np.pi) % (2 * np.pi) - np.pi + + # Check if we have violated joint limits + jl_valid = self._check_jl(ets, q) + + if not jl_valid and self.joint_limits: + # Abandon search and try again + found_with_limits = True + break + else: + return IKSolution( + q=q[ets.jindices], + success=True, + iterations=total_i + i, + searches=search + 1, + residual=E, + reason="Success", + ) + total_i += i + + # If we make it here, then we have failed + reason = "iteration and search limit reached" + + if linalg_error: + reason += f", {linalg_error} numpy.LinAlgError encountered" + + if found_with_limits: + reason += ", solution found but violates joint limits" + + return IKSolution( + q=q, + success=False, + iterations=total_i, + searches=self.slimit, + residual=E, + reason=reason, + ) + +
[docs] def error(self, Te: np.ndarray, Tep: np.ndarray) -> Tuple[np.ndarray, float]: + r""" + Calculates the error between Te and Tep + + Calculates the engle axis error between current end-effector pose Te and + the desired end-effector pose Tep. Also calulates the quadratic error E + which is weighted by the diagonal matrix We. + + .. math:: + + E = \frac{1}{2} \vec{e}^{\top} \mat{W}_e \vec{e} + + where :math:`\vec{e} \in \mathbb{R}^6` is the angle-axis error. + + Parameters + ---------- + Te + The current end-effector pose + Tep + The desired end-effector pose + + Returns + ------- + e + angle-axis error (6 vector) + E + The quadratic error weighted by We + + """ + e = rtb.angle_axis(Te, Tep) + E = 0.5 * e @ self.We @ e + + return e, E
+ +
[docs] @abstractmethod + def step( + self, ets: "rtb.ETS", Tep: np.ndarray, q: np.ndarray + ) -> Tuple[float, np.ndarray]: + """ + Abstract step method + + Superclasses will implement this method to perform a step of the + implemented IK algorithm + + Parameters + ---------- + ets + The ETS representing the manipulators kinematics + Tep + The desired end-effector pose + q + The current joint coordinate vector + + Raises + ------ + numpy.LinAlgError + If a step is impossible due to a linear algebra error + + Returns + ------- + E + The new error value + q + The new joint coordinate vector + + """ + pass # pragma: nocover
+ +
[docs] def _random_q(self, ets: "rtb.ETS", i: int = 1) -> np.ndarray: + """ + Generate a random valid joint configuration using a private RNG + + Generates a random q vector within the joint limits defined by + `ets.qlim`. + + Parameters + ---------- + ets + The ETS representing the manipulators kinematics + i + number of configurations to generate + + Returns + ------- + q + An `i x n` ndarray of random valid joint configurations, where n + is the number of joints in the `ets` + + """ + + if i == 1: + q = np.zeros((1, ets.n)) + + for i in range(ets.n): + q[0, i] = self._private_random.uniform(ets.qlim[0, i], ets.qlim[1, i]) + + else: + q = np.zeros((i, ets.n)) + + for j in range(i): + for i in range(ets.n): + q[j, i] = self._private_random.uniform( + ets.qlim[0, i], ets.qlim[1, i] + ) + + return q
+ +
[docs] def _check_jl(self, ets: "rtb.ETS", q: np.ndarray) -> bool: + """ + Checks if the joints are within their respective limits + + Parameters + ---------- + ets + the ETS + q + the current joint coordinate vector + + Returns + ------- + True if joints within feasible limits otherwise False + + """ + + # Loop through the joints in the ETS + for i in range(ets.n): + # Get the corresponding joint limits + ql0 = ets.qlim[0, i] + ql1 = ets.qlim[1, i] + + # Check if q exceeds the limits + if q[i] < ql0 or q[i] > ql1: + return False + + # If we make it here, all the joints are fine + return True
+ + +def _null_Σ(ets: "rtb.ETS", q: np.ndarray, ps: float, pi: Union[np.ndarray, float]): + """ + Formulates a relationship between joint limits and the joint velocity. + When this is projected into the null-space of the differential kinematics + to attempt to avoid exceeding joint limits + + :param q: The joint coordinates of the robot + :param ps: The minimum angle/distance (in radians or metres) in which the joint is + allowed to approach to its limit + :param pi: The influence angle/distance (in radians or metres) in which the velocity + damper becomes active + + :return: Σ + """ + + if isinstance(pi, float): + pi = pi * np.ones(ets.n) + + # Add cost to going in the direction of joint limits, if they are within + # the influence distance + Σ = np.zeros((ets.n, 1)) + + for i in range(ets.n): + qi = q[i] + ql0 = ets.qlim[0, i] + ql1 = ets.qlim[1, i] + + if qi - ql0 <= pi[i]: + Σ[i, 0] = -np.power(((qi - ql0) - pi[i]), 2) / np.power((ps - pi[i]), 2) + if ql1 - qi <= pi[i]: + Σ[i, 0] = np.power(((ql1 - qi) - pi[i]), 2) / np.power((ps - pi[i]), 2) + + return -Σ + + +def _calc_qnull( + ets: "rtb.ETS", + q: np.ndarray, + J: np.ndarray, + λΣ: float, + λm: float, + ps: float, + pi: Union[np.ndarray, float], +): + """ + Calculates the desired null-space motion according to the gains λΣ and λm. + This is a helper method that is used within the `step` method of an IK solver + + :return: qnull - the desired null-space motion + """ + + qnull_grad = np.zeros(ets.n) + qnull = np.zeros(ets.n) + + # Add the joint limit avoidance if the gain is above 0 + if λΣ > 0: + Σ = _null_Σ(ets, q, ps, pi) + qnull_grad += (1.0 / λΣ * Σ).flatten() + + # Add the manipulability maximisation if the gain is above 0 + if λm > 0: + Jm = ets.jacobm(q) + qnull_grad += (1.0 / λm * Jm).flatten() + + # Calculate the null-space motion + if λΣ > 0 or λΣ > 0: + null_space = np.eye(ets.n) - np.linalg.pinv(J) @ J + qnull = null_space @ qnull_grad + + return qnull.flatten() + + +
[docs]class IK_NR(IKSolver): + """ + Newton-Raphson Numerical Inverse Kinematics Solver + + A class which provides functionality to perform numerical inverse kinematics (IK) + using the Newton-Raphson method. See `step` method for mathematical description. + + Note + ---- + When using this class with redundant robots (>6 DoF), `pinv` must be set to `True` + + Parameters + ---------- + name + The name of the IK algorithm + ilimit + How many iterations are allowed within a search before a new search + is started + slimit + How many searches are allowed before being deemed unsuccessful + tol + Maximum allowed residual error E + mask + A 6 vector which assigns weights to Cartesian degrees-of-freedom + error priority + joint_limits + Reject solutions with joint limit violations + seed + A seed for the private RNG used to generate random joint coordinate + vectors + pinv + If True, will use the psuedoinverse in the `step` method instead of + the normal inverse + kq + The gain for joint limit avoidance. Setting to 0.0 will remove this + completely from the solution + km + The gain for maximisation. Setting to 0.0 will remove this completely + from the solution + ps + The minimum angle/distance (in radians or metres) in which the joint is + allowed to approach to its limit + pi + The influence angle/distance (in radians or metres) in null space motion + becomes active + + Examples + -------- + The following example gets the ``ets`` of a ``panda`` robot object, instantiates + the IK_NR solver class using default parameters, makes a goal pose ``Tep``, + and then solves for the joint coordinates which result in the pose ``Tep`` + using the ``solve`` method. + + .. runblock:: pycon + >>> import roboticstoolbox as rtb + >>> panda = rtb.models.Panda().ets() + >>> solver = rtb.IK_NR(pinv=True) + >>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854]) + >>> solver.solve(panda, Tep) + + Notes + ----- + When using the NR method, the initial joint coordinates :math:`q_0`, should correspond + to a non-singular manipulator pose, since it uses the manipulator Jacobian. When the + the problem is solvable, it converges very quickly. However, this method frequently + fails to converge on the goal. + + This class supports null-space motion to assist with maximising manipulability and + avoiding joint limits. These are enabled by setting kq and km to non-zero values. + + References + ---------- + - J. Haviland, and P. Corke. "Manipulator Differential Kinematics Part I: + Kinematics, Velocity, and Applications." arXiv preprint arXiv:2207.01796 (2022). + - J. Haviland, and P. Corke. "Manipulator Differential Kinematics Part II: + Acceleration and Advanced Applications." arXiv preprint arXiv:2207.01794 (2022). + + See Also + -------- + IKSolver + An abstract super class for numerical IK solvers + IK_GN + Implements the IKSolver class using the Gauss-Newton method + IK_LM + Implements the IKSolver class using the Levemberg-Marquadt method + IK_QP + Implements the IKSolver class using a quadratic programming approach + + + .. versionchanged:: 1.0.3 + Added the Newton-Raphson IK solver class + + """ # noqa + + def __init__( + self, + name: str = "IK Solver", + ilimit: int = 30, + slimit: int = 100, + tol: float = 1e-6, + mask: Union[ArrayLike, None] = None, + joint_limits: bool = True, + seed: Union[int, None] = None, + pinv: bool = False, + kq: float = 0.0, + km: float = 0.0, + ps: float = 0.0, + pi: Union[np.ndarray, float] = 0.3, + **kwargs, + ): + super().__init__( + name=name, + ilimit=ilimit, + slimit=slimit, + tol=tol, + mask=mask, + joint_limits=joint_limits, + seed=seed, + **kwargs, + ) + + self.pinv = pinv + self.kq = kq + self.km = km + self.ps = ps + self.pi = pi + + self.name = f"NR (pinv={pinv})" + + if self.kq > 0.0: + self.name += " Σ" + + if self.km > 0.0: + self.name += " Jm" + +
[docs] def step( + self, ets: "rtb.ETS", Tep: np.ndarray, q: np.ndarray + ) -> Tuple[float, np.ndarray]: + r""" + Performs a single iteration of the Newton-Raphson optimisation method + + .. math:: + + \vec{q}_{k+1} = \vec{q}_k + {^0\mat{J}(\vec{q}_k)}^{-1} \vec{e}_k + + Parameters + ---------- + ets + The ETS representing the manipulators kinematics + Tep + The desired end-effector pose + q + The current joint coordinate vector + + Raises + ------ + numpy.LinAlgError + If a step is impossible due to a linear algebra error + + Returns + ------- + E + The new error value + q + The new joint coordinate vector + + """ + + Te = ets.eval(q) + e, E = self.error(Te, Tep) + + J = ets.jacob0(q) + + # Null-space motion + qnull = _calc_qnull( + ets=ets, q=q, J=J, λΣ=self.kq, λm=self.km, ps=self.ps, pi=self.pi + ) + + if self.pinv: + q[ets.jindices] += np.linalg.pinv(J) @ e + qnull + else: + q[ets.jindices] += np.linalg.inv(J) @ e + qnull + + return E, q[ets.jindices]
+ + +
[docs]class IK_LM(IKSolver): + """ + Levemberg-Marquadt Numerical Inverse Kinematics Solver + + A class which provides functionality to perform numerical inverse kinematics (IK) + using the Levemberg-Marquadt method. See ``step`` method for mathematical description. + + Parameters + ---------- + name + The name of the IK algorithm + ilimit + How many iterations are allowed within a search before a new search + is started + slimit + How many searches are allowed before being deemed unsuccessful + tol + Maximum allowed residual error E + mask + A 6 vector which assigns weights to Cartesian degrees-of-freedom + error priority + joint_limits + Reject solutions with joint limit violations + seed + A seed for the private RNG used to generate random joint coordinate + vectors + k + Sets the gain value for the damping matrix Wn in the ``step`` method. See + notes + method + One of "chan", "sugihara" or "wampler". Defines which method is used + to calculate the damping matrix Wn in the ``step`` method + kq + The gain for joint limit avoidance. Setting to 0.0 will remove this + completely from the solution + km + The gain for maximisation. Setting to 0.0 will remove this completely + from the solution + ps + The minimum angle/distance (in radians or metres) in which the joint is + allowed to approach to its limit + pi + The influence angle/distance (in radians or metres) in null space motion + becomes active + + Examples + -------- + The following example gets the ``ets`` of a ``panda`` robot object, instantiates + the IK_LM solver class using default parameters, makes a goal pose ``Tep``, + and then solves for the joint coordinates which result in the pose ``Tep`` + using the `solve` method. + + .. runblock:: pycon + >>> import roboticstoolbox as rtb + >>> panda = rtb.models.Panda().ets() + >>> solver = rtb.IK_LM() + >>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854]) + >>> solver.solve(panda, Tep) + + Notes + ----- + The value for the ``k`` kwarg will depend on the ``method`` chosen and the arm you are + using. Use the following as a rough guide ``chan, k = 1.0 - 0.01``, + ``wampler, k = 0.01 - 0.0001``, and ``sugihara, k = 0.1 - 0.0001`` + + When using the this method, the initial joint coordinates :math:`q_0`, should correspond + to a non-singular manipulator pose, since it uses the manipulator Jacobian. + + This class supports null-space motion to assist with maximising manipulability and + avoiding joint limits. These are enabled by setting kq and km to non-zero values. + + References + ---------- + - J. Haviland, and P. Corke. "Manipulator Differential Kinematics Part I: + Kinematics, Velocity, and Applications." arXiv preprint arXiv:2207.01796 (2022). + - J. Haviland, and P. Corke. "Manipulator Differential Kinematics Part II: + Acceleration and Advanced Applications." arXiv preprint arXiv:2207.01794 (2022). + + See Also + -------- + IKSolver + An abstract super class for numerical IK solvers + IK_NR + Implements the IKSolver class using the Newton-Raphson method + IK_GN + Implements the IKSolver class using the Gauss-Newton method + IK_QP + Implements the IKSolver class using a quadratic programming approach + + + .. versionchanged:: 1.0.3 + Added the Levemberg-Marquadt IK solver class + + """ # noqa + + def __init__( + self, + name: str = "IK Solver", + ilimit: int = 30, + slimit: int = 100, + tol: float = 1e-6, + mask: Union[ArrayLike, None] = None, + joint_limits: bool = True, + seed: Union[int, None] = None, + k: float = 1.0, + method="chan", + kq: float = 0.0, + km: float = 0.0, + ps: float = 0.0, + pi: Union[np.ndarray, float] = 0.3, + **kwargs, + ): + super().__init__( + name=name, + ilimit=ilimit, + slimit=slimit, + tol=tol, + mask=mask, + joint_limits=joint_limits, + seed=seed, + **kwargs, + ) + + if method.lower().startswith("sugi"): + self.method = 1 + method_name = "Sugihara" + elif method.lower().startswith("wamp"): + self.method = 2 + method_name = "Wampler" + else: + self.method = 0 + method_name = "Chan" + + self.k = k + self.kq = kq + self.km = km + self.ps = ps + self.pi = pi + + self.name = f"LM ({method_name} λ={k})" + + if self.kq > 0.0: + self.name += " Σ" + + if self.km > 0.0: + self.name += " Jm" + +
[docs] def step(self, ets: "rtb.ETS", Tep: np.ndarray, q: np.ndarray): + r""" + Performs a single iteration of the Levenberg-Marquadt optimisation + + The operation is defined by the choice of `method` when instantiating the class. + + The next step is deined as + + .. math:: + \vec{q}_{k+1} + &= + \vec{q}_k + + \left( + \mat{A}_k + \right)^{-1} + \bf{g}_k \\ + % + \mat{A}_k + &= + {\mat{J}(\vec{q}_k)}^\top + \mat{W}_e \ + {\mat{J}(\vec{q}_k)} + + + \mat{W}_n + + where :math:`\mat{W}_n = \text{diag}(\vec{w_n})(\vec{w_n} \in \mathbb{R}^n_{>0})` is a + diagonal damping matrix. The damping matrix ensures that :math:`\mat{A}_k` is + non-singular and positive definite. The performance of the LM method largely depends + on the choice of :math:`\mat{W}_n`. + + **Chan's Method** + + Chan proposed + + .. math:: + \mat{W}_n + = + λ E_k \mat{1}_n + + where λ is a constant which reportedly does not have much influence on performance. + Use the kwarg `k` to adjust the weighting term λ. + + **Sugihara's Method** + + Sugihara proposed + + .. math:: + \mat{W}_n + = + E_k \mat{1}_n + \text{diag}(\hat{\vec{w}}_n) + + where :math:`\hat{\vec{w}}_n \in \mathbb{R}^n`, :math:`\hat{w}_{n_i} = l^2 \sim 0.01 l^2`, + and :math:`l` is the length of a typical link within the manipulator. We provide the + variable `k` as a kwarg to adjust the value of :math:`w_n`. + + **Wampler's Method** + + Wampler proposed :math:`\vec{w_n}` to be a constant. This is set through the `k` kwarg. + + Parameters + ---------- + ets + The ETS representing the manipulators kinematics + Tep + The desired end-effector pose + q + The current joint coordinate vector + + Raises + ------ + numpy.LinAlgError + If a step is impossible due to a linear algebra error + + Returns + ------- + E + The new error value + q + The new joint coordinate vector + + """ # noqa + + Te = ets.eval(q) + e, E = self.error(Te, Tep) + + if self.method == 1: + # Sugihara's method + Wn = E * np.eye(ets.n) + self.k * np.eye(ets.n) + elif self.method == 2: + # Wampler's method + Wn = self.k * np.eye(ets.n) + else: + # Chan's method + Wn = self.k * E * np.eye(ets.n) + + J = ets.jacob0(q) + g = J.T @ self.We @ e + + # Null-space motion + qnull = _calc_qnull( + ets=ets, q=q, J=J, λΣ=self.kq, λm=self.km, ps=self.ps, pi=self.pi + ) + + q[ets.jindices] += np.linalg.inv(J.T @ self.We @ J + Wn) @ g + qnull + + return E, q[ets.jindices]
+ + +
[docs]class IK_GN(IKSolver): + """ + Gauss-Newton Numerical Inverse Kinematics Solver + + A class which provides functionality to perform numerical inverse kinematics (IK) + using the Gauss-Newton method. See `step` method for mathematical description. + + Note + ---- + When using this class with redundant robots (>6 DoF), ``pinv`` must be set to ``True`` + + Parameters + ---------- + name + The name of the IK algorithm + ilimit + How many iterations are allowed within a search before a new search + is started + slimit + How many searches are allowed before being deemed unsuccessful + tol + Maximum allowed residual error E + mask + A 6 vector which assigns weights to Cartesian degrees-of-freedom + error priority + joint_limits + Reject solutions with joint limit violations + seed + A seed for the private RNG used to generate random joint coordinate + vectors + pinv + If True, will use the psuedoinverse in the `step` method instead of + the normal inverse + kq + The gain for joint limit avoidance. Setting to 0.0 will remove this + completely from the solution + km + The gain for maximisation. Setting to 0.0 will remove this completely + from the solution + ps + The minimum angle/distance (in radians or metres) in which the joint is + allowed to approach to its limit + pi + The influence angle/distance (in radians or metres) in null space motion + becomes active + + Examples + -------- + The following example gets the ``ets`` of a ``panda`` robot object, instantiates + the `IK_GN` solver class using default parameters, makes a goal pose ``Tep``, + and then solves for the joint coordinates which result in the pose ``Tep`` + using the `solve` method. + + .. runblock:: pycon + >>> import roboticstoolbox as rtb + >>> panda = rtb.models.Panda().ets() + >>> solver = rtb.IK_GN(pinv=True) + >>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854]) + >>> solver.solve(panda, Tep) + + Notes + ----- + When using the this method, the initial joint coordinates :math:`q_0`, should correspond + to a non-singular manipulator pose, since it uses the manipulator Jacobian. When the + the problem is solvable, it converges very quickly. + + This class supports null-space motion to assist with maximising manipulability and + avoiding joint limits. These are enabled by setting kq and km to non-zero values. + + References + ---------- + - J. Haviland, and P. Corke. "Manipulator Differential Kinematics Part I: + Kinematics, Velocity, and Applications." arXiv preprint arXiv:2207.01796 (2022). + - J. Haviland, and P. Corke. "Manipulator Differential Kinematics Part II: + Acceleration and Advanced Applications." arXiv preprint arXiv:2207.01794 (2022). + + See Also + -------- + IKSolver + An abstract super class for numerical IK solvers + IK_NR + Implements IKSolver using the Newton-Raphson method + IK_LM + Implements IKSolver using the Levemberg-Marquadt method + IK_QP + Implements IKSolver using a quadratic programming approach + + + .. versionchanged:: 1.0.3 + Added the Gauss-Newton IK solver class + + """ # noqa + + def __init__( + self, + name: str = "IK Solver", + ilimit: int = 30, + slimit: int = 100, + tol: float = 1e-6, + mask: Union[ArrayLike, None] = None, + joint_limits: bool = True, + seed: Union[int, None] = None, + pinv: bool = False, + kq: float = 0.0, + km: float = 0.0, + ps: float = 0.0, + pi: Union[np.ndarray, float] = 0.3, + **kwargs, + ): + super().__init__( + name=name, + ilimit=ilimit, + slimit=slimit, + tol=tol, + mask=mask, + joint_limits=joint_limits, + seed=seed, + **kwargs, + ) + + self.pinv = pinv + self.kq = kq + self.km = km + self.ps = ps + self.pi = pi + + self.name = f"GN (pinv={pinv})" + + if self.kq > 0.0: + self.name += " Σ" + + if self.km > 0.0: + self.name += " Jm" + +
[docs] def step( + self, ets: "rtb.ETS", Tep: np.ndarray, q: np.ndarray + ) -> Tuple[float, np.ndarray]: + r""" + Performs a single iteration of the Gauss-Newton optimisation method + + The next step is defined as + + .. math:: + + \vec{q}_{k+1} &= \vec{q}_k + + \left( + {\mat{J}(\vec{q}_k)}^\top + \mat{W}_e \ + {\mat{J}(\vec{q}_k)} + \right)^{-1} + \bf{g}_k \\ + \bf{g}_k &= + {\mat{J}(\vec{q}_k)}^\top + \mat{W}_e + \vec{e}_k + + where :math:`\mat{J} = {^0\mat{J}}` is the base-frame manipulator Jacobian. If + :math:`\mat{J}(\vec{q}_k)` is non-singular, and :math:`\mat{W}_e = \mat{1}_n`, then + the above provides the pseudoinverse solution. However, if :math:`\mat{J}(\vec{q}_k)` + is singular, the above can not be computed and the GN solution is infeasible. + + Parameters + ---------- + ets + The ETS representing the manipulators kinematics + Tep + The desired end-effector pose + q + The current joint coordinate vector + + Raises + ------ + numpy.LinAlgError + If a step is impossible due to a linear algebra error + + Returns + ------- + E + The new error value + q + The new joint coordinate vector + + """ # noqa + + Te = ets.eval(q) + e, E = self.error(Te, Tep) + + J = ets.jacob0(q) + + # Null-space motion + qnull = _calc_qnull( + ets=ets, q=q, J=J, λΣ=self.kq, λm=self.km, ps=self.ps, pi=self.pi + ) + + if self.pinv: + q[ets.jindices] += np.linalg.pinv(J) @ e + qnull + else: + q[ets.jindices] += np.linalg.inv(J) @ e + qnull + + return E, q[ets.jindices]
+ + +
[docs]class IK_QP(IKSolver): + """ + Quadratic Progamming Numerical Inverse Kinematics Solver + + A class which provides functionality to perform numerical inverse kinematics (IK) + using a quadratic progamming approach. See `step` method for mathematical + description. + + Parameters + ---------- + name + The name of the IK algorithm + ilimit + How many iterations are allowed within a search before a new search + is started + slimit + How many searches are allowed before being deemed unsuccessful + tol + Maximum allowed residual error E + mask + A 6 vector which assigns weights to Cartesian degrees-of-freedom + error priority + joint_limits + Reject solutions with joint limit violations + seed + A seed for the private RNG used to generate random joint coordinate + vectors + kj + A gain for joint velocity norm minimisation + ks + A gain which adjusts the cost of slack (intentional error) + kq + The gain for joint limit avoidance. Setting to 0.0 will remove this + completely from the solution + km + The gain for maximisation. Setting to 0.0 will remove this completely + from the solution + ps + The minimum angle/distance (in radians or metres) in which the joint is + allowed to approach to its limit + pi + The influence angle/distance (in radians or metres) in null space motion + becomes active + + Raises + ------ + ImportError + If the package ``qpsolvers`` is not installed + + Examples + -------- + The following example gets the ``ets`` of a ``panda`` robot object, instantiates + the `IK_QP` solver class using default parameters, makes a goal pose ``Tep``, + and then solves for the joint coordinates which result in the pose ``Tep`` + using the `solve` method. + + .. runblock:: pycon + >>> import roboticstoolbox as rtb + >>> panda = rtb.models.Panda().ets() + >>> solver = rtb.IK_QP() + >>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854]) + >>> solver.solve(panda, Tep) + + Notes + ----- + When using the this method, the initial joint coordinates :math:`q_0`, should correspond + to a non-singular manipulator pose, since it uses the manipulator Jacobian. When the + the problem is solvable, it converges very quickly. + + References + ---------- + - J. Haviland, and P. Corke. "Manipulator Differential Kinematics Part II: + Acceleration and Advanced Applications." arXiv preprint arXiv:2207.01794 (2022). + + See Also + -------- + IKSolver + An abstract super class for numerical IK solvers + IK_NR + Implements IKSolver class using the Newton-Raphson method + IK_GN + Implements IKSolver class using the Gauss-Newton method + IK_LM + Implements IKSolver class using the Levemberg-Marquadt method + + + .. versionchanged:: 1.0.3 + Added the Quadratic Programming IK solver class + + """ # noqa + + def __init__( + self, + name: str = "IK Solver", + ilimit: int = 30, + slimit: int = 100, + tol: float = 1e-6, + mask: Union[ArrayLike, None] = None, + joint_limits: bool = True, + seed: Union[int, None] = None, + kj=0.01, + ks=1.0, + kq: float = 0.0, + km: float = 0.0, + ps: float = 0.0, + pi: Union[np.ndarray, float] = 0.3, + **kwargs, + ): + if not _qp: # pragma: nocover + raise ImportError( + "the package qpsolvers is required for this class. \nInstall using 'pip" + " install qpsolvers'" + ) + + super().__init__( + name=name, + ilimit=ilimit, + slimit=slimit, + tol=tol, + mask=mask, + joint_limits=joint_limits, + seed=seed, + **kwargs, + ) + + self.kj = kj + self.ks = ks + self.kq = kq + self.km = km + self.ps = ps + self.pi = pi + + self.name = "QP)" + + if self.kq > 0.0: + self.name += " Σ" + + if self.km > 0.0: + self.name += " Jm" + +
[docs] def step( + self, ets: "rtb.ETS", Tep: np.ndarray, q: np.ndarray + ) -> Tuple[float, np.ndarray]: + r""" + Performs a single iteration of the Gauss-Newton optimisation method + + The next step is defined as + + .. math:: + + \vec{q}_{k+1} = \vec{q}_{k} + \dot{\vec{q}}. + + where the QP is defined as + + .. math:: + + \min_x \quad f_o(\vec{x}) &= \frac{1}{2} \vec{x}^\top \mathcal{Q} \vec{x}+ \mathcal{C}^\top \vec{x}, \\ + \text{subject to} \quad \mathcal{J} \vec{x} &= \vec{\nu}, \\ + \mathcal{A} \vec{x} &\leq \mathcal{B}, \\ + \vec{x}^- &\leq \vec{x} \leq \vec{x}^+ + + with + + .. math:: + + \vec{x} &= + \begin{pmatrix} + \dvec{q} \\ \vec{\delta} + \end{pmatrix} \in \mathbb{R}^{(n+6)} \\ + \mathcal{Q} &= + \begin{pmatrix} + \lambda_q \mat{1}_{n} & \mathbf{0}_{6 \times 6} \\ \mathbf{0}_{n \times n} & \lambda_\delta \mat{1}_{6} + \end{pmatrix} \in \mathbb{R}^{(n+6) \times (n+6)} \\ + \mathcal{J} &= + \begin{pmatrix} + \mat{J}(\vec{q}) & \mat{1}_{6} + \end{pmatrix} \in \mathbb{R}^{6 \times (n+6)} \\ + \mathcal{C} &= + \begin{pmatrix} + \mat{J}_m \\ \bf{0}_{6 \times 1} + \end{pmatrix} \in \mathbb{R}^{(n + 6)} \\ + \mathcal{A} &= + \begin{pmatrix} + \mat{1}_{n \times n + 6} \\ + \end{pmatrix} \in \mathbb{R}^{(l + n) \times (n + 6)} \\ + \mathcal{B} &= + \eta + \begin{pmatrix} + \frac{\rho_0 - \rho_s} + {\rho_i - \rho_s} \\ + \vdots \\ + \frac{\rho_n - \rho_s} + {\rho_i - \rho_s} + \end{pmatrix} \in \mathbb{R}^{n} \\ + \vec{x}^{-, +} &= + \begin{pmatrix} + \dvec{q}^{-, +} \\ + \vec{\delta}^{-, +} + \end{pmatrix} \in \mathbb{R}^{(n+6)}, + + where :math:`\vec{\delta} \in \mathbb{R}^6` is the slack vector, + :math:`\lambda_\delta \in \mathbb{R}^+` is a gain term which adjusts the + cost of the norm of the slack vector in the optimiser, + :math:`\dvec{q}^{-,+}` are the minimum and maximum joint velocities, and + :math:`\dvec{\delta}^{-,+}` are the minimum and maximum slack velocities. + + Parameters + ---------- + ets + The ETS representing the manipulators kinematics + Tep + The desired end-effector pose + q + The current joint coordinate vector + + Raises + ------ + numpy.LinAlgError + If a step is impossible due to a linear algebra error + + Returns + ------- + E + The new error value + q + The new joint coordinate vector + + """ # noqa + + Te = ets.eval(q) + e, E = self.error(Te, Tep) + J = ets.jacob0(q) + + if isinstance(self.pi, float): + self.pi = self.pi * np.ones(ets.n) + + # Quadratic component of objective function + Q = np.eye(ets.n + 6) + + # Joint velocity component of Q + Q[: ets.n, : ets.n] *= self.kj + + # Slack component of Q + Q[ets.n :, ets.n :] = self.ks * (1 / np.sum(np.abs(e))) * np.eye(6) + + # The equality contraints + Aeq = np.concatenate((J, np.eye(6)), axis=1) + beq = e.reshape((6,)) + + # The inequality constraints for joint limit avoidance + if self.kq > 0.0: + Ain = np.zeros((ets.n + 6, ets.n + 6)) + bin = np.zeros(ets.n + 6) + + # Form the joint limit velocity damper + Ain_l = np.zeros((ets.n, ets.n)) + Bin_l = np.zeros(ets.n) + + for i in range(ets.n): + ql0 = ets.qlim[0, i] + ql1 = ets.qlim[1, i] + + if ql1 - q[i] <= self.pi[i]: + Bin_l[i] = ((ql1 - q[i]) - self.ps) / (self.pi[i] - self.ps) + Ain_l[i, i] = 1 + + if q[i] - ql0 <= self.pi[i]: + Bin_l[i] = -(((ql0 - q[i]) + self.ps) / (self.pi[i] - self.ps)) + Ain_l[i, i] = -1 + + Ain[: ets.n, : ets.n] = Ain_l + bin[: ets.n] = (1.0 / self.kq) * Bin_l + else: + Ain = None + bin = None + + # Manipulability maximisation + if self.km > 0.0: + Jm = ets.jacobm(q).reshape((ets.n,)) + c = np.concatenate(((1.0 / self.km) * -Jm, np.zeros(6))) + else: + c = np.zeros(ets.n + 6) + + xd = qp.solve_qp(Q, c, Ain, bin, Aeq, beq, lb=None, ub=None, solver="quadprog") + + if xd is None: # pragma: nocover + raise np.linalg.LinAlgError("QP Unsolvable") + + q += xd[: ets.n] + + return E, q
+ + +if __name__ == "__main__": # pragma nocover + sol = IKSolution( + np.array([1, 2, 3]), success=True, iterations=10, searches=100, residual=0.1 + ) + + a, b, c, d, e = sol + + print(a, b, c, d, e) +
+ +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/robot/KinematicCache.html b/_modules/roboticstoolbox/robot/KinematicCache.html new file mode 100644 index 00000000..9ea20a14 --- /dev/null +++ b/_modules/roboticstoolbox/robot/KinematicCache.html @@ -0,0 +1,536 @@ + + + + + + roboticstoolbox.robot.KinematicCache — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + +
+ + +
+ +
+ +
+
+
+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/robot/Link.html b/_modules/roboticstoolbox/robot/Link.html new file mode 100644 index 00000000..3741b5ad --- /dev/null +++ b/_modules/roboticstoolbox/robot/Link.html @@ -0,0 +1,1826 @@ + + + + + + roboticstoolbox.robot.Link — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +

Source code for roboticstoolbox.robot.Link

+from copy import deepcopy
+from abc import ABC
+from typing_extensions import Self
+
+# from multiprocessing.sharedctypes import Value
+import numpy as np
+from functools import wraps
+from spatialmath.base import getvector, isscalar, isvector, ismatrix
+from spatialmath import SE3, SE2
+from ansitable import ANSITable, Column
+from spatialgeometry import Shape, SceneNode, SceneGroup
+from typing import List, Union, Tuple, overload
+
+import roboticstoolbox as rtb
+from roboticstoolbox.robot.ETS import ETS, ETS2
+from roboticstoolbox.robot.ET import ET, ET2
+from warnings import warn
+
+from roboticstoolbox.tools.types import ArrayLike, NDArray
+
+# A generic type variable representing any subclass of BaseETS
+# ETSType = TypeVar("ETSType", bound=BaseETS)
+# ETType = TypeVar("ETType", bound=BaseET)
+
+
+def _listen_dyn(func):
+    """
+    @_listen_dyn
+
+    Decorator for property setters
+
+    Use this decorator for any property setter that updates a parameter that
+    affects the result of inverse dynamics.  This allows the C version of the
+    parameters only having to be updated when they change, rather than on
+    every call.  This decorator signals the change by:
+
+    - invoking the ``.dynchanged()`` method of the robot that owns the link.
+      This assumes that the Link object is owned by a robot, this happens
+      when the Link object is passed to a robot constructor.
+    - setting the ``._hasdynamics`` attribute of the Link
+
+    Example::
+
+        @m.setter
+        @_listen_dyn
+        def m(self, m_new):
+            self._m = m_new
+
+    :seealso: :func:`DHLink._dyn_changed`
+    """
+
+    @wraps(func)
+    def wrapper_listen_dyn(*args):
+        if args[0]._robot is not None:
+            args[0]._robot.dynchanged()
+        args[0]._hasdynamics = True
+        return func(*args)
+
+    return wrapper_listen_dyn
+
+
+
+
+
+
+
+
+
[docs]class Link2(BaseLink): + def __init__(self, ets: ETS2 = ETS2(), jindex: Union[int, None] = None, **kwargs): + # process common options + super().__init__(ets=ets, **kwargs) + + # check we have an ETS + if not isinstance(self._ets, ETS2): # pragma: nocover + raise TypeError("The self._ets argument must be of type ETS2") + + # Set the jindex + if len(self._ets) > 0 and self._ets[-1].isjoint: + if jindex is not None: + self._ets[-1].jindex = jindex # pragma: nocover + +
[docs] def A(self, q: float = 0.0) -> SE2: + """ + Link transform matrix + + ``link.A(q)`` is an SE(2) matrix that describes the rigid-body + transformation from the previous to the current link frame to + the next, which depends on the joint coordinate ``q``. + + Parameters + ---------- + q + Joint coordinate (radians or metres). Not required for links + with no variable + + Returns + ------- + T + link frame transformation matrix + + """ + + if self.isjoint: + if self._Ts is not None: + return SE2(self._Ts @ self._ets[-1].A(q), check=False) + else: + return SE2(self._ets[-1].A(q), check=False) + + elif self._Ts is not None: + return SE2(self._Ts, check=False) + else: + return SE2()
+
+ +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/robot/Prismatic.html b/_modules/roboticstoolbox/robot/Prismatic.html new file mode 100644 index 00000000..e18f611b --- /dev/null +++ b/_modules/roboticstoolbox/robot/Prismatic.html @@ -0,0 +1,292 @@ + + + + + + + + + + roboticstoolbox.robot.Prismatic — Robotics Toolbox Python 0.3.7 + documentation + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + + +
+ + + + + +
+ +
+ + + + + + + + + + + + + + + + + +
+ +
    + +
  • »
  • + +
  • Module code »
  • + +
  • roboticstoolbox.robot.Prismatic
  • + + +
  • + +
  • + +
+ + +
+
+
+
+ +

Source code for roboticstoolbox.robot.Prismatic

+#!/usr/bin/env python
+"""
+@author: Jesse Haviland
+"""
+
+import numpy as np
+from roboticstoolbox.robot.Link import Link
+
+
+
[docs]class Prismatic(Link): + """ + A class for prismatic link types + + A subclass of the Link class for a prismatic joint: holds all information + related to a robot link such as kinematics parameters, rigid-body inertial + parameters, motor and transmission parameters. + + :param theta: kinematic: joint angle + :type theta: float + :param d: kinematic - link offset + :type d: float + :param alpha: kinematic - link twist + :type alpha: float + :param a: kinematic - link length + :type a: float + :param sigma: kinematic - 0 if revolute, 1 if prismatic + :type sigma: int + :param mdh: kinematic - 0 if standard D&H, else 1 + :type mdh: int + :param offset: kinematic - joint variable offset + :type offset: float + + :param qlim: joint variable limits [min max] + :type qlim: float ndarray(1,2) + :param flip: joint moves in opposite direction + :type flip: bool + + :param m: dynamic - link mass + :type m: float + :param r: dynamic - position of COM with respect to link frame + :type r: float ndarray(3) + :param I: dynamic - inertia of link with respect to COM + :type I: float ndarray(3,3) + :param Jm: dynamic - motor inertia + :type Jm: float + :param B: dynamic - motor viscous friction (1x1 or 2x1) + :type B: float + :param Tc: dynamic - motor Coulomb friction (1x2 or 2x1) + :type Tc: float ndarray(2) + :param G: dynamic - gear ratio + :type G: float + + :references: + - Robotics, Vision & Control, P. Corke, Springer 2011, Chap 7. + + """ + + def __init__( + self, + alpha=0.0, + theta=0.0, + a=0.0, + mdh=0.0, + offset=0.0, + qlim=np.zeros(2), + flip=False, + m=0.0, + r=np.zeros(3), + I=np.zeros((3, 3)), + Jm=0.0, + B=0.0, + Tc=np.zeros(2), + G=1.0): + + d = 0.0 + sigma = 1 + + super(Prismatic, self).__init__( + d, alpha, theta, a, sigma, mdh, offset, + qlim, flip, m, r, I, Jm, B, Tc, G)
+
+ +
+ +
+ + +
+
+ +
+ +
+ + + + + + + + + + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/robot/Revolute.html b/_modules/roboticstoolbox/robot/Revolute.html new file mode 100644 index 00000000..0b46055e --- /dev/null +++ b/_modules/roboticstoolbox/robot/Revolute.html @@ -0,0 +1,286 @@ + + + + + + + + + + roboticstoolbox.robot.Revolute — Robotics Toolbox Python 0.3.7 + documentation + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + + +
+ + + + + +
+ +
+ + + + + + + + + + + + + + + + + +
+ +
    + +
  • »
  • + +
  • Module code »
  • + +
  • roboticstoolbox.robot.Revolute
  • + + +
  • + +
  • + +
+ + +
+
+
+
+ +

Source code for roboticstoolbox.robot.Revolute

+#!/usr/bin/env python
+"""
+@author: Jesse Haviland
+"""
+
+import numpy as np
+from roboticstoolbox.robot.Link import Link
+
+
+
[docs]class Revolute(Link): + """ + A class for revolute link types + + :param d: kinematic - link offset + :type d: float + :param alpha: kinematic - link twist + :type alpha: float + :param a: kinematic - link length + :type a: float + :param sigma: kinematic - 0 if revolute, 1 if prismatic + :type sigma: int + :param mdh: kinematic - 0 if standard D&H, else 1 + :type mdh: int + :param offset: kinematic - joint variable offset + :type offset: float + + :param qlim: joint variable limits [min max] + :type qlim: float ndarray(1,2) + :param flip: joint moves in opposite direction + :type flip: bool + + :param m: dynamic - link mass + :type m: float + :param r: dynamic - position of COM with respect to link frame + :type r: float ndarray(3) + :param I: dynamic - inertia of link with respect to COM + :type I: float ndarray(3,3) + :param Jm: dynamic - motor inertia + :type Jm: float + :param B: dynamic - motor viscous friction (1x1 or 2x1) + :type B: float + :param Tc: dynamic - motor Coulomb friction (1x2 or 2x1) + :type Tc: float ndarray(2) + :param G: dynamic - gear ratio + :type G: float + + :references: + - Robotics, Vision & Control, P. Corke, Springer 2011, Chap 7. + + """ + + def __init__( + self, + d=0.0, + alpha=0.0, + a=0.0, + mdh=0.0, + offset=0.0, + qlim=np.zeros(2), + flip=False, + m=0.0, + r=np.zeros(3), + I=np.zeros((3, 3)), + Jm=0.0, + B=0.0, + Tc=np.zeros(2), + G=1.0): + + theta = 0.0 + sigma = 0 + + super(Revolute, self).__init__( + d, alpha, theta, a, sigma, mdh, offset, + qlim, flip, m, r, I, Jm, B, Tc, G)
+
+ +
+ +
+ + +
+
+ +
+ +
+ + + + + + + + + + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/robot/Robot.html b/_modules/roboticstoolbox/robot/Robot.html new file mode 100644 index 00000000..6e7640ec --- /dev/null +++ b/_modules/roboticstoolbox/robot/Robot.html @@ -0,0 +1,2130 @@ + + + + + + roboticstoolbox.robot.Robot — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +

Source code for roboticstoolbox.robot.Robot

+#!/usr/bin/env python
+
+"""
+@author: Jesse Haviland
+@author: Peter Corke
+"""
+
+# import sys
+from os.path import splitext
+from copy import deepcopy
+from warnings import warn
+from pathlib import PurePosixPath, Path
+from typing import (
+    List,
+    TypeVar,
+    Union,
+    Dict,
+    Tuple,
+    overload,
+)
+from typing_extensions import Literal as L
+
+
+import numpy as np
+
+import spatialmath.base as smb
+from spatialmath.base.argcheck import (
+    getvector,
+    getmatrix,
+    verifymatrix,
+)
+
+from spatialgeometry import Shape, CollisionShape, Cylinder
+
+from spatialmath import (
+    SE3,
+    SE2,
+    SpatialAcceleration,
+    SpatialVelocity,
+    SpatialInertia,
+    SpatialForce,
+)
+
+import roboticstoolbox as rtb
+from roboticstoolbox.robot.BaseRobot import BaseRobot
+from roboticstoolbox.robot.RobotKinematics import RobotKinematicsMixin
+from roboticstoolbox.robot.Gripper import Gripper
+from roboticstoolbox.robot.Link import BaseLink, Link, Link2
+from roboticstoolbox.robot.ETS import ETS, ETS2
+from roboticstoolbox.tools import xacro
+from roboticstoolbox.tools import URDF
+from roboticstoolbox.tools.types import ArrayLike, NDArray
+from roboticstoolbox.tools.data import rtb_path_to_datafile
+
+# A generic type variable representing any subclass of BaseLink
+LinkType = TypeVar("LinkType", bound=BaseLink)
+
+
+# ==================================================================================== #
+# ================= Robot Class ====================================================== #
+# ==================================================================================== #
+
+
+
[docs]class Robot(BaseRobot[Link], RobotKinematicsMixin): + _color = True + + def __init__( + self, + arg: Union[List[Link], ETS, "Robot"], + gripper_links: Union[Link, List[Link], None] = None, + name: str = "", + manufacturer: str = "", + comment: str = "", + base: Union[NDArray, SE3, None] = None, + tool: Union[NDArray, SE3, None] = None, + gravity: ArrayLike = [0, 0, -9.81], + keywords: Union[List[str], Tuple[str]] = [], + symbolic: bool = False, + configs: Union[Dict[str, NDArray], None] = None, + check_jindex: bool = True, + urdf_string: Union[str, None] = None, + urdf_filepath: Union[Path, PurePosixPath, None] = None, + ): + # Process links + if isinstance(arg, Robot): + # We're passed a Robot, clone it + # We need to preserve the parent link as we copy + + # Copy each link within the robot + links = [deepcopy(link) for link in arg.links] + gripper_links = [] + + for gripper in arg.grippers: + glinks = [] + for link in gripper.links: + glinks.append(deepcopy(link)) + + gripper_links.append(glinks[0]) + links = links + glinks + + # Sever parent connection, but save the string + # The constructor will piece this together for us + for link in links: + link._children = [] + if link.parent is not None: + link._parent_name = link.parent.name + link._parent = None + + super().__init__(links, gripper_links=gripper_links) + + for i, gripper in enumerate(self.grippers): + gripper.tool = arg.grippers[i].tool.copy() + + self._urdf_string = arg.urdf_string + self._urdf_filepath = arg.urdf_filepath + else: + if isinstance(arg, ETS): + # We're passed an ETS string + links = [] + # chop it up into segments, a link frame after every joint + parent = None + for j, ets_j in enumerate(arg.split()): + elink = Link(ETS(ets_j), parent=parent, name=f"link{j:d}") + if ( + elink.qlim is None + and elink.v is not None + and elink.v.qlim is not None + ): + elink.qlim = elink.v.qlim # pragma nocover + parent = elink + links.append(elink) + + elif smb.islistof(arg, Link): + links = arg + + else: + raise TypeError("arg was invalid, must be List[Link], ETS, or Robot") + + # Initialise Base Robot object + super().__init__( + links=links, + gripper_links=gripper_links, + name=name, + manufacturer=manufacturer, + comment=comment, + base=base, + tool=tool, + gravity=gravity, + keywords=keywords, + symbolic=symbolic, + configs=configs, + check_jindex=check_jindex, + ) + + # --------------------------------------------------------------------- # + # --------- Swift Methods --------------------------------------------- # + # --------------------------------------------------------------------- # + + def _to_dict(self, robot_alpha=1.0, collision_alpha=0.0): + ob = [] + + for link in self.links: + if robot_alpha > 0: + for gi in link.geometry: + gi.set_alpha(robot_alpha) + ob.append(gi.to_dict()) + if collision_alpha > 0: + for gi in link.collision: + gi.set_alpha(collision_alpha) + ob.append(gi.to_dict()) + + # Do the grippers now + for gripper in self.grippers: + for link in gripper.links: + if robot_alpha > 0: + for gi in link.geometry: + gi.set_alpha(robot_alpha) + ob.append(gi.to_dict()) + if collision_alpha > 0: + for gi in link.collision: + gi.set_alpha(collision_alpha) + ob.append(gi.to_dict()) + + # for o in ob: + # print(o) + + return ob + + def _fk_dict(self, robot_alpha=1.0, collision_alpha=0.0): + ob = [] + + # Do the robot + for link in self.links: + if robot_alpha > 0: + for gi in link.geometry: + ob.append(gi.fk_dict()) + if collision_alpha > 0: + for gi in link.collision: + ob.append(gi.fk_dict()) + + # Do the grippers now + for gripper in self.grippers: + for link in gripper.links: + if robot_alpha > 0: + for gi in link.geometry: + ob.append(gi.fk_dict()) + if collision_alpha > 0: + for gi in link.collision: + ob.append(gi.fk_dict()) + + return ob + + # --------------------------------------------------------------------- # + # --------- URDF Methods ---------------------------------------------- # + # --------------------------------------------------------------------- # + +
[docs] @staticmethod + def URDF_read( + file_path, tld=None, xacro_tld=None + ) -> Tuple[List[Link], str, str, Union[Path, PurePosixPath]]: + """ + Read a URDF file as Links + + File should be specified relative to ``RTBDATA/URDF/xacro`` + + Parameters + ---------- + file_path + File path relative to the xacro folder + tld + A custom top-level directory which holds the xacro data, + defaults to None + xacro_tld + A custom top-level within the xacro data, + defaults to None + + Returns + ------- + links + a list of links + name + the name of the robot + urdf + a string representing the URDF + file_path + a path to the original file + + Notes + ----- + If ``tld`` is not supplied, filepath pointing to xacro data should + be directly under ``RTBDATA/URDF/xacro`` OR under ``./xacro`` relative + to the model file calling this method. If ``tld`` is supplied, then + ```file_path``` needs to be relative to ``tld`` + + """ + + # Get the path to the class that defines the robot + if tld is None: + base_path = rtb_path_to_datafile("xacro") + else: + base_path = PurePosixPath(tld) + + # Add on relative path to get to the URDF or xacro file + # base_path = PurePath(classpath).parent.parent / 'URDF' / 'xacro' + file_path = base_path / PurePosixPath(file_path) + _, ext = splitext(file_path) + + if ext == ".xacro": + # it's a xacro file, preprocess it + if xacro_tld is not None: + xacro_tld = base_path / PurePosixPath(xacro_tld) + urdf_string = xacro.main(file_path, xacro_tld) + try: + urdf = URDF.loadstr(urdf_string, file_path, base_path) + except BaseException as e: # pragma nocover + print("error parsing URDF file", file_path) + raise e + else: # pragma nocover + urdf_string = open(file_path).read() + urdf = URDF.loadstr(urdf_string, file_path, base_path) + + if not isinstance(urdf_string, str): # pragma nocover + raise ValueError("Parsing failed, did not get valid URDF string back") + + return urdf.elinks, urdf.name, urdf_string, file_path
+ +
[docs] @classmethod + def URDF(cls, file_path: str, gripper: Union[int, str, None] = None): + """ + Construct a Robot object from URDF file + + Parameters + ---------- + file_path + the path to the URDF + gripper + index or name of the gripper link(s) + + Returns + ------- + If ``gripper`` is specified, links from that link outward are removed + from the rigid-body tree and folded into a ``Gripper`` object. + + """ + + links, name, urdf_string, urdf_filepath = Robot.URDF_read(file_path) + + gripperLink: Union[Link, None] = None + + if gripper is not None: + if isinstance(gripper, int): + gripperLink = links[gripper] + elif isinstance(gripper, str): + for link in links: + if link.name == gripper: + gripperLink = link + break + else: # pragma nocover + raise ValueError(f"no link named {gripper}") + else: # pragma nocover + raise TypeError("bad argument passed as gripper") + + # links, name, urdf_string, urdf_filepath = Robot.URDF_read(file_path) + + return cls( + links, + name=name, + gripper_links=gripperLink, + urdf_string=urdf_string, + urdf_filepath=urdf_filepath, + )
+ + # # --------------------------------------------------------------------- # + # # --------- Utility Methods ------------------------------------------- # + # # --------------------------------------------------------------------- # + + # def showgraph(self, display_graph: bool = True, **kwargs) -> Union[None, str]: + # """ + # Display a link transform graph in browser + + # ``robot.showgraph()`` displays a graph of the robot's link frames + # and the ETS between them. It uses GraphViz dot. + + # The nodes are: + # - Base is shown as a grey square. This is the world frame origin, + # but can be changed using the ``base`` attribute of the robot. + # - Link frames are indicated by circles + # - ETS transforms are indicated by rounded boxes + + # The edges are: + # - an arrow if `jtype` is False or the joint is fixed + # - an arrow with a round head if `jtype` is True and the joint is + # revolute + # - an arrow with a box head if `jtype` is True and the joint is + # prismatic + + # Edge labels or nodes in blue have a fixed transformation to the + # preceding link. + + # Parameters + # ---------- + # display_graph + # Open the graph in a browser if True. Otherwise will return the + # file path + # etsbox + # Put the link ETS in a box, otherwise an edge label + # jtype + # Arrowhead to node indicates revolute or prismatic type + # static + # Show static joints in blue and bold + + # Examples + # -------- + # >>> import roboticstoolbox as rtb + # >>> panda = rtb.models.URDF.Panda() + # >>> panda.showgraph() + + # .. image:: ../figs/panda-graph.svg + # :width: 600 + + # See Also + # -------- + # :func:`dotfile` + + # """ + + # # Lazy import + # import tempfile + # import subprocess + # import webbrowser + + # # create the temporary dotfile + # dotfile = tempfile.TemporaryFile(mode="w") + # self.dotfile(dotfile, **kwargs) + + # # rewind the dot file, create PDF file in the filesystem, run dot + # dotfile.seek(0) + # pdffile = tempfile.NamedTemporaryFile(suffix=".pdf", delete=False) + # subprocess.run("dot -Tpdf", shell=True, stdin=dotfile, stdout=pdffile) + + # # open the PDF file in browser (hopefully portable), then cleanup + # if display_graph: # pragma nocover + # webbrowser.open(f"file://{pdffile.name}") + # else: + # return pdffile.name + + # def dotfile( + # self, + # filename: Union[str, IO[str]], + # etsbox: bool = False, + # ets: L["full", "brief"] = "full", + # jtype: bool = False, + # static: bool = True, + # ): + # """ + # Write a link transform graph as a GraphViz dot file + + # The file can be processed using dot: + # % dot -Tpng -o out.png dotfile.dot + + # The nodes are: + # - Base is shown as a grey square. This is the world frame origin, + # but can be changed using the ``base`` attribute of the robot. + # - Link frames are indicated by circles + # - ETS transforms are indicated by rounded boxes + + # The edges are: + # - an arrow if `jtype` is False or the joint is fixed + # - an arrow with a round head if `jtype` is True and the joint is + # revolute + # - an arrow with a box head if `jtype` is True and the joint is + # prismatic + + # Edge labels or nodes in blue have a fixed transformation to the + # preceding link. + + # Note + # ---- + # If ``filename`` is a file object then the file will *not* + # be closed after the GraphViz model is written. + + # Parameters + # ---------- + # file + # Name of file to write to + # etsbox + # Put the link ETS in a box, otherwise an edge label + # ets + # Display the full ets with "full" or a brief version with "brief" + # jtype + # Arrowhead to node indicates revolute or prismatic type + # static + # Show static joints in blue and bold + + # See Also + # -------- + # :func:`showgraph` + + # """ + + # if isinstance(filename, str): + # file = open(filename, "w") + # else: + # file = filename + + # header = r"""digraph G { + # graph [rankdir=LR]; + # """ + + # def draw_edge(link, etsbox, jtype, static): + # # draw the edge + # if jtype: + # if link.isprismatic: + # edge_options = 'arrowhead="box", arrowtail="inv", dir="both"' + # elif link.isrevolute: + # edge_options = 'arrowhead="dot", arrowtail="inv", dir="both"' + # else: + # edge_options = 'arrowhead="normal"' + # else: + # edge_options = 'arrowhead="normal"' + + # if link.parent is None: + # parent = "BASE" + # else: + # parent = link.parent.name + + # if etsbox: + # # put the ets fragment in a box + # if not link.isjoint and static: + # node_options = ', fontcolor="blue"' + # else: + # node_options = "" + + # try: + # file.write( + # ' {}_ets [shape=box, style=rounded, label="{}"{}];\n'.format( + # link.name, + # link.ets.__str__(q=f"q{link.jindex}"), + # node_options, + # ) + # ) + # except UnicodeEncodeError: # pragma nocover + # file.write( + # ' {}_ets [shape=box, style=rounded, label="{}"{}];\n'.format( + # link.name, + # link.ets.__str__(q=f"q{link.jindex}") + # .encode("ascii", "ignore") + # .decode("ascii"), + # node_options, + # ) + # ) + + # file.write(" {} -> {}_ets;\n".format(parent, link.name)) + # file.write( + # " {}_ets -> {} [{}];\n".format(link.name, link.name, edge_options) + # ) + # else: + # # put the ets fragment as an edge label + # if not link.isjoint and static: + # edge_options += 'fontcolor="blue"' + # if ets == "full": + # estr = link.ets.__str__(q=f"q{link.jindex}") + # elif ets == "brief": + # if link.jindex is None: + # estr = "" + # else: + # estr = f"...q{link.jindex}" + # else: + # return + # try: + # file.write( + # ' {} -> {} [label="{}", {}];\n'.format( + # parent, + # link.name, + # estr, + # edge_options, + # ) + # ) + # except UnicodeEncodeError: # pragma nocover + # file.write( + # ' {} -> {} [label="{}", {}];\n'.format( + # parent, + # link.name, + # estr.encode("ascii", "ignore").decode("ascii"), + # edge_options, + # ) + # ) + + # file.write(header) + + # # add the base link + # file.write(" BASE [shape=square, style=filled, fillcolor=gray]\n") + + # # add the links + # for link in self: + # # draw the link frame node (circle) or ee node (doublecircle) + # if link in self.ee_links: + # # end-effector + # node_options = 'shape="doublecircle", color="blue", fontcolor="blue"' + # else: + # node_options = 'shape="circle"' + + # file.write(" {} [{}];\n".format(link.name, node_options)) + + # draw_edge(link, etsbox, jtype, static) + + # for gripper in self.grippers: + # for link in gripper.links: + # file.write(" {} [shape=cds];\n".format(link.name)) + # draw_edge(link, etsbox, jtype, static) + + # file.write("}\n") + + # if isinstance(filename, str): + # file.close() # noqa + + # --------------------------------------------------------------------- # + # --------- Kinematic Methods ----------------------------------------- # + # --------------------------------------------------------------------- # + + @property + def reach(self) -> float: + r""" + Reach of the robot + + A conservative estimate of the reach of the robot. It is computed as + the sum of the translational ETs that define the link transform. + + Note + ---- + Computed on the first access. If kinematic parameters + subsequently change this will not be reflected. + + Returns + ------- + reach + Maximum reach of the robot + + Notes + ----- + - Probably an overestimate of reach + - Used by numerical inverse kinematics to scale translational + error. + - For a prismatic joint, uses ``qlim`` if it is set + + """ + + # TODO + # This should be a start, end method and compute the reach based on the + # given ets. Then use an lru_cache to speed up return + + if self._reach is None: + d_all = [] + for link in self.ee_links: + d = 0 + while True: + for et in link.ets: + if et.istranslation: + if et.isjoint: + # the length of a prismatic joint depends on the + # joint limits. They might be set in the ET + # or in the Link depending on how the robot + # was constructed + if link.qlim is not None: + d += max(link.qlim) + elif et.qlim is not None: # pragma nocover + d += max(et.qlim) + else: + d += abs(et.eta) + link = link.parent + if link is None or isinstance(link, str): + d_all.append(d) + break + + self._reach = max(d_all) + return self._reach + +
[docs] def fkine_all(self, q: ArrayLike) -> SE3: + """ + Compute the pose of every link frame + + ``T = robot.fkine_all(q)`` is an SE3 instance with ``robot.nlinks + + 1`` values: + + - ``T[0]`` is the base transform + - ``T[i]`` is the pose of link whose ``number`` is ``i`` + + Parameters + ---------- + q + The joint configuration + + Returns + ------- + fkine_all + Pose of all links + + References + ---------- + - J. Haviland, and P. Corke. "Manipulator Differential Kinematics Part I: + Kinematics, Velocity, and Applications." arXiv preprint arXiv:2207.01796 (2022). + + """ # noqa + + q = getvector(q) + + Tbase = SE3(self.base) # add base, also sets the type + + linkframes = Tbase.__class__.Alloc(self.nlinks + 1) + linkframes[0] = Tbase + + def recurse(Tall, Tparent, q, link): + # if joint?? + T = Tparent + while True: + T *= SE3(link.A(q[link.jindex])) + + Tall[link.number] = T + + if link.nchildren == 0: + # no children + return + elif link.nchildren == 1: + # one child + if link in self.ee_links: # pragma nocover + # this link is an end-effector, go no further + return + link = link.children[0] + continue + else: + # multiple children + for child in link.children: + recurse(Tall, T, q, child) + return + + recurse(linkframes, Tbase, q, self.links[0]) + + return linkframes
+ + @overload + def manipulability( + self, + q: ArrayLike = ..., + J: None = None, + end: Union[str, Link, Gripper, None] = None, + start: Union[str, Link, Gripper, None] = None, + method: L["yoshikawa", "asada", "minsingular", "invcondition"] = "yoshikawa", + axes: Union[L["all", "trans", "rot"], List[bool]] = "all", + **kwargs, + ) -> Union[float, NDArray]: # pragma nocover + ... + + @overload + def manipulability( + self, + q: None = None, + J: NDArray = ..., + end: Union[str, Link, Gripper, None] = None, + start: Union[str, Link, Gripper, None] = None, + method: L["yoshikawa", "asada", "minsingular", "invcondition"] = "yoshikawa", + axes: Union[L["all", "trans", "rot"], List[bool]] = "all", + **kwargs, + ) -> Union[float, NDArray]: # pragma nocover + ... + +
[docs] def manipulability( + self, + q=None, + J=None, + end: Union[str, Link, Gripper, None] = None, + start: Union[str, Link, Gripper, None] = None, + method: L["yoshikawa", "asada", "minsingular", "invcondition"] = "yoshikawa", + axes: Union[L["all", "trans", "rot"], List[bool]] = "all", + **kwargs, + ): + """ + Manipulability measure + + ``manipulability(q)`` is the scalar manipulability index + for the robot at the joint configuration ``q``. It indicates + dexterity, that is, how well conditioned the robot is for motion + with respect to the 6 degrees of Cartesian motion. The values is + zero if the robot is at a singularity. + + Parameters + ---------- + q + Joint coordinates, one of J or q required + J + Jacobian in base frame if already computed, one of J or + q required + method + method to use, "yoshikawa" (default), "invcondition", + "minsingular" or "asada" + axes + Task space axes to consider: "all" [default], + "trans", or "rot" + + Returns + ------- + manipulability + + Synopsis + -------- + + Various measures are supported: + + | Measure | Description | + |-------------------|-------------------------------------------------| + | ``"yoshikawa"`` | Volume of the velocity ellipsoid, *distance* | + | | from singularity [Yoshikawa85]_ | + | ``"invcondition"``| Inverse condition number of Jacobian, isotropy | + | | of the velocity ellipsoid [Klein87]_ | + | ``"minsingular"`` | Minimum singular value of the Jacobian, | + | | *distance* from singularity [Klein87]_ | + | ``"asada"`` | Isotropy of the task-space acceleration | + | | ellipsoid which is a function of the Cartesian | + | | inertia matrix which depends on the inertial | + | | parameters [Asada83]_ | + + **Trajectory operation**: + + If ``q`` is a matrix (m,n) then the result (m,) is a vector of + manipulability indices for each joint configuration specified by a row + of ``q``. + + Notes + ----- + - Invokes the ``jacob0`` method of the robot if ``J`` is not passed + - The "all" option includes rotational and translational + dexterity, but this involves adding different units. It can be + more useful to look at the translational and rotational + manipulability separately. + - Examples in the RVC book (1st edition) can be replicated by + using the "all" option + - Asada's measure requires inertial a robot model with inertial + parameters. + + References + ---------- + .. [Yoshikawa85] Manipulability of Robotic Mechanisms. Yoshikawa T., + The International Journal of Robotics Research. + 1985;4(2):3-9. doi:10.1177/027836498500400201 + .. [Asada83] A geometrical representation of manipulator dynamics and + its application to arm design, H. Asada, + Journal of Dynamic Systems, Measurement, and Control, + vol. 105, p. 131, 1983. + .. [Klein87] Dexterity Measures for the Design and Control of + Kinematically Redundant Manipulators. Klein CA, Blaho BE. + The International Journal of Robotics Research. + 1987;6(2):72-83. doi:10.1177/027836498700600206 + - Robotics, Vision & Control, Chap 8, P. Corke, Springer 2011. + + + .. versionchanged:: 1.0.3 + Removed 'both' option for axes, added a custom list option. + + """ + + ets = self.ets(end, start) + + axes_list: List[bool] = [] + + if isinstance(axes, list): + axes_list = axes + elif axes == "all": + axes_list = [True, True, True, True, True, True] + elif axes.startswith("trans"): + axes_list = [True, True, True, False, False, False] + elif axes.startswith("rot"): + axes_list = [False, False, False, True, True, True] + elif axes == "both": + return ( + self.manipulability( + q=q, J=J, end=end, start=start, method=method, axes="trans" + ), + self.manipulability( + q=q, J=J, end=end, start=start, method=method, axes="rot" + ), + ) + else: + raise ValueError("axes must be all, trans, rot or both") + + def yoshikawa(robot, J, q, axes_list): + J = J[axes_list, :] + if J.shape[0] == J.shape[1]: + # simplified case for square matrix + return abs(np.linalg.det(J)) + else: + m2 = np.linalg.det(J @ J.T) + return np.sqrt(abs(m2)) + + def condition(robot, J, q, axes_list): + J = J[axes_list, :] + + # return 1/cond(J) + return 1 / np.linalg.cond(J) + + def minsingular(robot, J, q, axes_list): + J = J[axes_list, :] + s = np.linalg.svd(J, compute_uv=False) + + # return last/smallest singular value of J + return s[-1] + + def asada(robot, J, q, axes_list): + # dof = np.sum(axes_list) + if np.linalg.matrix_rank(J) < 6: + return 0 + Ji = np.linalg.pinv(J) + Mx = Ji.T @ robot.inertia(q) @ Ji + d = np.where(axes_list)[0] + Mx = Mx[d] + Mx = Mx[:, d.tolist()] + e, _ = np.linalg.eig(Mx) + return np.min(e) / np.max(e) + + # choose the handler function + if method.lower().startswith("yoshi"): + mfunc = yoshikawa + elif method.lower().startswith("invc"): + mfunc = condition + elif method.lower().startswith("mins"): + mfunc = minsingular + elif method.lower().startswith("asa"): + mfunc = asada + else: + raise ValueError("Invalid method chosen") + + # Calculate manipulability based on supplied Jacobian + if J is not None: + w = [mfunc(self, J, q, axes_list)] + + # Otherwise use the q vector/matrix + else: + q = np.array(getmatrix(q, (None, self.n))) + w = np.zeros(q.shape[0]) + + for k, qk in enumerate(q): + Jk = ets.jacob0(qk) + w[k] = mfunc(self, Jk, qk, axes_list) + + if len(w) == 1: + return w[0] + else: + return w
+ +
[docs] def jtraj( + self, + T1: Union[NDArray, SE3], + T2: Union[NDArray, SE3], + t: Union[NDArray, int], + **kwargs, + ): + """ + Joint-space trajectory between SE(3) poses + + The initial and final poses are mapped to joint space using inverse + kinematics: + + - if the object has an analytic solution ``ikine_a`` that will be used, + - otherwise the general numerical algorithm ``ikine_lm`` will be used. + + ``traj = obot.jtraj(T1, T2, t)`` is a trajectory object whose + attribute ``traj.q`` is a row-wise joint-space trajectory. + + Parameters + ---------- + T1 + initial end-effector pose + T2 + final end-effector pose + t + time vector or number of steps + kwargs + arguments passed to the IK solver + + Returns + ------- + trajectory + + """ # noqa + + if hasattr(self, "ikine_a"): + ik = self.ikine_a # type: ignore + else: + ik = self.ikine_LM + + q1 = ik(T1, **kwargs) + q2 = ik(T2, **kwargs) + + return rtb.jtraj(q1.q, q2.q, t)
+ + @overload + def jacob0_dot( + self, + q: ArrayLike, + qd: ArrayLike, + J0: None = None, + representation: Union[L["rpy/xyz", "rpy/zyx", "eul", "exp"], None] = None, + ) -> NDArray: # pragma no cover + ... + + @overload + def jacob0_dot( + self, + q: None, + qd: ArrayLike, + J0: NDArray = ..., + representation: Union[L["rpy/xyz", "rpy/zyx", "eul", "exp"], None] = None, + ) -> NDArray: # pragma no cover + ... + +
[docs] def jacob0_dot( + self, + q, + qd: ArrayLike, + J0=None, + representation: Union[L["rpy/xyz", "rpy/zyx", "eul", "exp"], None] = None, + ): + r""" + Derivative of Jacobian + + ``robot.jacob_dot(q, qd)`` computes the rate of change of the + Jacobian elements + + .. math:: + + \dmat{J} = \frac{d \mat{J}}{d \vec{q}} \frac{d \vec{q}}{dt} + + where the first term is the rank-3 Hessian. + + Parameters + ---------- + q + The joint configuration of the robot + qd + The joint velocity of the robot + J0 + Jacobian in {0} frame + representation + angular representation + + Returns + ------- + jdot + The derivative of the manipulator Jacobian + + Synopsis + -------- + + If ``J0`` is already calculated for the joint + coordinates ``q`` it can be passed in to to save computation time. + + It is computed as the mode-3 product of the Hessian tensor and the + velocity vector. + + The derivative of an analytical Jacobian can be obtained by setting + ``representation`` as + + |``representation`` | Rotational representation | + |---------------------|-------------------------------------| + |``'rpy/xyz'`` | RPY angular rates in XYZ order | + |``'rpy/zyx'`` | RPY angular rates in XYZ order | + |``'eul'`` | Euler angular rates in ZYZ order | + |``'exp'`` | exponential coordinate rates | + + + References + ---------- + - Kinematic Derivatives using the Elementary Transform + Sequence, J. Haviland and P. Corke + + See Also + -------- + :func:`jacob0` + :func:`hessian0` + + """ + + qd = np.array(qd) + + if representation is None: + if J0 is None: + J0 = self.jacob0(q) + H = self.hessian0(q, J0=J0) + + else: + # # determine analytic rotation + # T = self.fkine(q).A + # gamma = smb.r2x(smb.t2r(T), representation=representation) + + # # get transformation angular velocity to analytic velocity + # Ai = smb.rotvelxform( + # gamma, representation=representation, inverse=True, full=True + # ) + + # # get analytic rate from joint rates + # omega = J0[3:, :] @ qd + # gamma_dot = Ai[3:, 3:] @ omega + # Ai_dot = smb.rotvelxform_inv_dot(gamma, gamma_dot, full=True) + # Ai_dot = sp.linalg.block_diag(np.zeros((3, 3)), Ai_dot) + + # Jd = Ai_dot @ J0 + Ai @ Jd + + # not actually sure this can be written in closed form + + H = smb.numhess( + lambda q: self.jacob0_analytical(q, representation=representation), q + ) + + # Jd = Ai @ Jd + + # return Jd + + return np.tensordot(H, qd, (0, 0))
+ + @overload + def jacobm( + self, + q: ArrayLike = ..., + J: None = None, + H: None = None, + end: Union[str, Link, Gripper, None] = None, + start: Union[str, Link, Gripper, None] = None, + axes: Union[L["all", "trans", "rot"], List[bool]] = "all", + ) -> NDArray: # pragma no cover + ... + + @overload + def jacobm( + self, + q: None = None, + J: NDArray = ..., + H: NDArray = ..., + end: Union[str, Link, Gripper, None] = None, + start: Union[str, Link, Gripper, None] = None, + axes: Union[L["all", "trans", "rot"], List[bool]] = "all", + ) -> NDArray: # pragma no cover + ... + +
[docs] def jacobm( + self, + q=None, + J=None, + H=None, + end: Union[str, Link, Gripper, None] = None, + start: Union[str, Link, Gripper, None] = None, + axes: Union[L["all", "trans", "rot"], List[bool]] = "all", + ) -> NDArray: + r""" + The manipulability Jacobian + + This measure relates the rate of change of the manipulability to the + joint velocities of the robot. One of J or q is required. Supply J + and H if already calculated to save computation time + + Parameters + ---------- + q + The joint angles/configuration of the robot (Optional, + if not supplied will use the stored q values). + J + The manipulator Jacobian in any frame + H + The manipulator Hessian in any frame + end + the final link or Gripper which the Hessian represents + start + the first link which the Hessian represents + + Returns + ------- + jacobm + The manipulability Jacobian + + Synopsis + -------- + Yoshikawa's manipulability measure + + .. math:: + + m(\vec{q}) = \sqrt{\mat{J}(\vec{q}) \mat{J}(\vec{q})^T} + + This method returns its Jacobian with respect to configuration + + .. math:: + + \frac{\partial m(\vec{q})}{\partial \vec{q}} + + References + ---------- + - J. Haviland, and P. Corke. "Manipulator Differential Kinematics Part I: + Kinematics, Velocity, and Applications." arXiv preprint arXiv:2207.01796 (2022). + - J. Haviland, and P. Corke. "Manipulator Differential Kinematics Part II: + Acceleration and Advanced Applications." arXiv preprint arXiv:2207.01794 (2022). + + """ # noqa + + end, start, _ = self._get_limit_links(end, start) + + # + if not isinstance(axes, list): + if axes.startswith("all"): + axes = [True, True, True, True, True, True] + elif axes.startswith("trans"): + axes = [True, True, True, False, False, False] + elif axes.startswith("rot"): + axes = [False, False, False, True, True, True] + else: + raise ValueError("axes must be all, trans or rot") + + if J is None: + if q is None: + q = np.copy(self.q) + else: + q = getvector(q, self.n) + + J = self.jacob0(q, start=start, end=end) + else: + verifymatrix(J, (6, self.n)) + + n = J.shape[1] + + if H is None: + H = self.hessian0(J0=J, start=start, end=end) + else: + verifymatrix(H, (6, self.n, self.n)) + + manipulability = self.manipulability( + q, J=J, start=start, end=end, axes=axes # type: ignore + ) + + J = J[axes, :] # type: ignore + H = H[:, axes, :] # type: ignore + + b = np.linalg.inv(J @ np.transpose(J)) + Jm = np.zeros((n, 1)) + + for i in range(n): + c = J @ np.transpose(H[i, :, :]) + Jm[i, 0] = manipulability * np.transpose(c.flatten("F")) @ b.flatten("F") + + return Jm
+ + # --------------------------------------------------------------------- # + # --------- Collision Methods ----------------------------------------- # + # --------------------------------------------------------------------- # + +
[docs] def closest_point( + self, q: ArrayLike, shape: Shape, inf_dist: float = 1.0, skip: bool = False + ) -> Tuple[Union[int, None], Union[NDArray, None], Union[NDArray, None],]: + """ + Find the closest point between robot and shape + + ``closest_point(shape, inf_dist)`` returns the minimum euclidean + distance between this robot and shape, provided it is less than + inf_dist. It will also return the points on self and shape in the + world frame which connect the line of length distance between the + shapes. If the distance is negative then the shapes are collided. + + Attributes + ---------- + shape + The shape to compare distance to + inf_dist + The minimum distance within which to consider + the shape + skip + Skip setting all shape transforms based on q, use this + option if using this method in conjuction with Swift to save time + + Returns + ------- + d + distance between the robot and shape + p1 + [x, y, z] point on the robot (in the world frame) + p2 + [x, y, z] point on the shape (in the world frame) + + """ + + if not skip: + self._update_link_tf(q) + self._propogate_scene_tree() + shape._propogate_scene_tree() + + d = 10000 + p1 = None + p2 = None + + for link in self.links: + td, tp1, tp2 = link.closest_point(shape, inf_dist, skip=True) + + if td is not None and td < d: + d = td + p1 = tp1 + p2 = tp2 + + if d == 10000: + d = None + + return d, p1, p2
+ +
[docs] def iscollided(self, q, shape: Shape, skip: bool = False) -> bool: + """ + Check if the robot is in collision with a shape + + ``iscollided(shape)`` checks if this robot and shape have collided + + Attributes + ---------- + shape + The shape to compare distance to + skip + Skip setting all shape transforms based on q, use this + option if using this method in conjuction with Swift to save time + + Returns + ------- + iscollided + True if shapes have collided + + """ + + if not skip: + self._update_link_tf(q) + self._propogate_scene_tree() + shape._propogate_scene_tree() + + for link in self.links: + if link.iscollided(shape, skip=True): + return True + + if isinstance(self, rtb.Robot): + for gripper in self.grippers: + for link in gripper.links: + if link.iscollided(shape, skip=True): + return True + + return False
+ +
[docs] def collided(self, q, shape: Shape, skip: bool = False) -> bool: + """ + Check if the robot is in collision with a shape + + ``collided(shape)`` checks if this robot and shape have collided + + Attributes + ---------- + shape + The shape to compare distance to + skip + Skip setting all shape transforms based on q, use this + option if using this method in conjuction with Swift to save time + + Returns + ------- + collided + True if shapes have collided + + """ + + warn("method collided is deprecated, use iscollided instead", FutureWarning) + return self.iscollided(q, shape, skip=skip)
+ + # --------------------------------------------------------------------- # + # --------- Constraint Methods ---------------------------------------- # + # --------------------------------------------------------------------- # + +
[docs] def joint_velocity_damper( + self, + ps: float = 0.05, + pi: float = 0.1, + n: Union[int, None] = None, + gain: float = 1.0, + ) -> Tuple[NDArray, NDArray]: + """ + Compute the joint velocity damper for QP motion control + + Formulates an inequality contraint which, when optimised for will + make it impossible for the robot to run into joint limits. Requires + the joint limits of the robot to be specified. See examples/mmc.py + for use case + + Attributes + ---------- + ps + The minimum angle (in radians) in which the joint is + allowed to approach to its limit + pi + The influence angle (in radians) in which the velocity + damper becomes active + n + The number of joints to consider. Defaults to all joints + gain + The gain for the velocity damper + + Returns + ------- + Ain + A (6,) vector inequality contraint for an optisator + Bin + b (6,) vector inequality contraint for an optisator + + """ + + if n is None: + n = self.n + + Ain = np.zeros((n, n)) + Bin = np.zeros(n) + + for i in range(n): + if self.q[i] - self.qlim[0, i] <= pi: + Bin[i] = -gain * (((self.qlim[0, i] - self.q[i]) + ps) / (pi - ps)) + Ain[i, i] = -1 + if self.qlim[1, i] - self.q[i] <= pi: + Bin[i] = gain * ((self.qlim[1, i] - self.q[i]) - ps) / (pi - ps) + Ain[i, i] = 1 + + return Ain, Bin
+ + + +
[docs] def vision_collision_damper( + self, + shape: CollisionShape, + camera: Union["Robot", SE3, None] = None, + camera_n: int = 0, + q=None, + di=0.3, + ds=0.05, + xi=1.0, + end=None, + start=None, + collision_list=None, + ): # pragma nocover + """ + Compute a vision collision constrain for QP motion control + + Formulates an inequality contraint which, when optimised for will + make it impossible for the robot to run into a line of sight. + See examples/fetch_vision.py for use case + + Attributes + ---------- + camera + The camera link, either as a robotic link or SE3 + pose + camera_n + Degrees of freedom of the camera link + ds + The minimum distance in which a joint is allowed to + approach the collision object shape + di + The influence distance in which the velocity + damper becomes active + xi + The gain for the velocity damper + end + The end link of the robot to consider + start + The start link of the robot to consider + collision_list + A list of shapes to consider for collision + + Returns + ------- + Ain + A (6,) vector inequality contraint for an optisator + Bin + b (6,) vector inequality contraint for an optisator + """ + + end, start, _ = self._get_limit_links(start=start, end=end) + + links, n, _ = self.get_path(start=start, end=end) + + q = np.array(q) + j = 0 + Ain = None + bin = None + + def rotation_between_vectors(a, b): + a = a / np.linalg.norm(a) + b = b / np.linalg.norm(b) + + angle = np.arccos(np.dot(a, b)) + axis = np.cross(a, b) + + return SE3.AngleAxis(angle, axis) + + if isinstance(camera, rtb.BaseRobot): + wTcp = camera.fkine(camera.q).A[:3, 3] + elif isinstance(camera, SE3): + wTcp = camera.t + else: + raise TypeError("Camera must be a robotic link or SE3 pose") + + wTtp = shape.T[:3, -1] + + # Create line of sight object + los_mid = SE3((wTcp + wTtp) / 2) + los_orientation = rotation_between_vectors( + np.array([0.0, 0.0, 1.0]), wTcp - wTtp # type: ignore + ) + + los = Cylinder( + radius=0.001, + length=np.linalg.norm(wTcp - wTtp), # type: ignore + base=(los_mid * los_orientation), + ) + + def indiv_calculation(link: Link, link_col: CollisionShape, q: NDArray): + d, wTlp, wTvp = link_col.closest_point(los, di) + + if d is not None and wTlp is not None and wTvp is not None: + lpTvp = -wTlp + wTvp + + norm = lpTvp / d + norm_h = np.expand_dims( + np.concatenate((norm, [0.0, 0.0, 0.0])), axis=0 + ) # type: ignore + + tool = SE3( + (np.linalg.inv(self.fkine(q, end=link).A) @ SE3(wTlp).A)[:3, 3] + ) + + Je = self.jacob0(q, end=link, tool=tool.A) + Je[:3, :] = self._T[:3, :3] @ Je[:3, :] + n_dim = Je.shape[1] + + if isinstance(camera, "Robot"): + Jv = camera.jacob0(camera.q) + Jv[:3, :] = self._T[:3, :3] @ Jv[:3, :] + + Jv *= ( + np.linalg.norm(wTvp - shape.T[:3, -1]) / los.length + ) # type: ignore + + dpc = norm_h @ Jv + dpc = np.concatenate( + ( + dpc[0, :-camera_n], + np.zeros(self.n - (camera.n - camera_n)), + dpc[0, -camera_n:], + ) + ) + else: + dpc = np.zeros((1, self.n + camera_n)) + + dpt = norm_h @ shape.v + dpt *= np.linalg.norm(wTvp - wTcp) / los.length # type: ignore + + l_Ain = np.zeros((1, self.n + camera_n)) + l_Ain[0, :n_dim] = norm_h @ Je + l_Ain -= dpc + l_bin = (xi * (d - ds) / (di - ds)) + dpt + else: + l_Ain = None + l_bin = None + + return l_Ain, l_bin + + for link in links: + if link.isjoint: + j += 1 + + if collision_list is None: + col_list = link.collision + else: + col_list = collision_list[j - 1] + + for link_col in col_list: + l_Ain, l_bin = indiv_calculation(link, link_col, q) + + if l_Ain is not None and l_bin is not None: + if Ain is None: + Ain = l_Ain + else: + Ain = np.concatenate((Ain, l_Ain)) + + if bin is None: + bin = np.array(l_bin) + else: + bin = np.concatenate((bin, l_bin)) + + return Ain, bin
+ + # --------------------------------------------------------------------- # + # --------- Dynamics Methods ------------------------------------------ # + # --------------------------------------------------------------------- # + +
[docs] def rne( + self, + q: NDArray, + qd: NDArray, + qdd: NDArray, + symbolic: bool = False, + gravity: Union[ArrayLike, None] = None, + ): + """ + Compute inverse dynamics via recursive Newton-Euler formulation + + ``rne_dh(q, qd, qdd)`` where the arguments have shape (n,) where n is + the number of robot joints. The result has shape (n,). + + ``rne_dh(q, qd, qdd)`` where the arguments have shape (m,n) where n + is the number of robot joints and where m is the number of steps in + the joint trajectory. The result has shape (m,n). + + ``rne_dh(p)`` where the input is a 1D array ``p`` = [q, qd, qdd] with + shape (3n,), and the result has shape (n,). + + ``rne_dh(p)`` where the input is a 2D array ``p`` = [q, qd, qdd] with + shape (m,3n) and the result has shape (m,n). + + Parameters + ---------- + q + Joint coordinates + qd + Joint velocity + qdd + Joint acceleration + symbolic + If True, supports symbolic expressions + gravity + Gravitational acceleration, defaults to attribute + of self + + Returns + ------- + tau + Joint force/torques + + Notes + ----- + - This version supports symbolic model parameters + - Verified against MATLAB code + + """ + + n = self.n + + # allocate intermediate variables + Xup = SE3.Alloc(n) + Xtree = SE3.Alloc(n) + + v = SpatialVelocity.Alloc(n) + a = SpatialAcceleration.Alloc(n) + f = SpatialForce.Alloc(n) + I = SpatialInertia.Alloc(n) # noqa + s = [] # joint motion subspace + + # Handle trajectory case + q = getmatrix(q, (None, None)) + qd = getmatrix(qd, (None, None)) + qdd = getmatrix(qdd, (None, None)) + l, _ = q.shape # type: ignore + + if symbolic: # pragma: nocover + Q = np.empty((l, n), dtype="O") # joint torque/force + else: + Q = np.empty((l, n)) # joint torque/force + + # TODO Should the dynamic parameters of static links preceding joint be + # somehow merged with the joint? + + # A temp variable to handle static joints + Ts = SE3() + + # A counter through joints + j = 0 + + for k in range(l): + qk = q[k, :] + qdk = qd[k, :] + qddk = qdd[k, :] + + # initialize intermediate variables + for link in self.links: + if link.isjoint: + I[j] = SpatialInertia(m=link.m, r=link.r) + if symbolic and link.Ts is None: # pragma: nocover + Xtree[j] = SE3(np.eye(4, dtype="O"), check=False) + elif link.Ts is not None: + Xtree[j] = Ts * SE3(link.Ts, check=False) + + if link.v is not None: + s.append(link.v.s) + + # Increment the joint counter + j += 1 + + # Reset the Ts tracker + Ts = SE3() + else: # pragma nocover + # TODO Keep track of inertia and transform??? + if link.Ts is not None: + Ts *= SE3(link.Ts, check=False) + + if gravity is None: + a_grav = -SpatialAcceleration(self.gravity) + else: # pragma nocover + a_grav = -SpatialAcceleration(gravity) + + # forward recursion + for j in range(0, n): + vJ = SpatialVelocity(s[j] * qdk[j]) + + # transform from parent(j) to j + Xup[j] = SE3(self.links[j].A(qk[j])).inv() + + if self.links[j].parent is None: + v[j] = vJ + a[j] = Xup[j] * a_grav + SpatialAcceleration(s[j] * qddk[j]) + else: + jp = self.links[j].parent.jindex # type: ignore + v[j] = Xup[j] * v[jp] + vJ + a[j] = ( + Xup[j] * a[jp] + SpatialAcceleration(s[j] * qddk[j]) + v[j] @ vJ + ) + + f[j] = I[j] * a[j] + v[j] @ (I[j] * v[j]) + + # backward recursion + for j in reversed(range(0, n)): + # next line could be dot(), but fails for symbolic arguments + Q[k, j] = sum(f[j].A * s[j]) + + if self.links[j].parent is not None: + jp = self.links[j].parent.jindex # type: ignore + f[jp] = f[jp] + Xup[j] * f[j] + + if l == 1: + return Q[0] + else: # pragma nocover + return Q
+ + +# ============================================================================= # +# ================= Robot2 Class ============================================== # +# ============================================================================= # + + +
[docs]class Robot2(BaseRobot[Link2]): + def __init__(self, arg, **kwargs): + if isinstance(arg, ETS2): + # we're passed an ETS string + links = [] + # chop it up into segments, a link frame after every joint + parent = None + for j, ets_j in enumerate(arg.split()): + elink = Link2(ETS2(ets_j), parent=parent, name=f"link{j:d}") + parent = elink + if ( + elink.qlim is None + and elink.v is not None + and elink.v.qlim is not None + ): # pragma nocover + elink.qlim = elink.v.qlim + links.append(elink) + + elif smb.islistof(arg, Link2): + links = arg + + else: # pragma nocover + raise TypeError("constructor argument must be ETS2 or list of Link2") + + super().__init__(links, **kwargs) + + # Should just set it to None + self.base = SE2() # override superclass + + @property + def base(self) -> SE2: + """ + Get/set robot base transform (Robot superclass) + + ``robot.base`` is the robot base transform + + Returns + ------- + base + robot tool transform + + - ``robot.base = ...`` checks and sets the robot base transform + + Notes + ----- + - The private attribute ``_base`` will be None in the case of + no base transform, but this property will return ``SE3()`` which + is an identity matrix. + """ + if self._base is None: # pragma nocover + self._base = SE2() + + # return a copy, otherwise somebody with + # reference to the base can change it + return self._base.copy() + + @base.setter + def base(self, T): + if isinstance(T, SE2): + self._base = T + elif SE2.isvalid(T): # pragma nocover + self._tool = SE2(T, check=True) + +
[docs] def jacob0(self, q, start=None, end=None): + return self.ets(start, end).jacob0(q)
+ +
[docs] def jacobe(self, q, start=None, end=None): + return self.ets(start, end).jacobe(q)
+ +
[docs] def fkine(self, q, end=None, start=None): + return self.ets(start, end).fkine(q)
+ + @property + def reach(self) -> float: + r""" + Reach of the robot + + A conservative estimate of the reach of the robot. It is computed as + the sum of the translational ETs that define the link transform. + + Note + ---- + Computed on the first access. If kinematic parameters + subsequently change this will not be reflected. + + Returns + ------- + reach + Maximum reach of the robot + + Notes + ----- + - Probably an overestimate of reach + - Used by numerical inverse kinematics to scale translational + error. + - For a prismatic joint, uses ``qlim`` if it is set + + """ + + # TODO + # This should be a start, end method and compute the reach based on the + # given ets. Then use an lru_cache to speed up return + + if self._reach is None: + d_all = [] + for link in self.ee_links: + d = 0 + while True: + for et in link.ets: + if et.istranslation: + if et.isjoint: + # the length of a prismatic joint depends on the + # joint limits. They might be set in the ET + # or in the Link depending on how the robot + # was constructed + if link.qlim is not None: + d += max(link.qlim) + elif et.qlim is not None: # pragma nocover + d += max(et.qlim) + else: + d += abs(et.eta) + link = link.parent + if link is None or isinstance(link, str): + d_all.append(d) + break + + self._reach = max(d_all) + return self._reach + +
[docs] def fkine_all(self, q: ArrayLike) -> SE2: + """ + Compute the pose of every link frame + + ``T = robot.fkine_all(q)`` is an SE3 instance with ``robot.nlinks + + 1`` values: + + - ``T[0]`` is the base transform + - ``T[i]`` is the pose of link whose ``number`` is ``i`` + + Parameters + ---------- + q + The joint configuration + + Returns + ------- + fkine_all + Pose of all links + + References + ---------- + - J. Haviland, and P. Corke. "Manipulator Differential Kinematics Part I: + Kinematics, Velocity, and Applications." arXiv preprint arXiv:2207.01796 (2022). + + """ # noqa + + q = getvector(q) + + Tbase = SE2(self.base) # add base, also sets the type + + linkframes = Tbase.__class__.Alloc(self.nlinks + 1) + linkframes[0] = Tbase + + def recurse(Tall, Tparent, q, link): + # if joint?? + T = Tparent + while True: + T *= SE2(link.A(q[link.jindex])) + + Tall[link.number] = T + + if link.nchildren == 0: + # no children + return + elif link.nchildren == 1: + # one child + if link in self.ee_links: # pragma nocover + # this link is an end-effector, go no further + return + link = link.children[0] + continue + else: + # multiple children + for child in link.children: + recurse(Tall, T, q, child) + return + + recurse(linkframes, Tbase, q, self.links[0]) + + return linkframes
+
+ +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/robot/SerialLink.html b/_modules/roboticstoolbox/robot/SerialLink.html new file mode 100644 index 00000000..b7410849 --- /dev/null +++ b/_modules/roboticstoolbox/robot/SerialLink.html @@ -0,0 +1,3195 @@ + + + + + + + + + + roboticstoolbox.robot.SerialLink — Robotics Toolbox Python 0.3.7 + documentation + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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Source code for roboticstoolbox.robot.SerialLink

+#!/usr/bin/env python
+"""
+@author Jesse Haviland
+"""
+
+import numpy as np
+from functools import wraps
+from roboticstoolbox.robot.Link import Link
+from roboticstoolbox.tools.null import null
+from spatialmath.base.argcheck import \
+    getvector, ismatrix, isscalar, verifymatrix
+from spatialmath.base.transforms3d import tr2delta, tr2eul
+from spatialmath import SE3
+from scipy.optimize import minimize, Bounds, LinearConstraint
+from frne import init, frne, delete
+from roboticstoolbox.backend.PyPlot.functions import \
+    _plot, _teach, _fellipse, _vellipse, _plot_ellipse, \
+    _plot2, _teach2
+from roboticstoolbox.robot.Dynamics import Dynamics
+
+
+
+        # except ModuleNotFoundError:
+        #     print(
+        #         'Could not find matplotlib.'
+        #         ' Matplotlib required for this function')
+
+ +
+ +
+ + +
+
+ +
+ +
+ + + + + + + + + + + + + + + \ No newline at end of file diff --git a/_modules/roboticstoolbox/tools/trajectory.html b/_modules/roboticstoolbox/tools/trajectory.html new file mode 100644 index 00000000..36e07b81 --- /dev/null +++ b/_modules/roboticstoolbox/tools/trajectory.html @@ -0,0 +1,1274 @@ + + + + + + roboticstoolbox.tools.trajectory — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+
    +
  • + + +
  • +
  • +
+
+
+
+
+ +

Source code for roboticstoolbox.tools.trajectory

+import numpy as np
+import math
+import warnings
+from collections import namedtuple
+import matplotlib.pyplot as plt
+from spatialmath.base.argcheck import (
+    isvector,
+    getvector,
+    # assertmatrix,
+    getvector,
+    isscalar,
+)
+
+
+
[docs]class Trajectory: + """ + A container class for trajectory data. + """ + +
[docs] def __init__(self, name, t, s, sd=None, sdd=None, istime=False): + """ + Construct a new trajectory instance + + :param name: name of the function that created the trajectory + :type name: str + :param t: independent variable, eg. time or step + :type t: ndarray(m) + :param s: position + :type s: ndarray(m) or ndarray(m,n) + :param sd: velocity + :type sd: ndarray(m) or ndarray(m,n) + :param sdd: acceleration + :type sdd: ndarray(m) or ndarray(m,n) + :param istime: ``t`` is time, otherwise step number + :type istime: bool + :param tblend: blend duration (``trapezoidal`` only) + :type istime: float + + The object has attributes: + + - ``t`` the independent variable + - ``s`` the position + - ``sd`` the velocity + - ``sdd`` the acceleration + + If ``t`` is time, ie. ``istime`` is True, then the units of ``sd`` and + ``sdd`` are :math:`s^{-1}` and :math:`s^{-2}` respectively, otherwise + with respect to ``t``. + + .. note:: Data is stored with timesteps as rows and axes as columns. + + :References: + + - Robotics, Vision & Control in Python, 3e, P. Corke, Springer 2023, Chap 3. + """ + self.name = name + self.t = t + self.s = s + self.sd = sd + self.sdd = sdd + self.istime = istime
+ +
[docs] def __str__(self): + s = ( + f"Trajectory created by {self.name}: {len(self)} time steps x" + f" {self.naxes} axes" + ) + return s
+ +
[docs] def __repr__(self): + return str(self)
+ +
[docs] def __len__(self): + """ + Length of trajectory + + :return: number of steps in the trajectory + :rtype: int + """ + return self.s.shape[0]
+ + @property + def q(self): + """ + Position trajectory + + :return: trajectory with one row per timestep, one column per axis + :rtype: ndarray(n,m) + + .. note:: This is a synonym for ``.s``, for compatibility with other + applications. + """ + return self.s + + @property + def qd(self): + """ + Velocity trajectory + + :return: trajectory velocity with one row per timestep, one column per axis + :rtype: ndarray(n,m) + + .. note:: This is a synonym for ``.sd``, for compatibility with other + applications. + """ + return self.sd + + @property + def qdd(self): + """ + Acceleration trajectory + + :return: trajectory acceleration with one row per timestep, one column per axis + :rtype: ndarray(n,m) + + .. note:: This is a synonym for ``.sdd``, for compatibility with other + applications. + """ + return self.sdd + + # @property + # def t(self): + # """ + # Trajectory time + + # :return: trajectory time vector + # :rtype: ndarray(n) + + # .. note:: This is a synonym for ``.t``, for compatibility with other + # applications. + # """ + # return self.x + + @property + def naxes(self): + """ + Number of axes in the trajectory + + :return: number of axes or dimensions + :rtype: int + """ + if self.s.ndim == 1: + return 1 + else: + return self.s.shape[1] + +
[docs] def plot(self, block=False, plotargs=None, textargs=None): + """ + Plot trajectory + + :param block: wait till plot is dismissed + :type block: bool + + + Plot the position, velocity and acceleration data. The format of the + plot depends on the function that created it. + + - ``quintic`` and ``trapezoidal`` show the individual points with markers + - ``trapezoidal`` color code the different motion phases + - ``jtraj`` general m-axis trajectory, show legend + + :seealso: :func:`~quintic`, :func:`~trapezoidal` + """ + + plotopts = {"marker": "o", "markersize": 3} + if plotargs is not None: + plotopts = {**plotopts, **plotargs} + textopts = {"fontsize": 12} + if textargs is not None: + textopts = {**textopts, **textargs} + + nplots = 3 + if self.name == "mstraj": + nplots = 1 + + plt.figure() + ax = plt.subplot(nplots, 1, 1) + + # plot position + if self.name in "quintic": + ax.plot(self.t, self.s, **plotopts) + + elif self.name == "trapezoidal": + # accel phase + tf = self.t[-1] + k = self.t <= self.tblend + ax.plot(self.t[k], self.s[k], color="red", label="acceleration", **plotopts) + + # coast phase + k = (self.t > self.tblend) & (self.t <= (tf - self.tblend)) + ax.plot(self.t[k], self.s[k], color="green", **plotopts) + k = np.where(k)[0][0] + ax.plot( + self.t[k - 1 : k + 1], + self.s[k - 1 : k + 1], + color="green", + label="linear", + **plotopts, + ) + + # decel phase + k = self.t > (tf - self.tblend) + ax.plot(self.t[k], self.s[k], color="blue", **plotopts) + k = np.where(k)[0][0] + ax.plot( + self.t[k - 1 : k + 1], + self.s[k - 1 : k + 1], + color="blue", + label="deceleration", + **plotopts, + ) + + ax.grid(True) + else: + ax.plot(self.t, self.s, **plotopts) + + if self.s.ndim > 1: + ax.legend([f"q{i+1}" for i in range(self.naxes)]) + + ax.grid(True) + ax.set_xlim(0, max(self.t)) + + if self.istime: + ax.set_ylabel("$q(t)$", **textopts) + else: + ax.set_ylabel("$q(k)$", **textopts) + + if nplots > 1: + # plot velocity + ax = plt.subplot(3, 1, 2) + ax.plot(self.t, self.sd, **plotopts) + ax.grid(True) + ax.set_xlim(0, max(self.t)) + + if self.istime: + ax.set_ylabel("$\dot{{q}}(t)$", **textopts) + else: + ax.set_ylabel("$dq/dk$", **textopts) + + # plot acceleration + ax = plt.subplot(3, 1, 3) + ax.plot(self.t, self.sdd, **plotopts) + ax.grid(True) + ax.set_xlim(0, max(self.t)) + + if self.istime: + ax.set_ylabel(f"$\ddot{{q}}(t)$", **textopts) + ax.set_xlabel("t (seconds)") + else: + ax.set_ylabel("$d^2q/dk^2$", **textopts) + ax.set_xlabel("k (step)") + + plt.show(block=block)
+ +
[docs] def qplot(self, **kwargs): + """ + Plot multi-axis trajectory + + :param **kwargs: optional arguments passed to ``qplot`` + + Plots a multi-axis trajectory, held within the object, as position against time. + + :seealso: :func:`qplot` + """ + xplot(self.t, self.q, **kwargs)
+ + +# -------------------------------------------------------------------------- # + + +
[docs]def quintic(q0, qf, t, qd0=0, qdf=0): + """ + Generate scalar polynomial trajectory + + :param q0: initial value + :type q0: float + :param qf: final value + :type qf: float + :param t: time + :type t: int or array_like(m) + :param qd0: initial velocity, optional + :type q0: float + :param qdf: final velocity, optional + :type q0: float + :return: trajectory + :rtype: :class:`Trajectory` instance + + - ``tg = quintic(q0, q1, m)`` is a scalar trajectory (Mx1) that varies + smoothly from ``q0`` to ``qf`` using a quintic polynomial. The initial + and final velocity and acceleration are zero. ``m`` is an integer scalar, + indicating the total number of timesteps and + + - Velocity is in units of distance per trajectory step, not per + second. + - Acceleration is in units of distance per trajectory step squared, + *not* per second squared. + + - ``tg = quintic(q0, q1, t)`` as above but ``t`` is a uniformly-spaced time + vector + + - Velocity is in units of distance per second. + - Acceleration is in units of distance per second squared. + + The return value is an object that contains position, velocity and + acceleration data. + + Example: + + .. runblock:: pycon + + >>> from roboticstoolbox import quintic + >>> tg = quintic(1, 2, 10) + >>> tg + >>> len(tg) + >>> tg.q + >>> tg.plot() + + .. plot:: + + from roboticstoolbox import quintic + tg = quintic(1, 2, 10) + tg.plot() + + + .. note:: The time vector T is assumed to be monotonically increasing, and + time scaling is based on the first and last element. + + :References: + + - Robotics, Vision & Control in Python, 3e, P. Corke, Springer 2023, Chap 3. + + + :seealso: :func:`trapezoidal`, :func:`mtraj`. + """ + if isinstance(t, int): + t = np.arange(0, t) + istime = False + elif isvector(t): + t = getvector(t) + istime = True + else: + raise TypeError("bad argument for time, must be int or vector") + tf = max(t) + + polyfunc = quintic_func(q0, qf, tf, qd0, qdf) + + # evaluate the polynomials + traj = polyfunc(t) + p = traj[0] + pd = traj[1] + pdd = traj[2] + + return Trajectory("quintic", t, p, pd, pdd, istime)
+ + +
[docs]def quintic_func(q0, qf, T, qd0=0, qdf=0): + """ + Quintic scalar polynomial as a function + + :param q0: initial value + :type q0: float + :param qf: final value + :type qf: float + :param T: trajectory time + :type T: float + :param qd0: initial velocity, defaults to 0 + :type q0: float, optional + :param qdf: final velocity, defaults to 0 + :type q0: float, optional + :return: polynomial function :math:`f: t \mapsto (q(t), \dot{q}(t), \ddot{q}(t))` + :rtype: callable + + Returns a function which computes the specific quintic polynomial, and its + derivatives, as described by the parameters. + + Example: + + .. runblock:: pycon + + >>> from roboticstoolbox import quintic_func + >>> f = quintic_func(1, 2, 5) + >>> f(0) + >>> f(5) + >>> f(2.5) + + :seealso: :func:`quintic` :func:`trapezoidal_func` + """ + + # solve for the polynomial coefficients using least squares + # fmt: off + X = [ + [ 0.0, 0.0, 0.0, 0.0, 0.0, 1.0], + [ T**5, T**4, T**3, T**2, T, 1.0], + [ 0.0, 0.0, 0.0, 0.0, 1.0, 0.0], + [ 5.0 * T**4, 4.0 * T**3, 3.0 * T**2, 2.0 * T, 1.0, 0.0], + [ 0.0, 0.0, 0.0, 2.0, 0.0, 0.0], + [20.0 * T**3, 12.0 * T**2, 6.0 * T, 2.0, 0.0, 0.0], + ] + # fmt: on + coeffs, resid, rank, s = np.linalg.lstsq( + X, np.r_[q0, qf, qd0, qdf, 0, 0], rcond=None + ) + + # coefficients of derivatives + coeffs_d = coeffs[0:5] * np.arange(5, 0, -1) + coeffs_dd = coeffs_d[0:4] * np.arange(4, 0, -1) + + return lambda x: ( + np.polyval(coeffs, x), + np.polyval(coeffs_d, x), + np.polyval(coeffs_dd, x), + )
+ + +# -------------------------------------------------------------------------- # + + +
[docs]def lspb(*args, **kwargs): + """ + .. warning:: Deprecated, use ``trapezoidal`` instead. + """ + warnings.warn("lsp is deprecated, use trapezoidal", FutureWarning) + return trapezoidal(*args, **kwargs)
+ + +# -------------------------------------------------------------------------- # + + +
[docs]def trapezoidal(q0, qf, t, V=None): + """ + Scalar trapezoidal trajectory + + :param q0: initial value + :type q0: float + :param qf: final value + :type qf: float + :param t: time + :type t: float or array_like + :param V: velocity of linear segment, optional + :type V: float + :return: trajectory + :rtype: :class:`Trajectory` instance + + Computes a trapezoidal trajectory, which has a linear motion segment with + parabolic blends. + + - ``tg = trapezoidal(q0, qf, t)`` is a scalar trajectory (Mx1) that varies + smoothly from ``q0`` to ``qf`` in M steps using a constant velocity + segment and parabolic blends. Time ``t`` can be either: + + * an integer scalar, indicating the total number of timesteps + + - Velocity is in units of distance per trajectory step, not per + second. + - Acceleration is in units of distance per trajectory step squared, + *not* per second squared. + + * an array_like, containing the time steps. + + - Results are scaled to units of time. + + - ``tg = trapezoidal(q0, q1, t, V)`` as above but specifies the velocity of the + linear segment which is normally computed automatically. + + The return value is an object that contains position, velocity and + acceleration data. + + Example: + + .. runblock:: pycon + + >>> from roboticstoolbox import trapezoidal + >>> tg = trapezoidal(1, 2, 10) + >>> tg + >>> len(tg) + >>> tg.q + + .. plot:: + + from roboticstoolbox import trapezoidal + tg = trapezoidal(1, 2, 10) + tg.plot() + + .. note:: + + - For some values of ``V`` no solution is possible and an error is flagged. + - The time vector, if given, is assumed to be monotonically increasing, + and time scaling is based on the first and last element. + - ``tg`` has an extra attribute ``xblend`` which is the blend duration. + + :References: + + - Robotics, Vision & Control in Python, 3e, P. Corke, Springer 2023, Chap 3. + + :seealso: :func:`quintic`, :func:`trapezoidal_func` :func:`mtraj`. + """ + + if isinstance(t, int): + t = np.arange(0, t) + istime = False + elif isvector(t): + t = getvector(t) + istime = True + else: + raise TypeError("bad argument for time, must be int or vector") + + tf = max(t) + + trapezoidalfunc = trapezoidal_func(q0, qf, tf, V) + + # evaluate the polynomials + traj = trapezoidalfunc(t) + p = traj[0] + pd = traj[1] + pdd = traj[2] + + traj = Trajectory("trapezoidal", t, p, pd, pdd, istime) + traj.tblend = trapezoidalfunc.tb + return traj
+ + +
[docs]def trapezoidal_func(q0, qf, T, V=None): + r""" + Trapezoidal scalar profile as a function + + :param q0: initial value + :type q0: float + :param qf: final value + :type qf: float + :param T: maximum time + :type T: float + :param V: velocity of linear segment + :type V: float, optional + :return: trapezoidal profile function :math:`f: t \mapsto (q(t), \dot{q}(t), \ddot{q}(t))` + :rtype: callable + + Returns a function which computes the specific trapezoidal profile, and its + derivatives, as described by the parameters. + + Example: + + .. runblock:: pycon + + >>> from roboticstoolbox import trapezoidal_func + >>> f = trapezoidal_func(1, 2, 5) + >>> f(0) + >>> f(5) + >>> f(2.5) + """ + + if V is None: + # if velocity not specified, compute it + V = (qf - q0) / T * 1.5 + else: + V = abs(V) * np.sign(qf - q0) + if abs(V) < (abs(qf - q0) / T): + raise ValueError("V too small") + elif abs(V) > (2 * abs(qf - q0) / T): + raise ValueError("V too big") + + if V == 0: + tb = np.inf + a = 0 + else: + tb = (q0 - qf + V * T) / V + a = V / tb + + def trapezoidalfunc(t): + + p = [] + pd = [] + pdd = [] + + if isscalar(t): + t = [t] + for tk in t: + if tk < 0: + pk = q0 + pdk = 0 + pddk = 0 + elif tk <= tb: + # initial blend + pk = q0 + a / 2 * tk**2 + pdk = a * tk + pddk = a + elif tk <= (T - tb): + # linear motion + pk = (qf + q0 - V * T) / 2 + V * tk + pdk = V + pddk = 0 + elif tk <= T: + # final blend + pk = qf - a / 2 * T**2 + a * T * tk - a / 2 * tk**2 + pdk = a * T - a * tk + pddk = -a + else: + pk = qf + pdk = 0 + pddk = 0 + p.append(pk) + pd.append(pdk) + pdd.append(pddk) + return (np.array(p), np.array(pd), np.array(pdd)) + + # return the function, but add some computed parameters as attributes + # as a way of returning extra values without a tuple return + func = trapezoidalfunc + func.tb = tb + func.V = V + + return func
+ + +# -------------------------------------------------------------------------- # + + +
[docs]def mtraj(tfunc, q0, qf, t): + """ + Multi-axis trajectory + + :param tfunc: a 1D trajectory function, eg. :func:`quintic` or :func:`trapezoidal` + :type tfunc: callable + :param q0: initial configuration + :type q0: ndarray(m) + :param qf: final configuration + :type qf: ndarray(m) + :param t: time vector or number of steps + :type t: array_like or int + :raises TypeError: ``tfunc`` is not callable + :raises ValueError: length of ``q0`` and ``qf`` are different + :return: trajectory + :rtype: :class:`Trajectory` instance + + - ``tg = mtraj(func, q0, qf, n)`` is a multi-axis trajectory varying + from configuration ``q0`` (M) to ``qf`` (M) according to the scalar trajectory + function ``tfunc`` in ``n`` steps. + + - ``tg = mtraj(func, q0, qf, t)`` as above but ``t`` is a uniformly-spaced time + vector + + The scalar trajectory function is applied to each axis:: + + tg = tfunc(s0, sF, n) + + and possible values of ``tfunc`` include ``trapezoidal`` for a trapezoidal trajectory, or + ``quintic`` for a polynomial trajectory. + + The return value is an object that contains position, velocity and + acceleration data. + + .. note:: The time vector, if given, is assumed to be monotonically increasing, and + time scaling is based on the first and last element. + + :References: + - Robotics, Vision & Control in Python, 3e, P. Corke, Springer 2023, Chap 3. + + :seealso: :func:`quintic`, :func:`trapezoidal` + """ + + if not callable(tfunc): + raise TypeError("first argument must be a function reference") + + q0 = getvector(q0) + qf = getvector(qf) + if len(q0) != len(qf): + raise ValueError("must be same number of elements in q0 and qf") + + traj = [] + for i in range(len(q0)): + # for each axis + traj.append(tfunc(q0[i], qf[i], t)) + + x = traj[0].t + y = np.array([tg.s for tg in traj]).T + yd = np.array([tg.sd for tg in traj]).T + ydd = np.array([tg.sdd for tg in traj]).T + + istime = traj[0].istime + + return Trajectory("mtraj", x, y, yd, ydd, istime)
+ + +# -------------------------------------------------------------------------- # + + +
[docs]def jtraj(q0, qf, t, qd0=None, qd1=None): + """ + Compute a joint-space trajectory + + :param q0: initial joint coordinate + :type q0: array_like(n) + :param qf: final joint coordinate + :type qf: array_like(n) + :param t: time vector or number of steps + :type t: array_like or int + :param qd0: initial velocity, defaults to zero + :type qd0: array_like(n), optional + :param qd1: final velocity, defaults to zero + :type qd1: array_like(n), optional + :return: trajectory + :rtype: :class:`Trajectory` instance + + - ``tg = jtraj(q0, qf, N)`` is a joint space trajectory where the joint + coordinates vary from ``q0`` (M) to ``qf`` (M). A quintic (5th order) + polynomial is used with default zero boundary conditions for velocity and + acceleration. Time is assumed to vary from 0 to 1 in ``N`` steps. + + - ``tg = jtraj(q0, qf, t)`` as above but ``t`` is a uniformly-spaced time + vector + + The return value is an object that contains position, velocity and + acceleration data. + + .. note:: The time vector, if given, scales the velocity and acceleration outputs + assuming that the time vector starts at zero and increases linearly. + + :References: + + - Robotics, Vision & Control in Python, 3e, P. Corke, Springer 2023, Chap 3. + + :seealso: :func:`ctraj`, :func:`qplot`, :func:`~SerialLink.jtraj` + """ + # print(f" --- jtraj: {q0} --> {q1} in {tv}") + if isinstance(t, int): + tscal = 1.0 + ts = np.linspace(0, 1, t) # normalized time from 0 -> 1 + tv = ts * t + else: + tv = t.flatten() + tscal = max(t) + ts = t.flatten() / tscal + + q0 = getvector(q0) + qf = getvector(qf) + + if not len(q0) == len(qf): + raise ValueError("q0 and q1 must be same size") + + if qd0 is None: + qd0 = np.zeros(q0.shape) + else: + qd0 = getvector(qd0) + if not len(qd0) == len(q0): + raise ValueError("qd0 has wrong size") + if qd1 is None: + qd1 = np.zeros(q0.shape) + else: + qd1 = getvector(qd1) + if not len(qd1) == len(q0): + raise ValueError("qd1 has wrong size") + + # compute the polynomial coefficients + A = 6 * (qf - q0) - 3 * (qd1 + qd0) * tscal + B = -15 * (qf - q0) + (8 * qd0 + 7 * qd1) * tscal + C = 10 * (qf - q0) - (6 * qd0 + 4 * qd1) * tscal + E = qd0 * tscal # as the t vector has been normalized + F = q0 + + # n = len(q0) + + tt = np.array([ts**5, ts**4, ts**3, ts**2, ts, np.ones(ts.shape)]).T + coeffs = np.array([A, B, C, np.zeros(A.shape), E, F]) # 6xN + + qt = tt @ coeffs + + # compute velocity + coeffs = np.array([np.zeros(A.shape), 5 * A, 4 * B, 3 * C, np.zeros(A.shape), E]) + qdt = tt @ coeffs / tscal + + # compute acceleration + coeffs = np.array( + [np.zeros(A.shape), np.zeros(A.shape), 20 * A, 12 * B, 6 * C, np.zeros(A.shape)] + ) + qddt = tt @ coeffs / tscal**2 + + return Trajectory("jtraj", tv, qt, qdt, qddt, istime=True)
+ + +# -------------------------------------------------------------------------- # + + +
[docs]def ctraj(T0, T1, t=None, s=None): + """ + Cartesian trajectory between two poses + + :param T0: initial pose + :type T0: SE3 + :param T1: final pose + :type T1: SE3 + :param t: number of samples or time vector + :type t: int or ndarray(n) + :param s: array of distance along the path, in the interval [0, 1] + :type s: ndarray(s) + :return T0: smooth path from ``T0`` to ``T1`` + :rtype: SE3 + + ``ctraj(T0, T1, n)`` is a Cartesian trajectory from SE3 pose ``T0`` to + ``T1`` with ``n`` points that follow a trapezoidal velocity profile along + the path. The Cartesian trajectory is an SE3 instance containing ``n`` + values. + + ``ctraj(T0, T1, t)`` as above but the trajectory is sampled at + the points in the array ``t``. + + ``ctraj(T0, T1, s=s)`` as above but the elements of ``s`` specify the + fractional distance along the path, and these values are in the + range [0 1]. The i'th point corresponds to a distance ``s[i]`` along + the path. + + Examples:: + + >>> tg = ctraj(SE3.Rand(), SE3.Rand(), 20) + >>> len(tg) + 20 + + .. note:: + + - In the second case ``s`` could be generated by a scalar trajectory + generator such as ``quintic`` or ``trapezoidal`` (default). + - Orientation interpolation is performed using unit-quaternion + interpolation. + + :References: + - Robotics, Vision & Control in Python, 3e, P. Corke, Springer 2023, Chap 3. + + :seealso: :func:`~roboticstoolbox.trajectory.trapezoidal`, + :func:`~spatialmath.unitquaternion.interp` + """ + + if isinstance(t, int): + s = trapezoidal(0, 1, t).s + elif isvector(t): + t = getvector(t) + s = trapezoidal(0, 1, t / np.max(t)).s + elif isvector(s): + s = getvector(s) + else: + raise TypeError("bad argument for time, must be int or vector") + + return T0.interp(T1, s)
+ + +
[docs]def cmstraj(): + # Cartesian version of mstraj, via points are SE3 + # perhaps refactor mstraj to allow interpolation of any type + pass
+ + +# -------------------------------------------------------------------------- # + + +
[docs]def mstraj( + viapoints, + dt, + tacc, + qdmax=None, + tsegment=None, + q0=None, + qd0=None, + qdf=None, + verbose=False, +): + """ + Multi-segment multi-axis trajectory + + :param viapoints: A set of viapoints, one per row + :type viapoints: ndarray(m,n) + :param dt: time step + :type dt: float (seconds) + :param tacc: acceleration time (seconds) + :type tacc: float + :param qdmax: maximum speed, defaults to None + :type qdmax: array_like(n) or float, optional + :param tsegment: maximum time of each motion segment (seconds), defaults + to None + :type tsegment: array_like, optional + :param q0: initial coordinates, defaults to first row of viapoints + :type q0: array_like(n), optional + :param qd0: inital velocity, defaults to zero + :type qd0: array_like(n), optional + :param qdf: final velocity, defaults to zero + :type qdf: array_like(n), optional + :param verbose: print debug information, defaults to False + :type verbose: bool, optional + :return: trajectory + :rtype: :class:`Trajectory` instance + + Computes a trajectory for N axes moving smoothly through a set of + viapoints. + The motion comprises M segments: + + - The initial coordinates are the first row of ``viapoints`` or ``q0`` if + provided. + - The final coordinates are the last row of ``viapoints`` + - Each segment is linear motion and polynomial blends connect the + viapoints. + - All joints arrive at each via point at the same time, ie. the motion is + coordinated across axes + + The time of the segments can be specified in two different ways: + + #. In terms of segment time where ``tsegment`` is an array of segment times + which is the number of via points minus one:: + + traj = mstraj(viapoints, dt, tacc, tsegment=TS) + + #. Governed by the speed of the slowest axis for the segment. The axis + speed is a scalar (all axes have the same speed) or an N-vector of speed + per axis:: + + traj = mstraj(viapoints, dt, tacc, qdmax=SPEED) + + The return value is a namedtuple (named ``mstraj``) with elements: + + - ``t`` the time coordinate as a numpy ndarray, shape=(K,) + - ``q`` the axis values as a numpy ndarray, shape=(K,N) + - ``arrive`` a list of arrival times for each segment + - ``info`` a list of named tuples, one per segment that describe the + slowest axis, segment time, and time stamp + - ``via`` the passed set of via points + + The trajectory proper is (``traj.t``, ``traj.q``). The trajectory is a + matrix has one row per time step, and one column per axis. + + .. note:: + + - Only one of ``qdmag`` or ``tsegment`` can be specified + - If ``tacc`` is greater than zero then the path smoothly accelerates + between segments using a polynomial blend. This means that the the via + point is not actually reached. + - The path length K is a function of the number of via + points and the time or velocity limits that apply. + - Can be used to create joint space trajectories where each axis is a + joint coordinate. + - Can be used to create Cartesian trajectories where the "axes" + correspond to translation and orientation in RPY or Euler angle form. + - If ``qdmax`` is a scalar then all axes are assumed to have the same + maximum speed. + - ``tg`` has extra attributes ``arrive``, ``info`` and ``via`` + + :References: + - Robotics, Vision & Control in Python, 3e, P. Corke, Springer 2023, Chap 3. + + :seealso: :func:`trapezoidal`, :func:`ctraj`, :func:`mtraj` + """ + + if q0 is None: + q0 = viapoints[0, :] + viapoints = viapoints[1:, :] + else: + q0 = getvector(q0) + if not viapoints.shape[1] == len(q0): + raise ValueError("WP and Q0 must have same number of columns") + + ns, nj = viapoints.shape + Tacc = tacc + + if qdmax is not None and tsegment is not None: + raise ValueError("cannot specify both qdmax and tsegment") + + if qdmax is None: + if tsegment is None: + raise ValueError("tsegment must be given if qdmax is not") + + if not len(tsegment) == ns: + raise ValueError("Length of TSEG does not match number of viapoints") + + if tsegment is None: + + # This is unreachable, left just in case + if qdmax is None: # pragma nocover + raise ValueError("qdmax must be given if tsegment is not") + + if isinstance(qdmax, (int, float)): + # if qdmax is a scalar assume all axes have the same speed + qdmax = np.tile(qdmax, (nj,)) + else: + qdmax = getvector(qdmax) + + if not len(qdmax) == nj: + raise ValueError("Length of QDMAX does not match number of axes") + + if isinstance(Tacc, (int, float)): + Tacc = np.tile(Tacc, (ns,)) + else: + if not len(Tacc) == ns: + raise ValueError("Tacc is wrong size") + if qd0 is None: + qd0 = np.zeros((nj,)) + else: + if not len(qd0) == len(q0): + raise ValueError("qd0 is wrong size") + if qdf is None: + qdf = np.zeros((nj,)) + else: + if not len(qdf) == len(q0): + raise ValueError("qdf is wrong size") + + # set the initial conditions + q_prev = q0 + qd_prev = qd0 + + clock = 0 # keep track of time + arrive = np.zeros((ns,)) # record planned time of arrival at via points + tg = np.zeros((0, nj)) + infolist = [] + info = namedtuple("mstraj_info", "slowest segtime clock") + + def mrange(start, stop, step): + """ + mrange(start, stop, step) behaves like MATLAB start:step:stop + and includes the final value unlike range() or np.arange() + """ + # ret = [] + istart = round(start / step) + istop = round(stop / step) + return np.arange(istart, istop + 1) * step + + for seg in range(0, ns): + q_next = viapoints[seg, :] # current target + + if verbose: # pragma nocover + print(f"------- segment {seg}: {q_prev} --> {q_next}") + + # set the blend time, just half an interval for the first segment + + tacc = Tacc[seg] + + tacc = math.ceil(tacc / dt) * dt + tacc2 = math.ceil(tacc / 2 / dt) * dt + if seg == 0: + taccx = tacc2 + else: + taccx = tacc + + # estimate travel time + # could better estimate distance travelled during the blend + + dq = q_next - q_prev # total distance to move this segment + + # probably should iterate over the next section to get qb right... + # while 1 + # qd_next = (qnextnext - qnext) + # tb = abs(qd_next - qd) ./ qddmax; + # qb = f(tb, max acceleration) + # dq = q_next - q_prev - qb + # tl = abs(dq) ./ qdmax; + + if qdmax is not None: + # qdmax is specified, compute slowest axis + + # qb = taccx * qdmax / 2 # distance moved during blend + tb = taccx + + # convert to time + tl = abs(dq) / qdmax + # tl = abs(dq - qb) / qdmax + tl = np.ceil(tl / dt) * dt + + # find the total time and slowest axis + tt = tb + tl + slowest = np.argmax(tt) + tseg = tt[slowest] + + # best if there is some linear motion component + if tseg <= 2 * tacc: + tseg = 2 * tacc + + elif tsegment is not None: + # segment time specified, use that + tseg = tsegment[seg] + slowest = math.nan + + infolist.append(info(slowest, tseg, clock)) + + # log the planned arrival time + arrive[seg] = clock + tseg + if seg > 0: + arrive[seg] += tacc2 + + if verbose: # pragma nocover + print( + f"seg {seg}, distance {dq}, " + "slowest axis {slowest}, time required {tseg}" + ) + + # create the trajectories for this segment + + # linear velocity from qprev to qnext + qd = dq / tseg + + # add the blend polynomial + if taccx > 0: + qb = jtraj( + q0, q_prev + tacc2 * qd, mrange(0, taccx, dt), qd0=qd_prev, qd1=qd + ).s + if verbose: # pragma nocover + print(qb) + tg = np.vstack([tg, qb[1:, :]]) + + clock = clock + taccx # update the clock + + # add the linear part, from tacc/2+dt to tseg-tacc/2 + for t in mrange(tacc2 + dt, tseg - tacc2, dt): + s = t / tseg + q0 = (1 - s) * q_prev + s * q_next # linear step + if verbose: # pragma nocover + print(t, s, q0) + tg = np.vstack([tg, q0]) + clock += dt + + q_prev = q_next # next target becomes previous target + qd_prev = qd + + # add the final blend + if tacc2 > 0: + qb = jtraj(q0, q_next, mrange(0, tacc2, dt), qd0=qd_prev, qd1=qdf).s + tg = np.vstack([tg, qb[1:, :]]) + + infolist.append(info(None, tseg, clock)) + + traj = Trajectory("mstraj", dt * np.arange(0, tg.shape[0]), tg) + traj.arrive = arrive + traj.info = infolist + traj.via = viapoints + + return traj
+ + +if __name__ == "__main__": + + # t = quintic(0, 1, 50) + # t.plot() + + # t = quintic(0, 1, np.linspace(0, 1, 50)) + # t.plot() + + # t = trapezoidal(0, 1, 50) + # t.plot() + # t = trapezoidal(0, 1, np.linspace(0, 1, 50)) + # t.plot(block=True) + + from roboticstoolbox import * + from spatialmath import SO2 + + # puma = models.DH.Puma560() + + # traj = jtraj(puma.qz, puma.qr, 100) + # traj.plot(block=True) + + via = SO2(30, unit="deg") * np.array([[-1, 1, 1, -1, -1], [1, 1, -1, -1, 1]]) + traj0 = mstraj(via.T, dt=0.2, tacc=0.5, qdmax=[2, 1]) + xplot(traj0.q[:, 0], traj0.q[:, 1], color="red") + traj0.plot(block=True) +
+ +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/_sources/IK/ik.rst.txt b/_sources/IK/ik.rst.txt new file mode 100644 index 00000000..87529ba8 --- /dev/null +++ b/_sources/IK/ik.rst.txt @@ -0,0 +1,261 @@ +.. _IK: + + +Inverse Kinematics +================== + + +The Robotics Toolbox supports an extensive set of numerical inverse kinematics (IK) tools and we will demonstrate the different ways these IK tools can be interacted with in this document. + +For a **tutorial** on numerical IK, see `here `_. + +Within the Toolbox, we have two sets of solvers: solvers written in C++ and solvers written in Python. However, within all of our solvers there are several common arguments: + +.. rubric:: Tep + +``Tep`` represent the desired end-effector pose. + +A note on the semantics of the above variable: + +* **T** represents an SE(3) (a homogeneous tranformation matrix in 3 dimensions, a 4x4 matrix) +* *e* is short for end-effector referring to the end of the kinematic chain +* *p* is short for prime or desired +* Since there is no letter to the left of the **T**, the world or base reference frame is implied + +Therefore, ``Tep`` refers to the desired end-effector pose in the base robot frame represented as an SE(3). + +.. rubric:: ilimit + +The ``ilimit`` specifies how many iterations are allowed within a single search. After ``ilimit`` is reached, either, a new attempt is made or the IK solution has failed depending on ``slimit`` + +.. rubric:: slimit + +The ``slimit`` specifies how many searches are allowed before the problem is deemed unsolvable. The maximum number of iterations allowed is therefore ``ilimit`` x ``slimit``. By having ``slimit`` > 1, a global search is performed. Since finding a solution with numerical IK heavily depends on the initial choice of ``q0``, performing a global search where ``slimit`` >> 1 will provide a far greater chance of success. + +.. rubric:: q0 + +``q0`` is the inital joint coordinate vector. If ``q0`` is 1 dimensional (, ``n``), then ``q0`` is only used for the first attempt, after which a new random valid initial joint coordinate vector will be generated. If ``q0`` is 2 dimensional (``slimit``, ``n``), then the next vector within ``q0`` will be used for the next search. + +.. rubric:: tol + +``tol`` sets the error tolerance before the solution is deemed successful. The error is typically set by some quadratic error function + +.. math:: + + E = \frac{1}{2} \vec{e}^{\top} \mat{W}_e \vec{e} + +where :math:`\vec{e} \in \mathbb{R}^6` is the angle-axis error, and :math:`\mat{W}_e` assigns weights to Cartesian degrees-of-freedom + +.. rubric:: mask + +``mask`` is a (,6) array that sets :math:`\mat{W}_e` in error equation above. The vector has six elements that correspond to translation in X, Y and Z, and rotation about X, Y and Z respectively. The value can be 0 (for ignore) or above to assign a priority relative to other Cartesian DoF. + +For the case where the manipulator has fewer than 6 DoF the solution space has more dimensions than can be spanned by the manipulator joint coordinates. + +In this case we use the ``mask`` option where the ``mask`` vector specifies the Cartesian DOF that will be ignored in reaching a solution. The number of non-zero elements must equal the number of manipulator DOF. + +For example when using a 3 DOF manipulator tool orientation might be unimportant, in which case use the option ``mask=[1, 1, 1, 0, 0, 0]``. + +.. rubric:: joint_limits + +setting ``joint_limits = True`` will reject solutions with joint limit violations. Note that finding a solution with valid joint coordinates is likely to take longer than without. + +.. rubric:: Others + +There are other arguments which may be unique to the solver, so check the documentation of the solver you wish to use for a complete list and explanation of arguments. + +C++ Solvers +----------- + +These solvers are written in high performance C++ and wrapped in Python methods. The methods are made available within the :py:class:`~roboticstoolbox.robot.ETS.ETS` and :py:class:`~roboticstoolbox.robot.Robot.Robot` classes. Being written in C++, these solvers are extraordinarily fast and typically take 30 to 90 µs. However, these solvers are hard to extend or modify. + +These methods have been written purely for speed so they do not contain the niceties of the Python alternative. For example, if you give the incorrect length for the ``q0`` vector, you could end up with a ``seg-fault`` or other undetermined behaviour. Therefore, when using these methods it is very important that you understand each of the parameters and the parameters passed are of the correct type and length. + +The C++ solvers return a tuple with the following members: + +============== ========= ===================================================================================================== +Element Type Description +============== ========= ===================================================================================================== +``q`` `ndarray` The joint coordinates of the solution. Note that these will not be valid if failed to find a solution +``success`` `bool` True if a valid solution was found +``iterations`` `int` How many iterations were performed +``searches`` `int` How many searches were performed +``residual`` `float` The final error value from the cost function +============== ========= ===================================================================================================== + +The C++ solvers can be identified as methods which start with ``ik_``. + +.. rubric:: ETS C++ IK Methods + +.. autosummary:: + :toctree: stubs + + ~roboticstoolbox.robot.ETS.ETS.ik_LM + ~roboticstoolbox.robot.ETS.ETS.ik_GN + ~roboticstoolbox.robot.ETS.ETS.ik_NR + +.. rubric:: Robot C++ IK Methods + +.. autosummary:: + :toctree: stubs + + ~roboticstoolbox.robot.Robot.Robot.ik_LM + ~roboticstoolbox.robot.Robot.Robot.ik_GN + ~roboticstoolbox.robot.Robot.Robot.ik_NR + +In the following example, we create a :py:class:`~roboticstoolbox.models.URDF.Panda` robot and one of the fast IK solvers available within the :py:class:`~roboticstoolbox.robot.Robot.Robot` class. + +.. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> # Make a Panda robot + >>> panda = rtb.models.Panda() + >>> # Make a goal pose + >>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854]) + >>> # Solve the IK problem + >>> panda.ik_LM(Tep) + +In the following example, we create a :py:class:`~roboticstoolbox.models.URDF.Panda` robot and and then get the :py:class:`~roboticstoolbox.robot.ETS.ETS` representation. Subsequently, we use one of the fast IK solvers available within the :py:class:`~roboticstoolbox.robot.ETS.ETS` class. + +.. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> # Make a Panda robot + >>> panda = rtb.models.Panda() + >>> # Get the ETS + >>> ets = panda.ets() + >>> # Make a goal pose + >>> Tep = ets.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854]) + >>> # Solve the IK problem + >>> ets.ik_LM(Tep) + + + + + +Python Solvers +-------------- + +These solvers are Python classes which extend the abstract base class :py:class:`~roboticstoolbox.robot.IK.IKSolver` and the :py:meth:`~roboticstoolbox.robot.IK.IKSolver.solve` method returns an :py:class:`~roboticstoolbox.robot.IK.IKSolution`. These solvers are slow and will typically take 100 - 1000 ms. However, these solvers are easy to extend and modify. + +.. rubric:: The Abstract Base Class + +.. toctree:: + :maxdepth: 1 + + iksolver + +The :py:class:`~roboticstoolbox.robot.IK.IKSolver` provides basic functionality for performing numerical IK. Superclasses can inherit this class and must implement the :py:meth:`~roboticstoolbox.robot.IK.IKSolver.solve` method. Additionally a superclass redefine any other methods necessary such as :py:meth:`~roboticstoolbox.robot.IK.IKSolver.error` to provide a custom error function. + +.. rubric:: The Solution DataClass + +.. toctree:: + :maxdepth: 1 + + iksolution + +The :py:class:`~roboticstoolbox.robot.IK.IKSolution` is a :py:class:`dataclasses.dataclass` instance with the following members. + +============== ========= ===================================================================================================== +Element Type Description +============== ========= ===================================================================================================== +``q`` `ndarray` The joint coordinates of the solution. Note that these will not be valid if failed to find a solution +``success`` `bool` True if a valid solution was found +``iterations`` `int` How many iterations were performed +``searches`` `int` How many searches were performed +``residual`` `float` The final error value from the cost function +``reason`` `str` The reason the IK problem failed if applicable +============== ========= ===================================================================================================== + +.. rubric:: The Implemented IK Solvers + +These solvers can be identified as a :py:class:`Class` starting with ``IK_``. + +.. toctree:: + :maxdepth: 1 + + ik_lm + ik_qp + ik_gn + ik_nr + +.. rubric:: Example + +In the following example, we create an IK Solver class and pass an :py:class:`~roboticstoolbox.robot.ETS.ETS` to it to solve the problem. This style may be preferable to experiments where you wish to compare the same solver on different robots. + +.. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> # Make a Panda robot + >>> panda = rtb.models.Panda() + >>> # Get the ETS of the Panda + >>> ets = panda.ets() + >>> # Make an IK solver + >>> solver = rtb.IK_LM() + >>> # Make a goal pose + >>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854]) + >>> # Solve the IK problem + >>> solver.solve(ets, Tep) + + + +.. IK Solvers Available with an ETS +.. ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +Additionally, these :py:class:`Class` based solvers have been implemented as methods within the :py:class:`~roboticstoolbox.robot.ETS.ETS` and :py:class:`~roboticstoolbox.robot.Robot.Robot` classes. The method names start with ``ikine_``. + + +.. toctree: + :caption: IK Solvers from an ETS + + +.. rubric:: ETS Python IK Methods + + +.. autosummary:: + :toctree: stubs + + ~roboticstoolbox.robot.ETS.ETS.ikine_LM + ~roboticstoolbox.robot.ETS.ETS.ikine_QP + ~roboticstoolbox.robot.ETS.ETS.ikine_GN + ~roboticstoolbox.robot.ETS.ETS.ikine_NR + + +.. rubric:: Robot Python IK Methods + +.. autosummary:: + :toctree: stubs + + ~roboticstoolbox.robot.Robot.Robot.ikine_LM + ~roboticstoolbox.robot.Robot.Robot.ikine_QP + ~roboticstoolbox.robot.Robot.Robot.ikine_GN + ~roboticstoolbox.robot.Robot.Robot.ikine_NR + + +.. rubric:: Example + +In the following example, we create a :py:class:`~roboticstoolbox.models.URDF.Panda` robot and one of the IK solvers available within the :py:class:`~roboticstoolbox.robot.Robot.Robot` class. This style is far more convenient than the above example. + +.. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> # Make a Panda robot + >>> panda = rtb.models.Panda() + >>> # Make a goal pose + >>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854]) + >>> # Solve the IK problem + >>> panda.ikine_LM(Tep) + +In the following example, we create a :py:class:`~roboticstoolbox.models.URDF.Panda` robot and and then get the :py:class:`~roboticstoolbox.robot.ETS.ETS` representation. Subsequently, we use one of the IK solvers available within the :py:class:`~roboticstoolbox.robot.ETS.ETS` class. + +.. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> # Make a Panda robot + >>> panda = rtb.models.Panda() + >>> # Get the ETS + >>> ets = panda.ets() + >>> # Make a goal pose + >>> Tep = ets.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854]) + >>> # Solve the IK problem + >>> ets.ikine_LM(Tep) diff --git a/_sources/IK/ik_gn.rst.txt b/_sources/IK/ik_gn.rst.txt new file mode 100644 index 00000000..9bea7bd0 --- /dev/null +++ b/_sources/IK/ik_gn.rst.txt @@ -0,0 +1,25 @@ +IK_GN - Gauss-Newton Numerical IK +--------------------------------- + +.. currentmodule:: roboticstoolbox.robot.IK + +.. autoclass:: IK_GN + :show-inheritance: + + +.. rubric:: Methods + +.. autosummary:: + :toctree: stubs + + ~IK_GN.step + ~IK_GN.solve + ~IK_GN.error + +.. rubric:: Private Methods + +.. autosummary:: + :toctree: stubs + + ~IK_GN._random_q + ~IK_GN._check_jl diff --git a/_sources/IK/ik_lm.rst.txt b/_sources/IK/ik_lm.rst.txt new file mode 100644 index 00000000..41d337ad --- /dev/null +++ b/_sources/IK/ik_lm.rst.txt @@ -0,0 +1,25 @@ +IK_LM - Levemberg-Marquadt Numerical IK +--------------------------------------- + +.. currentmodule:: roboticstoolbox.robot.IK + +.. autoclass:: IK_LM + :show-inheritance: + + +.. rubric:: Methods + +.. autosummary:: + :toctree: stubs + + ~IK_LM.step + ~IK_LM.solve + ~IK_LM.error + +.. rubric:: Private Methods + +.. autosummary:: + :toctree: stubs + + ~IK_LM._random_q + ~IK_LM._check_jl diff --git a/_sources/IK/ik_nr.rst.txt b/_sources/IK/ik_nr.rst.txt new file mode 100644 index 00000000..5aa34548 --- /dev/null +++ b/_sources/IK/ik_nr.rst.txt @@ -0,0 +1,25 @@ +IK_NR - Newton-Raphson Numerical IK +----------------------------------- + +.. currentmodule:: roboticstoolbox.robot.IK + +.. autoclass:: IK_NR + :show-inheritance: + + +.. rubric:: Methods + +.. autosummary:: + :toctree: stubs + + ~IK_NR.step + ~IK_NR.solve + ~IK_NR.error + +.. rubric:: Private Methods + +.. autosummary:: + :toctree: stubs + + ~IK_NR._random_q + ~IK_NR._check_jl diff --git a/_sources/IK/ik_qp.rst.txt b/_sources/IK/ik_qp.rst.txt new file mode 100644 index 00000000..06767812 --- /dev/null +++ b/_sources/IK/ik_qp.rst.txt @@ -0,0 +1,25 @@ +IK_QP - Numerical IK with Quadratic Programming +----------------------------------------------- + +.. currentmodule:: roboticstoolbox.robot.IK + +.. autoclass:: IK_QP + :show-inheritance: + + +.. rubric:: Methods + +.. autosummary:: + :toctree: stubs + + ~IK_QP.step + ~IK_QP.solve + ~IK_QP.error + +.. rubric:: Private Methods + +.. autosummary:: + :toctree: stubs + + ~IK_QP._random_q + ~IK_QP._check_jl diff --git a/_sources/IK/iksolution.rst.txt b/_sources/IK/iksolution.rst.txt new file mode 100644 index 00000000..2f8ff05b --- /dev/null +++ b/_sources/IK/iksolution.rst.txt @@ -0,0 +1,4 @@ +IKSolution - as IK Solution dataclass +------------------------ + +.. autoclass:: roboticstoolbox.robot.IK.IKSolution diff --git a/_sources/IK/iksolver.rst.txt b/_sources/IK/iksolver.rst.txt new file mode 100644 index 00000000..ce3563c8 --- /dev/null +++ b/_sources/IK/iksolver.rst.txt @@ -0,0 +1,25 @@ +IKSolver - IK Solver Abstract Base Class +----------------------------------------- + +.. currentmodule:: roboticstoolbox.robot.IK + +.. autoclass:: IKSolver + :show-inheritance: + + +.. rubric:: Methods + +.. autosummary:: + :toctree: stubs + + ~IKSolver.step + ~IKSolver.solve + ~IKSolver.error + +.. rubric:: Private Methods + +.. autosummary:: + :toctree: stubs + + ~IKSolver._random_q + ~IKSolver._check_jl diff --git a/_sources/IK/stubs/roboticstoolbox.robot.ERobot.ERobot.ik_gn.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.ERobot.ERobot.ik_gn.rst.txt new file mode 100644 index 00000000..fdf679e5 --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.ERobot.ERobot.ik_gn.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +ERobot.ik\_gn +====== + +.. currentmodule:: roboticstoolbox.robot.ERobot + +.. automethod:: roboticstoolbox.robot.ERobot.ERobot.ik_gn \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.ERobot.ERobot.ik_lm_chan.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.ERobot.ERobot.ik_lm_chan.rst.txt new file mode 100644 index 00000000..3bd7825a --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.ERobot.ERobot.ik_lm_chan.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +ERobot.ik\_lm\_chan +============ + +.. currentmodule:: roboticstoolbox.robot.ERobot + +.. automethod:: roboticstoolbox.robot.ERobot.ERobot.ik_lm_chan \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.ERobot.ERobot.ik_lm_sugihara.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.ERobot.ERobot.ik_lm_sugihara.rst.txt new file mode 100644 index 00000000..dd1d4885 --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.ERobot.ERobot.ik_lm_sugihara.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +ERobot.ik\_lm\_sugihara +================ + +.. currentmodule:: roboticstoolbox.robot.ERobot + +.. automethod:: roboticstoolbox.robot.ERobot.ERobot.ik_lm_sugihara \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.ERobot.ERobot.ik_lm_wampler.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.ERobot.ERobot.ik_lm_wampler.rst.txt new file mode 100644 index 00000000..0dd62f6b --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.ERobot.ERobot.ik_lm_wampler.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +ERobot.ik\_lm\_wampler +=============== + +.. currentmodule:: roboticstoolbox.robot.ERobot + +.. automethod:: roboticstoolbox.robot.ERobot.ERobot.ik_lm_wampler \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.ERobot.ERobot.ik_nr.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.ERobot.ERobot.ik_nr.rst.txt new file mode 100644 index 00000000..66c3b2b9 --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.ERobot.ERobot.ik_nr.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +ERobot.ik\_nr +====== + +.. currentmodule:: roboticstoolbox.robot.ERobot + +.. automethod:: roboticstoolbox.robot.ERobot.ERobot.ik_nr \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.ERobot.ERobot.ikine_LM.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.ERobot.ERobot.ikine_LM.rst.txt new file mode 100644 index 00000000..3e9cd371 --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.ERobot.ERobot.ikine_LM.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +ERobot.ikine\_LM +========= + +.. currentmodule:: roboticstoolbox.robot.ERobot + +.. automethod:: roboticstoolbox.robot.ERobot.ERobot.ikine_LM \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_GN.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_GN.rst.txt new file mode 100644 index 00000000..ca4e3e0e --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_GN.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +ETS.ik\_GN +====== + +.. currentmodule:: roboticstoolbox.robot.ETS + +.. automethod:: roboticstoolbox.robot.ETS.ETS.ik_GN \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_LM.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_LM.rst.txt new file mode 100644 index 00000000..5cfdf9ca --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_LM.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +ETS.ik\_LM +====== + +.. currentmodule:: roboticstoolbox.robot.ETS + +.. automethod:: roboticstoolbox.robot.ETS.ETS.ik_LM \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_NR.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_NR.rst.txt new file mode 100644 index 00000000..18feaa8b --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_NR.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +ETS.ik\_NR +====== + +.. currentmodule:: roboticstoolbox.robot.ETS + +.. automethod:: roboticstoolbox.robot.ETS.ETS.ik_NR \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_gn.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_gn.rst.txt new file mode 100644 index 00000000..ca4e3e0e --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_gn.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +ETS.ik\_GN +====== + +.. currentmodule:: roboticstoolbox.robot.ETS + +.. automethod:: roboticstoolbox.robot.ETS.ETS.ik_GN \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_lm_chan.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_lm_chan.rst.txt new file mode 100644 index 00000000..774638c3 --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_lm_chan.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +ETS.ik\_lm\_chan +============ + +.. currentmodule:: roboticstoolbox.robot.ETS + +.. automethod:: roboticstoolbox.robot.ETS.ETS.ik_lm_chan \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_lm_sugihara.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_lm_sugihara.rst.txt new file mode 100644 index 00000000..a0964807 --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_lm_sugihara.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +ETS.ik\_lm\_sugihara +================ + +.. currentmodule:: roboticstoolbox.robot.ETS + +.. automethod:: roboticstoolbox.robot.ETS.ETS.ik_lm_sugihara \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_lm_wampler.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_lm_wampler.rst.txt new file mode 100644 index 00000000..f0ef8c41 --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_lm_wampler.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +ETS.ik\_lm\_wampler +=============== + +.. currentmodule:: roboticstoolbox.robot.ETS + +.. automethod:: roboticstoolbox.robot.ETS.ETS.ik_lm_wampler \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_nr.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_nr.rst.txt new file mode 100644 index 00000000..18feaa8b --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.ETS.ETS.ik_nr.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +ETS.ik\_NR +====== + +.. currentmodule:: roboticstoolbox.robot.ETS + +.. automethod:: roboticstoolbox.robot.ETS.ETS.ik_NR \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.ETS.ETS.ikine_GN.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.ETS.ETS.ikine_GN.rst.txt new file mode 100644 index 00000000..8af50e3d --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.ETS.ETS.ikine_GN.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +ETS.ikine\_GN +========= + +.. currentmodule:: roboticstoolbox.robot.ETS + +.. automethod:: roboticstoolbox.robot.ETS.ETS.ikine_GN \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.ETS.ETS.ikine_LM.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.ETS.ETS.ikine_LM.rst.txt new file mode 100644 index 00000000..d803faf5 --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.ETS.ETS.ikine_LM.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +ETS.ikine\_LM +========= + +.. currentmodule:: roboticstoolbox.robot.ETS + +.. automethod:: roboticstoolbox.robot.ETS.ETS.ikine_LM \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.ETS.ETS.ikine_NR.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.ETS.ETS.ikine_NR.rst.txt new file mode 100644 index 00000000..1bff1032 --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.ETS.ETS.ikine_NR.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +ETS.ikine\_NR +========= + +.. currentmodule:: roboticstoolbox.robot.ETS + +.. automethod:: roboticstoolbox.robot.ETS.ETS.ikine_NR \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.ETS.ETS.ikine_QP.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.ETS.ETS.ikine_QP.rst.txt new file mode 100644 index 00000000..f2a06048 --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.ETS.ETS.ikine_QP.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +ETS.ikine\_QP +========= + +.. currentmodule:: roboticstoolbox.robot.ETS + +.. automethod:: roboticstoolbox.robot.ETS.ETS.ikine_QP \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.IK.IKSolution.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.IK.IKSolution.rst.txt new file mode 100644 index 00000000..813c4992 --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.IK.IKSolution.rst.txt @@ -0,0 +1,33 @@ +roboticstoolbox.robot.IK.IKSolution +=================================== + +.. currentmodule:: roboticstoolbox.robot.IK + +.. autoclass:: IKSolution + + + .. automethod:: __init__ + + + .. rubric:: Methods + + .. autosummary:: + + ~IKSolution.__init__ + + + + + + .. rubric:: Attributes + + .. autosummary:: + + ~IKSolution.iterations + ~IKSolution.reason + ~IKSolution.residual + ~IKSolution.searches + ~IKSolution.q + ~IKSolution.success + + \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.IK.IKSolver._check_jl.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.IK.IKSolver._check_jl.rst.txt new file mode 100644 index 00000000..a44d0046 --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.IK.IKSolver._check_jl.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +IKSolver.\_check\_jl +=========== + +.. currentmodule:: roboticstoolbox.robot.IK + +.. automethod:: roboticstoolbox.robot.IK.IKSolver._check_jl \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.IK.IKSolver._random_q.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.IK.IKSolver._random_q.rst.txt new file mode 100644 index 00000000..e2ac97a5 --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.IK.IKSolver._random_q.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +IKSolver.\_random\_q +=========== + +.. currentmodule:: roboticstoolbox.robot.IK + +.. automethod:: roboticstoolbox.robot.IK.IKSolver._random_q \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.IK.IKSolver.error.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.IK.IKSolver.error.rst.txt new file mode 100644 index 00000000..1ecf6e39 --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.IK.IKSolver.error.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +IKSolver.error +===== + +.. currentmodule:: roboticstoolbox.robot.IK + +.. automethod:: roboticstoolbox.robot.IK.IKSolver.error \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.IK.IKSolver.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.IK.IKSolver.rst.txt new file mode 100644 index 00000000..9f4fba00 --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.IK.IKSolver.rst.txt @@ -0,0 +1,25 @@ +roboticstoolbox.robot.IK.IKSolver +================================= + +.. currentmodule:: roboticstoolbox.robot.IK + +.. autoclass:: IKSolver + + + .. automethod:: __init__ + + + .. rubric:: Methods + + .. autosummary:: + + ~IKSolver.__init__ + ~IKSolver.error + ~IKSolver.solve + ~IKSolver.step + + + + + + \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.IK.IKSolver.solve.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.IK.IKSolver.solve.rst.txt new file mode 100644 index 00000000..657efe5a --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.IK.IKSolver.solve.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +IKSolver.solve +===== + +.. currentmodule:: roboticstoolbox.robot.IK + +.. automethod:: roboticstoolbox.robot.IK.IKSolver.solve \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.IK.IKSolver.step.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.IK.IKSolver.step.rst.txt new file mode 100644 index 00000000..a920c8b4 --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.IK.IKSolver.step.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +IKSolver.step +==== + +.. currentmodule:: roboticstoolbox.robot.IK + +.. automethod:: roboticstoolbox.robot.IK.IKSolver.step \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_GN._check_jl.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_GN._check_jl.rst.txt new file mode 100644 index 00000000..f6a04505 --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_GN._check_jl.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +IK_GN.\_check\_jl +=========== + +.. currentmodule:: roboticstoolbox.robot.IK + +.. automethod:: roboticstoolbox.robot.IK.IK_GN._check_jl \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_GN._random_q.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_GN._random_q.rst.txt new file mode 100644 index 00000000..2f53e86b --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_GN._random_q.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +IK_GN.\_random\_q +=========== + +.. currentmodule:: roboticstoolbox.robot.IK + +.. automethod:: roboticstoolbox.robot.IK.IK_GN._random_q \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_GN.error.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_GN.error.rst.txt new file mode 100644 index 00000000..b5470417 --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_GN.error.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +IK_GN.error +===== + +.. currentmodule:: roboticstoolbox.robot.IK + +.. automethod:: roboticstoolbox.robot.IK.IK_GN.error \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_GN.solve.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_GN.solve.rst.txt new file mode 100644 index 00000000..cd10a778 --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_GN.solve.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +IK_GN.solve +===== + +.. currentmodule:: roboticstoolbox.robot.IK + +.. automethod:: roboticstoolbox.robot.IK.IK_GN.solve \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_GN.step.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_GN.step.rst.txt new file mode 100644 index 00000000..06478d52 --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_GN.step.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +IK_GN.step +==== + +.. currentmodule:: roboticstoolbox.robot.IK + +.. automethod:: roboticstoolbox.robot.IK.IK_GN.step \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_LM._check_jl.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_LM._check_jl.rst.txt new file mode 100644 index 00000000..f49f0296 --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_LM._check_jl.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +IK_LM.\_check\_jl +=========== + +.. currentmodule:: roboticstoolbox.robot.IK + +.. automethod:: roboticstoolbox.robot.IK.IK_LM._check_jl \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_LM._random_q.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_LM._random_q.rst.txt new file mode 100644 index 00000000..51717cba --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_LM._random_q.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +IK_LM.\_random\_q +=========== + +.. currentmodule:: roboticstoolbox.robot.IK + +.. automethod:: roboticstoolbox.robot.IK.IK_LM._random_q \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_LM.error.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_LM.error.rst.txt new file mode 100644 index 00000000..0892ec42 --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_LM.error.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +IK_LM.error +===== + +.. currentmodule:: roboticstoolbox.robot.IK + +.. automethod:: roboticstoolbox.robot.IK.IK_LM.error \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_LM.solve.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_LM.solve.rst.txt new file mode 100644 index 00000000..26a7075a --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_LM.solve.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +IK_LM.solve +===== + +.. currentmodule:: roboticstoolbox.robot.IK + +.. automethod:: roboticstoolbox.robot.IK.IK_LM.solve \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_LM.step.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_LM.step.rst.txt new file mode 100644 index 00000000..c830e597 --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_LM.step.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +IK_LM.step +==== + +.. currentmodule:: roboticstoolbox.robot.IK + +.. automethod:: roboticstoolbox.robot.IK.IK_LM.step \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_NR._check_jl.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_NR._check_jl.rst.txt new file mode 100644 index 00000000..a92f4642 --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_NR._check_jl.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +IK_NR.\_check\_jl +=========== + +.. currentmodule:: roboticstoolbox.robot.IK + +.. automethod:: roboticstoolbox.robot.IK.IK_NR._check_jl \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_NR._random_q.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_NR._random_q.rst.txt new file mode 100644 index 00000000..84dd879b --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_NR._random_q.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +IK_NR.\_random\_q +=========== + +.. currentmodule:: roboticstoolbox.robot.IK + +.. automethod:: roboticstoolbox.robot.IK.IK_NR._random_q \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_NR.error.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_NR.error.rst.txt new file mode 100644 index 00000000..cd58f936 --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_NR.error.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +IK_NR.error +===== + +.. currentmodule:: roboticstoolbox.robot.IK + +.. automethod:: roboticstoolbox.robot.IK.IK_NR.error \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_NR.solve.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_NR.solve.rst.txt new file mode 100644 index 00000000..689014fd --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_NR.solve.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +IK_NR.solve +===== + +.. currentmodule:: roboticstoolbox.robot.IK + +.. automethod:: roboticstoolbox.robot.IK.IK_NR.solve \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_NR.step.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_NR.step.rst.txt new file mode 100644 index 00000000..2b5cbbab --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_NR.step.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +IK_NR.step +==== + +.. currentmodule:: roboticstoolbox.robot.IK + +.. automethod:: roboticstoolbox.robot.IK.IK_NR.step \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_QP._check_jl.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_QP._check_jl.rst.txt new file mode 100644 index 00000000..6dfb4df3 --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_QP._check_jl.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +IK_QP.\_check\_jl +=========== + +.. currentmodule:: roboticstoolbox.robot.IK + +.. automethod:: roboticstoolbox.robot.IK.IK_QP._check_jl \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_QP._random_q.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_QP._random_q.rst.txt new file mode 100644 index 00000000..efc847e8 --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_QP._random_q.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +IK_QP.\_random\_q +=========== + +.. currentmodule:: roboticstoolbox.robot.IK + +.. automethod:: roboticstoolbox.robot.IK.IK_QP._random_q \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_QP.error.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_QP.error.rst.txt new file mode 100644 index 00000000..43f37145 --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_QP.error.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +IK_QP.error +===== + +.. currentmodule:: roboticstoolbox.robot.IK + +.. automethod:: roboticstoolbox.robot.IK.IK_QP.error \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_QP.solve.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_QP.solve.rst.txt new file mode 100644 index 00000000..44083aaf --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_QP.solve.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +IK_QP.solve +===== + +.. currentmodule:: roboticstoolbox.robot.IK + +.. automethod:: roboticstoolbox.robot.IK.IK_QP.solve \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_QP.step.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_QP.step.rst.txt new file mode 100644 index 00000000..469e4749 --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.IK.IK_QP.step.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +IK_QP.step +==== + +.. currentmodule:: roboticstoolbox.robot.IK + +.. automethod:: roboticstoolbox.robot.IK.IK_QP.step \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.Robot.Robot.ik_GN.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.Robot.Robot.ik_GN.rst.txt new file mode 100644 index 00000000..407d8871 --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.Robot.Robot.ik_GN.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +Robot.ik\_GN +====== + +.. currentmodule:: roboticstoolbox.robot.Robot + +.. automethod:: roboticstoolbox.robot.Robot.Robot.ik_GN \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.Robot.Robot.ik_LM.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.Robot.Robot.ik_LM.rst.txt new file mode 100644 index 00000000..1a1e970c --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.Robot.Robot.ik_LM.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +Robot.ik\_LM +====== + +.. currentmodule:: roboticstoolbox.robot.Robot + +.. automethod:: roboticstoolbox.robot.Robot.Robot.ik_LM \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.Robot.Robot.ik_NR.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.Robot.Robot.ik_NR.rst.txt new file mode 100644 index 00000000..8238f56a --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.Robot.Robot.ik_NR.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +Robot.ik\_NR +====== + +.. currentmodule:: roboticstoolbox.robot.Robot + +.. automethod:: roboticstoolbox.robot.Robot.Robot.ik_NR \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.Robot.Robot.ikine_GN.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.Robot.Robot.ikine_GN.rst.txt new file mode 100644 index 00000000..908b999b --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.Robot.Robot.ikine_GN.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +Robot.ikine\_GN +========= + +.. currentmodule:: roboticstoolbox.robot.Robot + +.. automethod:: roboticstoolbox.robot.Robot.Robot.ikine_GN \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.Robot.Robot.ikine_LM.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.Robot.Robot.ikine_LM.rst.txt new file mode 100644 index 00000000..8be1903f --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.Robot.Robot.ikine_LM.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +Robot.ikine\_LM +========= + +.. currentmodule:: roboticstoolbox.robot.Robot + +.. automethod:: roboticstoolbox.robot.Robot.Robot.ikine_LM \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.Robot.Robot.ikine_NR.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.Robot.Robot.ikine_NR.rst.txt new file mode 100644 index 00000000..1192f7df --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.Robot.Robot.ikine_NR.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +Robot.ikine\_NR +========= + +.. currentmodule:: roboticstoolbox.robot.Robot + +.. automethod:: roboticstoolbox.robot.Robot.Robot.ikine_NR \ No newline at end of file diff --git a/_sources/IK/stubs/roboticstoolbox.robot.Robot.Robot.ikine_QP.rst.txt b/_sources/IK/stubs/roboticstoolbox.robot.Robot.Robot.ikine_QP.rst.txt new file mode 100644 index 00000000..b766ab3b --- /dev/null +++ b/_sources/IK/stubs/roboticstoolbox.robot.Robot.Robot.ikine_QP.rst.txt @@ -0,0 +1,8 @@ +:orphan: + +Robot.ikine\_QP +========= + +.. currentmodule:: roboticstoolbox.robot.Robot + +.. automethod:: roboticstoolbox.robot.Robot.Robot.ikine_QP \ No newline at end of file diff --git a/docs/source/PGraph.rst b/_sources/PGraph.rst.txt similarity index 100% rename from docs/source/PGraph.rst rename to _sources/PGraph.rst.txt diff --git a/_sources/arm.rst.txt b/_sources/arm.rst.txt new file mode 100644 index 00000000..035ab26f --- /dev/null +++ b/_sources/arm.rst.txt @@ -0,0 +1,11 @@ +********** +Robot Arms +********** + +.. toctree:: + :maxdepth: 4 + + arm_models + .. arm_backend + arm_trajectory + IK/ik \ No newline at end of file diff --git a/_sources/arm_backend.rst.txt b/_sources/arm_backend.rst.txt new file mode 100644 index 00000000..e14c48bc --- /dev/null +++ b/_sources/arm_backend.rst.txt @@ -0,0 +1,10 @@ +Backends +======== + +The Robotics Toolbox supports a number of backends, through a standard API, to communicate with simulators or +physical robots. + +.. toctree:: + :maxdepth: 2 + + arm_backend_pyplot diff --git a/_sources/arm_backend_pyplot.rst.txt b/_sources/arm_backend_pyplot.rst.txt new file mode 100644 index 00000000..0175ae20 --- /dev/null +++ b/_sources/arm_backend_pyplot.rst.txt @@ -0,0 +1,8 @@ +PyPlot (matplotlib) +------------------- + +.. automodule:: roboticstoolbox.backends.PyPlot + :members: + :undoc-members: + :show-inheritance: + :inherited-members: diff --git a/_sources/arm_backend_swift.rst.txt b/_sources/arm_backend_swift.rst.txt new file mode 100644 index 00000000..02f7e752 --- /dev/null +++ b/_sources/arm_backend_swift.rst.txt @@ -0,0 +1,8 @@ +Swift +----- + +.. automodule:: roboticstoolbox.backends.Swift + :members: + :undoc-members: + :show-inheritance: + :inherited-members: \ No newline at end of file diff --git a/_sources/arm_backend_vpython.rst.txt b/_sources/arm_backend_vpython.rst.txt new file mode 100644 index 00000000..550474b2 --- /dev/null +++ b/_sources/arm_backend_vpython.rst.txt @@ -0,0 +1,9 @@ +VPython +------- + +.. automodule:: roboticstoolbox.backends.VPython + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + \ No newline at end of file diff --git a/_sources/arm_dh.rst.txt b/_sources/arm_dh.rst.txt new file mode 100644 index 00000000..4535007d --- /dev/null +++ b/_sources/arm_dh.rst.txt @@ -0,0 +1,85 @@ +Denavit-Hartenberg models +========================= + +.. codeauthor:: Jesse Haviland + +This class is used to model robots defined in terms of standard or modified +Denavit-Hartenberg notation. + +.. note:: These classes provide similar functionality and notation to MATLAB Toolbox ``SerialLink`` and + ``Link`` classes. + +DHRobot +------- + +.. inheritance-diagram:: roboticstoolbox.DHRobot + :top-classes: roboticstoolbox.Robot + :parts: 2 + +The various :ref:`DH Models` all subclass this class. + +.. autoclass:: roboticstoolbox.robot.DHRobot.DHRobot + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + +DHLink +------ + +The ``DHRobot`` is defined by a list of ``DHLink`` subclass objects. + +.. inheritance-diagram:: roboticstoolbox.RevoluteDH roboticstoolbox.PrismaticDH roboticstoolbox.RevoluteMDH roboticstoolbox.PrismaticMDH + :top-classes: roboticstoolbox.robot.Link + :parts: 2 + + +.. autoclass:: roboticstoolbox.robot.DHLink + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + +Revolute - standard DH +^^^^^^^^^^^^^^^^^^^^^^ + +.. autoclass:: roboticstoolbox.robot.DHLink.RevoluteDH + :show-inheritance: + +Prismatic - standard DH +^^^^^^^^^^^^^^^^^^^^^^^ + +.. autoclass:: roboticstoolbox.robot.DHLink.PrismaticDH + :show-inheritance: + + +Revolute - modified DH +^^^^^^^^^^^^^^^^^^^^^^ + +.. autoclass:: roboticstoolbox.robot.DHLink.RevoluteMDH + :show-inheritance: + +Prismatic - modified DH +^^^^^^^^^^^^^^^^^^^^^^^ + +.. autoclass:: roboticstoolbox.robot.DHLink.PrismaticMDH + :show-inheritance: + + +.. _DH Models: + +Models +------ + +A number of models are defined in terms of Denavit-Hartenberg parameters, either +standard or modified. They can be listed by: + +.. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> rtb.models.list(mtype="DH") + +.. automodule:: roboticstoolbox.models.DH + :members: + :undoc-members: + :show-inheritance: \ No newline at end of file diff --git a/_sources/arm_erobot.rst.txt b/_sources/arm_erobot.rst.txt new file mode 100644 index 00000000..80401a94 --- /dev/null +++ b/_sources/arm_erobot.rst.txt @@ -0,0 +1,73 @@ +ERobot models +============= + +.. codeauthor:: Jesse Haviland + + +ERobot +------ + +.. inheritance-diagram:: roboticstoolbox.ERobot + :top-classes: roboticstoolbox.Robot + :parts: 2 + +The various models :ref:`E Models` all subclass this class. + +.. automodule:: roboticstoolbox.robot.ERobot + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + +ELink +------- + +The ``ERobot`` is defined by a tree of ``ELink`` subclass objects. + +.. inheritance-diagram:: roboticstoolbox.ELink + :top-classes: roboticstoolbox.robot.Link + :parts: 2 + +.. automodule:: roboticstoolbox.robot.ELink + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + + +.. _E Models: + +ERobot models +------------- + +Defined using ETS +^^^^^^^^^^^^^^^^^ + +A number of models are defined in terms of elementary transform sequences. +They can be listed by: + +.. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> rtb.models.list(mtype="ETS") + +.. automodule:: roboticstoolbox.models.ETS + :members: + :undoc-members: + :show-inheritance: + +Defined from URDF +^^^^^^^^^^^^^^^^^ + +A number of models are defined in terms of Denavit-Hartenberg parameters, either +standard or modified. They can be listed by: + +.. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> rtb.models.list(mtype="URDF") + +.. automodule:: roboticstoolbox.models.URDF + :members: + :undoc-members: + :show-inheritance: \ No newline at end of file diff --git a/_sources/arm_ets.rst.txt b/_sources/arm_ets.rst.txt new file mode 100644 index 00000000..251b4449 --- /dev/null +++ b/_sources/arm_ets.rst.txt @@ -0,0 +1,85 @@ +Elementary transform sequence (ETS) models +========================================== + +.. codeauthor:: Jesse Haviland + +Elementary transforms are canonic rotations or translations about, or along, +the x-, y- or z-axes. The amount of rotation or translation can be a constant +or a variable. A variable amount corresponds to a joint. + +Consider the example: + +.. runblock:: pycon + :linenos: + + >>> from roboticstoolbox import ET + >>> ET.Rx(45, 'deg') + >>> ET.tz(0.75) + >>> e = ET.Rx(0.3) * ET.tz(0.75) + >>> print(e) + >>> e.fkine([]) + +In lines 2-5 we defined two elementary transforms. Line 2 defines a rotation +of 45° about the x-axis, and line 4 defines a translation of 0.75m along the z-axis. +Compounding them in line 6, the result is the two elementary transforms in a +sequence - an elementary transform sequence (ETS). An ETS can be arbitrarily +long. + +Line 8 *evaluates* the forward kinematics of the sequence, substituting in values, +and the result is an SE(3) matrix encapsulated in an ``4x4`` numpy array. + +The real power comes from having variable arguments to the elementary transforms +as shown in this example where we define a simple two link planar manipulator. + + +.. runblock:: pycon + :linenos: + + >>> from roboticstoolbox import ET + >>> e = ET.Rz() * ET.tx(1) * ET.Rz() * ET.tx(1) + >>> print(e) + >>> len(e) + >>> e[1:3] + >>> e.fkine([0, 0]) + >>> e.fkine([1.57, -1.57]) + +Line 2 succinctly describes the kinematics in terms of elementary transforms: a +rotation around the z-axis by the first joint angle, then a translation in the +x-direction, then a rotation around the z-axis by the second joint angle, and +finally a translation in the x-direction. + +Line 3 creates the elementary transform sequence, with variable transforms. +``e`` is a single object that encapsulates a list of elementary transforms, and list like +methods such as ``len`` as well as indexing and slicing as shown in lines 5-8. + +Lines 9-18 *evaluate* the sequence, and substitutes elements from the passed +arrays as the joint variables. + +This approach is general enough to be able to describe any serial-link robot +manipulator. For a branched manipulator we can use ETS to describe the +connections between every parent and child link pair. + +The ETS inherits list-like properties and has methods like ``reverse`` and ``pop``. + +**Reference:** + + - `A simple and systematic approach to assigning Denavit-Hartenberg parameters `_. + Peter I. Corke, IEEE Transactions on Robotics, 23(3), pp 590-594, June 2007. + +ETS - 3D +-------- + +.. autoclass:: roboticstoolbox.robot.ETS.ETS + :members: __str__, __repr__, __mul__, __getitem__, n, m, structure, joints, jointset, split, inv, compile, insert, fkine, jacob0, jacobe, hessian0, hessiane + :undoc-members: + :show-inheritance: + :member-order: bysource + +ETS - 2D +-------- + +.. autoclass:: roboticstoolbox.robot.ETS.ETS2 + :members: __str__, __repr__, __mul__, __getitem__, n, m, structure, joints, jointset, split, inv, compile, insert, fkine, jacob0, jacobe + :undoc-members: + :show-inheritance: + diff --git a/_sources/arm_models.rst.txt b/_sources/arm_models.rst.txt new file mode 100644 index 00000000..decca10a --- /dev/null +++ b/_sources/arm_models.rst.txt @@ -0,0 +1,10 @@ +Manipulator models +================== + +.. toctree:: + :maxdepth: 2 + + arm_dh + arm_ets + arm_erobot + arm_superclass \ No newline at end of file diff --git a/_sources/arm_superclass.rst.txt b/_sources/arm_superclass.rst.txt new file mode 100644 index 00000000..6fecf35c --- /dev/null +++ b/_sources/arm_superclass.rst.txt @@ -0,0 +1,44 @@ +Superclasses +============ + + +Robot +------- + +.. inheritance-diagram:: roboticstoolbox.ERobot + :top-classes: roboticstoolbox.Robot + :parts: 2 + +The various models :ref:`E Models` all subclass this class. + +.. automodule:: roboticstoolbox.robot.Robot + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :special-members: __getitem__ + +Kinematic cache +--------------- + +.. automodule:: roboticstoolbox.robot.KinematicCache + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :special-members: __init__ + +Link +---- + +The ``ERobot`` is defined by a tree of ``ELink`` subclass objects. + +.. inheritance-diagram:: roboticstoolbox.Link + :top-classes: roboticstoolbox.robot.Link + :parts: 2 + +.. automodule:: roboticstoolbox.robot.Link + :members: + :undoc-members: + :show-inheritance: + :inherited-members: diff --git a/_sources/arm_trajectory.rst.txt b/_sources/arm_trajectory.rst.txt new file mode 100644 index 00000000..253d37af --- /dev/null +++ b/_sources/arm_trajectory.rst.txt @@ -0,0 +1,10 @@ +Trajectories +************ + +.. automodule:: roboticstoolbox.tools.trajectory + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :special-members: + :exclude-members: __dict__, __module__, __weakref__ \ No newline at end of file diff --git a/_sources/blocks-arm.rst.txt b/_sources/blocks-arm.rst.txt new file mode 100644 index 00000000..16139a2e --- /dev/null +++ b/_sources/blocks-arm.rst.txt @@ -0,0 +1,13 @@ +Robot manipulator blocks +======================== + +.. image:: https://raw.githubusercontent.com/petercorke/bdsim/master/figs/BDSimLogo_NoBackgnd@2x.png + :width: 300 + +.. automodule:: roboticstoolbox.blocks.arm + :members: + :undoc-members: + :show-inheritance: + :special-members: __init__ + :exclude-members: output, reset, step, start, done, deriv, nin, nout, inlabels, outlabels + diff --git a/_sources/blocks-mobile.rst.txt b/_sources/blocks-mobile.rst.txt new file mode 100644 index 00000000..b0bbdcdb --- /dev/null +++ b/_sources/blocks-mobile.rst.txt @@ -0,0 +1,14 @@ +Mobile robot blocks +=================== + +.. image:: https://raw.githubusercontent.com/petercorke/bdsim/master/figs/BDSimLogo_NoBackgnd@2x.png + :width: 300 + +.. automodule:: roboticstoolbox.blocks.mobile + :members: + :undoc-members: + :show-inheritance: + :special-members: __init__ + :exclude-members: output, reset, step, start, done, deriv, nin, nout, inlabels, outlabels + + diff --git a/_sources/blocks-spatial.rst.txt b/_sources/blocks-spatial.rst.txt new file mode 100644 index 00000000..2a44ce62 --- /dev/null +++ b/_sources/blocks-spatial.rst.txt @@ -0,0 +1,13 @@ +Spatial maths blocks +==================== + +.. image:: https://raw.githubusercontent.com/petercorke/bdsim/master/figs/BDSimLogo_NoBackgnd@2x.png + :width: 300 + +.. automodule:: roboticstoolbox.blocks.spatial + :members: + :undoc-members: + :show-inheritance: + :special-members: __init__ + :exclude-members: output, reset, step, start, done, deriv, nin, nout, inlabels, outlabels + diff --git a/_sources/blocks-uav.rst.txt b/_sources/blocks-uav.rst.txt new file mode 100644 index 00000000..60fd78b8 --- /dev/null +++ b/_sources/blocks-uav.rst.txt @@ -0,0 +1,13 @@ +UAV blocks +========== + +.. image:: https://raw.githubusercontent.com/petercorke/bdsim/master/figs/BDSimLogo_NoBackgnd@2x.png + :width: 300 + +.. automodule:: roboticstoolbox.blocks.uav + :members: + :undoc-members: + :show-inheritance: + :special-members: __init__ + :exclude-members: output, reset, step, start, done, deriv, nin, nout, inlabels, outlabels + diff --git a/_sources/blocks.rst.txt b/_sources/blocks.rst.txt new file mode 100644 index 00000000..0457e311 --- /dev/null +++ b/_sources/blocks.rst.txt @@ -0,0 +1,17 @@ +bdsim blocks +============ + +.. image:: https://raw.githubusercontent.com/petercorke/bdsim/master/figs/BDSimLogo_NoBackgnd@2x.png + :width: 400 + +A set of block definitions that add robotic capability to the `bdsim `_ +block diagram simulation environment. + +.. toctree:: + :maxdepth: 2 + + blocks-spatial + blocks-arm + blocks-mobile + blocks-uav + diff --git a/docs/source/dgraph.rst b/_sources/dgraph.rst.txt similarity index 100% rename from docs/source/dgraph.rst rename to _sources/dgraph.rst.txt diff --git a/docs/source/directed.rst b/_sources/directed.rst.txt similarity index 100% rename from docs/source/directed.rst rename to _sources/directed.rst.txt diff --git a/docs/source/dvertex.rst b/_sources/dvertex.rst.txt similarity index 100% rename from docs/source/dvertex.rst rename to _sources/dvertex.rst.txt diff --git a/docs/source/edge.rst b/_sources/edge.rst.txt similarity index 100% rename from docs/source/edge.rst rename to _sources/edge.rst.txt diff --git a/docs/source/index.rst b/_sources/index.rst.txt similarity index 100% rename from docs/source/index.rst rename to _sources/index.rst.txt diff --git a/_sources/intro.rst.txt b/_sources/intro.rst.txt new file mode 100644 index 00000000..569f553b --- /dev/null +++ b/_sources/intro.rst.txt @@ -0,0 +1,780 @@ +************ +Introduction +************ + + +Introduction +============ + +This is a modified version of a paper accepted to ICRA2021 [corke21a]_. + +The Robotics Toolbox for MATLAB® (RTB-M) was created around 1991 to support +Peter Corke’s PhD research and was first published in 1995-6 [Corke95]_ +[Corke96]_. It has evolved over 25 years to track changes and improvements to +the MATLAB language and ecosystem, such as the addition of structures, objects, +lists (cell arrays) and strings, myriad of other improvements to the language, +new graphics and new tools such as IDE, debugger, notebooks (LiveScripts), apps +and continuous integration. An adverse consequence is that many poor (in +retrospect) early design decisions hinder development. + +Over time additional functionality was added, in particular for vision, and two +major refactorings led to the current state of three toolboxes: Robotics Toolbox +for MATLAB and Machine Vision Toolbox for MATLAB (1999) both of which are built +on the Spatial Math Toolbox for MATLAB (SMTB-M) in 2019. + +The code was formally open sourced to support its use for the third edition of +John Craig’s book [Craig2005]_. It was hosted on ftp sites, personal web +servers, Google code and currently GitHub and maintained under a succession of +version control tools including rcs, cvs, svn and git. A support forum on +Google Groups was established in 2008 and as of 2020 has over 1400 members. + + +A Python version +================ + +The imperative for a Python version has long existed and the first port was +started in 2008 but ultimately failed for lack of ongoing resources to complete +a sufficient subset of functionality. Subsequent attempts have all met the same +fate. + +The design goals of this version can be summarised as new functionality: + +* A superset of the MATLAB Toolbox functionality +* Build on the Spatial Math Toolbox for Python [SMTB-P]_ which provides objects to + represent rotations as SO(2) and SE(3) matrices as well as unit-quaternions; + rigid-body motions as SE(2) and SE(3) matrices or twists in + se(2) and se(3); and Featherstone’s spatial vectors [Featherstone87]_. +* Support models expressed using Denavit-Hartenberg notation (standard and + modified), elementary transform sequences [Corke07]_ [Haviland20]_, and URDF-style + rigid-body trees. Support branched, but not closed-loop or parallel, robots +* Collision checking + +and improved software engineering: + +* Use Python 3 (3.6 and greater) +* Utilize WebGL and Javascript graphics technologies +* Documentation in ReStructured Text using Sphinx and delivered via GitHub pages. +* Hosted on GitHub with continuous integration using GitHub actions +* High code-quality metrics for test coverage and automated code review and security analysis +* As few dependencies as possible, in particular being able to work with ROS but not be dependent on ROS. This sidesteps ROS constraints on operating system and Python versions. +* Modular approach to interfacing to different graphics libraries, simulators and physical robots. +* Support Python notebooks which allows publication of static notebooks (for example via GitHub) and interactive online notebooks (`MyBinder.org `_). +* Use of UniCode characters to make console output easier to read + + +Spatial math layer +================== + +Robotics and computer vision require us to describe position, orientation and +pose in 3D space. Mobile robotics has the same requirement, but generally for 2D +space. We therefore need tools to represent quantities such as rigid-body +transformations (matrices :math:`\in \SE{n}` or twists :math:`\in \se{n}`), +rotations (matrices :math:`\in \SO{n}` or :math:`\so{n}`, Euler or roll-pitch-yaw +angles, or unit quaternions :math:`\in \mathrm{S}^3`). Such capability is amongst the oldest in +RTB-M and the equivalent functionality exists in RTB-P which makes use of the +Spatial Maths Toolbox for Python (SMTB-P) [SMTB-P]_. For example: + +.. runblock:: pycon + + >>> from spatialmath.base import * + >>> T = transl(0.5, 0.0, 0.0) @ rpy2tr(0.1, 0.2, 0.3, order='xyz') @ trotx(-90, 'deg') + >>> print(T) + +There is strong similarity to the equivalent MATLAB case apart from the use of +the ``@`` operator, the use of keyword arguments instead of keyword-value pairs, +and the format of the printed array. All the *classic* RTB-M functions are +provided in the ``spatialmath.base`` package as well as additional functions for +quaternions, vectors, twists and argument handling. There are also functions to +perform interpolation, plot and animate coordinate frames, and create movies, +using matplotlib. The underlying datatypes in all cases are 1D and 2D NumPy +arrays. + +.. warning:: For a user transitioning from MATLAB the most significant difference is + the use of 1D arrays -- all MATLAB arrays have two dimensions, even if one of + them is equal to one. + +However some challenges arise when using arrays, whether native MATLAB matrices +or NumPy arrays as in this case. Firstly, arrays are not typed and for example a +:math:`3 \times 3` array could be an element of :math:`\SE{2}` or +:math:`\SO{3}` or an arbitrary matrix. + +Secondly, the operators we need for poses are a subset of those available for +matrices, and some operators may need to be redefined in a specific way. For +example, :math:`\SE{3} * \SE{3} \rightarrow \SE{3}` but :math:`\SE{3} + \SE{3} \rightarrow \mathbb{R}^{4 \times 4}`, and equality testing for a +unit-quaternion has to respect the double mapping. + +Thirdly, in robotics we often need to represent time sequences of poses. We +could add an extra dimension to the matrices representing rigid-body +transformations or unit-quaternions, or place them in a list. The first +approach is cumbersome and reduces code clarity, while the second cannot ensure +that all elements of the list have the same type. + +We use classes and data encapsulation to address all these issues. SMTB-P +provides abstraction classes ``SE3``, ``Twist3``, ``SO3``, ``UnitQuaternion``, +``SE2``, ``Twist2`` and ``SO2``. For example, the previous example could be written +as: + +.. runblock:: pycon + :linenos: + + >>> from spatialmath import * + >>> T = SE3(0.5, 0.0, 0.0) * SE3.RPY([0.1, 0.2, 0.3], order='xyz') * SE3.Rx(-90, unit='deg') + >>> print(T) + >>> T.eul() + >>> T.R + >>> T.t + +where composition is denoted by the ``*`` operator and the matrix is printed more elegantly (and elements are color +coded at the console or in ipython). +``SE3.RPY()`` is a class method that acts like a constructor, creating an ``SE3`` instance from a set of roll-pitch-yaw angles, +and ``SE3.Rx()`` creates an ``SE3`` instance from a pure rotation about the x-axis. +Attempts to compose with a non ``SE3`` instance would result in a ``TypeError``. + +The orientation of the new coordinate frame may be expressed in terms of Euler angles (line 9) +and components can be extracted such as the rotation submatrix (line 11) and translation (line 15). + +The pose ``T`` can also be displayed as a 3D coordinate frame:: + + >>> T.plot(color='red', label='2') + + +Rotation can also be represented by a unit quaternion + +.. runblock:: pycon + + >>> from spatialmath import UnitQuaternion + >>> print(UnitQuaternion.Rx(0.3)) + >>> print(UnitQuaternion.AngVec(0.3, [1, 0, 0])) + +which again demonstrates several alternative constructors. + + + +Multiple values +^^^^^^^^^^^^^^^ + +To support sequences of values each of these types inherits list properties from ``collections.UserList`` + +.. figure:: ../figs/pose-values.png + :width: 600 + :alt: Any of the SMTB-P pose classes can contain a list of values + + Any of the SMTB-P pose classes can contain a list of values + +We can index the values, iterate over the values, assign to values. +Some constructors take an array-like argument allowing creation of multi-valued pose objects, +for example: + +.. runblock:: pycon + + >>> from spatialmath import SE3 + >>> import numpy as np + >>> R = SE3.Rx(np.linspace(0, np.pi/2, num=100)) + >>> len(R) + +where the instance ``R`` contains a sequence of 100 rotation matrices. +Composition with a single-valued (scalar) pose instance broadcasts the scalar +across the sequence + +.. figure:: ../figs/broadcasting.png + :alt: Overloaded operators support broadcasting + + Overloaded operators support broadcasting + +Common constructors +^^^^^^^^^^^^^^^^^^^ + +The Toolboxes classes are somewhat polymorphic and share many "variant constructors" that allow object construction: + +- with orientation expressed in terms of canonic axis rotations, Euler vectors, angle-vector pair, + Euler or roll-pitch-yaw angles or orientation- and approach-vectors. +- from random values ``.Rand()`` +- ``SE3``, ``SE2``, ``SO3`` and ``SO2`` also support a matrix exponential constructor where the argument is the + corresponding Lie algebra element. +- empty, i.e. having no values or a length of 0 ``.Empty()`` +- an array of ``N`` values initialized to the object's identity value ``.Alloc(N)`` + +Common methods and operators +^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +The types all have an inverse method ``.inv()`` and support composition with the inverse using the ``/`` operator +and integer exponentiation (repeated composition) using the ``**`` operator. +Other overloaded operators include ``*``, ``*=``, ``**``, ``**=``, ``/``, ``/=``, ``==``, ``!=``, ``+``, ``-``. + +All of this allows for concise and readable code. +The use of classes ensures type safety and that the matrices abstracted by the class are always valid members of +the group. +Operations such as addition, which are not group operations, yield a NumPy array rather than a class instance. + +Performance +^^^^^^^^^^^ + +These benefits come at a price in terms of execution time due to the overhead of +constructors, methods which wrap base functions, and type checking. The +Toolbox supports SymPy which provides powerful symbolic support for Python and +it works well in conjunction with NumPy, ie. a NumPy array can contain symbolic +elements. Many the Toolbox methods and functions contain extra logic to ensure +that symbolic operations work as expected. While this adds to the overhead it +means that for the user, working with symbols is as easy as working with +numbers. + + +.. table:: Performance on a 3.6GHz Intel Core i9 + + =================== ============== + Function/method Execution time + =================== ============== + ``base.rotx()`` 4.07 μs + ``base.trotx()`` 5.79 μs + ``SE3.Rx()`` 12.3 μs + ``SE3 * SE3`` 4.69 μs + ``4x4 @`` 0.986 μs + ``SE3.inv()`` 7.62 μs + ``base.trinv()`` 4.19 μs + ``np.linalg.inv()`` 4.49 μs + =================== ============== + + +Robotics Toolbox +================ + +Robot models +^^^^^^^^^^^^ + +The Toolbox ships with over 30 robot models, most of which are purely kinematic +but some have inertial and frictional parameters. Kinematic models can be +specified in a variety of ways: standard or modified Denavit-Hartenberg (DH, +MDH) notation, as an ETS string [Corke07]_, as a rigid-body tree, or from a URDF +file. + + +Denavit-Hartenberg parameters +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +To specify a kinematic model using DH notation, we create a new subclass of ``DHRobot`` and pass the superclass constructor +a list of link objects. For example, a Puma560 is simply:: + + >>> robot = DHRobot( + [ + RevoluteDH(alpha=pi/2), + RevoluteDH(a=0.4318), + RevoluteDH(d=0.15005, a=0.0203, alpha=-pi/2), + RevoluteDH(d=0.4318, alpha=pi/2) + RevoluteDH(alpha=-pi/2), + RevoluteDH() + ], name="Puma560") + + +where only the non-zero parameters need to be specified. In this case we used +``RevoluteDH`` objects for a revolute joint described using standard DH +conventions. Other classes available are ``PrismaticDH``, ``RevoluteMDH`` and +``PrismaticMDH``. Other parameters such as mass, CoG, link inertia, motor +inertia, viscous friction, Coulomb friction, and joint limits can also be +specified using additional keyword arguments. + +The toolbox provides such definitions wrapped as class definitions, for example:: + + class Puma560(DHRobot): + + def __init__(self): + super().__init__( + [ + RevoluteDH(alpha=pi/2), + RevoluteDH(a=0.4318), + RevoluteDH(d=0.15005, a=0.0203, alpha=-pi/2), + RevoluteDH(d=0.4318, alpha=pi/2) + RevoluteDH(alpha=-pi/2), + RevoluteDH() + ], name="Puma560" + ) + +We can now easily perform standard kinematic operations + +.. runblock:: pycon + :linenos: + + >>> import roboticstoolbox as rtb + >>> puma = rtb.models.DH.Puma560() # instantiate robot model + >>> print(puma) + >>> print(puma.qr) + >>> T = puma.fkine([0.1, 0.2, 0.3, 0.4, 0.5, 0.6]) # forward kinematics + >>> print(T) + >>> sol = puma.ikine_LM(T) # inverse kinematics + >>> print(sol) + +The Toolbox supports named joint configurations and these are shown in the table +at lines 16-22. + +``ikine_LM`` is a generalised iterative numerical solution based on +Levenberg-Marquadt minimization, and additional status results are also +returned as part of a named tuple. + +The default plot method:: + + >>> puma.plot(q) + +uses matplotlib to produce a "noodle robot" plot like + +.. figure:: ../figs/noodle+ellipsoid.png + :width: 600 + :alt: Puma560, with a velocity ellipsoid, rendered using the default matplotlib visualizer + + Puma560, with a velocity ellipsoid, rendered using the default matplotlib visualizer. + +and we can use the mouse to rotate and zoom the plot. + +.. note:: The initial joint configuration for the inverse-kinematic solution may be specified, but + defaults to zero, and affects both the search time and the solution found, since in general + a manipulator may have several multiple joint configurations which result in the same end-effector pose. + For a redundant manipulator, a solution will be found but there is no + explicit control over the null-space. For a manipulator with :math:`n < 6` DOF + an additional argument is required to indicate which of the + :math:`6-n` Cartesian DOF are to be unconstrained in the solution. + +.. note:: A solution is not possible if the specified transform describes + a point out of reach of the manipulator -- in such a case the function will + return with an error. + +The inverse kinematic procedure for most robots can +be derived symbolically +and an efficient closed-form solution obtained. +Some provided robot models have an analytical solution coded, for example: + +.. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> puma = rtb.models.DH.Puma560() # instantiate robot model + >>> T = puma.fkine([0.1, 0.2, 0.3, 0.4, 0.5, 0.6]) + >>> puma.ikine_a(T, config="lun") # analytic inverse kinematics + +where we have specified a left-handed, elbow up and wrist no-flip configuration. + + +ETS notation +^^^^^^^^^^^^ + +A Puma robot can also be specified in ETS format [Corke07]_ as a sequence of simple rigid-body transformations -- pure translation +or pure rotation -- each with either a constant parameter or a free parameter which is a joint variable. + +.. runblock:: pycon + :linenos: + + >>> from roboticstoolbox import ET + >>> import roboticstoolbox as rtb + >>> # Puma dimensions (m), see RVC2 Fig. 7.4 for details + >>> l1 = 0.672; l2 = -0.2337; l3 = 0.4318; l4 = 0.0203; l5 = 0.0837; l6 = 0.4318 + >>> e = ET.tz(l1) * ET.Rz() * ET.ty(l2) * ET.Ry() * ET.tz(l3) * ET.tx(l4) * ET.ty(l5) * ET.Ry() * ET.tz(l6) * ET.Rz() * ET.Ry() * ET.Rz() + >>> print(e) + >>> robot = rtb.Robot(e) + >>> print(robot) + +Line 3 defines the unique lengths of the Puma robot, and line 4 defines the kinematic chain in +terms of elementary transforms. +In line 7 we pass this to the constructor for a ``Robot`` which partitions the +elementary transform sequence into a series of links and joints -- link frames are declared +after each joint variable as well as the start and end of the sequence. +The ``Robot`` can represent single-branched robots with any combination of revolute and prismatic joints, but +can also represent more general branched mechanisms. + + +Robot: rigid-body tree and URDF import +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +The final approach to manipulator modeling is to an import a URDF file. The Toolbox includes a parser with built-in xacro processor +which makes many models from the ROS universe available. + +Provided models, such as for Panda or Puma, are again encapsulated as classes: + +.. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> panda = rtb.models.DH.Panda() + >>> print(panda) + >>> T = panda.fkine(panda.qz) + >>> print(T) + +and kinematic operations are performed using methods with the same name +as discussed above. +For branched robots, with multiple end-effectors, the name of the frame of interest must be provided. + +Some URDF models have multiple end-effectors, in which case the particular +end-effector must be specified. + +.. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> panda = rtb.models.URDF.Panda() + >>> print(panda) + >>> T = panda.fkine(panda.qz, end='panda_hand') + >>> print(T) + +In the table above we see the end-effectors indicated by @ (determined automatically +from the URDF file), so we specify one of these. We can also specify any +other link in order to determine the pose of that link's coordinate frame. + +This URDF model comes with meshes provided as Collada file which provide +detailed geometry and color. This can be visualized using the Swift simulator: + + >>> panda.plot(qz, backend="swift") + +which produces the 3-D plot + +.. figure:: ../figs/swift.png + :width: 600 + :alt: Panda robot rendered using the Toolbox’s Swift visualizer + + Panda robot rendered using the Toolbox’s Swift visualizer. + +Swift is a web-based visualizer using three.js to provide high-quality 3D animations. +It can produce vivid 3D effects using anaglyphs viewed with colored glasses. +Animations can be recorded as MP4 files or animated GIF files which are useful for inclusion in GitHub markdown documents. + +Trajectories +============ + +A joint-space trajectory for the Puma robot from its zero angle +pose to the upright (or READY) pose in 100 steps is + +.. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> puma = rtb.models.DH.Puma560() + >>> traj = rtb.jtraj(puma.qz, puma.qr, 100) + >>> traj.q.shape + +where ``puma.qr`` is an example of a named joint configuration. +``traj`` is named tuple with elements ``q`` = :math:`\vec{q}_k`, ``qd`` = :math:`\dvec{q}_k` and ``qdd`` = :math:`\ddvec{q}_k`. +Each element is an array with one row per time step, and each row is a joint coordinate vector. +The trajectory is a fifth order polynomial which has continuous jerk. +By default, the initial and final velocities are zero, but these may be specified by additional +arguments. + +We could plot the joint coordinates as a function of time using the convenience +function:: + + >>> rtb.qplot(traj.q) + + +Straight line (Cartesian) paths can be generated in a similar way between +two points specified by a pair of poses in :math:`\SE{3}` + +.. .. runblock:: pycon +.. :linenos: + +.. >>> import numpy as np +.. >>> from spatialmath import SE3 +.. >>> import roboticstoolbox as rtb +.. >>> puma = rtb.models.DH.Puma560() +.. >>> t = np.arange(0, 2, 0.010) +.. >>> T0 = SE3(0.6, -0.5, 0.0) +.. >>> T1 = SE3(0.4, 0.5, 0.2) +.. >>> Ts = rtb.tools.trajectory.ctraj(T0, T1, len(t)) +.. >>> len(Ts) +.. >>> sol = puma.ikine_LM(Ts) # named tuple of arrays +.. >>> sol.q.shape + +At line 9 we see that the resulting trajectory, ``Ts``, is an ``SE3`` instance with 200 values. + +At line 11 we compute the inverse kinematics of each pose in the trajectory +using a single call to the ``ikine_LM`` method. +The result is a list of named tuples, which gives the IK success status for +each time step. +At line 12 we convert this into an array, with one row per time step, and each +row is a joint coordinate. +The starting +joint coordinates for each inverse kinematic solution +is taken as the result of the solution at the previous time step. + + +Symbolic manipulation +===================== + +As mentioned earlier, the Toolbox supports symbolic manipulation using SymPy. For example: + +.. runblock:: pycon + + >>> import spatialmath.base as base + >>> phi, theta, psi = base.sym.symbol('φ, ϴ, ψ') + >>> base.rpy2r(phi, theta, psi) + +The capability extends to forward kinematics + +.. runblock:: pycon + :linenos: + + >>> import roboticstoolbox as rtb + >>> import spatialmath.base as base + >>> puma = rtb.models.DH.Puma560(symbolic=True) + >>> q = base.sym.symbol("q_:6") # q = (q_1, q_2, ... q_5) + >>> T = puma.fkine(q) + >>> T.t[0] + +If we display the value of ``puma`` we see that the :math:`\alpha_j` values are +now displayed in red to indicate that they are symbolic constants. The +x-coordinate of the end-effector is given by line 7. + + +SymPy allows any expression to be converted to LaTeX or a variety of languages +including C, Python and Octave/MATLAB. + +Differential kinematics +======================= + +The Toolbox computes Jacobians:: + + + >>> J = puma.jacob0(q) + >>> J = puma.jacobe(q) + +in the base or end-effector frames respectively, as NumPy arrays. +At a singular configuration + +.. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> puma = rtb.models.DH.Puma560() + >>> J = puma.jacob0(puma.qr) + >>> np.linalg.matrix_rank(J) + >>> rtb.jsingu(J) + +Jacobians can also be computed for symbolic joint variables as for forward kinematics above. + +For ``Robot`` instances we can also compute the Hessians:: + + >>> H = puma.hessian0(q) + >>> H = puma.hessiane(q) + +in the base or end-effector frames respectively, as 3D NumPy arrays in :math:`\mathbb{R}^{6 \times n \times n}`. + +For all robot classes we can compute manipulability + +.. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> puma = rtb.models.DH.Puma560() + >>> m = puma.manipulability(puma.qn) + >>> print("Yoshikawa manipulability is", m) + >>> m = puma.manipulability(puma.qn, method="asada") + >>> print("Asada manipulability is", m) + +for the Yoshikawa and Asada measures respectively, and + +.. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> puma = rtb.models.DH.Puma560() + >>> m = puma.manipulability(puma.qn, axes="trans") + >>> print("Yoshikawa manipulability is", m) + +is the Yoshikawa measure computed for just the task space translational degrees +of freedom. +For ``Robot`` instances we can also compute the manipulability +Jacobian:: + + >>> Jm = puma.manipm(q, J, H) + +such that :math:`\dot{m} = \mat{J}_m(\vec{q}) \dvec{q}`. + +Dynamics +^^^^^^^^ + +The Python Toolbox supports several approaches to computing dynamics. +For models defined using standard or modified DH notation we use a classical version of the recursive Newton-Euler +algorithm implemented in Python or C. + +.. note:: The same C code as used by RTB-M is called directly from Python, and does not use NumPy. + +For example, the inverse dynamics + +.. runblock:: pycon + + >>> import roboticstoolbox as rtb + >>> puma = rtb.models.DH.Puma560() + >>> tau = puma.rne(puma.qn, np.zeros((6,)), np.zeros((6,))) + >>> print(tau) + +is the gravity torque for the robot in the configuration ``qn``. + +Inertia, Coriolis/centripetal and gravity terms are computed by:: + + >>> puma.inertia(q) + >>> puma.coriolis(q, qd) + >>> puma.gravload(q) + +respectively, using the method of Orin and Walker from the inverse dynamics. These values include the effect of motor inertia and friction. + +Forward dynamics are given by:: + + >>> qdd = puma.accel(q, tau, qd) + +We can integrate this over time by:: + + >>> q = puma.fdyn(5, q0, mycontrol, ...) + +which uses an RK45 numerical integration from the SciPy package to solve for the joint trajectory ``q`` given the +optional control function called as:: + + tau = mycontrol(robot, t, q, qd, **args) + +The fast C implementation is not capable of symbolic operation so a Python +version of RNE ``rne_python`` has been implemented as well. For a 6- or 7-DoF +manipulator the torque expressions have thousands of terms yet are computed in +less than a second. However, subsequent expression manipulation is slow. + +For the Puma560 robot the C version of inverse dynamics takes 23μs while the +Python version takes 1.5ms (:math:`65\times` slower). With symbolic operands it +takes 170ms (:math:`113\times` slower) to produce the unsimplified torque +expressions. + +For ``Robot`` subclasses there is also an implementation of Featherstone's spatial vector +method, ``rne()``, and SMTB-P provides a set of classes for spatial +velocity, acceleration, momentum, force and inertia. + + +New capability +============== + +There are several areas of innovation compared to the MATLAB version of the Toolbox. + +Branched mechanisms +^^^^^^^^^^^^^^^^^^^ + +The RTB-M ``SerialLink`` class had no option to express branching. In RTB-P the +equivalent class is ``DHRobot`` is similarly limited, but a new class ``ERobot`` +is more general and allows for branching (but not closed kinematic loops). The +robot is described by a set of ``ELink`` objects, each of which points to its +parent link. The ``ERobot`` has references to the root and leaf ``ELink`` objects. This +structure closely mirrors the URDF representation, allowing for easy import of +URDF models. + +Collision checking +^^^^^^^^^^^^^^^^^^ + +RTB-M had a simple, contributed but unsupported, collision checking capability. +This is dramatically improved in the Python version using [PyBullet]_ +which supports primitive shapes such as Cylinders, Spheres and Boxes as well as +mesh objects. Every robot link can have a collision shape in addition to the shape +used for rendering. +We can conveniently perform collision checks between links as well as between +whole robots, discrete links, and objects in the world. For example a :math:`1 +\times 1 \times 1` box centered at :math:`(1,0,0)` can be tested against all, or +just one link, of the robot by:: + + >>> panda = rtb.models.Panda() + >>> obstacle = rtb.Box([1, 1, 1], SE3(1, 0, 0)) + >>> iscollision = panda.collided(obstacle) # boolean + >>> iscollision = panda.links[0].collided(obstacle) + + +Additionally, we can compute the minimum Euclidean distance between whole +robots, discrete links, or objects. Each distance is the length of a line +segment defined by two points in the world frame + + +>>> d, p1, p2 = panda.closest_point(obstacle) +>>> d, p1, p2 = panda.links[0].closest_point(obstacle) + + +Interfaces +^^^^^^^^^^ + +RTB-M could only animate a robot in a figure, and there was limited but +not-well-supported ability to interface to V-REP and a physical robot. The +Python version supports a simple, but universal API to a robot inspired by the +simplicity and expressiveness of the OpenAI Gym API which was designed as a +toolkit for developing and comparing reinforcement learning algorithms. Whether +simulating a robot or controlling a real physical robot, the API operates in the +same manner, providing users with a common interface which is not found among +other robotics packages. + +By default the Toolbox behaves like the MATLAB version with a plot method:: + +>>> puma.plot(q) + +which will plot the robot at the specified joint configurmation, or animate it if ``q`` is an :math:`m \times 6` matrix, using +the default ``PyPlot`` backend which draws a "noodle robot" using the PyPlot backend. + +The more general solution, and what is implemented inside ``plot`` in the example above, is:: + + >>> pyplot = roboticstoolbox.backends.PyPlot() + >>> pyplot.launch() + >>> pyplot.add(puma) + >>> puma.q = q + >>> puma.step() + +This makes it possible to animate multiple robots in the one graphical window, or the one robot in various environments either graphical +or real. + +.. figure:: ../figs/vp_ss_0000-2.png + :width: 600 + :alt: Puma560 rendered using the web-based VPython visualizer. + + Puma560 rendered using the web-based VPython visualizer. + +The VPython backend provides browser-based 3D graphics based on WebGL. This is advantageous for displaying on mobile +devices. Still frames and animations can be recorded. + + +Code engineering +^^^^^^^^^^^^^^^^ + +The code is implemented in Python :math:`\ge 3.6` and all code is hosted on GitHub and +unit-testing is performed using GitHub-actions. Test coverage is uploaded to +``codecov.io`` for visualization and trending, and we use ``lgtm.com`` to perform +automated code review. The code is documented with ReStructured Text format +docstrings which provides powerful markup including cross-referencing, +equations, class inheritance diagrams and figures -- all of which is converted +to HTML documentation whenever a change is pushed, and this is accessible via +GitHub pages. Issues can be reported via GitHub issues or patches submitted as +pull requests. + +RTB-P, and its dependencies, can be installed simply by either of:: + + $ pip install roboticstoolbox-python + + $ conda install -c conda-forge roboticstoolbox-python + +which includes basic visualization using matplotlib. +Options such as ``vpython`` can be used to specify additional dependencies to be installed. +The Toolbox adopts a "when needed" approach to many dependencies and will only attempt +to import them if the user attempts to exploit a functionality that requires it. + +If a dependency is not installed, a warning provides instructions on how to install it using ``pip``. +More details are given on the project home page. +This applies to the visualizers Vpython and Swift, as well as pybullet and ROS. +The Toolbox provides capability to import URDF-xacro files without ROS. +The backend architecture allows a user to connect to a ROS environment if required, and only then does ROS have to be +installed. + + +Conclusion +========== + +This article has introduced and demonstrated in tutorial form the principle +features of the Robotics Toolbox for Python which runs on Mac, Windows and Linux +using Python 3.6 or better. The code is free and open, and released under the +MIT licence. It provides many of the essential tools necessary for robotic +manipulator modelling, simulation and control which is essential for robotics +education and research. It is familiar yet new, and we hope it will serve the +community well for the next 25 years. + +A high-performance reactive motion controller, NEO, is based on this toolbox +[neo]_. Currently under development are backend interfaces for CoppeliaSim, +Dynamixel servo chains, and ROS; symbolic dynamics, simplification and code +generation; mobile robotics motion models, planners, EKF localization, map +making and SLAM; and a minimalist block-diagram simulation tool [bdsim]_. + +References +========== + +.. [Corke95] `P. Corke. "A computer tool for simulation and analysis: the Robotics Toolbox for MATLAB". In Proc. National Conf. Australian Robot Association, pages 319–330, Melbourne, July 1995. `_ +.. [Corke96] `P. Corke. "A robotics toolbox for MATLAB". IEEE Robotics and Automation Magazine, 3(1):24–32, Sept. 1996. `_ +.. [Craig2005] J. Craig, "Introduction to Robotics", Wiley, 2005. +.. [Featherstone87] R. Featherstone, Robot Dynamics Algorithms. Kluwer Academic, 1987. +.. [Corke07] P. Corke, `“A simple and systematic approach to assigning Denavit- Hartenberg parameters,” IEEE transactions on robotics, vol. 23, no. 3, pp. 590–594, 2007, DOI 10.1109/TRO.2007.896765. `_. +.. [Haviland20] `J. Haviland and P. Corke, “A systematic approach to computing the manipulator Jacobian and Hessian using the elementary transform sequence,” arXiv preprint, 2020. `_ +.. [PyBullet] `PyBullet `_ +.. [SMTB-P] `Spatial Math Toolbox for Python `_ +.. [bdsim] `Block diagram simulator for Python `_ +.. [neo] `NEO: A Novel Expeditious Optimisation Algorithm for Reactive Motion Control of Manipulatorshttps://jhavl.github.io/neo>`_ +.. [corke21a] P. Corke and J. Haviland, "Not your grandmother’s toolbox – the Robotics Toolbox reinvented for Python", Proc. ICRA 2021. \ No newline at end of file diff --git a/docs/source/matrices.rst b/_sources/matrices.rst.txt similarity index 100% rename from docs/source/matrices.rst rename to _sources/matrices.rst.txt diff --git a/_sources/mobile-SLAM.rst.txt b/_sources/mobile-SLAM.rst.txt new file mode 100644 index 00000000..d75cd3b7 --- /dev/null +++ b/_sources/mobile-SLAM.rst.txt @@ -0,0 +1,59 @@ +Localization and Mapping +======================== + +These classes support simulation of vehicle and map estimation in a simple +planar world with point landmarks. + + +State estimation +---------------- + +Two state estimators are included. + +Extended Kalman filter +^^^^^^^^^^^^^^^^^^^^^^ + +The EKF is capable of vehicle localization, map estimation or SLAM. + +.. autoclass:: roboticstoolbox.mobile.EKF + :undoc-members: + :show-inheritance: + :inherited-members: + :special-members: __init__ + +Particle filter +^^^^^^^^^^^^^^^ + +The particle filter is capable of map-based vehicle localization. + +.. autoclass:: roboticstoolbox.mobile.ParticleFilter + :undoc-members: + :show-inheritance: + :inherited-members: + :special-members: __init__ + +Sensor models +------------- + +.. autoclass:: roboticstoolbox.mobile.RangeBearingSensor + :undoc-members: + :show-inheritance: + :inherited-members: + :special-members: __init__ + +.. autoclass:: roboticstoolbox.mobile.SensorBase + :undoc-members: + :show-inheritance: + :inherited-members: + :special-members: __init__ + +Map models +---------- + +.. automodule:: roboticstoolbox.mobile.landmarkmap + :undoc-members: + :show-inheritance: + :inherited-members: + :special-members: __init__ + + diff --git a/_sources/mobile-drivers.rst.txt b/_sources/mobile-drivers.rst.txt new file mode 100644 index 00000000..950a59d1 --- /dev/null +++ b/_sources/mobile-drivers.rst.txt @@ -0,0 +1,26 @@ +Drivers +======= + +These classes define agents thta can drive a mobile robot around a workplace, and are +useful for testing localization algorithms. + +Random Path +^^^^^^^^^^^ + + .. autoclass:: roboticstoolbox.mobile.drivers.RandomPath + :members: + :undoc-members: + :inherited-members: + :special-members: __init__ + :show-inheritance: + +Superclass +^^^^^^^^^^ + + .. autoclass:: roboticstoolbox.mobile.drivers.VehicleDriverBase + :members: + :undoc-members: + :inherited-members: + :special-members: __init__ + :show-inheritance: + diff --git a/_sources/mobile-planner-base.rst.txt b/_sources/mobile-planner-base.rst.txt new file mode 100644 index 00000000..9808ee04 --- /dev/null +++ b/_sources/mobile-planner-base.rst.txt @@ -0,0 +1,26 @@ +Supporting classes +================== + +Planner superclass +------------------ + +.. autoclass:: roboticstoolbox.mobile.PlannerBase + :members: + :undoc-members: + :show-inheritance: + + +Occupancy grid base classes +--------------------------- + +.. autoclass:: roboticstoolbox.mobile.OccGrid.BaseMap + :members: + :undoc-members: + :show-inheritance: + +.. autoclass:: roboticstoolbox.mobile.OccGrid.BaseOccupancyGrid + :members: + :undoc-members: + :show-inheritance: + + diff --git a/_sources/mobile-planner-continuous.rst.txt b/_sources/mobile-planner-continuous.rst.txt new file mode 100644 index 00000000..90bfeaed --- /dev/null +++ b/_sources/mobile-planner-continuous.rst.txt @@ -0,0 +1,80 @@ +Continuous configuration-space planners +======================================= + + +========================================================= ==================== =================== =================== +Planner Plans in Discrete/Continuous Obstacle avoidance +========================================================= ==================== =================== =================== +:class:`~roboticstoolbox.mobile.DubinsPlanner` :math:`\SE{2}` continuous no +:class:`~roboticstoolbox.mobile.ReedsSheppPlanner` :math:`\SE{2}` continuous no +:class:`~roboticstoolbox.mobile.CurvaturePolyPlanner` :math:`\SE{2}` continuous no +:class:`~roboticstoolbox.mobile.QuinticPolyPlanner` :math:`\SE{2}` continuous no +:class:`~roboticstoolbox.mobile.RRTPlanner` :math:`\SE{2}` continuous yes +========================================================= ==================== =================== =================== + + +These planners do not support planning around obstacles, but allow for the +start and goal configuration :math:`(x, y, \theta)` to be specified. + +PRM planner +----------- + +.. autoclass:: roboticstoolbox.mobile.PRMPlanner + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :exclude-members: isoccupied, random, random_init, progress_start, progress_end, progress_next, message, plot_bg + + +Dubins path planner +------------------- + +.. autoclass:: roboticstoolbox.mobile.DubinsPlanner + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :exclude-members: isoccupied, random, random_init, progress_start, progress_end, progress_next, message, plot_bg + + +Reeds-Shepp path planner +------------------------ + +.. autoclass:: roboticstoolbox.mobile.ReedsSheppPlanner + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :exclude-members: isoccupied, random, random_init, progress_start, progress_end, progress_next, message, plot_bg + +Curvature-polynomial planner +---------------------------- + +.. autoclass:: roboticstoolbox.mobile.CurvaturePolyPlanner + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :exclude-members: isoccupied, random, random_init, progress_start, progress_end, progress_next, message, plot_bg + +Quintic-polynomial planner +-------------------------- + +.. autoclass:: roboticstoolbox.mobile.QuinticPolyPlanner + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :exclude-members: isoccupied, random, random_init, progress_start, progress_end, progress_next, message, plot_bg + +RRT planner +----------- + +.. autoclass:: roboticstoolbox.mobile.RRTPlanner + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :exclude-members: isoccupied, random, random_init, progress_start, progress_end, progress_next, message, plot_bg + diff --git a/_sources/mobile-planner-discrete.rst.txt b/_sources/mobile-planner-discrete.rst.txt new file mode 100644 index 00000000..eb430270 --- /dev/null +++ b/_sources/mobile-planner-discrete.rst.txt @@ -0,0 +1,100 @@ +Discrete (Grid-based) planners +============================== + + +========================================================= ==================== =================== =================== +Planner Plans in Discrete/Continuous Obstacle avoidance +========================================================= ==================== =================== =================== +:class:`~roboticstoolbox.mobile.Bug2` :math:`\mathbb{R}^2` discrete yes +:class:`~roboticstoolbox.mobile.DistanceTransformPlanner` :math:`\mathbb{R}^2` discrete yes +:class:`~roboticstoolbox.mobile.DstarPlanner` :math:`\mathbb{R}^2` discrete yes +========================================================= ==================== =================== =================== + + +Distance transform planner +-------------------------- + +.. autoclass:: roboticstoolbox.mobile.DistanceTransformPlanner + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :exclude-members: random, random_init, progress_start, progress_end, progress_next, message + +D* planner +---------- + +.. autoclass:: roboticstoolbox.mobile.DstarPlanner + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :exclude-members: next, isoccupied, random, random_init, progress_start, progress_end, progress_next, message + + +Lattice planner +--------------- + +.. autoclass:: roboticstoolbox.mobile.LatticePlanner + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :exclude-members: isoccupied, random, random_init, progress_start, progress_end, progress_next, message, plot_bg + +Continuous configuration-space planners +======================================= + +These planners do not support planning around obstacles, but allow for the +start and goal configuration :math:`(x, y, \theta)` to be specified. + +Dubins path planner +------------------- + +.. autoclass:: roboticstoolbox.mobile.DubinsPlanner + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :exclude-members: isoccupied, random, random_init, progress_start, progress_end, progress_next, message, plot_bg + + +Reeds-Shepp path planner +------------------------ + +.. autoclass:: roboticstoolbox.mobile.ReedsSheppPlanner + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :exclude-members: isoccupied, random, random_init, progress_start, progress_end, progress_next, message, plot_bg + +Curvature-polynomial planner +---------------------------- + +.. autoclass:: roboticstoolbox.mobile.CurvaturePolyPlanner + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :exclude-members: isoccupied, random, random_init, progress_start, progress_end, progress_next, message, plot_bg + +Quintic-polynomial planner +-------------------------- + +.. autoclass:: roboticstoolbox.mobile.QuinticPolyPlanner + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :exclude-members: isoccupied, random, random_init, progress_start, progress_end, progress_next, message, plot_bg + +RRT planner +----------- + +.. autoclass:: roboticstoolbox.mobile.RRTPlanner + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :exclude-members: isoccupied, random, random_init, progress_start, progress_end, progress_next, message, plot_bg diff --git a/_sources/mobile-planner-map.rst.txt b/_sources/mobile-planner-map.rst.txt new file mode 100644 index 00000000..638b9773 --- /dev/null +++ b/_sources/mobile-planner-map.rst.txt @@ -0,0 +1,36 @@ +Map classes +=========== + +Occupancy grid classes +---------------------- + +Binary Occupancy grid +^^^^^^^^^^^^^^^^^^^^^ + +.. autoclass:: roboticstoolbox.mobile.BinaryOccupancyGrid + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :special-members: __init__ + + +Occupancy grid +^^^^^^^^^^^^^^ + +.. autoclass:: roboticstoolbox.mobile.OccupancyGrid + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :special-members: __init__ + +Polygon map +----------- + +.. autoclass:: roboticstoolbox.mobile.PolygonMap + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :special-members: __init__ diff --git a/_sources/mobile-planner.rst.txt b/_sources/mobile-planner.rst.txt new file mode 100644 index 00000000..5eb30fae --- /dev/null +++ b/_sources/mobile-planner.rst.txt @@ -0,0 +1,36 @@ +Path planning +************* + +A set of path planners for robots operating in planar environments with +configuration :math:`\vec{q} \in \mathbb{R}^2` or :math:`\vec{q} \in \mathbb{R}^2 \times S^1 \sim \SE{2}`. +All inherit from :class:`PlannerBase`. + +Some planners are based on code from the PathPlanning category of +`PythonRobotics `_ by Atsushi Sakai. + +.. inheritance-diagram:: roboticstoolbox.mobile.DistanceTransformPlanner roboticstoolbox.mobile.DstarPlanner roboticstoolbox.mobile.DubinsPlanner roboticstoolbox.mobile.ReedsSheppPlanner roboticstoolbox.mobile.QuinticPolyPlanner roboticstoolbox.mobile.CurvaturePolyPlanner roboticstoolbox.mobile.RRTPlanner + :parts: 1 + +========================================================= ==================== =================== =================== +Planner Plans in Discrete/Continuous Obstacle avoidance +========================================================= ==================== =================== =================== +:class:`~roboticstoolbox.mobile.Bug2` :math:`\mathbb{R}^2` discrete yes +:class:`~roboticstoolbox.mobile.DistanceTransformPlanner` :math:`\mathbb{R}^2` discrete yes +:class:`~roboticstoolbox.mobile.DstarPlanner` :math:`\mathbb{R}^2` discrete yes +:class:`~roboticstoolbox.mobile.PRMPlanner` :math:`\mathbb{R}^2` continuous yes +:class:`~roboticstoolbox.mobile.LatticePlanner` :math:`\SE{2}` discrete yes +:class:`~roboticstoolbox.mobile.DubinsPlanner` :math:`\SE{2}` continuous no +:class:`~roboticstoolbox.mobile.ReedsSheppPlanner` :math:`\SE{2}` continuous no +:class:`~roboticstoolbox.mobile.CurvaturePolyPlanner` :math:`\SE{2}` continuous no +:class:`~roboticstoolbox.mobile.QuinticPolyPlanner` :math:`\SE{2}` continuous no +:class:`~roboticstoolbox.mobile.RRTPlanner` :math:`\SE{2}` continuous yes +========================================================= ==================== =================== =================== + + +.. toctree:: + :maxdepth: 2 + + mobile-planner-continuous + mobile-planner-discrete + mobile-planner-map + mobile-planner-base \ No newline at end of file diff --git a/_sources/mobile-reactive.rst.txt b/_sources/mobile-reactive.rst.txt new file mode 100644 index 00000000..10f8978f --- /dev/null +++ b/_sources/mobile-reactive.rst.txt @@ -0,0 +1,12 @@ +Reactive navigation +=================== + +These algorithms work with an occupancy grid representation of the world. Start +and goal are specified by 2D :math:`(x, y)` coordinates in the plane. + +.. autoclass:: roboticstoolbox.mobile.Bug2 + :members: query + :undoc-members: + :show-inheritance: + :inherited-members: + :exclude-members: query, plan, next \ No newline at end of file diff --git a/_sources/mobile-vehicle-animation.rst.txt b/_sources/mobile-vehicle-animation.rst.txt new file mode 100644 index 00000000..1a61e422 --- /dev/null +++ b/_sources/mobile-vehicle-animation.rst.txt @@ -0,0 +1,42 @@ +Animations +---------- + +These classes create a graphical object that can be animated to show vehicle +position or pose. + +.. inheritance-diagram:: roboticstoolbox.VehicleMarker roboticstoolbox.VehiclePolygon roboticstoolbox.VehicleIcon + :parts: 1 + :top-classes: roboticstoolbox.VehicleAnimationBase + +.. autoclass:: roboticstoolbox.mobile.VehicleAnimationBase + :exclude-members: add, update, plot + +Simple marker +^^^^^^^^^^^^^ + +.. autoclass:: roboticstoolbox.mobile.VehicleMarker + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :special-members: __init__ + +Polygon shape +^^^^^^^^^^^^^ + +.. autoclass:: roboticstoolbox.mobile.VehiclePolygon + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :special-members: __init__ + +Image icon +^^^^^^^^^^ + +.. autoclass:: roboticstoolbox.mobile.VehicleIcon + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :special-members: __init__ \ No newline at end of file diff --git a/_sources/mobile-vehicle-base.rst.txt b/_sources/mobile-vehicle-base.rst.txt new file mode 100644 index 00000000..4405e145 --- /dev/null +++ b/_sources/mobile-vehicle-base.rst.txt @@ -0,0 +1,8 @@ +Vehicle abstract baseclass +-------------------------- + +.. autoclass:: roboticstoolbox.mobile.VehicleBase + :members: + :undoc-members: + :inherited-members: + :special-members: __init__ diff --git a/_sources/mobile-vehicle-bicycle.rst.txt b/_sources/mobile-vehicle-bicycle.rst.txt new file mode 100644 index 00000000..483d7844 --- /dev/null +++ b/_sources/mobile-vehicle-bicycle.rst.txt @@ -0,0 +1,9 @@ +Bicycle +------- + +.. autoclass:: roboticstoolbox.mobile.Bicycle + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :special-members: __init__ diff --git a/_sources/mobile-vehicle-diffsteer.rst.txt b/_sources/mobile-vehicle-diffsteer.rst.txt new file mode 100644 index 00000000..b8a345d8 --- /dev/null +++ b/_sources/mobile-vehicle-diffsteer.rst.txt @@ -0,0 +1,9 @@ +Differential steering +--------------------- + +.. autoclass:: roboticstoolbox.mobile.DiffSteer + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :special-members: __init__ diff --git a/_sources/mobile-vehicle-unicycle.rst.txt b/_sources/mobile-vehicle-unicycle.rst.txt new file mode 100644 index 00000000..da6b6268 --- /dev/null +++ b/_sources/mobile-vehicle-unicycle.rst.txt @@ -0,0 +1,9 @@ +Unicycle +-------- + +.. autoclass:: roboticstoolbox.mobile.Unicycle + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :special-members: __init__ diff --git a/_sources/mobile-vehicle-vehicle.rst.txt b/_sources/mobile-vehicle-vehicle.rst.txt new file mode 100644 index 00000000..7f281478 --- /dev/null +++ b/_sources/mobile-vehicle-vehicle.rst.txt @@ -0,0 +1,11 @@ +Vehicle superclass +------------------ + +The various vehicle models all subclass this class. + +.. autoclass:: roboticstoolbox.mobile.Vehicle + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :special-members: __init__ \ No newline at end of file diff --git a/_sources/mobile-vehicle.rst.txt b/_sources/mobile-vehicle.rst.txt new file mode 100644 index 00000000..ecaa1b50 --- /dev/null +++ b/_sources/mobile-vehicle.rst.txt @@ -0,0 +1,17 @@ +Motion models +============= + +These classes describe vehicle motion models, sometimes referred to as +kinematic or kino-dynamic models. They provide methods to: + +* predict new configuration based on odometry +* compute configuration derivative +* simulate and animate motion +* compute Jacobians + +.. toctree:: + + mobile-vehicle-bicycle + mobile-vehicle-unicycle + mobile-vehicle-diffsteer + mobile-vehicle-base diff --git a/_sources/mobile.rst.txt b/_sources/mobile.rst.txt new file mode 100644 index 00000000..f6545021 --- /dev/null +++ b/_sources/mobile.rst.txt @@ -0,0 +1,17 @@ +************* +Mobile robots +************* + + +The Robotics Toolbox supports kino-dynamic vehicle models, animation, planning +and state estimation. + +.. toctree:: + :maxdepth: 2 + + mobile-vehicle + mobile-vehicle-animation + mobile-drivers + mobile-planner + mobile-reactive + mobile-SLAM diff --git a/_sources/mobile_SLAM.rst.txt b/_sources/mobile_SLAM.rst.txt new file mode 100644 index 00000000..d75cd3b7 --- /dev/null +++ b/_sources/mobile_SLAM.rst.txt @@ -0,0 +1,59 @@ +Localization and Mapping +======================== + +These classes support simulation of vehicle and map estimation in a simple +planar world with point landmarks. + + +State estimation +---------------- + +Two state estimators are included. + +Extended Kalman filter +^^^^^^^^^^^^^^^^^^^^^^ + +The EKF is capable of vehicle localization, map estimation or SLAM. + +.. autoclass:: roboticstoolbox.mobile.EKF + :undoc-members: + :show-inheritance: + :inherited-members: + :special-members: __init__ + +Particle filter +^^^^^^^^^^^^^^^ + +The particle filter is capable of map-based vehicle localization. + +.. autoclass:: roboticstoolbox.mobile.ParticleFilter + :undoc-members: + :show-inheritance: + :inherited-members: + :special-members: __init__ + +Sensor models +------------- + +.. autoclass:: roboticstoolbox.mobile.RangeBearingSensor + :undoc-members: + :show-inheritance: + :inherited-members: + :special-members: __init__ + +.. autoclass:: roboticstoolbox.mobile.SensorBase + :undoc-members: + :show-inheritance: + :inherited-members: + :special-members: __init__ + +Map models +---------- + +.. automodule:: roboticstoolbox.mobile.landmarkmap + :undoc-members: + :show-inheritance: + :inherited-members: + :special-members: __init__ + + diff --git a/_sources/mobile_drivers.rst.txt b/_sources/mobile_drivers.rst.txt new file mode 100644 index 00000000..079998fa --- /dev/null +++ b/_sources/mobile_drivers.rst.txt @@ -0,0 +1,26 @@ +Mobile robot drivers +==================== + +These classes can drive a mobile robot around a workplace. Useful for testing +localization algorithms. + +Random Path +^^^^^^^^^^^ + + .. autoclass:: roboticstoolbox.mobile.drivers.RandomPath + :members: + :undoc-members: + :inherited-members: + :special-members: __init__ + :show-inheritance: + +Superclass +^^^^^^^^^^ + + .. autoclass:: roboticstoolbox.mobile.drivers.VehicleDriverBase + :members: + :undoc-members: + :inherited-members: + :special-members: __init__ + :show-inheritance: + diff --git a/_sources/mobile_planner.rst.txt b/_sources/mobile_planner.rst.txt new file mode 100644 index 00000000..bb162e5d --- /dev/null +++ b/_sources/mobile_planner.rst.txt @@ -0,0 +1,196 @@ +Mobile robot path planning +************************** + +A set of path planners for robots operating in planar environments with +configuration :math:`\vec{q} \in \mathbb{R}^2` or :math:`\vec{q} \in \SE{2}`. +All inherit from :class:`PlannerBase`. + +Some planners are based on code from the PathPlanning category of +`PythonRobotics `_ by Atsushi Sakai. + +.. inheritance-diagram:: roboticstoolbox.mobile.DistanceTransformPlanner roboticstoolbox.mobile.DstarPlanner roboticstoolbox.mobile.DubinsPlanner roboticstoolbox.mobile.ReedsSheppPlanner roboticstoolbox.mobile.QuinticPolyPlanner roboticstoolbox.mobile.CurvaturePolyPlanner roboticstoolbox.mobile.RRTPlanner + :parts: 1 + +========================================================= ==================== =================== =================== +Planner Plans in Discrete/Continuous Obstacle avoidance +========================================================= ==================== =================== =================== +:class:`~roboticstoolbox.mobile.Bug2` :math:`\mathbb{R}^2` discrete yes +:class:`~roboticstoolbox.mobile.DistanceTransformPlanner` :math:`\mathbb{R}^2` discrete yes +:class:`~roboticstoolbox.mobile.DstarPlanner` :math:`\mathbb{R}^2` discrete yes +:class:`~roboticstoolbox.mobile.PRMPlanner` :math:`\mathbb{R}^2` continuous yes +:class:`~roboticstoolbox.mobile.LatticePlanner` :math:`\SE{2}` discrete yes +:class:`~roboticstoolbox.mobile.DubinsPlanner` :math:`\SE{2}` continuous no +:class:`~roboticstoolbox.mobile.ReedsSheppPlanner` :math:`\SE{2}` continuous no +:class:`~roboticstoolbox.mobile.CurvaturePolyPlanner` :math:`\SE{2}` continuous no +:class:`~roboticstoolbox.mobile.QuinticPolyPlanner` :math:`\SE{2}` continuous no +:class:`~roboticstoolbox.mobile.RRTPlanner` :math:`\SE{2}` continuous yes +========================================================= ==================== =================== =================== + + + +Discrete (Grid-based) planners +============================== + + +Distance transform planner +-------------------------- + +.. autoclass:: roboticstoolbox.mobile.DistanceTransformPlanner + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :exclude-members: random, random_init, progress_start, progress_end, progress_next, message + +D* planner +---------- + +.. autoclass:: roboticstoolbox.mobile.DstarPlanner + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :exclude-members: next, isoccupied, random, random_init, progress_start, progress_end, progress_next, message + +PRM planner +----------- + +.. autoclass:: roboticstoolbox.mobile.PRMPlanner + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :exclude-members: isoccupied, random, random_init, progress_start, progress_end, progress_next, message, plot_bg + + +Lattice planner +--------------- + +.. autoclass:: roboticstoolbox.mobile.LatticePlanner + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :exclude-members: isoccupied, random, random_init, progress_start, progress_end, progress_next, message, plot_bg + +Continuous configuration-space planners +======================================= + +These planners do not support planning around obstacles, but allow for the +start and goal configuration :math:`(x, y, \theta)` to be specified. + +Dubins path planner +------------------- + +.. autoclass:: roboticstoolbox.mobile.DubinsPlanner + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :exclude-members: isoccupied, random, random_init, progress_start, progress_end, progress_next, message, plot_bg + + +Reeds-Shepp path planner +------------------------ + +.. autoclass:: roboticstoolbox.mobile.ReedsSheppPlanner + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :exclude-members: isoccupied, random, random_init, progress_start, progress_end, progress_next, message, plot_bg + +Curvature-polynomial planner +---------------------------- + +.. autoclass:: roboticstoolbox.mobile.CurvaturePolyPlanner + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :exclude-members: isoccupied, random, random_init, progress_start, progress_end, progress_next, message, plot_bg + +Quintic-polynomial planner +-------------------------- + +.. autoclass:: roboticstoolbox.mobile.QuinticPolyPlanner + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :exclude-members: isoccupied, random, random_init, progress_start, progress_end, progress_next, message, plot_bg + +RRT planner +----------- + +.. autoclass:: roboticstoolbox.mobile.RRTPlanner + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :exclude-members: isoccupied, random, random_init, progress_start, progress_end, progress_next, message, plot_bg + +Map classes +=========== + +Occupancy grid classes +---------------------- + +Binary Occupancy grid +^^^^^^^^^^^^^^^^^^^^^ + +.. autoclass:: roboticstoolbox.mobile.BinaryOccupancyGrid + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :special-members: __init__ + + +Occupancy grid +^^^^^^^^^^^^^^ + +.. autoclass:: roboticstoolbox.mobile.OccupancyGrid + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :special-members: __init__ + +Polygon map +----------- + +.. autoclass:: roboticstoolbox.mobile.PolygonMap + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :special-members: __init__ + + +Supporting classes +================== + +Planner superclass +------------------ + +.. autoclass:: roboticstoolbox.mobile.PlannerBase + :members: + :undoc-members: + :show-inheritance: + + +Occupancy grid base classes +--------------------------- + +.. autoclass:: roboticstoolbox.mobile.OccGrid.BaseMap + :members: + :undoc-members: + :show-inheritance: + +.. autoclass:: roboticstoolbox.mobile.OccGrid.BaseOccupancyGrid + :members: + :undoc-members: + :show-inheritance: + + diff --git a/_sources/mobile_reactive.rst.txt b/_sources/mobile_reactive.rst.txt new file mode 100644 index 00000000..949c60ee --- /dev/null +++ b/_sources/mobile_reactive.rst.txt @@ -0,0 +1,12 @@ +Reactive navigation +=================== + +These planners work with an occupancy grid representation of the world. Start +and goal are specified by 2D :math:`(x, y)` coordinates in the plane. + +.. autoclass:: roboticstoolbox.mobile.Bug2 + :members: query + :undoc-members: + :show-inheritance: + :inherited-members: + :exclude-members: query, plan, next \ No newline at end of file diff --git a/_sources/mobile_vehicle.rst.txt b/_sources/mobile_vehicle.rst.txt new file mode 100644 index 00000000..406e13cd --- /dev/null +++ b/_sources/mobile_vehicle.rst.txt @@ -0,0 +1,40 @@ +Mobile robot kinematic models +============================= + +These vehicle kinematic classes have methods to: + +* predict new configuration based on odometry +* compute configuration derivative +* simulate and animate motion +* compute Jacobians + +Bicycle model +^^^^^^^^^^^^^ + + .. autoclass:: roboticstoolbox.mobile.Bicycle + :members: + :undoc-members: + :inherited-members: + :show-inheritance: + :special-members: __init__ + +Unicycle model +^^^^^^^^^^^^^^ + +.. autoclass:: roboticstoolbox.mobile.Unicycle + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :special-members: __init__ + + +Superclass +^^^^^^^^^^ + +.. autoclass:: roboticstoolbox.mobile.VehicleBase + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :special-members: __init__ \ No newline at end of file diff --git a/_sources/mobile_vehicle_animation.rst.txt b/_sources/mobile_vehicle_animation.rst.txt new file mode 100644 index 00000000..2e3341e2 --- /dev/null +++ b/_sources/mobile_vehicle_animation.rst.txt @@ -0,0 +1,42 @@ +Mobile robot animations +----------------------- + +These classes create a graphical object that can be animated to show vehicle +position or pose. + +.. inheritance-diagram:: roboticstoolbox.VehicleMarker roboticstoolbox.VehiclePolygon roboticstoolbox.VehicleIcon + :parts: 1 + :top-classes: roboticstoolbox.VehicleAnimationBase + +.. autoclass:: roboticstoolbox.mobile.VehicleAnimationBase + :exclude-members: add, update, plot + +Simple marker +^^^^^^^^^^^^^ + +.. autoclass:: roboticstoolbox.mobile.VehicleMarker + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :special-members: __init__ + +Polygon shape +^^^^^^^^^^^^^ + +.. autoclass:: roboticstoolbox.mobile.VehiclePolygon + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :special-members: __init__ + +Image icon +^^^^^^^^^^ + +.. autoclass:: roboticstoolbox.mobile.VehicleIcon + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :special-members: __init__ \ No newline at end of file diff --git a/_sources/mobile_vehicle_bicycle.rst.txt b/_sources/mobile_vehicle_bicycle.rst.txt new file mode 100644 index 00000000..5522aacf --- /dev/null +++ b/_sources/mobile_vehicle_bicycle.rst.txt @@ -0,0 +1,8 @@ +Bicycle model +------------- + +.. autoclass:: roboticstoolbox.mobile.Bicycle + :members: + :undoc-members: + :inherited-members: + :special-members: __init__ diff --git a/_sources/mobile_vehicle_unicycle.rst.txt b/_sources/mobile_vehicle_unicycle.rst.txt new file mode 100644 index 00000000..56152941 --- /dev/null +++ b/_sources/mobile_vehicle_unicycle.rst.txt @@ -0,0 +1,9 @@ +Unicycle model +-------------- + +.. autoclass:: roboticstoolbox.mobile.Unicycle + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :special-members: __init__ diff --git a/_sources/mobile_vehicle_vehicle.rst.txt b/_sources/mobile_vehicle_vehicle.rst.txt new file mode 100644 index 00000000..7f281478 --- /dev/null +++ b/_sources/mobile_vehicle_vehicle.rst.txt @@ -0,0 +1,11 @@ +Vehicle superclass +------------------ + +The various vehicle models all subclass this class. + +.. autoclass:: roboticstoolbox.mobile.Vehicle + :members: + :undoc-members: + :show-inheritance: + :inherited-members: + :special-members: __init__ \ No newline at end of file diff --git a/docs/source/modules.rst b/_sources/modules.rst.txt similarity index 100% rename from docs/source/modules.rst rename to _sources/modules.rst.txt diff --git a/docs/source/planning.rst b/_sources/planning.rst.txt similarity index 100% rename from docs/source/planning.rst rename to _sources/planning.rst.txt diff --git a/docs/source/ugraph.rst b/_sources/ugraph.rst.txt similarity index 100% rename from docs/source/ugraph.rst rename to _sources/ugraph.rst.txt diff --git a/docs/source/undirected.rst b/_sources/undirected.rst.txt similarity index 100% rename from docs/source/undirected.rst rename to _sources/undirected.rst.txt diff --git a/docs/source/uvertex.rst b/_sources/uvertex.rst.txt similarity index 100% rename from docs/source/uvertex.rst rename to _sources/uvertex.rst.txt diff --git a/_static/RobToolBox_RoundLogoB.png b/_static/RobToolBox_RoundLogoB.png new file mode 100644 index 00000000..e122f084 Binary files /dev/null and b/_static/RobToolBox_RoundLogoB.png differ diff --git a/_static/_sphinx_javascript_frameworks_compat.js b/_static/_sphinx_javascript_frameworks_compat.js new file mode 100644 index 00000000..81415803 --- 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translated[Documentation.PLURAL_EXPR(n)]; + return n === 1 ? singular : plural; + }, + + addTranslations: (catalog) => { + Object.assign(Documentation.TRANSLATIONS, catalog.messages); + Documentation.PLURAL_EXPR = new Function( + "n", + `return (${catalog.plural_expr})` + ); + Documentation.LOCALE = catalog.locale; + }, + + /** + * helper function to focus on search bar + */ + focusSearchBar: () => { + document.querySelectorAll("input[name=q]")[0]?.focus(); + }, + + /** + * Initialise the domain index toggle buttons + */ + initDomainIndexTable: () => { + const toggler = (el) => { + const idNumber = el.id.substr(7); + const toggledRows = document.querySelectorAll(`tr.cg-${idNumber}`); + if (el.src.substr(-9) === "minus.png") { + el.src = `${el.src.substr(0, el.src.length - 9)}plus.png`; + toggledRows.forEach((el) => (el.style.display = "none")); + } else { + el.src = `${el.src.substr(0, el.src.length - 8)}minus.png`; + toggledRows.forEach((el) => (el.style.display = "")); + } + }; + + 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prevLink.href) { + window.location.href = prevLink.href; + event.preventDefault(); + } + break; + case "ArrowRight": + if (!DOCUMENTATION_OPTIONS.NAVIGATION_WITH_KEYS) break; + + const nextLink = document.querySelector('link[rel="next"]'); + if (nextLink && nextLink.href) { + window.location.href = nextLink.href; + event.preventDefault(); + } + break; + } + } + + // some keyboard layouts may need Shift to get / + switch (event.key) { + case "/": + if (!DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS) break; + Documentation.focusSearchBar(); + event.preventDefault(); + } + }); + }, +}; + +// quick alias for translations +const _ = Documentation.gettext; + +_ready(Documentation.init); diff --git a/_static/documentation_options.js b/_static/documentation_options.js new file mode 100644 index 00000000..7e4c114f --- /dev/null +++ b/_static/documentation_options.js @@ -0,0 +1,13 @@ +const DOCUMENTATION_OPTIONS = { + VERSION: '', + LANGUAGE: 'en', + COLLAPSE_INDEX: false, + BUILDER: 'html', + 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+ * jQuery JavaScript Library v3.5.1 + * https://jquery.com/ + * + * Includes Sizzle.js + * https://sizzlejs.com/ + * + * Copyright JS Foundation and other contributors + * Released under the MIT license + * https://jquery.org/license + * + * Date: 2020-05-04T22:49Z + */ +( function( global, factory ) { + + "use strict"; + + if ( typeof module === "object" && typeof module.exports === "object" ) { + + // For CommonJS and CommonJS-like environments where a proper `window` + // is present, execute the factory and get jQuery. + // For environments that do not have a `window` with a `document` + // (such as Node.js), expose a factory as module.exports. + // This accentuates the need for the creation of a real `window`. + // e.g. var jQuery = require("jquery")(window); + // See ticket #14549 for more info. + module.exports = global.document ? + factory( global, true ) : + function( w ) { + if ( !w.document ) { + throw new Error( "jQuery requires a window with a document" ); + } + return factory( w ); + }; + } else { + factory( global ); + } + +// Pass this if window is not defined yet +} )( typeof window !== "undefined" ? window : this, function( window, noGlobal ) { + +// Edge <= 12 - 13+, Firefox <=18 - 45+, IE 10 - 11, Safari 5.1 - 9+, iOS 6 - 9.1 +// throw exceptions when non-strict code (e.g., ASP.NET 4.5) accesses strict mode +// arguments.callee.caller (trac-13335). But as of jQuery 3.0 (2016), strict mode should be common +// enough that all such attempts are guarded in a try block. +"use strict"; + +var arr = []; + +var getProto = Object.getPrototypeOf; + +var slice = arr.slice; + +var flat = arr.flat ? function( array ) { + return arr.flat.call( array ); +} : function( array ) { + return arr.concat.apply( [], array ); +}; + + +var push = arr.push; + +var indexOf = arr.indexOf; + +var class2type = {}; + +var toString = class2type.toString; + +var hasOwn = class2type.hasOwnProperty; + +var fnToString = hasOwn.toString; + +var ObjectFunctionString = fnToString.call( Object ); + +var support = {}; + +var isFunction = function isFunction( obj ) { + + // Support: Chrome <=57, Firefox <=52 + // In some browsers, typeof returns "function" for HTML elements + // (i.e., `typeof document.createElement( "object" ) === "function"`). + // We don't want to classify *any* DOM node as a function. + return typeof obj === "function" && typeof obj.nodeType !== "number"; + }; + + +var isWindow = function isWindow( obj ) { + return obj != null && obj === obj.window; + }; + + +var document = window.document; + + + + var preservedScriptAttributes = { + type: true, + src: true, + nonce: true, + noModule: true + }; + + function DOMEval( code, node, doc ) { + doc = doc || document; + + var i, val, + script = doc.createElement( "script" ); + + script.text = code; + if ( node ) { + for ( i in preservedScriptAttributes ) { + + // Support: Firefox 64+, Edge 18+ + // Some browsers don't support the "nonce" property on scripts. + // On the other hand, just using `getAttribute` is not enough as + // the `nonce` attribute is reset to an empty string whenever it + // becomes browsing-context connected. + // See https://github.com/whatwg/html/issues/2369 + // See https://html.spec.whatwg.org/#nonce-attributes + // The `node.getAttribute` check was added for the sake of + // `jQuery.globalEval` so that it can fake a nonce-containing node + // via an object. + val = node[ i ] || node.getAttribute && node.getAttribute( i ); + if ( val ) { + script.setAttribute( i, val ); + } + } + } + doc.head.appendChild( script ).parentNode.removeChild( script ); + } + + +function toType( obj ) { + if ( obj == null ) { + return obj + ""; + } + + // Support: Android <=2.3 only (functionish RegExp) + return typeof obj === "object" || typeof obj === "function" ? + class2type[ toString.call( obj ) ] || "object" : + typeof obj; +} +/* global Symbol */ +// Defining this global in .eslintrc.json would create a danger of using the global +// unguarded in another place, it seems safer to define global only for this module + + + +var + version = "3.5.1", + + // Define a local copy of jQuery + jQuery = function( selector, context ) { + + // The jQuery object is actually just the init constructor 'enhanced' + // Need init if jQuery is called (just allow error to be thrown if not included) + return new jQuery.fn.init( selector, context ); + }; + +jQuery.fn = jQuery.prototype = { + + // The current version of jQuery being used + jquery: version, + + constructor: jQuery, + + // The default length of a jQuery object is 0 + length: 0, + + toArray: function() { + return slice.call( this ); + }, + + // Get the Nth element in the matched element set OR + // Get the whole matched element set as a clean array + get: function( num ) { + + // Return all the elements in a clean array + if ( num == null ) { + return slice.call( this ); + } + + // Return just the one element from the set + return num < 0 ? this[ num + this.length ] : this[ num ]; + }, + + // Take an array of elements and push it onto the stack + // (returning the new matched element set) + pushStack: function( elems ) { + + // Build a new jQuery matched element set + var ret = jQuery.merge( this.constructor(), elems ); + + // Add the old object onto the stack (as a reference) + ret.prevObject = this; + + // Return the newly-formed element set + return ret; + }, + + // Execute a callback for every element in the matched set. + each: function( callback ) { + return jQuery.each( this, callback ); + }, + + map: function( callback ) { + return this.pushStack( jQuery.map( this, function( elem, i ) { + return callback.call( elem, i, elem ); + } ) ); + }, + + slice: function() { + return this.pushStack( slice.apply( this, arguments ) ); + }, + + first: function() { + return this.eq( 0 ); + }, + + last: function() { + return this.eq( -1 ); + }, + + even: function() { + return this.pushStack( jQuery.grep( this, function( _elem, i ) { + return ( i + 1 ) % 2; + } ) ); + }, + + odd: function() { + return this.pushStack( jQuery.grep( this, function( _elem, i ) { + return i % 2; + } ) ); + }, + + eq: function( i ) { + var len = this.length, + j = +i + ( i < 0 ? len : 0 ); + return this.pushStack( j >= 0 && j < len ? [ this[ j ] ] : [] ); + }, + + end: function() { + return this.prevObject || this.constructor(); + }, + + // For internal use only. + // Behaves like an Array's method, not like a jQuery method. + push: push, + sort: arr.sort, + splice: arr.splice +}; + +jQuery.extend = jQuery.fn.extend = function() { + var options, name, src, copy, copyIsArray, clone, + target = arguments[ 0 ] || {}, + i = 1, + length = arguments.length, + deep = false; + + // Handle a deep copy situation + if ( typeof target === "boolean" ) { + deep = target; + + // Skip the boolean and the target + target = arguments[ i ] || {}; + i++; + } + + // Handle case when target is a string or something (possible in deep copy) + if ( typeof target !== "object" && !isFunction( target ) ) { + target = {}; + } + + // Extend jQuery itself if only one argument is passed + if ( i === length ) { + target = this; + i--; + } + + for ( ; i < length; i++ ) { + + // Only deal with non-null/undefined values + if ( ( options = arguments[ i ] ) != null ) { + + // Extend the base object + for ( name in options ) { + copy = options[ name ]; + + // Prevent Object.prototype pollution + // Prevent never-ending loop + if ( name === "__proto__" || target === copy ) { + continue; + } + + // Recurse if we're merging plain objects or arrays + if ( deep && copy && ( jQuery.isPlainObject( copy ) || + ( copyIsArray = Array.isArray( copy ) ) ) ) { + src = target[ name ]; + + // Ensure proper type for the source value + if ( copyIsArray && !Array.isArray( src ) ) { + clone = []; + } else if ( !copyIsArray && !jQuery.isPlainObject( src ) ) { + clone = {}; + } else { + clone = src; + } + copyIsArray = false; + + // Never move original objects, clone them + target[ name ] = jQuery.extend( deep, clone, copy ); + + // Don't bring in undefined values + } else if ( copy !== undefined ) { + target[ name ] = copy; + } + } + } + } + + // Return the modified object + return target; +}; + +jQuery.extend( { + + // Unique for each copy of jQuery on the page + expando: "jQuery" + ( version + Math.random() ).replace( /\D/g, "" ), + + // Assume jQuery is ready without the ready module + isReady: true, + + error: function( msg ) { + throw new Error( msg ); + }, + + noop: function() {}, + + isPlainObject: function( obj ) { + var proto, Ctor; + + // Detect obvious negatives + // Use toString instead of jQuery.type to catch host objects + if ( !obj || toString.call( obj ) !== "[object Object]" ) { + return false; + } + + proto = getProto( obj ); + + // Objects with no prototype (e.g., `Object.create( null )`) are plain + if ( !proto ) { + return true; + } + + // Objects with prototype are plain iff they were constructed by a global Object function + Ctor = hasOwn.call( proto, "constructor" ) && proto.constructor; + return typeof Ctor === "function" && fnToString.call( Ctor ) === ObjectFunctionString; + }, + + isEmptyObject: function( obj ) { + var name; + + for ( name in obj ) { + return false; + } + return true; + }, + + // Evaluates a script in a provided context; falls back to the global one + // if not specified. + globalEval: function( code, options, doc ) { + DOMEval( code, { nonce: options && options.nonce }, doc ); + }, + + each: function( obj, callback ) { + var length, i = 0; + + if ( isArrayLike( obj ) ) { + length = obj.length; + for ( ; i < length; i++ ) { + if ( callback.call( obj[ i ], i, obj[ i ] ) === false ) { + break; + } + } + } else { + for ( i in obj ) { + if ( callback.call( obj[ i ], i, obj[ i ] ) === false ) { + break; + } + } + } + + return obj; + }, + + // results is for internal usage only + makeArray: function( arr, results ) { + var ret = results || []; + + if ( arr != null ) { + if ( isArrayLike( Object( arr ) ) ) { + jQuery.merge( ret, + typeof arr === "string" ? + [ arr ] : arr + ); + } else { + push.call( ret, arr ); + } + } + + return ret; + }, + + inArray: function( elem, arr, i ) { + return arr == null ? -1 : indexOf.call( arr, elem, i ); + }, + + // Support: Android <=4.0 only, PhantomJS 1 only + // push.apply(_, arraylike) throws on ancient WebKit + merge: function( first, second ) { + var len = +second.length, + j = 0, + i = first.length; + + for ( ; j < len; j++ ) { + first[ i++ ] = second[ j ]; + } + + first.length = i; + + return first; + }, + + grep: function( elems, callback, invert ) { + var callbackInverse, + matches = [], + i = 0, + length = elems.length, + callbackExpect = !invert; + + // Go through the array, only saving the items + // that pass the validator function + for ( ; i < length; i++ ) { + callbackInverse = !callback( elems[ i ], i ); + if ( callbackInverse !== callbackExpect ) { + matches.push( elems[ i ] ); + } + } + + return matches; + }, + + // arg is for internal usage only + map: function( elems, callback, arg ) { + var length, value, + i = 0, + ret = []; + + // Go through the array, translating each of the items to their new values + if ( isArrayLike( elems ) ) { + length = elems.length; + for ( ; i < length; i++ ) { + value = callback( elems[ i ], i, arg ); + + if ( value != null ) { + ret.push( value ); + } + } + + // Go through every key on the object, + } else { + for ( i in elems ) { + value = callback( elems[ i ], i, arg ); + + if ( value != null ) { + ret.push( value ); + } + } + } + + // Flatten any nested arrays + return flat( ret ); + }, + + // A global GUID counter for objects + guid: 1, + + // jQuery.support is not used in Core but other projects attach their + // properties to it so it needs to exist. + support: support +} ); + +if ( typeof Symbol === "function" ) { + jQuery.fn[ Symbol.iterator ] = arr[ Symbol.iterator ]; +} + +// Populate the class2type map +jQuery.each( "Boolean Number String Function Array Date RegExp Object Error Symbol".split( " " ), +function( _i, name ) { + class2type[ "[object " + name + "]" ] = name.toLowerCase(); +} ); + +function isArrayLike( obj ) { + + // Support: real iOS 8.2 only (not reproducible in simulator) + // `in` check used to prevent JIT error (gh-2145) + // hasOwn isn't used here due to false negatives + // regarding Nodelist length in IE + var length = !!obj && "length" in obj && obj.length, + type = toType( obj ); + + if ( isFunction( obj ) || isWindow( obj ) ) { + return false; + } + + return type === "array" || length === 0 || + typeof length === "number" && length > 0 && ( length - 1 ) in obj; +} +var Sizzle = +/*! + * Sizzle CSS Selector Engine v2.3.5 + * https://sizzlejs.com/ + * + * Copyright JS Foundation and other contributors + * Released under the MIT license + * https://js.foundation/ + * + * Date: 2020-03-14 + */ +( function( window ) { +var i, + support, + Expr, + getText, + isXML, + tokenize, + compile, + select, + outermostContext, + sortInput, + hasDuplicate, + + // Local document vars + setDocument, + document, + docElem, + documentIsHTML, + rbuggyQSA, + rbuggyMatches, + matches, + contains, + + // Instance-specific data + expando = "sizzle" + 1 * new Date(), + preferredDoc = window.document, + dirruns = 0, + done = 0, + classCache = createCache(), + tokenCache = createCache(), + compilerCache = createCache(), + nonnativeSelectorCache = createCache(), + sortOrder = function( a, b ) { + if ( a === b ) { + hasDuplicate = true; + } + return 0; + }, + + // Instance methods + hasOwn = ( {} ).hasOwnProperty, + arr = [], + pop = arr.pop, + pushNative = arr.push, + push = arr.push, + slice = arr.slice, + + // Use a stripped-down indexOf as it's faster than native + // https://jsperf.com/thor-indexof-vs-for/5 + indexOf = function( list, elem ) { + var i = 0, + len = list.length; + for ( ; i < len; i++ ) { + if ( list[ i ] === elem ) { + return i; + } + } + return -1; + }, + + booleans = "checked|selected|async|autofocus|autoplay|controls|defer|disabled|hidden|" + + "ismap|loop|multiple|open|readonly|required|scoped", + + // Regular expressions + + // http://www.w3.org/TR/css3-selectors/#whitespace + whitespace = "[\\x20\\t\\r\\n\\f]", + + // https://www.w3.org/TR/css-syntax-3/#ident-token-diagram + identifier = "(?:\\\\[\\da-fA-F]{1,6}" + whitespace + + "?|\\\\[^\\r\\n\\f]|[\\w-]|[^\0-\\x7f])+", + + // Attribute selectors: http://www.w3.org/TR/selectors/#attribute-selectors + attributes = "\\[" + whitespace + "*(" + identifier + ")(?:" + whitespace + + + // Operator (capture 2) + "*([*^$|!~]?=)" + whitespace + + + // "Attribute values must be CSS identifiers [capture 5] + // or strings [capture 3 or capture 4]" + "*(?:'((?:\\\\.|[^\\\\'])*)'|\"((?:\\\\.|[^\\\\\"])*)\"|(" + identifier + "))|)" + + whitespace + "*\\]", + + pseudos = ":(" + identifier + ")(?:\\((" + + + // To reduce the number of selectors needing tokenize in the preFilter, prefer arguments: + // 1. quoted (capture 3; capture 4 or capture 5) + "('((?:\\\\.|[^\\\\'])*)'|\"((?:\\\\.|[^\\\\\"])*)\")|" + + + // 2. simple (capture 6) + "((?:\\\\.|[^\\\\()[\\]]|" + attributes + ")*)|" + + + // 3. anything else (capture 2) + ".*" + + ")\\)|)", + + // Leading and non-escaped trailing whitespace, capturing some non-whitespace characters preceding the latter + rwhitespace = new RegExp( whitespace + "+", "g" ), + rtrim = new RegExp( "^" + whitespace + "+|((?:^|[^\\\\])(?:\\\\.)*)" + + whitespace + "+$", "g" ), + + rcomma = new RegExp( "^" + whitespace + "*," + whitespace + "*" ), + rcombinators = new RegExp( "^" + whitespace + "*([>+~]|" + whitespace + ")" + whitespace + + "*" ), + rdescend = new RegExp( whitespace + "|>" ), + + rpseudo = new RegExp( pseudos ), + ridentifier = new RegExp( "^" + identifier + "$" ), + + matchExpr = { + "ID": new RegExp( "^#(" + identifier + ")" ), + "CLASS": new RegExp( "^\\.(" + identifier + ")" ), + "TAG": new RegExp( "^(" + identifier + "|[*])" ), + "ATTR": new RegExp( "^" + attributes ), + "PSEUDO": new RegExp( "^" + pseudos ), + "CHILD": new RegExp( "^:(only|first|last|nth|nth-last)-(child|of-type)(?:\\(" + + whitespace + "*(even|odd|(([+-]|)(\\d*)n|)" + whitespace + "*(?:([+-]|)" + + whitespace + "*(\\d+)|))" + whitespace + "*\\)|)", "i" ), + "bool": new RegExp( "^(?:" + booleans + ")$", "i" ), + + // For use in libraries implementing .is() + // We use this for POS matching in `select` + "needsContext": new RegExp( "^" + whitespace + + "*[>+~]|:(even|odd|eq|gt|lt|nth|first|last)(?:\\(" + whitespace + + "*((?:-\\d)?\\d*)" + whitespace + "*\\)|)(?=[^-]|$)", "i" ) + }, + + rhtml = /HTML$/i, + rinputs = /^(?:input|select|textarea|button)$/i, + rheader = /^h\d$/i, + + rnative = /^[^{]+\{\s*\[native \w/, + + // Easily-parseable/retrievable ID or TAG or CLASS selectors + rquickExpr = /^(?:#([\w-]+)|(\w+)|\.([\w-]+))$/, + + rsibling = /[+~]/, + + // CSS escapes + // http://www.w3.org/TR/CSS21/syndata.html#escaped-characters + runescape = new RegExp( "\\\\[\\da-fA-F]{1,6}" + whitespace + "?|\\\\([^\\r\\n\\f])", "g" ), + funescape = function( escape, nonHex ) { + var high = "0x" + escape.slice( 1 ) - 0x10000; + + return nonHex ? + + // Strip the backslash prefix from a non-hex escape sequence + nonHex : + + // Replace a hexadecimal escape sequence with the encoded Unicode code point + // Support: IE <=11+ + // For values outside the Basic Multilingual Plane (BMP), manually construct a + // surrogate pair + high < 0 ? + String.fromCharCode( high + 0x10000 ) : + String.fromCharCode( high >> 10 | 0xD800, high & 0x3FF | 0xDC00 ); + }, + + // CSS string/identifier serialization + // https://drafts.csswg.org/cssom/#common-serializing-idioms + rcssescape = /([\0-\x1f\x7f]|^-?\d)|^-$|[^\0-\x1f\x7f-\uFFFF\w-]/g, + fcssescape = function( ch, asCodePoint ) { + if ( asCodePoint ) { + + // U+0000 NULL becomes U+FFFD REPLACEMENT CHARACTER + if ( ch === "\0" ) { + return "\uFFFD"; + } + + // Control characters and (dependent upon position) numbers get escaped as code points + return ch.slice( 0, -1 ) + "\\" + + ch.charCodeAt( ch.length - 1 ).toString( 16 ) + " "; + } + + // Other potentially-special ASCII characters get backslash-escaped + return "\\" + ch; + }, + + // Used for iframes + // See setDocument() + // Removing the function wrapper causes a "Permission Denied" + // error in IE + unloadHandler = function() { + setDocument(); + }, + + inDisabledFieldset = addCombinator( + function( elem ) { + return elem.disabled === true && elem.nodeName.toLowerCase() === "fieldset"; + }, + { dir: "parentNode", next: "legend" } + ); + +// Optimize for push.apply( _, NodeList ) +try { + push.apply( + ( arr = slice.call( preferredDoc.childNodes ) ), + preferredDoc.childNodes + ); + + // Support: Android<4.0 + // Detect silently failing push.apply + // eslint-disable-next-line no-unused-expressions + arr[ preferredDoc.childNodes.length ].nodeType; +} catch ( e ) { + push = { apply: arr.length ? + + // Leverage slice if possible + function( target, els ) { + pushNative.apply( target, slice.call( els ) ); + } : + + // Support: IE<9 + // Otherwise append directly + function( target, els ) { + var j = target.length, + i = 0; + + // Can't trust NodeList.length + while ( ( target[ j++ ] = els[ i++ ] ) ) {} + target.length = j - 1; + } + }; +} + +function Sizzle( selector, context, results, seed ) { + var m, i, elem, nid, match, groups, newSelector, + newContext = context && context.ownerDocument, + + // nodeType defaults to 9, since context defaults to document + nodeType = context ? context.nodeType : 9; + + results = results || []; + + // Return early from calls with invalid selector or context + if ( typeof selector !== "string" || !selector || + nodeType !== 1 && nodeType !== 9 && nodeType !== 11 ) { + + return results; + } + + // Try to shortcut find operations (as opposed to filters) in HTML documents + if ( !seed ) { + setDocument( context ); + context = context || document; + + if ( documentIsHTML ) { + + // If the selector is sufficiently simple, try using a "get*By*" DOM method + // (excepting DocumentFragment context, where the methods don't exist) + if ( nodeType !== 11 && ( match = rquickExpr.exec( selector ) ) ) { + + // ID selector + if ( ( m = match[ 1 ] ) ) { + + // Document context + if ( nodeType === 9 ) { + if ( ( elem = context.getElementById( m ) ) ) { + + // Support: IE, Opera, Webkit + // TODO: identify versions + // getElementById can match elements by name instead of ID + if ( elem.id === m ) { + results.push( elem ); + return results; + } + } else { + return results; + } + + // Element context + } else { + + // Support: IE, Opera, Webkit + // TODO: identify versions + // getElementById can match elements by name instead of ID + if ( newContext && ( elem = newContext.getElementById( m ) ) && + contains( context, elem ) && + elem.id === m ) { + + results.push( elem ); + return results; + } + } + + // Type selector + } else if ( match[ 2 ] ) { + push.apply( results, context.getElementsByTagName( selector ) ); + return results; + + // Class selector + } else if ( ( m = match[ 3 ] ) && support.getElementsByClassName && + context.getElementsByClassName ) { + + push.apply( results, context.getElementsByClassName( m ) ); + return results; + } + } + + // Take advantage of querySelectorAll + if ( support.qsa && + !nonnativeSelectorCache[ selector + " " ] && + ( !rbuggyQSA || !rbuggyQSA.test( selector ) ) && + + // Support: IE 8 only + // Exclude object elements + ( nodeType !== 1 || context.nodeName.toLowerCase() !== "object" ) ) { + + newSelector = selector; + newContext = context; + + // qSA considers elements outside a scoping root when evaluating child or + // descendant combinators, which is not what we want. + // In such cases, we work around the behavior by prefixing every selector in the + // list with an ID selector referencing the scope context. + // The technique has to be used as well when a leading combinator is used + // as such selectors are not recognized by querySelectorAll. + // Thanks to Andrew Dupont for this technique. + if ( nodeType === 1 && + ( rdescend.test( selector ) || rcombinators.test( selector ) ) ) { + + // Expand context for sibling selectors + newContext = rsibling.test( selector ) && testContext( context.parentNode ) || + context; + + // We can use :scope instead of the ID hack if the browser + // supports it & if we're not changing the context. + if ( newContext !== context || !support.scope ) { + + // Capture the context ID, setting it first if necessary + if ( ( nid = context.getAttribute( "id" ) ) ) { + nid = nid.replace( rcssescape, fcssescape ); + } else { + context.setAttribute( "id", ( nid = expando ) ); + } + } + + // Prefix every selector in the list + groups = tokenize( selector ); + i = groups.length; + while ( i-- ) { + groups[ i ] = ( nid ? "#" + nid : ":scope" ) + " " + + toSelector( groups[ i ] ); + } + newSelector = groups.join( "," ); + } + + try { + push.apply( results, + newContext.querySelectorAll( newSelector ) + ); + return results; + } catch ( qsaError ) { + nonnativeSelectorCache( selector, true ); + } finally { + if ( nid === expando ) { + context.removeAttribute( "id" ); + } + } + } + } + } + + // All others + return select( selector.replace( rtrim, "$1" ), context, results, seed ); +} + +/** + * Create key-value caches of limited size + * @returns {function(string, object)} Returns the Object data after storing it on itself with + * property name the (space-suffixed) string and (if the cache is larger than Expr.cacheLength) + * deleting the oldest entry + */ +function createCache() { + var keys = []; + + function cache( key, value ) { + + // Use (key + " ") to avoid collision with native prototype properties (see Issue #157) + if ( keys.push( key + " " ) > Expr.cacheLength ) { + + // Only keep the most recent entries + delete cache[ keys.shift() ]; + } + return ( cache[ key + " " ] = value ); + } + return cache; +} + +/** + * Mark a function for special use by Sizzle + * @param {Function} fn The function to mark + */ +function markFunction( fn ) { + fn[ expando ] = true; + return fn; +} + +/** + * Support testing using an element + * @param {Function} fn Passed the created element and returns a boolean result + */ +function assert( fn ) { + var el = document.createElement( "fieldset" ); + + try { + return !!fn( el ); + } catch ( e ) { + return false; + } finally { + + // Remove from its parent by default + if ( el.parentNode ) { + el.parentNode.removeChild( el ); + } + + // release memory in IE + el = null; + } +} + +/** + * Adds the same handler for all of the specified attrs + * @param {String} attrs Pipe-separated list of attributes + * @param {Function} handler The method that will be applied + */ +function addHandle( attrs, handler ) { + var arr = attrs.split( "|" ), + i = arr.length; + + while ( i-- ) { + Expr.attrHandle[ arr[ i ] ] = handler; + } +} + +/** + * Checks document order of two siblings + * @param {Element} a + * @param {Element} b + * @returns {Number} Returns less than 0 if a precedes b, greater than 0 if a follows b + */ +function siblingCheck( a, b ) { + var cur = b && a, + diff = cur && a.nodeType === 1 && b.nodeType === 1 && + a.sourceIndex - b.sourceIndex; + + // Use IE sourceIndex if available on both nodes + if ( diff ) { + return diff; + } + + // Check if b follows a + if ( cur ) { + while ( ( cur = cur.nextSibling ) ) { + if ( cur === b ) { + return -1; + } + } + } + + return a ? 1 : -1; +} + +/** + * Returns a function to use in pseudos for input types + * @param {String} type + */ +function createInputPseudo( type ) { + return function( elem ) { + var name = elem.nodeName.toLowerCase(); + return name === "input" && elem.type === type; + }; +} + +/** + * Returns a function to use in pseudos for buttons + * @param {String} type + */ +function createButtonPseudo( type ) { + return function( elem ) { + var name = elem.nodeName.toLowerCase(); + return ( name === "input" || name === "button" ) && elem.type === type; + }; +} + +/** + * Returns a function to use in pseudos for :enabled/:disabled + * @param {Boolean} disabled true for :disabled; false for :enabled + */ +function createDisabledPseudo( disabled ) { + + // Known :disabled false positives: fieldset[disabled] > legend:nth-of-type(n+2) :can-disable + return function( elem ) { + + // Only certain elements can match :enabled or :disabled + // https://html.spec.whatwg.org/multipage/scripting.html#selector-enabled + // https://html.spec.whatwg.org/multipage/scripting.html#selector-disabled + if ( "form" in elem ) { + + // Check for inherited disabledness on relevant non-disabled elements: + // * listed form-associated elements in a disabled fieldset + // https://html.spec.whatwg.org/multipage/forms.html#category-listed + // https://html.spec.whatwg.org/multipage/forms.html#concept-fe-disabled + // * option elements in a disabled optgroup + // https://html.spec.whatwg.org/multipage/forms.html#concept-option-disabled + // All such elements have a "form" property. + if ( elem.parentNode && elem.disabled === false ) { + + // Option elements defer to a parent optgroup if present + if ( "label" in elem ) { + if ( "label" in elem.parentNode ) { + return elem.parentNode.disabled === disabled; + } else { + return elem.disabled === disabled; + } + } + + // Support: IE 6 - 11 + // Use the isDisabled shortcut property to check for disabled fieldset ancestors + return elem.isDisabled === disabled || + + // Where there is no isDisabled, check manually + /* jshint -W018 */ + elem.isDisabled !== !disabled && + inDisabledFieldset( elem ) === disabled; + } + + return elem.disabled === disabled; + + // Try to winnow out elements that can't be disabled before trusting the disabled property. + // Some victims get caught in our net (label, legend, menu, track), but it shouldn't + // even exist on them, let alone have a boolean value. + } else if ( "label" in elem ) { + return elem.disabled === disabled; + } + + // Remaining elements are neither :enabled nor :disabled + return false; + }; +} + +/** + * Returns a function to use in pseudos for positionals + * @param {Function} fn + */ +function createPositionalPseudo( fn ) { + return markFunction( function( argument ) { + argument = +argument; + return markFunction( function( seed, matches ) { + var j, + matchIndexes = fn( [], seed.length, argument ), + i = matchIndexes.length; + + // Match elements found at the specified indexes + while ( i-- ) { + if ( seed[ ( j = matchIndexes[ i ] ) ] ) { + seed[ j ] = !( matches[ j ] = seed[ j ] ); + } + } + } ); + } ); +} + +/** + * Checks a node for validity as a Sizzle context + * @param {Element|Object=} context + * @returns {Element|Object|Boolean} The input node if acceptable, otherwise a falsy value + */ +function testContext( context ) { + return context && typeof context.getElementsByTagName !== "undefined" && context; +} + +// Expose support vars for convenience +support = Sizzle.support = {}; + +/** + * Detects XML nodes + * @param {Element|Object} elem An element or a document + * @returns {Boolean} True iff elem is a non-HTML XML node + */ +isXML = Sizzle.isXML = function( elem ) { + var namespace = elem.namespaceURI, + docElem = ( elem.ownerDocument || elem ).documentElement; + + // Support: IE <=8 + // Assume HTML when documentElement doesn't yet exist, such as inside loading iframes + // https://bugs.jquery.com/ticket/4833 + return !rhtml.test( namespace || docElem && docElem.nodeName || "HTML" ); +}; + +/** + * Sets document-related variables once based on the current document + * @param {Element|Object} [doc] An element or document object to use to set the document + * @returns {Object} Returns the current document + */ +setDocument = Sizzle.setDocument = function( node ) { + var hasCompare, subWindow, + doc = node ? node.ownerDocument || node : preferredDoc; + + // Return early if doc is invalid or already selected + // Support: IE 11+, Edge 17 - 18+ + // IE/Edge sometimes throw a "Permission denied" error when strict-comparing + // two documents; shallow comparisons work. + // eslint-disable-next-line eqeqeq + if ( doc == document || doc.nodeType !== 9 || !doc.documentElement ) { + return document; + } + + // Update global variables + document = doc; + docElem = document.documentElement; + documentIsHTML = !isXML( document ); + + // Support: IE 9 - 11+, Edge 12 - 18+ + // Accessing iframe documents after unload throws "permission denied" errors (jQuery #13936) + // Support: IE 11+, Edge 17 - 18+ + // IE/Edge sometimes throw a "Permission denied" error when strict-comparing + // two documents; shallow comparisons work. + // eslint-disable-next-line eqeqeq + if ( preferredDoc != document && + ( subWindow = document.defaultView ) && subWindow.top !== subWindow ) { + + // Support: IE 11, Edge + if ( subWindow.addEventListener ) { + subWindow.addEventListener( "unload", unloadHandler, false ); + + // Support: IE 9 - 10 only + } else if ( subWindow.attachEvent ) { + subWindow.attachEvent( "onunload", unloadHandler ); + } + } + + // Support: IE 8 - 11+, Edge 12 - 18+, Chrome <=16 - 25 only, Firefox <=3.6 - 31 only, + // Safari 4 - 5 only, Opera <=11.6 - 12.x only + // IE/Edge & older browsers don't support the :scope pseudo-class. + // Support: Safari 6.0 only + // Safari 6.0 supports :scope but it's an alias of :root there. + support.scope = assert( function( el ) { + docElem.appendChild( el ).appendChild( document.createElement( "div" ) ); + return typeof el.querySelectorAll !== "undefined" && + !el.querySelectorAll( ":scope fieldset div" ).length; + } ); + + /* Attributes + ---------------------------------------------------------------------- */ + + // Support: IE<8 + // Verify that getAttribute really returns attributes and not properties + // (excepting IE8 booleans) + support.attributes = assert( function( el ) { + el.className = "i"; + return !el.getAttribute( "className" ); + } ); + + /* getElement(s)By* + ---------------------------------------------------------------------- */ + + // Check if getElementsByTagName("*") returns only elements + support.getElementsByTagName = assert( function( el ) { + el.appendChild( document.createComment( "" ) ); + return !el.getElementsByTagName( "*" ).length; + } ); + + // Support: IE<9 + support.getElementsByClassName = rnative.test( document.getElementsByClassName ); + + // Support: IE<10 + // Check if getElementById returns elements by name + // The broken getElementById methods don't pick up programmatically-set names, + // so use a roundabout getElementsByName test + support.getById = assert( function( el ) { + docElem.appendChild( el ).id = expando; + return !document.getElementsByName || !document.getElementsByName( expando ).length; + } ); + + // ID filter and find + if ( support.getById ) { + Expr.filter[ "ID" ] = function( id ) { + var attrId = id.replace( runescape, funescape ); + return function( elem ) { + return elem.getAttribute( "id" ) === attrId; + }; + }; + Expr.find[ "ID" ] = function( id, context ) { + if ( typeof context.getElementById !== "undefined" && documentIsHTML ) { + var elem = context.getElementById( id ); + return elem ? [ elem ] : []; + } + }; + } else { + Expr.filter[ "ID" ] = function( id ) { + var attrId = id.replace( runescape, funescape ); + return function( elem ) { + var node = typeof elem.getAttributeNode !== "undefined" && + elem.getAttributeNode( "id" ); + return node && node.value === attrId; + }; + }; + + // Support: IE 6 - 7 only + // getElementById is not reliable as a find shortcut + Expr.find[ "ID" ] = function( id, context ) { + if ( typeof context.getElementById !== "undefined" && documentIsHTML ) { + var node, i, elems, + elem = context.getElementById( id ); + + if ( elem ) { + + // Verify the id attribute + node = elem.getAttributeNode( "id" ); + if ( node && node.value === id ) { + return [ elem ]; + } + + // Fall back on getElementsByName + elems = context.getElementsByName( id ); + i = 0; + while ( ( elem = elems[ i++ ] ) ) { + node = elem.getAttributeNode( "id" ); + if ( node && node.value === id ) { + return [ elem ]; + } + } + } + + return []; + } + }; + } + + // Tag + Expr.find[ "TAG" ] = support.getElementsByTagName ? + function( tag, context ) { + if ( typeof context.getElementsByTagName !== "undefined" ) { + return context.getElementsByTagName( tag ); + + // DocumentFragment nodes don't have gEBTN + } else if ( support.qsa ) { + return context.querySelectorAll( tag ); + } + } : + + function( tag, context ) { + var elem, + tmp = [], + i = 0, + + // By happy coincidence, a (broken) gEBTN appears on DocumentFragment nodes too + results = context.getElementsByTagName( tag ); + + // Filter out possible comments + if ( tag === "*" ) { + while ( ( elem = results[ i++ ] ) ) { + if ( elem.nodeType === 1 ) { + tmp.push( elem ); + } + } + + return tmp; + } + return results; + }; + + // Class + Expr.find[ "CLASS" ] = support.getElementsByClassName && function( className, context ) { + if ( typeof context.getElementsByClassName !== "undefined" && documentIsHTML ) { + return context.getElementsByClassName( className ); + } + }; + + /* QSA/matchesSelector + ---------------------------------------------------------------------- */ + + // QSA and matchesSelector support + + // matchesSelector(:active) reports false when true (IE9/Opera 11.5) + rbuggyMatches = []; + + // qSa(:focus) reports false when true (Chrome 21) + // We allow this because of a bug in IE8/9 that throws an error + // whenever `document.activeElement` is accessed on an iframe + // So, we allow :focus to pass through QSA all the time to avoid the IE error + // See https://bugs.jquery.com/ticket/13378 + rbuggyQSA = []; + + if ( ( support.qsa = rnative.test( document.querySelectorAll ) ) ) { + + // Build QSA regex + // Regex strategy adopted from Diego Perini + assert( function( el ) { + + var input; + + // Select is set to empty string on purpose + // This is to test IE's treatment of not explicitly + // setting a boolean content attribute, + // since its presence should be enough + // https://bugs.jquery.com/ticket/12359 + docElem.appendChild( el ).innerHTML = "" + + ""; + + // Support: IE8, Opera 11-12.16 + // Nothing should be selected when empty strings follow ^= or $= or *= + // The test attribute must be unknown in Opera but "safe" for WinRT + // https://msdn.microsoft.com/en-us/library/ie/hh465388.aspx#attribute_section + if ( el.querySelectorAll( "[msallowcapture^='']" ).length ) { + rbuggyQSA.push( "[*^$]=" + whitespace + "*(?:''|\"\")" ); + } + + // Support: IE8 + // Boolean attributes and "value" are not treated correctly + if ( !el.querySelectorAll( "[selected]" ).length ) { + rbuggyQSA.push( "\\[" + whitespace + "*(?:value|" + booleans + ")" ); + } + + // Support: Chrome<29, Android<4.4, Safari<7.0+, iOS<7.0+, PhantomJS<1.9.8+ + if ( !el.querySelectorAll( "[id~=" + expando + "-]" ).length ) { + rbuggyQSA.push( "~=" ); + } + + // Support: IE 11+, Edge 15 - 18+ + // IE 11/Edge don't find elements on a `[name='']` query in some cases. + // Adding a temporary attribute to the document before the selection works + // around the issue. + // Interestingly, IE 10 & older don't seem to have the issue. + input = document.createElement( "input" ); + input.setAttribute( "name", "" ); + el.appendChild( input ); + if ( !el.querySelectorAll( "[name='']" ).length ) { + rbuggyQSA.push( "\\[" + whitespace + "*name" + whitespace + "*=" + + whitespace + "*(?:''|\"\")" ); + } + + // Webkit/Opera - :checked should return selected option elements + // http://www.w3.org/TR/2011/REC-css3-selectors-20110929/#checked + // IE8 throws error here and will not see later tests + if ( !el.querySelectorAll( ":checked" ).length ) { + rbuggyQSA.push( ":checked" ); + } + + // Support: Safari 8+, iOS 8+ + // https://bugs.webkit.org/show_bug.cgi?id=136851 + // In-page `selector#id sibling-combinator selector` fails + if ( !el.querySelectorAll( "a#" + expando + "+*" ).length ) { + rbuggyQSA.push( ".#.+[+~]" ); + } + + // Support: Firefox <=3.6 - 5 only + // Old Firefox doesn't throw on a badly-escaped identifier. + el.querySelectorAll( "\\\f" ); + rbuggyQSA.push( "[\\r\\n\\f]" ); + } ); + + assert( function( el ) { + el.innerHTML = "" + + ""; + + // Support: Windows 8 Native Apps + // The type and name attributes are restricted during .innerHTML assignment + var input = document.createElement( "input" ); + input.setAttribute( "type", "hidden" ); + el.appendChild( input ).setAttribute( "name", "D" ); + + // Support: IE8 + // Enforce case-sensitivity of name attribute + if ( el.querySelectorAll( "[name=d]" ).length ) { + rbuggyQSA.push( "name" + whitespace + "*[*^$|!~]?=" ); + } + + // FF 3.5 - :enabled/:disabled and hidden elements (hidden elements are still enabled) + // IE8 throws error here and will not see later tests + if ( el.querySelectorAll( ":enabled" ).length !== 2 ) { + rbuggyQSA.push( ":enabled", ":disabled" ); + } + + // Support: IE9-11+ + // IE's :disabled selector does not pick up the children of disabled fieldsets + docElem.appendChild( el ).disabled = true; + if ( el.querySelectorAll( ":disabled" ).length !== 2 ) { + rbuggyQSA.push( ":enabled", ":disabled" ); + } + + // Support: Opera 10 - 11 only + // Opera 10-11 does not throw on post-comma invalid pseudos + el.querySelectorAll( "*,:x" ); + rbuggyQSA.push( ",.*:" ); + } ); + } + + if ( ( support.matchesSelector = rnative.test( ( matches = docElem.matches || + docElem.webkitMatchesSelector || + docElem.mozMatchesSelector || + docElem.oMatchesSelector || + docElem.msMatchesSelector ) ) ) ) { + + assert( function( el ) { + + // Check to see if it's possible to do matchesSelector + // on a disconnected node (IE 9) + support.disconnectedMatch = matches.call( el, "*" ); + + // This should fail with an exception + // Gecko does not error, returns false instead + matches.call( el, "[s!='']:x" ); + rbuggyMatches.push( "!=", pseudos ); + } ); + } + + rbuggyQSA = rbuggyQSA.length && new RegExp( rbuggyQSA.join( "|" ) ); + rbuggyMatches = rbuggyMatches.length && new RegExp( rbuggyMatches.join( "|" ) ); + + /* Contains + ---------------------------------------------------------------------- */ + hasCompare = rnative.test( docElem.compareDocumentPosition ); + + // Element contains another + // Purposefully self-exclusive + // As in, an element does not contain itself + contains = hasCompare || rnative.test( docElem.contains ) ? + function( a, b ) { + var adown = a.nodeType === 9 ? a.documentElement : a, + bup = b && b.parentNode; + return a === bup || !!( bup && bup.nodeType === 1 && ( + adown.contains ? + adown.contains( bup ) : + a.compareDocumentPosition && a.compareDocumentPosition( bup ) & 16 + ) ); + } : + function( a, b ) { + if ( b ) { + while ( ( b = b.parentNode ) ) { + if ( b === a ) { + return true; + } + } + } + return false; + }; + + /* Sorting + ---------------------------------------------------------------------- */ + + // Document order sorting + sortOrder = hasCompare ? + function( a, b ) { + + // Flag for duplicate removal + if ( a === b ) { + hasDuplicate = true; + return 0; + } + + // Sort on method existence if only one input has compareDocumentPosition + var compare = !a.compareDocumentPosition - !b.compareDocumentPosition; + if ( compare ) { + return compare; + } + + // Calculate position if both inputs belong to the same document + // Support: IE 11+, Edge 17 - 18+ + // IE/Edge sometimes throw a "Permission denied" error when strict-comparing + // two documents; shallow comparisons work. + // eslint-disable-next-line eqeqeq + compare = ( a.ownerDocument || a ) == ( b.ownerDocument || b ) ? + a.compareDocumentPosition( b ) : + + // Otherwise we know they are disconnected + 1; + + // Disconnected nodes + if ( compare & 1 || + ( !support.sortDetached && b.compareDocumentPosition( a ) === compare ) ) { + + // Choose the first element that is related to our preferred document + // Support: IE 11+, Edge 17 - 18+ + // IE/Edge sometimes throw a "Permission denied" error when strict-comparing + // two documents; shallow comparisons work. + // eslint-disable-next-line eqeqeq + if ( a == document || a.ownerDocument == preferredDoc && + contains( preferredDoc, a ) ) { + return -1; + } + + // Support: IE 11+, Edge 17 - 18+ + // IE/Edge sometimes throw a "Permission denied" error when strict-comparing + // two documents; shallow comparisons work. + // eslint-disable-next-line eqeqeq + if ( b == document || b.ownerDocument == preferredDoc && + contains( preferredDoc, b ) ) { + return 1; + } + + // Maintain original order + return sortInput ? + ( indexOf( sortInput, a ) - indexOf( sortInput, b ) ) : + 0; + } + + return compare & 4 ? -1 : 1; + } : + function( a, b ) { + + // Exit early if the nodes are identical + if ( a === b ) { + hasDuplicate = true; + return 0; + } + + var cur, + i = 0, + aup = a.parentNode, + bup = b.parentNode, + ap = [ a ], + bp = [ b ]; + + // Parentless nodes are either documents or disconnected + if ( !aup || !bup ) { + + // Support: IE 11+, Edge 17 - 18+ + // IE/Edge sometimes throw a "Permission denied" error when strict-comparing + // two documents; shallow comparisons work. + /* eslint-disable eqeqeq */ + return a == document ? -1 : + b == document ? 1 : + /* eslint-enable eqeqeq */ + aup ? -1 : + bup ? 1 : + sortInput ? + ( indexOf( sortInput, a ) - indexOf( sortInput, b ) ) : + 0; + + // If the nodes are siblings, we can do a quick check + } else if ( aup === bup ) { + return siblingCheck( a, b ); + } + + // Otherwise we need full lists of their ancestors for comparison + cur = a; + while ( ( cur = cur.parentNode ) ) { + ap.unshift( cur ); + } + cur = b; + while ( ( cur = cur.parentNode ) ) { + bp.unshift( cur ); + } + + // Walk down the tree looking for a discrepancy + while ( ap[ i ] === bp[ i ] ) { + i++; + } + + return i ? + + // Do a sibling check if the nodes have a common ancestor + siblingCheck( ap[ i ], bp[ i ] ) : + + // Otherwise nodes in our document sort first + // Support: IE 11+, Edge 17 - 18+ + // IE/Edge sometimes throw a "Permission denied" error when strict-comparing + // two documents; shallow comparisons work. + /* eslint-disable eqeqeq */ + ap[ i ] == preferredDoc ? -1 : + bp[ i ] == preferredDoc ? 1 : + /* eslint-enable eqeqeq */ + 0; + }; + + return document; +}; + +Sizzle.matches = function( expr, elements ) { + return Sizzle( expr, null, null, elements ); +}; + +Sizzle.matchesSelector = function( elem, expr ) { + setDocument( elem ); + + if ( support.matchesSelector && documentIsHTML && + !nonnativeSelectorCache[ expr + " " ] && + ( !rbuggyMatches || !rbuggyMatches.test( expr ) ) && + ( !rbuggyQSA || !rbuggyQSA.test( expr ) ) ) { + + try { + var ret = matches.call( elem, expr ); + + // IE 9's matchesSelector returns false on disconnected nodes + if ( ret || support.disconnectedMatch || + + // As well, disconnected nodes are said to be in a document + // fragment in IE 9 + elem.document && elem.document.nodeType !== 11 ) { + return ret; + } + } catch ( e ) { + nonnativeSelectorCache( expr, true ); + } + } + + return Sizzle( expr, document, null, [ elem ] ).length > 0; +}; + +Sizzle.contains = function( context, elem ) { + + // Set document vars if needed + // Support: IE 11+, Edge 17 - 18+ + // IE/Edge sometimes throw a "Permission denied" error when strict-comparing + // two documents; shallow comparisons work. + // eslint-disable-next-line eqeqeq + if ( ( context.ownerDocument || context ) != document ) { + setDocument( context ); + } + return contains( context, elem ); +}; + +Sizzle.attr = function( elem, name ) { + + // Set document vars if needed + // Support: IE 11+, Edge 17 - 18+ + // IE/Edge sometimes throw a "Permission denied" error when strict-comparing + // two documents; shallow comparisons work. + // eslint-disable-next-line eqeqeq + if ( ( elem.ownerDocument || elem ) != document ) { + setDocument( elem ); + } + + var fn = Expr.attrHandle[ name.toLowerCase() ], + + // Don't get fooled by Object.prototype properties (jQuery #13807) + val = fn && hasOwn.call( Expr.attrHandle, name.toLowerCase() ) ? + fn( elem, name, !documentIsHTML ) : + undefined; + + return val !== undefined ? + val : + support.attributes || !documentIsHTML ? + elem.getAttribute( name ) : + ( val = elem.getAttributeNode( name ) ) && val.specified ? + val.value : + null; +}; + +Sizzle.escape = function( sel ) { + return ( sel + "" ).replace( rcssescape, fcssescape ); +}; + +Sizzle.error = function( msg ) { + throw new Error( "Syntax error, unrecognized expression: " + msg ); +}; + +/** + * Document sorting and removing duplicates + * @param {ArrayLike} results + */ +Sizzle.uniqueSort = function( results ) { + var elem, + duplicates = [], + j = 0, + i = 0; + + // Unless we *know* we can detect duplicates, assume their presence + hasDuplicate = !support.detectDuplicates; + sortInput = !support.sortStable && results.slice( 0 ); + results.sort( sortOrder ); + + if ( hasDuplicate ) { + while ( ( elem = results[ i++ ] ) ) { + if ( elem === results[ i ] ) { + j = duplicates.push( i ); + } + } + while ( j-- ) { + results.splice( duplicates[ j ], 1 ); + } + } + + // Clear input after sorting to release objects + // See https://github.com/jquery/sizzle/pull/225 + sortInput = null; + + return results; +}; + +/** + * Utility function for retrieving the text value of an array of DOM nodes + * @param {Array|Element} elem + */ +getText = Sizzle.getText = function( elem ) { + var node, + ret = "", + i = 0, + nodeType = elem.nodeType; + + if ( !nodeType ) { + + // If no nodeType, this is expected to be an array + while ( ( node = elem[ i++ ] ) ) { + + // Do not traverse comment nodes + ret += getText( node ); + } + } else if ( nodeType === 1 || nodeType === 9 || nodeType === 11 ) { + + // Use textContent for elements + // innerText usage removed for consistency of new lines (jQuery #11153) + if ( typeof elem.textContent === "string" ) { + return elem.textContent; + } else { + + // Traverse its children + for ( elem = elem.firstChild; elem; elem = elem.nextSibling ) { + ret += getText( elem ); + } + } + } else if ( nodeType === 3 || nodeType === 4 ) { + return elem.nodeValue; + } + + // Do not include comment or processing instruction nodes + + return ret; +}; + +Expr = Sizzle.selectors = { + + // Can be adjusted by the user + cacheLength: 50, + + createPseudo: markFunction, + + match: matchExpr, + + attrHandle: {}, + + find: {}, + + relative: { + ">": { dir: "parentNode", first: true }, + " ": { dir: "parentNode" }, + "+": { dir: "previousSibling", first: true }, + "~": { dir: "previousSibling" } + }, + + preFilter: { + "ATTR": function( match ) { + match[ 1 ] = match[ 1 ].replace( runescape, funescape ); + + // Move the given value to match[3] whether quoted or unquoted + match[ 3 ] = ( match[ 3 ] || match[ 4 ] || + match[ 5 ] || "" ).replace( runescape, funescape ); + + if ( match[ 2 ] === "~=" ) { + match[ 3 ] = " " + match[ 3 ] + " "; + } + + return match.slice( 0, 4 ); + }, + + "CHILD": function( match ) { + + /* matches from matchExpr["CHILD"] + 1 type (only|nth|...) + 2 what (child|of-type) + 3 argument (even|odd|\d*|\d*n([+-]\d+)?|...) + 4 xn-component of xn+y argument ([+-]?\d*n|) + 5 sign of xn-component + 6 x of xn-component + 7 sign of y-component + 8 y of y-component + */ + match[ 1 ] = match[ 1 ].toLowerCase(); + + if ( match[ 1 ].slice( 0, 3 ) === "nth" ) { + + // nth-* requires argument + if ( !match[ 3 ] ) { + Sizzle.error( match[ 0 ] ); + } + + // numeric x and y parameters for Expr.filter.CHILD + // remember that false/true cast respectively to 0/1 + match[ 4 ] = +( match[ 4 ] ? + match[ 5 ] + ( match[ 6 ] || 1 ) : + 2 * ( match[ 3 ] === "even" || match[ 3 ] === "odd" ) ); + match[ 5 ] = +( ( match[ 7 ] + match[ 8 ] ) || match[ 3 ] === "odd" ); + + // other types prohibit arguments + } else if ( match[ 3 ] ) { + Sizzle.error( match[ 0 ] ); + } + + return match; + }, + + "PSEUDO": function( match ) { + var excess, + unquoted = !match[ 6 ] && match[ 2 ]; + + if ( matchExpr[ "CHILD" ].test( match[ 0 ] ) ) { + return null; + } + + // Accept quoted arguments as-is + if ( match[ 3 ] ) { + match[ 2 ] = match[ 4 ] || match[ 5 ] || ""; + + // Strip excess characters from unquoted arguments + } else if ( unquoted && rpseudo.test( unquoted ) && + + // Get excess from tokenize (recursively) + ( excess = tokenize( unquoted, true ) ) && + + // advance to the next closing parenthesis + ( excess = unquoted.indexOf( ")", unquoted.length - excess ) - unquoted.length ) ) { + + // excess is a negative index + match[ 0 ] = match[ 0 ].slice( 0, excess ); + match[ 2 ] = unquoted.slice( 0, excess ); + } + + // Return only captures needed by the pseudo filter method (type and argument) + return match.slice( 0, 3 ); + } + }, + + filter: { + + "TAG": function( nodeNameSelector ) { + var nodeName = nodeNameSelector.replace( runescape, funescape ).toLowerCase(); + return nodeNameSelector === "*" ? + function() { + return true; + } : + function( elem ) { + return elem.nodeName && elem.nodeName.toLowerCase() === nodeName; + }; + }, + + "CLASS": function( className ) { + var pattern = classCache[ className + " " ]; + + return pattern || + ( pattern = new RegExp( "(^|" + whitespace + + ")" + className + "(" + whitespace + "|$)" ) ) && classCache( + className, function( elem ) { + return pattern.test( + typeof elem.className === "string" && elem.className || + typeof elem.getAttribute !== "undefined" && + elem.getAttribute( "class" ) || + "" + ); + } ); + }, + + "ATTR": function( name, operator, check ) { + return function( elem ) { + var result = Sizzle.attr( elem, name ); + + if ( result == null ) { + return operator === "!="; + } + if ( !operator ) { + return true; + } + + result += ""; + + /* eslint-disable max-len */ + + return operator === "=" ? result === check : + operator === "!=" ? result !== check : + operator === "^=" ? check && result.indexOf( check ) === 0 : + operator === "*=" ? check && result.indexOf( check ) > -1 : + operator === "$=" ? check && result.slice( -check.length ) === check : + operator === "~=" ? ( " " + result.replace( rwhitespace, " " ) + " " ).indexOf( check ) > -1 : + operator === "|=" ? result === check || result.slice( 0, check.length + 1 ) === check + "-" : + false; + /* eslint-enable max-len */ + + }; + }, + + "CHILD": function( type, what, _argument, first, last ) { + var simple = type.slice( 0, 3 ) !== "nth", + forward = type.slice( -4 ) !== "last", + ofType = what === "of-type"; + + return first === 1 && last === 0 ? + + // Shortcut for :nth-*(n) + function( elem ) { + return !!elem.parentNode; + } : + + function( elem, _context, xml ) { + var cache, uniqueCache, outerCache, node, nodeIndex, start, + dir = simple !== forward ? "nextSibling" : "previousSibling", + parent = elem.parentNode, + name = ofType && elem.nodeName.toLowerCase(), + useCache = !xml && !ofType, + diff = false; + + if ( parent ) { + + // :(first|last|only)-(child|of-type) + if ( simple ) { + while ( dir ) { + node = elem; + while ( ( node = node[ dir ] ) ) { + if ( ofType ? + node.nodeName.toLowerCase() === name : + node.nodeType === 1 ) { + + return false; + } + } + + // Reverse direction for :only-* (if we haven't yet done so) + start = dir = type === "only" && !start && "nextSibling"; + } + return true; + } + + start = [ forward ? parent.firstChild : parent.lastChild ]; + + // non-xml :nth-child(...) stores cache data on `parent` + if ( forward && useCache ) { + + // Seek `elem` from a previously-cached index + + // ...in a gzip-friendly way + node = parent; + outerCache = node[ expando ] || ( node[ expando ] = {} ); + + // Support: IE <9 only + // Defend against cloned attroperties (jQuery gh-1709) + uniqueCache = outerCache[ node.uniqueID ] || + ( outerCache[ node.uniqueID ] = {} ); + + cache = uniqueCache[ type ] || []; + nodeIndex = cache[ 0 ] === dirruns && cache[ 1 ]; + diff = nodeIndex && cache[ 2 ]; + node = nodeIndex && parent.childNodes[ nodeIndex ]; + + while ( ( node = ++nodeIndex && node && node[ dir ] || + + // Fallback to seeking `elem` from the start + ( diff = nodeIndex = 0 ) || start.pop() ) ) { + + // When found, cache indexes on `parent` and break + if ( node.nodeType === 1 && ++diff && node === elem ) { + uniqueCache[ type ] = [ dirruns, nodeIndex, diff ]; + break; + } + } + + } else { + + // Use previously-cached element index if available + if ( useCache ) { + + // ...in a gzip-friendly way + node = elem; + outerCache = node[ expando ] || ( node[ expando ] = {} ); + + // Support: IE <9 only + // Defend against cloned attroperties (jQuery gh-1709) + uniqueCache = outerCache[ node.uniqueID ] || + ( outerCache[ node.uniqueID ] = {} ); + + cache = uniqueCache[ type ] || []; + nodeIndex = cache[ 0 ] === dirruns && cache[ 1 ]; + diff = nodeIndex; + } + + // xml :nth-child(...) + // or :nth-last-child(...) or :nth(-last)?-of-type(...) + if ( diff === false ) { + + // Use the same loop as above to seek `elem` from the start + while ( ( node = ++nodeIndex && node && node[ dir ] || + ( diff = nodeIndex = 0 ) || start.pop() ) ) { + + if ( ( ofType ? + node.nodeName.toLowerCase() === name : + node.nodeType === 1 ) && + ++diff ) { + + // Cache the index of each encountered element + if ( useCache ) { + outerCache = node[ expando ] || + ( node[ expando ] = {} ); + + // Support: IE <9 only + // Defend against cloned attroperties (jQuery gh-1709) + uniqueCache = outerCache[ node.uniqueID ] || + ( outerCache[ node.uniqueID ] = {} ); + + uniqueCache[ type ] = [ dirruns, diff ]; + } + + if ( node === elem ) { + break; + } + } + } + } + } + + // Incorporate the offset, then check against cycle size + diff -= last; + return diff === first || ( diff % first === 0 && diff / first >= 0 ); + } + }; + }, + + "PSEUDO": function( pseudo, argument ) { + + // pseudo-class names are case-insensitive + // http://www.w3.org/TR/selectors/#pseudo-classes + // Prioritize by case sensitivity in case custom pseudos are added with uppercase letters + // Remember that setFilters inherits from pseudos + var args, + fn = Expr.pseudos[ pseudo ] || Expr.setFilters[ pseudo.toLowerCase() ] || + Sizzle.error( "unsupported pseudo: " + pseudo ); + + // The user may use createPseudo to indicate that + // arguments are needed to create the filter function + // just as Sizzle does + if ( fn[ expando ] ) { + return fn( argument ); + } + + // But maintain support for old signatures + if ( fn.length > 1 ) { + args = [ pseudo, pseudo, "", argument ]; + return Expr.setFilters.hasOwnProperty( pseudo.toLowerCase() ) ? + markFunction( function( seed, matches ) { + var idx, + matched = fn( seed, argument ), + i = matched.length; + while ( i-- ) { + idx = indexOf( seed, matched[ i ] ); + seed[ idx ] = !( matches[ idx ] = matched[ i ] ); + } + } ) : + function( elem ) { + return fn( elem, 0, args ); + }; + } + + return fn; + } + }, + + pseudos: { + + // Potentially complex pseudos + "not": markFunction( function( selector ) { + + // Trim the selector passed to compile + // to avoid treating leading and trailing + // spaces as combinators + var input = [], + results = [], + matcher = compile( selector.replace( rtrim, "$1" ) ); + + return matcher[ expando ] ? + markFunction( function( seed, matches, _context, xml ) { + var elem, + unmatched = matcher( seed, null, xml, [] ), + i = seed.length; + + // Match elements unmatched by `matcher` + while ( i-- ) { + if ( ( elem = unmatched[ i ] ) ) { + seed[ i ] = !( matches[ i ] = elem ); + } + } + } ) : + function( elem, _context, xml ) { + input[ 0 ] = elem; + matcher( input, null, xml, results ); + + // Don't keep the element (issue #299) + input[ 0 ] = null; + return !results.pop(); + }; + } ), + + "has": markFunction( function( selector ) { + return function( elem ) { + return Sizzle( selector, elem ).length > 0; + }; + } ), + + "contains": markFunction( function( text ) { + text = text.replace( runescape, funescape ); + return function( elem ) { + return ( elem.textContent || getText( elem ) ).indexOf( text ) > -1; + }; + } ), + + // "Whether an element is represented by a :lang() selector + // is based solely on the element's language value + // being equal to the identifier C, + // or beginning with the identifier C immediately followed by "-". + // The matching of C against the element's language value is performed case-insensitively. + // The identifier C does not have to be a valid language name." + // http://www.w3.org/TR/selectors/#lang-pseudo + "lang": markFunction( function( lang ) { + + // lang value must be a valid identifier + if ( !ridentifier.test( lang || "" ) ) { + Sizzle.error( "unsupported lang: " + lang ); + } + lang = lang.replace( runescape, funescape ).toLowerCase(); + return function( elem ) { + var elemLang; + do { + if ( ( elemLang = documentIsHTML ? + elem.lang : + elem.getAttribute( "xml:lang" ) || elem.getAttribute( "lang" ) ) ) { + + elemLang = elemLang.toLowerCase(); + return elemLang === lang || elemLang.indexOf( lang + "-" ) === 0; + } + } while ( ( elem = elem.parentNode ) && elem.nodeType === 1 ); + return false; + }; + } ), + + // Miscellaneous + "target": function( elem ) { + var hash = window.location && window.location.hash; + return hash && hash.slice( 1 ) === elem.id; + }, + + "root": function( elem ) { + return elem === docElem; + }, + + "focus": function( elem ) { + return elem === document.activeElement && + ( !document.hasFocus || document.hasFocus() ) && + !!( elem.type || elem.href || ~elem.tabIndex ); + }, + + // Boolean properties + "enabled": createDisabledPseudo( false ), + "disabled": createDisabledPseudo( true ), + + "checked": function( elem ) { + + // In CSS3, :checked should return both checked and selected elements + // http://www.w3.org/TR/2011/REC-css3-selectors-20110929/#checked + var nodeName = elem.nodeName.toLowerCase(); + return ( nodeName === "input" && !!elem.checked ) || + ( nodeName === "option" && !!elem.selected ); + }, + + "selected": function( elem ) { + + // Accessing this property makes selected-by-default + // options in Safari work properly + if ( elem.parentNode ) { + // eslint-disable-next-line no-unused-expressions + elem.parentNode.selectedIndex; + } + + return elem.selected === true; + }, + + // Contents + "empty": function( elem ) { + + // http://www.w3.org/TR/selectors/#empty-pseudo + // :empty is negated by element (1) or content nodes (text: 3; cdata: 4; entity ref: 5), + // but not by others (comment: 8; processing instruction: 7; etc.) + // nodeType < 6 works because attributes (2) do not appear as children + for ( elem = elem.firstChild; elem; elem = elem.nextSibling ) { + if ( elem.nodeType < 6 ) { + return false; + } + } + return true; + }, + + "parent": function( elem ) { + return !Expr.pseudos[ "empty" ]( elem ); + }, + + // Element/input types + "header": function( elem ) { + return rheader.test( elem.nodeName ); + }, + + "input": function( elem ) { + return rinputs.test( elem.nodeName ); + }, + + "button": function( elem ) { + var name = elem.nodeName.toLowerCase(); + return name === "input" && elem.type === "button" || name === "button"; + }, + + "text": function( elem ) { + var attr; + return elem.nodeName.toLowerCase() === "input" && + elem.type === "text" && + + // Support: IE<8 + // New HTML5 attribute values (e.g., "search") appear with elem.type === "text" + ( ( attr = elem.getAttribute( "type" ) ) == null || + attr.toLowerCase() === "text" ); + }, + + // Position-in-collection + "first": createPositionalPseudo( function() { + return [ 0 ]; + } ), + + "last": createPositionalPseudo( function( _matchIndexes, length ) { + return [ length - 1 ]; + } ), + + "eq": createPositionalPseudo( function( _matchIndexes, length, argument ) { + return [ argument < 0 ? argument + length : argument ]; + } ), + + "even": createPositionalPseudo( function( matchIndexes, length ) { + var i = 0; + for ( ; i < length; i += 2 ) { + matchIndexes.push( i ); + } + return matchIndexes; + } ), + + "odd": createPositionalPseudo( function( matchIndexes, length ) { + var i = 1; + for ( ; i < length; i += 2 ) { + matchIndexes.push( i ); + } + return matchIndexes; + } ), + + "lt": createPositionalPseudo( function( matchIndexes, length, argument ) { + var i = argument < 0 ? + argument + length : + argument > length ? + length : + argument; + for ( ; --i >= 0; ) { + matchIndexes.push( i ); + } + return matchIndexes; + } ), + + "gt": createPositionalPseudo( function( matchIndexes, length, argument ) { + var i = argument < 0 ? argument + length : argument; + for ( ; ++i < length; ) { + matchIndexes.push( i ); + } + return matchIndexes; + } ) + } +}; + +Expr.pseudos[ "nth" ] = Expr.pseudos[ "eq" ]; + +// Add button/input type pseudos +for ( i in { radio: true, checkbox: true, file: true, password: true, image: true } ) { + Expr.pseudos[ i ] = createInputPseudo( i ); +} +for ( i in { submit: true, reset: true } ) { + Expr.pseudos[ i ] = createButtonPseudo( i ); +} + +// Easy API for creating new setFilters +function setFilters() {} +setFilters.prototype = Expr.filters = Expr.pseudos; +Expr.setFilters = new setFilters(); + +tokenize = Sizzle.tokenize = function( selector, parseOnly ) { + var matched, match, tokens, type, + soFar, groups, preFilters, + cached = tokenCache[ selector + " " ]; + + if ( cached ) { + return parseOnly ? 0 : cached.slice( 0 ); + } + + soFar = selector; + groups = []; + preFilters = Expr.preFilter; + + while ( soFar ) { + + // Comma and first run + if ( !matched || ( match = rcomma.exec( soFar ) ) ) { + if ( match ) { + + // Don't consume trailing commas as valid + soFar = soFar.slice( match[ 0 ].length ) || soFar; + } + groups.push( ( tokens = [] ) ); + } + + matched = false; + + // Combinators + if ( ( match = rcombinators.exec( soFar ) ) ) { + matched = match.shift(); + tokens.push( { + value: matched, + + // Cast descendant combinators to space + type: match[ 0 ].replace( rtrim, " " ) + } ); + soFar = soFar.slice( matched.length ); + } + + // Filters + for ( type in Expr.filter ) { + if ( ( match = matchExpr[ type ].exec( soFar ) ) && ( !preFilters[ type ] || + ( match = preFilters[ type ]( match ) ) ) ) { + matched = match.shift(); + tokens.push( { + value: matched, + type: type, + matches: match + } ); + soFar = soFar.slice( matched.length ); + } + } + + if ( !matched ) { + break; + } + } + + // Return the length of the invalid excess + // if we're just parsing + // Otherwise, throw an error or return tokens + return parseOnly ? + soFar.length : + soFar ? + Sizzle.error( selector ) : + + // Cache the tokens + tokenCache( selector, groups ).slice( 0 ); +}; + +function toSelector( tokens ) { + var i = 0, + len = tokens.length, + selector = ""; + for ( ; i < len; i++ ) { + selector += tokens[ i ].value; + } + return selector; +} + +function addCombinator( matcher, combinator, base ) { + var dir = combinator.dir, + skip = combinator.next, + key = skip || dir, + checkNonElements = base && key === "parentNode", + doneName = done++; + + return combinator.first ? + + // Check against closest ancestor/preceding element + function( elem, context, xml ) { + while ( ( elem = elem[ dir ] ) ) { + if ( elem.nodeType === 1 || checkNonElements ) { + return matcher( elem, context, xml ); + } + } + return false; + } : + + // Check against all ancestor/preceding elements + function( elem, context, xml ) { + var oldCache, uniqueCache, outerCache, + newCache = [ dirruns, doneName ]; + + // We can't set arbitrary data on XML nodes, so they don't benefit from combinator caching + if ( xml ) { + while ( ( elem = elem[ dir ] ) ) { + if ( elem.nodeType === 1 || checkNonElements ) { + if ( matcher( elem, context, xml ) ) { + return true; + } + } + } + } else { + while ( ( elem = elem[ dir ] ) ) { + if ( elem.nodeType === 1 || checkNonElements ) { + outerCache = elem[ expando ] || ( elem[ expando ] = {} ); + + // Support: IE <9 only + // Defend against cloned attroperties (jQuery gh-1709) + uniqueCache = outerCache[ elem.uniqueID ] || + ( outerCache[ elem.uniqueID ] = {} ); + + if ( skip && skip === elem.nodeName.toLowerCase() ) { + elem = elem[ dir ] || elem; + } else if ( ( oldCache = uniqueCache[ key ] ) && + oldCache[ 0 ] === dirruns && oldCache[ 1 ] === doneName ) { + + // Assign to newCache so results back-propagate to previous elements + return ( newCache[ 2 ] = oldCache[ 2 ] ); + } else { + + // Reuse newcache so results back-propagate to previous elements + uniqueCache[ key ] = newCache; + + // A match means we're done; a fail means we have to keep checking + if ( ( newCache[ 2 ] = matcher( elem, context, xml ) ) ) { + return true; + } + } + } + } + } + return false; + }; +} + +function elementMatcher( matchers ) { + return matchers.length > 1 ? + function( elem, context, xml ) { + var i = matchers.length; + while ( i-- ) { + if ( !matchers[ i ]( elem, context, xml ) ) { + return false; + } + } + return true; + } : + matchers[ 0 ]; +} + +function multipleContexts( selector, contexts, results ) { + var i = 0, + len = contexts.length; + for ( ; i < len; i++ ) { + Sizzle( selector, contexts[ i ], results ); + } + return results; +} + +function condense( unmatched, map, filter, context, xml ) { + var elem, + newUnmatched = [], + i = 0, + len = unmatched.length, + mapped = map != null; + + for ( ; i < len; i++ ) { + if ( ( elem = unmatched[ i ] ) ) { + if ( !filter || filter( elem, context, xml ) ) { + newUnmatched.push( elem ); + if ( mapped ) { + map.push( i ); + } + } + } + } + + return newUnmatched; +} + +function setMatcher( preFilter, selector, matcher, postFilter, postFinder, postSelector ) { + if ( postFilter && !postFilter[ expando ] ) { + postFilter = setMatcher( postFilter ); + } + if ( postFinder && !postFinder[ expando ] ) { + postFinder = setMatcher( postFinder, postSelector ); + } + return markFunction( function( seed, results, context, xml ) { + var temp, i, elem, + preMap = [], + postMap = [], + preexisting = results.length, + + // Get initial elements from seed or context + elems = seed || multipleContexts( + selector || "*", + context.nodeType ? [ context ] : context, + [] + ), + + // Prefilter to get matcher input, preserving a map for seed-results synchronization + matcherIn = preFilter && ( seed || !selector ) ? + condense( elems, preMap, preFilter, context, xml ) : + elems, + + matcherOut = matcher ? + + // If we have a postFinder, or filtered seed, or non-seed postFilter or preexisting results, + postFinder || ( seed ? preFilter : preexisting || postFilter ) ? + + // ...intermediate processing is necessary + [] : + + // ...otherwise use results directly + results : + matcherIn; + + // Find primary matches + if ( matcher ) { + matcher( matcherIn, matcherOut, context, xml ); + } + + // Apply postFilter + if ( postFilter ) { + temp = condense( matcherOut, postMap ); + postFilter( temp, [], context, xml ); + + // Un-match failing elements by moving them back to matcherIn + i = temp.length; + while ( i-- ) { + if ( ( elem = temp[ i ] ) ) { + matcherOut[ postMap[ i ] ] = !( matcherIn[ postMap[ i ] ] = elem ); + } + } + } + + if ( seed ) { + if ( postFinder || preFilter ) { + if ( postFinder ) { + + // Get the final matcherOut by condensing this intermediate into postFinder contexts + temp = []; + i = matcherOut.length; + while ( i-- ) { + if ( ( elem = matcherOut[ i ] ) ) { + + // Restore matcherIn since elem is not yet a final match + temp.push( ( matcherIn[ i ] = elem ) ); + } + } + postFinder( null, ( matcherOut = [] ), temp, xml ); + } + + // Move matched elements from seed to results to keep them synchronized + i = matcherOut.length; + while ( i-- ) { + if ( ( elem = matcherOut[ i ] ) && + ( temp = postFinder ? indexOf( seed, elem ) : preMap[ i ] ) > -1 ) { + + seed[ temp ] = !( results[ temp ] = elem ); + } + } + } + + // Add elements to results, through postFinder if defined + } else { + matcherOut = condense( + matcherOut === results ? + matcherOut.splice( preexisting, matcherOut.length ) : + matcherOut + ); + if ( postFinder ) { + postFinder( null, results, matcherOut, xml ); + } else { + push.apply( results, matcherOut ); + } + } + } ); +} + +function matcherFromTokens( tokens ) { + var checkContext, matcher, j, + len = tokens.length, + leadingRelative = Expr.relative[ tokens[ 0 ].type ], + implicitRelative = leadingRelative || Expr.relative[ " " ], + i = leadingRelative ? 1 : 0, + + // The foundational matcher ensures that elements are reachable from top-level context(s) + matchContext = addCombinator( function( elem ) { + return elem === checkContext; + }, implicitRelative, true ), + matchAnyContext = addCombinator( function( elem ) { + return indexOf( checkContext, elem ) > -1; + }, implicitRelative, true ), + matchers = [ function( elem, context, xml ) { + var ret = ( !leadingRelative && ( xml || context !== outermostContext ) ) || ( + ( checkContext = context ).nodeType ? + matchContext( elem, context, xml ) : + matchAnyContext( elem, context, xml ) ); + + // Avoid hanging onto element (issue #299) + checkContext = null; + return ret; + } ]; + + for ( ; i < len; i++ ) { + if ( ( matcher = Expr.relative[ tokens[ i ].type ] ) ) { + matchers = [ addCombinator( elementMatcher( matchers ), matcher ) ]; + } else { + matcher = Expr.filter[ tokens[ i ].type ].apply( null, tokens[ i ].matches ); + + // Return special upon seeing a positional matcher + if ( matcher[ expando ] ) { + + // Find the next relative operator (if any) for proper handling + j = ++i; + for ( ; j < len; j++ ) { + if ( Expr.relative[ tokens[ j ].type ] ) { + break; + } + } + return setMatcher( + i > 1 && elementMatcher( matchers ), + i > 1 && toSelector( + + // If the preceding token was a descendant combinator, insert an implicit any-element `*` + tokens + .slice( 0, i - 1 ) + .concat( { value: tokens[ i - 2 ].type === " " ? "*" : "" } ) + ).replace( rtrim, "$1" ), + matcher, + i < j && matcherFromTokens( tokens.slice( i, j ) ), + j < len && matcherFromTokens( ( tokens = tokens.slice( j ) ) ), + j < len && toSelector( tokens ) + ); + } + matchers.push( matcher ); + } + } + + return elementMatcher( matchers ); +} + +function matcherFromGroupMatchers( elementMatchers, setMatchers ) { + var bySet = setMatchers.length > 0, + byElement = elementMatchers.length > 0, + superMatcher = function( seed, context, xml, results, outermost ) { + var elem, j, matcher, + matchedCount = 0, + i = "0", + unmatched = seed && [], + setMatched = [], + contextBackup = outermostContext, + + // We must always have either seed elements or outermost context + elems = seed || byElement && Expr.find[ "TAG" ]( "*", outermost ), + + // Use integer dirruns iff this is the outermost matcher + dirrunsUnique = ( dirruns += contextBackup == null ? 1 : Math.random() || 0.1 ), + len = elems.length; + + if ( outermost ) { + + // Support: IE 11+, Edge 17 - 18+ + // IE/Edge sometimes throw a "Permission denied" error when strict-comparing + // two documents; shallow comparisons work. + // eslint-disable-next-line eqeqeq + outermostContext = context == document || context || outermost; + } + + // Add elements passing elementMatchers directly to results + // Support: IE<9, Safari + // Tolerate NodeList properties (IE: "length"; Safari: ) matching elements by id + for ( ; i !== len && ( elem = elems[ i ] ) != null; i++ ) { + if ( byElement && elem ) { + j = 0; + + // Support: IE 11+, Edge 17 - 18+ + // IE/Edge sometimes throw a "Permission denied" error when strict-comparing + // two documents; shallow comparisons work. + // eslint-disable-next-line eqeqeq + if ( !context && elem.ownerDocument != document ) { + setDocument( elem ); + xml = !documentIsHTML; + } + while ( ( matcher = elementMatchers[ j++ ] ) ) { + if ( matcher( elem, context || document, xml ) ) { + results.push( elem ); + break; + } + } + if ( outermost ) { + dirruns = dirrunsUnique; + } + } + + // Track unmatched elements for set filters + if ( bySet ) { + + // They will have gone through all possible matchers + if ( ( elem = !matcher && elem ) ) { + matchedCount--; + } + + // Lengthen the array for every element, matched or not + if ( seed ) { + unmatched.push( elem ); + } + } + } + + // `i` is now the count of elements visited above, and adding it to `matchedCount` + // makes the latter nonnegative. + matchedCount += i; + + // Apply set filters to unmatched elements + // NOTE: This can be skipped if there are no unmatched elements (i.e., `matchedCount` + // equals `i`), unless we didn't visit _any_ elements in the above loop because we have + // no element matchers and no seed. + // Incrementing an initially-string "0" `i` allows `i` to remain a string only in that + // case, which will result in a "00" `matchedCount` that differs from `i` but is also + // numerically zero. + if ( bySet && i !== matchedCount ) { + j = 0; + while ( ( matcher = setMatchers[ j++ ] ) ) { + matcher( unmatched, setMatched, context, xml ); + } + + if ( seed ) { + + // Reintegrate element matches to eliminate the need for sorting + if ( matchedCount > 0 ) { + while ( i-- ) { + if ( !( unmatched[ i ] || setMatched[ i ] ) ) { + setMatched[ i ] = pop.call( results ); + } + } + } + + // Discard index placeholder values to get only actual matches + setMatched = condense( setMatched ); + } + + // Add matches to results + push.apply( results, setMatched ); + + // Seedless set matches succeeding multiple successful matchers stipulate sorting + if ( outermost && !seed && setMatched.length > 0 && + ( matchedCount + setMatchers.length ) > 1 ) { + + Sizzle.uniqueSort( results ); + } + } + + // Override manipulation of globals by nested matchers + if ( outermost ) { + dirruns = dirrunsUnique; + outermostContext = contextBackup; + } + + return unmatched; + }; + + return bySet ? + markFunction( superMatcher ) : + superMatcher; +} + +compile = Sizzle.compile = function( selector, match /* Internal Use Only */ ) { + var i, + setMatchers = [], + elementMatchers = [], + cached = compilerCache[ selector + " " ]; + + if ( !cached ) { + + // Generate a function of recursive functions that can be used to check each element + if ( !match ) { + match = tokenize( selector ); + } + i = match.length; + while ( i-- ) { + cached = matcherFromTokens( match[ i ] ); + if ( cached[ expando ] ) { + setMatchers.push( cached ); + } else { + elementMatchers.push( cached ); + } + } + + // Cache the compiled function + cached = compilerCache( + selector, + matcherFromGroupMatchers( elementMatchers, setMatchers ) + ); + + // Save selector and tokenization + cached.selector = selector; + } + return cached; +}; + +/** + * A low-level selection function that works with Sizzle's compiled + * selector functions + * @param {String|Function} selector A selector or a pre-compiled + * selector function built with Sizzle.compile + * @param {Element} context + * @param {Array} [results] + * @param {Array} [seed] A set of elements to match against + */ +select = Sizzle.select = function( selector, context, results, seed ) { + var i, tokens, token, type, find, + compiled = typeof selector === "function" && selector, + match = !seed && tokenize( ( selector = compiled.selector || selector ) ); + + results = results || []; + + // Try to minimize operations if there is only one selector in the list and no seed + // (the latter of which guarantees us context) + if ( match.length === 1 ) { + + // Reduce context if the leading compound selector is an ID + tokens = match[ 0 ] = match[ 0 ].slice( 0 ); + if ( tokens.length > 2 && ( token = tokens[ 0 ] ).type === "ID" && + context.nodeType === 9 && documentIsHTML && Expr.relative[ tokens[ 1 ].type ] ) { + + context = ( Expr.find[ "ID" ]( token.matches[ 0 ] + .replace( runescape, funescape ), context ) || [] )[ 0 ]; + if ( !context ) { + return results; + + // Precompiled matchers will still verify ancestry, so step up a level + } else if ( compiled ) { + context = context.parentNode; + } + + selector = selector.slice( tokens.shift().value.length ); + } + + // Fetch a seed set for right-to-left matching + i = matchExpr[ "needsContext" ].test( selector ) ? 0 : tokens.length; + while ( i-- ) { + token = tokens[ i ]; + + // Abort if we hit a combinator + if ( Expr.relative[ ( type = token.type ) ] ) { + break; + } + if ( ( find = Expr.find[ type ] ) ) { + + // Search, expanding context for leading sibling combinators + if ( ( seed = find( + token.matches[ 0 ].replace( runescape, funescape ), + rsibling.test( tokens[ 0 ].type ) && testContext( context.parentNode ) || + context + ) ) ) { + + // If seed is empty or no tokens remain, we can return early + tokens.splice( i, 1 ); + selector = seed.length && toSelector( tokens ); + if ( !selector ) { + push.apply( results, seed ); + return results; + } + + break; + } + } + } + } + + // Compile and execute a filtering function if one is not provided + // Provide `match` to avoid retokenization if we modified the selector above + ( compiled || compile( selector, match ) )( + seed, + context, + !documentIsHTML, + results, + !context || rsibling.test( selector ) && testContext( context.parentNode ) || context + ); + return results; +}; + +// One-time assignments + +// Sort stability +support.sortStable = expando.split( "" ).sort( sortOrder ).join( "" ) === expando; + +// Support: Chrome 14-35+ +// Always assume duplicates if they aren't passed to the comparison function +support.detectDuplicates = !!hasDuplicate; + +// Initialize against the default document +setDocument(); + +// Support: Webkit<537.32 - Safari 6.0.3/Chrome 25 (fixed in Chrome 27) +// Detached nodes confoundingly follow *each other* +support.sortDetached = assert( function( el ) { + + // Should return 1, but returns 4 (following) + return el.compareDocumentPosition( document.createElement( "fieldset" ) ) & 1; +} ); + +// Support: IE<8 +// Prevent attribute/property "interpolation" +// https://msdn.microsoft.com/en-us/library/ms536429%28VS.85%29.aspx +if ( !assert( function( el ) { + el.innerHTML = ""; + return el.firstChild.getAttribute( "href" ) === "#"; +} ) ) { + addHandle( "type|href|height|width", function( elem, name, isXML ) { + if ( !isXML ) { + return elem.getAttribute( name, name.toLowerCase() === "type" ? 1 : 2 ); + } + } ); +} + +// Support: IE<9 +// Use defaultValue in place of getAttribute("value") +if ( !support.attributes || !assert( function( el ) { + el.innerHTML = ""; + el.firstChild.setAttribute( "value", "" ); + return el.firstChild.getAttribute( "value" ) === ""; +} ) ) { + addHandle( "value", function( elem, _name, isXML ) { + if ( !isXML && elem.nodeName.toLowerCase() === "input" ) { + return elem.defaultValue; + } + } ); +} + +// Support: IE<9 +// Use getAttributeNode to fetch booleans when getAttribute lies +if ( !assert( function( el ) { + return el.getAttribute( "disabled" ) == null; +} ) ) { + addHandle( booleans, function( elem, name, isXML ) { + var val; + if ( !isXML ) { + return elem[ name ] === true ? name.toLowerCase() : + ( val = elem.getAttributeNode( name ) ) && val.specified ? + val.value : + null; + } + } ); +} + +return Sizzle; + +} )( window ); + + + +jQuery.find = Sizzle; +jQuery.expr = Sizzle.selectors; + +// Deprecated +jQuery.expr[ ":" ] = jQuery.expr.pseudos; +jQuery.uniqueSort = jQuery.unique = Sizzle.uniqueSort; +jQuery.text = Sizzle.getText; +jQuery.isXMLDoc = Sizzle.isXML; +jQuery.contains = Sizzle.contains; +jQuery.escapeSelector = Sizzle.escape; + + + + +var dir = function( elem, dir, until ) { + var matched = [], + truncate = until !== undefined; + + while ( ( elem = elem[ dir ] ) && elem.nodeType !== 9 ) { + if ( elem.nodeType === 1 ) { + if ( truncate && jQuery( elem ).is( until ) ) { + break; + } + matched.push( elem ); + } + } + return matched; +}; + + +var siblings = function( n, elem ) { + var matched = []; + + for ( ; n; n = n.nextSibling ) { + if ( n.nodeType === 1 && n !== elem ) { + matched.push( n ); + } + } + + return matched; +}; + + +var rneedsContext = jQuery.expr.match.needsContext; + + + +function nodeName( elem, name ) { + + return elem.nodeName && elem.nodeName.toLowerCase() === name.toLowerCase(); + +}; +var rsingleTag = ( /^<([a-z][^\/\0>:\x20\t\r\n\f]*)[\x20\t\r\n\f]*\/?>(?:<\/\1>|)$/i ); + + + +// Implement the identical functionality for filter and not +function winnow( elements, qualifier, not ) { + if ( isFunction( qualifier ) ) { + return jQuery.grep( elements, function( elem, i ) { + return !!qualifier.call( elem, i, elem ) !== not; + } ); + } + + // Single element + if ( qualifier.nodeType ) { + return jQuery.grep( elements, function( elem ) { + return ( elem === qualifier ) !== not; + } ); + } + + // Arraylike of elements (jQuery, arguments, Array) + if ( typeof qualifier !== "string" ) { + return jQuery.grep( elements, function( elem ) { + return ( indexOf.call( qualifier, elem ) > -1 ) !== not; + } ); + } + + // Filtered directly for both simple and complex selectors + return jQuery.filter( qualifier, elements, not ); +} + +jQuery.filter = function( expr, elems, not ) { + var elem = elems[ 0 ]; + + if ( not ) { + expr = ":not(" + expr + ")"; + } + + if ( elems.length === 1 && elem.nodeType === 1 ) { + return jQuery.find.matchesSelector( elem, expr ) ? [ elem ] : []; + } + + return jQuery.find.matches( expr, jQuery.grep( elems, function( elem ) { + return elem.nodeType === 1; + } ) ); +}; + +jQuery.fn.extend( { + find: function( selector ) { + var i, ret, + len = this.length, + self = this; + + if ( typeof selector !== "string" ) { + return this.pushStack( jQuery( selector ).filter( function() { + for ( i = 0; i < len; i++ ) { + if ( jQuery.contains( self[ i ], this ) ) { + return true; + } + } + } ) ); + } + + ret = this.pushStack( [] ); + + for ( i = 0; i < len; i++ ) { + jQuery.find( selector, self[ i ], ret ); + } + + return len > 1 ? jQuery.uniqueSort( ret ) : ret; + }, + filter: function( selector ) { + return this.pushStack( winnow( this, selector || [], false ) ); + }, + not: function( selector ) { + return this.pushStack( winnow( this, selector || [], true ) ); + }, + is: function( selector ) { + return !!winnow( + this, + + // If this is a positional/relative selector, check membership in the returned set + // so $("p:first").is("p:last") won't return true for a doc with two "p". + typeof selector === "string" && rneedsContext.test( selector ) ? + jQuery( selector ) : + selector || [], + false + ).length; + } +} ); + + +// Initialize a jQuery object + + +// A central reference to the root jQuery(document) +var rootjQuery, + + // A simple way to check for HTML strings + // Prioritize #id over to avoid XSS via location.hash (#9521) + // Strict HTML recognition (#11290: must start with <) + // Shortcut simple #id case for speed + rquickExpr = /^(?:\s*(<[\w\W]+>)[^>]*|#([\w-]+))$/, + + init = jQuery.fn.init = function( selector, context, root ) { + var match, elem; + + // HANDLE: $(""), $(null), $(undefined), $(false) + if ( !selector ) { + return this; + } + + // Method init() accepts an alternate rootjQuery + // so migrate can support jQuery.sub (gh-2101) + root = root || rootjQuery; + + // Handle HTML strings + if ( typeof selector === "string" ) { + if ( selector[ 0 ] === "<" && + selector[ selector.length - 1 ] === ">" && + selector.length >= 3 ) { + + // Assume that strings that start and end with <> are HTML and skip the regex check + match = [ null, selector, null ]; + + } else { + match = rquickExpr.exec( selector ); + } + + // Match html or make sure no context is specified for #id + if ( match && ( match[ 1 ] || !context ) ) { + + // HANDLE: $(html) -> $(array) + if ( match[ 1 ] ) { + context = context instanceof jQuery ? context[ 0 ] : context; + + // Option to run scripts is true for back-compat + // Intentionally let the error be thrown if parseHTML is not present + jQuery.merge( this, jQuery.parseHTML( + match[ 1 ], + context && context.nodeType ? context.ownerDocument || context : document, + true + ) ); + + // HANDLE: $(html, props) + if ( rsingleTag.test( match[ 1 ] ) && jQuery.isPlainObject( context ) ) { + for ( match in context ) { + + // Properties of context are called as methods if possible + if ( isFunction( this[ match ] ) ) { + this[ match ]( context[ match ] ); + + // ...and otherwise set as attributes + } else { + this.attr( match, context[ match ] ); + } + } + } + + return this; + + // HANDLE: $(#id) + } else { + elem = document.getElementById( match[ 2 ] ); + + if ( elem ) { + + // Inject the element directly into the jQuery object + this[ 0 ] = elem; + this.length = 1; + } + return this; + } + + // HANDLE: $(expr, $(...)) + } else if ( !context || context.jquery ) { + return ( context || root ).find( selector ); + + // HANDLE: $(expr, context) + // (which is just equivalent to: $(context).find(expr) + } else { + return this.constructor( context ).find( selector ); + } + + // HANDLE: $(DOMElement) + } else if ( selector.nodeType ) { + this[ 0 ] = selector; + this.length = 1; + return this; + + // HANDLE: $(function) + // Shortcut for document ready + } else if ( isFunction( selector ) ) { + return root.ready !== undefined ? + root.ready( selector ) : + + // Execute immediately if ready is not present + selector( jQuery ); + } + + return jQuery.makeArray( selector, this ); + }; + +// Give the init function the jQuery prototype for later instantiation +init.prototype = jQuery.fn; + +// Initialize central reference +rootjQuery = jQuery( document ); + + +var rparentsprev = /^(?:parents|prev(?:Until|All))/, + + // Methods guaranteed to produce a unique set when starting from a unique set + guaranteedUnique = { + children: true, + contents: true, + next: true, + prev: true + }; + +jQuery.fn.extend( { + has: function( target ) { + var targets = jQuery( target, this ), + l = targets.length; + + return this.filter( function() { + var i = 0; + for ( ; i < l; i++ ) { + if ( jQuery.contains( this, targets[ i ] ) ) { + return true; + } + } + } ); + }, + + closest: function( selectors, context ) { + var cur, + i = 0, + l = this.length, + matched = [], + targets = typeof selectors !== "string" && jQuery( selectors ); + + // Positional selectors never match, since there's no _selection_ context + if ( !rneedsContext.test( selectors ) ) { + for ( ; i < l; i++ ) { + for ( cur = this[ i ]; cur && cur !== context; cur = cur.parentNode ) { + + // Always skip document fragments + if ( cur.nodeType < 11 && ( targets ? + targets.index( cur ) > -1 : + + // Don't pass non-elements to Sizzle + cur.nodeType === 1 && + jQuery.find.matchesSelector( cur, selectors ) ) ) { + + matched.push( cur ); + break; + } + } + } + } + + return this.pushStack( matched.length > 1 ? jQuery.uniqueSort( matched ) : matched ); + }, + + // Determine the position of an element within the set + index: function( elem ) { + + // No argument, return index in parent + if ( !elem ) { + return ( this[ 0 ] && this[ 0 ].parentNode ) ? this.first().prevAll().length : -1; + } + + // Index in selector + if ( typeof elem === "string" ) { + return indexOf.call( jQuery( elem ), this[ 0 ] ); + } + + // Locate the position of the desired element + return indexOf.call( this, + + // If it receives a jQuery object, the first element is used + elem.jquery ? elem[ 0 ] : elem + ); + }, + + add: function( selector, context ) { + return this.pushStack( + jQuery.uniqueSort( + jQuery.merge( this.get(), jQuery( selector, context ) ) + ) + ); + }, + + addBack: function( selector ) { + return this.add( selector == null ? + this.prevObject : this.prevObject.filter( selector ) + ); + } +} ); + +function sibling( cur, dir ) { + while ( ( cur = cur[ dir ] ) && cur.nodeType !== 1 ) {} + return cur; +} + +jQuery.each( { + parent: function( elem ) { + var parent = elem.parentNode; + return parent && parent.nodeType !== 11 ? parent : null; + }, + parents: function( elem ) { + return dir( elem, "parentNode" ); + }, + parentsUntil: function( elem, _i, until ) { + return dir( elem, "parentNode", until ); + }, + next: function( elem ) { + return sibling( elem, "nextSibling" ); + }, + prev: function( elem ) { + return sibling( elem, "previousSibling" ); + }, + nextAll: function( elem ) { + return dir( elem, "nextSibling" ); + }, + prevAll: function( elem ) { + return dir( elem, "previousSibling" ); + }, + nextUntil: function( elem, _i, until ) { + return dir( elem, "nextSibling", until ); + }, + prevUntil: function( elem, _i, until ) { + return dir( elem, "previousSibling", until ); + }, + siblings: function( elem ) { + return siblings( ( elem.parentNode || {} ).firstChild, elem ); + }, + children: function( elem ) { + return siblings( elem.firstChild ); + }, + contents: function( elem ) { + if ( elem.contentDocument != null && + + // Support: IE 11+ + // elements with no `data` attribute has an object + // `contentDocument` with a `null` prototype. + getProto( elem.contentDocument ) ) { + + return elem.contentDocument; + } + + // Support: IE 9 - 11 only, iOS 7 only, Android Browser <=4.3 only + // Treat the template element as a regular one in browsers that + // don't support it. + if ( nodeName( elem, "template" ) ) { + elem = elem.content || elem; + } + + return jQuery.merge( [], elem.childNodes ); + } +}, function( name, fn ) { + jQuery.fn[ name ] = function( until, selector ) { + var matched = jQuery.map( this, fn, until ); + + if ( name.slice( -5 ) !== "Until" ) { + selector = until; + } + + if ( selector && typeof selector === "string" ) { + matched = jQuery.filter( selector, matched ); + } + + if ( this.length > 1 ) { + + // Remove duplicates + if ( !guaranteedUnique[ name ] ) { + jQuery.uniqueSort( matched ); + } + + // Reverse order for parents* and prev-derivatives + if ( rparentsprev.test( name ) ) { + matched.reverse(); + } + } + + return this.pushStack( matched ); + }; +} ); +var rnothtmlwhite = ( /[^\x20\t\r\n\f]+/g ); + + + +// Convert String-formatted options into Object-formatted ones +function createOptions( options ) { + var object = {}; + jQuery.each( options.match( rnothtmlwhite ) || [], function( _, flag ) { + object[ flag ] = true; + } ); + return object; +} + +/* + * Create a callback list using the following parameters: + * + * options: an optional list of space-separated options that will change how + * the callback list behaves or a more traditional option object + * + * By default a callback list will act like an event callback list and can be + * "fired" multiple times. + * + * Possible options: + * + * once: will ensure the callback list can only be fired once (like a Deferred) + * + * memory: will keep track of previous values and will call any callback added + * after the list has been fired right away with the latest "memorized" + * values (like a Deferred) + * + * unique: will ensure a callback can only be added once (no duplicate in the list) + * + * stopOnFalse: interrupt callings when a callback returns false + * + */ +jQuery.Callbacks = function( options ) { + + // Convert options from String-formatted to Object-formatted if needed + // (we check in cache first) + options = typeof options === "string" ? + createOptions( options ) : + jQuery.extend( {}, options ); + + var // Flag to know if list is currently firing + firing, + + // Last fire value for non-forgettable lists + memory, + + // Flag to know if list was already fired + fired, + + // Flag to prevent firing + locked, + + // Actual callback list + list = [], + + // Queue of execution data for repeatable lists + queue = [], + + // Index of currently firing callback (modified by add/remove as needed) + firingIndex = -1, + + // Fire callbacks + fire = function() { + + // Enforce single-firing + locked = locked || options.once; + + // Execute callbacks for all pending executions, + // respecting firingIndex overrides and runtime changes + fired = firing = true; + for ( ; queue.length; firingIndex = -1 ) { + memory = queue.shift(); + while ( ++firingIndex < list.length ) { + + // Run callback and check for early termination + if ( list[ firingIndex ].apply( memory[ 0 ], memory[ 1 ] ) === false && + options.stopOnFalse ) { + + // Jump to end and forget the data so .add doesn't re-fire + firingIndex = list.length; + memory = false; + } + } + } + + // Forget the data if we're done with it + if ( !options.memory ) { + memory = false; + } + + firing = false; + + // Clean up if we're done firing for good + if ( locked ) { + + // Keep an empty list if we have data for future add calls + if ( memory ) { + list = []; + + // Otherwise, this object is spent + } else { + list = ""; + } + } + }, + + // Actual Callbacks object + self = { + + // Add a callback or a collection of callbacks to the list + add: function() { + if ( list ) { + + // If we have memory from a past run, we should fire after adding + if ( memory && !firing ) { + firingIndex = list.length - 1; + queue.push( memory ); + } + + ( function add( args ) { + jQuery.each( args, function( _, arg ) { + if ( isFunction( arg ) ) { + if ( !options.unique || !self.has( arg ) ) { + list.push( arg ); + } + } else if ( arg && arg.length && toType( arg ) !== "string" ) { + + // Inspect recursively + add( arg ); + } + } ); + } )( arguments ); + + if ( memory && !firing ) { + fire(); + } + } + return this; + }, + + // Remove a callback from the list + remove: function() { + jQuery.each( arguments, function( _, arg ) { + var index; + while ( ( index = jQuery.inArray( arg, list, index ) ) > -1 ) { + list.splice( index, 1 ); + + // Handle firing indexes + if ( index <= firingIndex ) { + firingIndex--; + } + } + } ); + return this; + }, + + // Check if a given callback is in the list. + // If no argument is given, return whether or not list has callbacks attached. + has: function( fn ) { + return fn ? + jQuery.inArray( fn, list ) > -1 : + list.length > 0; + }, + + // Remove all callbacks from the list + empty: function() { + if ( list ) { + list = []; + } + return this; + }, + + // Disable .fire and .add + // Abort any current/pending executions + // Clear all callbacks and values + disable: function() { + locked = queue = []; + list = memory = ""; + return this; + }, + disabled: function() { + return !list; + }, + + // Disable .fire + // Also disable .add unless we have memory (since it would have no effect) + // Abort any pending executions + lock: function() { + locked = queue = []; + if ( !memory && !firing ) { + list = memory = ""; + } + return this; + }, + locked: function() { + return !!locked; + }, + + // Call all callbacks with the given context and arguments + fireWith: function( context, args ) { + if ( !locked ) { + args = args || []; + args = [ context, args.slice ? args.slice() : args ]; + queue.push( args ); + if ( !firing ) { + fire(); + } + } + return this; + }, + + // Call all the callbacks with the given arguments + fire: function() { + self.fireWith( this, arguments ); + return this; + }, + + // To know if the callbacks have already been called at least once + fired: function() { + return !!fired; + } + }; + + return self; +}; + + +function Identity( v ) { + return v; +} +function Thrower( ex ) { + throw ex; +} + +function adoptValue( value, resolve, reject, noValue ) { + var method; + + try { + + // Check for promise aspect first to privilege synchronous behavior + if ( value && isFunction( ( method = value.promise ) ) ) { + method.call( value ).done( resolve ).fail( reject ); + + // Other thenables + } else if ( value && isFunction( ( method = value.then ) ) ) { + method.call( value, resolve, reject ); + + // Other non-thenables + } else { + + // Control `resolve` arguments by letting Array#slice cast boolean `noValue` to integer: + // * false: [ value ].slice( 0 ) => resolve( value ) + // * true: [ value ].slice( 1 ) => resolve() + resolve.apply( undefined, [ value ].slice( noValue ) ); + } + + // For Promises/A+, convert exceptions into rejections + // Since jQuery.when doesn't unwrap thenables, we can skip the extra checks appearing in + // Deferred#then to conditionally suppress rejection. + } catch ( value ) { + + // Support: Android 4.0 only + // Strict mode functions invoked without .call/.apply get global-object context + reject.apply( undefined, [ value ] ); + } +} + +jQuery.extend( { + + Deferred: function( func ) { + var tuples = [ + + // action, add listener, callbacks, + // ... .then handlers, argument index, [final state] + [ "notify", "progress", jQuery.Callbacks( "memory" ), + jQuery.Callbacks( "memory" ), 2 ], + [ "resolve", "done", jQuery.Callbacks( "once memory" ), + jQuery.Callbacks( "once memory" ), 0, "resolved" ], + [ "reject", "fail", jQuery.Callbacks( "once memory" ), + jQuery.Callbacks( "once memory" ), 1, "rejected" ] + ], + state = "pending", + promise = { + state: function() { + return state; + }, + always: function() { + deferred.done( arguments ).fail( arguments ); + return this; + }, + "catch": function( fn ) { + return promise.then( null, fn ); + }, + + // Keep pipe for back-compat + pipe: function( /* fnDone, fnFail, fnProgress */ ) { + var fns = arguments; + + return jQuery.Deferred( function( newDefer ) { + jQuery.each( tuples, function( _i, tuple ) { + + // Map tuples (progress, done, fail) to arguments (done, fail, progress) + var fn = isFunction( fns[ tuple[ 4 ] ] ) && fns[ tuple[ 4 ] ]; + + // deferred.progress(function() { bind to newDefer or newDefer.notify }) + // deferred.done(function() { bind to newDefer or newDefer.resolve }) + // deferred.fail(function() { bind to newDefer or newDefer.reject }) + deferred[ tuple[ 1 ] ]( function() { + var returned = fn && fn.apply( this, arguments ); + if ( returned && isFunction( returned.promise ) ) { + returned.promise() + .progress( newDefer.notify ) + .done( newDefer.resolve ) + .fail( newDefer.reject ); + } else { + newDefer[ tuple[ 0 ] + "With" ]( + this, + fn ? [ returned ] : arguments + ); + } + } ); + } ); + fns = null; + } ).promise(); + }, + then: function( onFulfilled, onRejected, onProgress ) { + var maxDepth = 0; + function resolve( depth, deferred, handler, special ) { + return function() { + var that = this, + args = arguments, + mightThrow = function() { + var returned, then; + + // Support: Promises/A+ section 2.3.3.3.3 + // https://promisesaplus.com/#point-59 + // Ignore double-resolution attempts + if ( depth < maxDepth ) { + return; + } + + returned = handler.apply( that, args ); + + // Support: Promises/A+ section 2.3.1 + // https://promisesaplus.com/#point-48 + if ( returned === deferred.promise() ) { + throw new TypeError( "Thenable self-resolution" ); + } + + // Support: Promises/A+ sections 2.3.3.1, 3.5 + // https://promisesaplus.com/#point-54 + // https://promisesaplus.com/#point-75 + // Retrieve `then` only once + then = returned && + + // Support: Promises/A+ section 2.3.4 + // https://promisesaplus.com/#point-64 + // Only check objects and functions for thenability + ( typeof returned === "object" || + typeof returned === "function" ) && + returned.then; + + // Handle a returned thenable + if ( isFunction( then ) ) { + + // Special processors (notify) just wait for resolution + if ( special ) { + then.call( + returned, + resolve( maxDepth, deferred, Identity, special ), + resolve( maxDepth, deferred, Thrower, special ) + ); + + // Normal processors (resolve) also hook into progress + } else { + + // ...and disregard older resolution values + maxDepth++; + + then.call( + returned, + resolve( maxDepth, deferred, Identity, special ), + resolve( maxDepth, deferred, Thrower, special ), + resolve( maxDepth, deferred, Identity, + deferred.notifyWith ) + ); + } + + // Handle all other returned values + } else { + + // Only substitute handlers pass on context + // and multiple values (non-spec behavior) + if ( handler !== Identity ) { + that = undefined; + args = [ returned ]; + } + + // Process the value(s) + // Default process is resolve + ( special || deferred.resolveWith )( that, args ); + } + }, + + // Only normal processors (resolve) catch and reject exceptions + process = special ? + mightThrow : + function() { + try { + mightThrow(); + } catch ( e ) { + + if ( jQuery.Deferred.exceptionHook ) { + jQuery.Deferred.exceptionHook( e, + process.stackTrace ); + } + + // Support: Promises/A+ section 2.3.3.3.4.1 + // https://promisesaplus.com/#point-61 + // Ignore post-resolution exceptions + if ( depth + 1 >= maxDepth ) { + + // Only substitute handlers pass on context + // and multiple values (non-spec behavior) + if ( handler !== Thrower ) { + that = undefined; + args = [ e ]; + } + + deferred.rejectWith( that, args ); + } + } + }; + + // Support: Promises/A+ section 2.3.3.3.1 + // https://promisesaplus.com/#point-57 + // Re-resolve promises immediately to dodge false rejection from + // subsequent errors + if ( depth ) { + process(); + } else { + + // Call an optional hook to record the stack, in case of exception + // since it's otherwise lost when execution goes async + if ( jQuery.Deferred.getStackHook ) { + process.stackTrace = jQuery.Deferred.getStackHook(); + } + window.setTimeout( process ); + } + }; + } + + return jQuery.Deferred( function( newDefer ) { + + // progress_handlers.add( ... ) + tuples[ 0 ][ 3 ].add( + resolve( + 0, + newDefer, + isFunction( onProgress ) ? + onProgress : + Identity, + newDefer.notifyWith + ) + ); + + // fulfilled_handlers.add( ... ) + tuples[ 1 ][ 3 ].add( + resolve( + 0, + newDefer, + isFunction( onFulfilled ) ? + onFulfilled : + Identity + ) + ); + + // rejected_handlers.add( ... ) + tuples[ 2 ][ 3 ].add( + resolve( + 0, + newDefer, + isFunction( onRejected ) ? + onRejected : + Thrower + ) + ); + } ).promise(); + }, + + // Get a promise for this deferred + // If obj is provided, the promise aspect is added to the object + promise: function( obj ) { + return obj != null ? jQuery.extend( obj, promise ) : promise; + } + }, + deferred = {}; + + // Add list-specific methods + jQuery.each( tuples, function( i, tuple ) { + var list = tuple[ 2 ], + stateString = tuple[ 5 ]; + + // promise.progress = list.add + // promise.done = list.add + // promise.fail = list.add + promise[ tuple[ 1 ] ] = list.add; + + // Handle state + if ( stateString ) { + list.add( + function() { + + // state = "resolved" (i.e., fulfilled) + // state = "rejected" + state = stateString; + }, + + // rejected_callbacks.disable + // fulfilled_callbacks.disable + tuples[ 3 - i ][ 2 ].disable, + + // rejected_handlers.disable + // fulfilled_handlers.disable + tuples[ 3 - i ][ 3 ].disable, + + // progress_callbacks.lock + tuples[ 0 ][ 2 ].lock, + + // progress_handlers.lock + tuples[ 0 ][ 3 ].lock + ); + } + + // progress_handlers.fire + // fulfilled_handlers.fire + // rejected_handlers.fire + list.add( tuple[ 3 ].fire ); + + // deferred.notify = function() { deferred.notifyWith(...) } + // deferred.resolve = function() { deferred.resolveWith(...) } + // deferred.reject = function() { deferred.rejectWith(...) } + deferred[ tuple[ 0 ] ] = function() { + deferred[ tuple[ 0 ] + "With" ]( this === deferred ? undefined : this, arguments ); + return this; + }; + + // deferred.notifyWith = list.fireWith + // deferred.resolveWith = list.fireWith + // deferred.rejectWith = list.fireWith + deferred[ tuple[ 0 ] + "With" ] = list.fireWith; + } ); + + // Make the deferred a promise + promise.promise( deferred ); + + // Call given func if any + if ( func ) { + func.call( deferred, deferred ); + } + + // All done! + return deferred; + }, + + // Deferred helper + when: function( singleValue ) { + var + + // count of uncompleted subordinates + remaining = arguments.length, + + // count of unprocessed arguments + i = remaining, + + // subordinate fulfillment data + resolveContexts = Array( i ), + resolveValues = slice.call( arguments ), + + // the master Deferred + master = jQuery.Deferred(), + + // subordinate callback factory + updateFunc = function( i ) { + return function( value ) { + resolveContexts[ i ] = this; + resolveValues[ i ] = arguments.length > 1 ? slice.call( arguments ) : value; + if ( !( --remaining ) ) { + master.resolveWith( resolveContexts, resolveValues ); + } + }; + }; + + // Single- and empty arguments are adopted like Promise.resolve + if ( remaining <= 1 ) { + adoptValue( singleValue, master.done( updateFunc( i ) ).resolve, master.reject, + !remaining ); + + // Use .then() to unwrap secondary thenables (cf. gh-3000) + if ( master.state() === "pending" || + isFunction( resolveValues[ i ] && resolveValues[ i ].then ) ) { + + return master.then(); + } + } + + // Multiple arguments are aggregated like Promise.all array elements + while ( i-- ) { + adoptValue( resolveValues[ i ], updateFunc( i ), master.reject ); + } + + return master.promise(); + } +} ); + + +// These usually indicate a programmer mistake during development, +// warn about them ASAP rather than swallowing them by default. +var rerrorNames = /^(Eval|Internal|Range|Reference|Syntax|Type|URI)Error$/; + +jQuery.Deferred.exceptionHook = function( error, stack ) { + + // Support: IE 8 - 9 only + // Console exists when dev tools are open, which can happen at any time + if ( window.console && window.console.warn && error && rerrorNames.test( error.name ) ) { + window.console.warn( "jQuery.Deferred exception: " + error.message, error.stack, stack ); + } +}; + + + + +jQuery.readyException = function( error ) { + window.setTimeout( function() { + throw error; + } ); +}; + + + + +// The deferred used on DOM ready +var readyList = jQuery.Deferred(); + +jQuery.fn.ready = function( fn ) { + + readyList + .then( fn ) + + // Wrap jQuery.readyException in a function so that the lookup + // happens at the time of error handling instead of callback + // registration. + .catch( function( error ) { + jQuery.readyException( error ); + } ); + + return this; +}; + +jQuery.extend( { + + // Is the DOM ready to be used? Set to true once it occurs. + isReady: false, + + // A counter to track how many items to wait for before + // the ready event fires. See #6781 + readyWait: 1, + + // Handle when the DOM is ready + ready: function( wait ) { + + // Abort if there are pending holds or we're already ready + if ( wait === true ? --jQuery.readyWait : jQuery.isReady ) { + return; + } + + // Remember that the DOM is ready + jQuery.isReady = true; + + // If a normal DOM Ready event fired, decrement, and wait if need be + if ( wait !== true && --jQuery.readyWait > 0 ) { + return; + } + + // If there are functions bound, to execute + readyList.resolveWith( document, [ jQuery ] ); + } +} ); + +jQuery.ready.then = readyList.then; + +// The ready event handler and self cleanup method +function completed() { + document.removeEventListener( "DOMContentLoaded", completed ); + window.removeEventListener( "load", completed ); + jQuery.ready(); +} + +// Catch cases where $(document).ready() is called +// after the browser event has already occurred. +// Support: IE <=9 - 10 only +// Older IE sometimes signals "interactive" too soon +if ( document.readyState === "complete" || + ( document.readyState !== "loading" && !document.documentElement.doScroll ) ) { + + // Handle it asynchronously to allow scripts the opportunity to delay ready + window.setTimeout( jQuery.ready ); + +} else { + + // Use the handy event callback + document.addEventListener( "DOMContentLoaded", completed ); + + // A fallback to window.onload, that will always work + window.addEventListener( "load", completed ); +} + + + + +// Multifunctional method to get and set values of a collection +// The value/s can optionally be executed if it's a function +var access = function( elems, fn, key, value, chainable, emptyGet, raw ) { + var i = 0, + len = elems.length, + bulk = key == null; + + // Sets many values + if ( toType( key ) === "object" ) { + chainable = true; + for ( i in key ) { + access( elems, fn, i, key[ i ], true, emptyGet, raw ); + } + + // Sets one value + } else if ( value !== undefined ) { + chainable = true; + + if ( !isFunction( value ) ) { + raw = true; + } + + if ( bulk ) { + + // Bulk operations run against the entire set + if ( raw ) { + fn.call( elems, value ); + fn = null; + + // ...except when executing function values + } else { + bulk = fn; + fn = function( elem, _key, value ) { + return bulk.call( jQuery( elem ), value ); + }; + } + } + + if ( fn ) { + for ( ; i < len; i++ ) { + fn( + elems[ i ], key, raw ? + value : + value.call( elems[ i ], i, fn( elems[ i ], key ) ) + ); + } + } + } + + if ( chainable ) { + return elems; + } + + // Gets + if ( bulk ) { + return fn.call( elems ); + } + + return len ? fn( elems[ 0 ], key ) : emptyGet; +}; + + +// Matches dashed string for camelizing +var rmsPrefix = /^-ms-/, + rdashAlpha = /-([a-z])/g; + +// Used by camelCase as callback to replace() +function fcamelCase( _all, letter ) { + return letter.toUpperCase(); +} + +// Convert dashed to camelCase; used by the css and data modules +// Support: IE <=9 - 11, Edge 12 - 15 +// Microsoft forgot to hump their vendor prefix (#9572) +function camelCase( string ) { + return string.replace( rmsPrefix, "ms-" ).replace( rdashAlpha, fcamelCase ); +} +var acceptData = function( owner ) { + + // Accepts only: + // - Node + // - Node.ELEMENT_NODE + // - Node.DOCUMENT_NODE + // - Object + // - Any + return owner.nodeType === 1 || owner.nodeType === 9 || !( +owner.nodeType ); +}; + + + + +function Data() { + this.expando = jQuery.expando + Data.uid++; +} + +Data.uid = 1; + +Data.prototype = { + + cache: function( owner ) { + + // Check if the owner object already has a cache + var value = owner[ this.expando ]; + + // If not, create one + if ( !value ) { + value = {}; + + // We can accept data for non-element nodes in modern browsers, + // but we should not, see #8335. + // Always return an empty object. + if ( acceptData( owner ) ) { + + // If it is a node unlikely to be stringify-ed or looped over + // use plain assignment + if ( owner.nodeType ) { + owner[ this.expando ] = value; + + // Otherwise secure it in a non-enumerable property + // configurable must be true to allow the property to be + // deleted when data is removed + } else { + Object.defineProperty( owner, this.expando, { + value: value, + configurable: true + } ); + } + } + } + + return value; + }, + set: function( owner, data, value ) { + var prop, + cache = this.cache( owner ); + + // Handle: [ owner, key, value ] args + // Always use camelCase key (gh-2257) + if ( typeof data === "string" ) { + cache[ camelCase( data ) ] = value; + + // Handle: [ owner, { properties } ] args + } else { + + // Copy the properties one-by-one to the cache object + for ( prop in data ) { + cache[ camelCase( prop ) ] = data[ prop ]; + } + } + return cache; + }, + get: function( owner, key ) { + return key === undefined ? + this.cache( owner ) : + + // Always use camelCase key (gh-2257) + owner[ this.expando ] && owner[ this.expando ][ camelCase( key ) ]; + }, + access: function( owner, key, value ) { + + // In cases where either: + // + // 1. No key was specified + // 2. A string key was specified, but no value provided + // + // Take the "read" path and allow the get method to determine + // which value to return, respectively either: + // + // 1. The entire cache object + // 2. The data stored at the key + // + if ( key === undefined || + ( ( key && typeof key === "string" ) && value === undefined ) ) { + + return this.get( owner, key ); + } + + // When the key is not a string, or both a key and value + // are specified, set or extend (existing objects) with either: + // + // 1. An object of properties + // 2. A key and value + // + this.set( owner, key, value ); + + // Since the "set" path can have two possible entry points + // return the expected data based on which path was taken[*] + return value !== undefined ? value : key; + }, + remove: function( owner, key ) { + var i, + cache = owner[ this.expando ]; + + if ( cache === undefined ) { + return; + } + + if ( key !== undefined ) { + + // Support array or space separated string of keys + if ( Array.isArray( key ) ) { + + // If key is an array of keys... + // We always set camelCase keys, so remove that. + key = key.map( camelCase ); + } else { + key = camelCase( key ); + + // If a key with the spaces exists, use it. + // Otherwise, create an array by matching non-whitespace + key = key in cache ? + [ key ] : + ( key.match( rnothtmlwhite ) || [] ); + } + + i = key.length; + + while ( i-- ) { + delete cache[ key[ i ] ]; + } + } + + // Remove the expando if there's no more data + if ( key === undefined || jQuery.isEmptyObject( cache ) ) { + + // Support: Chrome <=35 - 45 + // Webkit & Blink performance suffers when deleting properties + // from DOM nodes, so set to undefined instead + // https://bugs.chromium.org/p/chromium/issues/detail?id=378607 (bug restricted) + if ( owner.nodeType ) { + owner[ this.expando ] = undefined; + } else { + delete owner[ this.expando ]; + } + } + }, + hasData: function( owner ) { + var cache = owner[ this.expando ]; + return cache !== undefined && !jQuery.isEmptyObject( cache ); + } +}; +var dataPriv = new Data(); + +var dataUser = new Data(); + + + +// Implementation Summary +// +// 1. Enforce API surface and semantic compatibility with 1.9.x branch +// 2. Improve the module's maintainability by reducing the storage +// paths to a single mechanism. +// 3. Use the same single mechanism to support "private" and "user" data. +// 4. _Never_ expose "private" data to user code (TODO: Drop _data, _removeData) +// 5. Avoid exposing implementation details on user objects (eg. expando properties) +// 6. Provide a clear path for implementation upgrade to WeakMap in 2014 + +var rbrace = /^(?:\{[\w\W]*\}|\[[\w\W]*\])$/, + rmultiDash = /[A-Z]/g; + +function getData( data ) { + if ( data === "true" ) { + return true; + } + + if ( data === "false" ) { + return false; + } + + if ( data === "null" ) { + return null; + } + + // Only convert to a number if it doesn't change the string + if ( data === +data + "" ) { + return +data; + } + + if ( rbrace.test( data ) ) { + return JSON.parse( data ); + } + + return data; +} + +function dataAttr( elem, key, data ) { + var name; + + // If nothing was found internally, try to fetch any + // data from the HTML5 data-* attribute + if ( data === undefined && elem.nodeType === 1 ) { + name = "data-" + key.replace( rmultiDash, "-$&" ).toLowerCase(); + data = elem.getAttribute( name ); + + if ( typeof data === "string" ) { + try { + data = getData( data ); + } catch ( e ) {} + + // Make sure we set the data so it isn't changed later + dataUser.set( elem, key, data ); + } else { + data = undefined; + } + } + return data; +} + +jQuery.extend( { + hasData: function( elem ) { + return dataUser.hasData( elem ) || dataPriv.hasData( elem ); + }, + + data: function( elem, name, data ) { + return dataUser.access( elem, name, data ); + }, + + removeData: function( elem, name ) { + dataUser.remove( elem, name ); + }, + + // TODO: Now that all calls to _data and _removeData have been replaced + // with direct calls to dataPriv methods, these can be deprecated. + _data: function( elem, name, data ) { + return dataPriv.access( elem, name, data ); + }, + + _removeData: function( elem, name ) { + dataPriv.remove( elem, name ); + } +} ); + +jQuery.fn.extend( { + data: function( key, value ) { + var i, name, data, + elem = this[ 0 ], + attrs = elem && elem.attributes; + + // Gets all values + if ( key === undefined ) { + if ( this.length ) { + data = dataUser.get( elem ); + + if ( elem.nodeType === 1 && !dataPriv.get( elem, "hasDataAttrs" ) ) { + i = attrs.length; + while ( i-- ) { + + // Support: IE 11 only + // The attrs elements can be null (#14894) + if ( attrs[ i ] ) { + name = attrs[ i ].name; + if ( name.indexOf( "data-" ) === 0 ) { + name = camelCase( name.slice( 5 ) ); + dataAttr( elem, name, data[ name ] ); + } + } + } + dataPriv.set( elem, "hasDataAttrs", true ); + } + } + + return data; + } + + // Sets multiple values + if ( typeof key === "object" ) { + return this.each( function() { + dataUser.set( this, key ); + } ); + } + + return access( this, function( value ) { + var data; + + // The calling jQuery object (element matches) is not empty + // (and therefore has an element appears at this[ 0 ]) and the + // `value` parameter was not undefined. An empty jQuery object + // will result in `undefined` for elem = this[ 0 ] which will + // throw an exception if an attempt to read a data cache is made. + if ( elem && value === undefined ) { + + // Attempt to get data from the cache + // The key will always be camelCased in Data + data = dataUser.get( elem, key ); + if ( data !== undefined ) { + return data; + } + + // Attempt to "discover" the data in + // HTML5 custom data-* attrs + data = dataAttr( elem, key ); + if ( data !== undefined ) { + return data; + } + + // We tried really hard, but the data doesn't exist. + return; + } + + // Set the data... + this.each( function() { + + // We always store the camelCased key + dataUser.set( this, key, value ); + } ); + }, null, value, arguments.length > 1, null, true ); + }, + + removeData: function( key ) { + return this.each( function() { + dataUser.remove( this, key ); + } ); + } +} ); + + +jQuery.extend( { + queue: function( elem, type, data ) { + var queue; + + if ( elem ) { + type = ( type || "fx" ) + "queue"; + queue = dataPriv.get( elem, type ); + + // Speed up dequeue by getting out quickly if this is just a lookup + if ( data ) { + if ( !queue || Array.isArray( data ) ) { + queue = dataPriv.access( elem, type, jQuery.makeArray( data ) ); + } else { + queue.push( data ); + } + } + return queue || []; + } + }, + + dequeue: function( elem, type ) { + type = type || "fx"; + + var queue = jQuery.queue( elem, type ), + startLength = queue.length, + fn = queue.shift(), + hooks = jQuery._queueHooks( elem, type ), + next = function() { + jQuery.dequeue( elem, type ); + }; + + // If the fx queue is dequeued, always remove the progress sentinel + if ( fn === "inprogress" ) { + fn = queue.shift(); + startLength--; + } + + if ( fn ) { + + // Add a progress sentinel to prevent the fx queue from being + // automatically dequeued + if ( type === "fx" ) { + queue.unshift( "inprogress" ); + } + + // Clear up the last queue stop function + delete hooks.stop; + fn.call( elem, next, hooks ); + } + + if ( !startLength && hooks ) { + hooks.empty.fire(); + } + }, + + // Not public - generate a queueHooks object, or return the current one + _queueHooks: function( elem, type ) { + var key = type + "queueHooks"; + return dataPriv.get( elem, key ) || dataPriv.access( elem, key, { + empty: jQuery.Callbacks( "once memory" ).add( function() { + dataPriv.remove( elem, [ type + "queue", key ] ); + } ) + } ); + } +} ); + +jQuery.fn.extend( { + queue: function( type, data ) { + var setter = 2; + + if ( typeof type !== "string" ) { + data = type; + type = "fx"; + setter--; + } + + if ( arguments.length < setter ) { + return jQuery.queue( this[ 0 ], type ); + } + + return data === undefined ? + this : + this.each( function() { + var queue = jQuery.queue( this, type, data ); + + // Ensure a hooks for this queue + jQuery._queueHooks( this, type ); + + if ( type === "fx" && queue[ 0 ] !== "inprogress" ) { + jQuery.dequeue( this, type ); + } + } ); + }, + dequeue: function( type ) { + return this.each( function() { + jQuery.dequeue( this, type ); + } ); + }, + clearQueue: function( type ) { + return this.queue( type || "fx", [] ); + }, + + // Get a promise resolved when queues of a certain type + // are emptied (fx is the type by default) + promise: function( type, obj ) { + var tmp, + count = 1, + defer = jQuery.Deferred(), + elements = this, + i = this.length, + resolve = function() { + if ( !( --count ) ) { + defer.resolveWith( elements, [ elements ] ); + } + }; + + if ( typeof type !== "string" ) { + obj = type; + type = undefined; + } + type = type || "fx"; + + while ( i-- ) { + tmp = dataPriv.get( elements[ i ], type + "queueHooks" ); + if ( tmp && tmp.empty ) { + count++; + tmp.empty.add( resolve ); + } + } + resolve(); + return defer.promise( obj ); + } +} ); +var pnum = ( /[+-]?(?:\d*\.|)\d+(?:[eE][+-]?\d+|)/ ).source; + +var rcssNum = new RegExp( "^(?:([+-])=|)(" + pnum + ")([a-z%]*)$", "i" ); + + +var cssExpand = [ "Top", "Right", "Bottom", "Left" ]; + +var documentElement = document.documentElement; + + + + var isAttached = function( elem ) { + return jQuery.contains( elem.ownerDocument, elem ); + }, + composed = { composed: true }; + + // Support: IE 9 - 11+, Edge 12 - 18+, iOS 10.0 - 10.2 only + // Check attachment across shadow DOM boundaries when possible (gh-3504) + // Support: iOS 10.0-10.2 only + // Early iOS 10 versions support `attachShadow` but not `getRootNode`, + // leading to errors. We need to check for `getRootNode`. + if ( documentElement.getRootNode ) { + isAttached = function( elem ) { + return jQuery.contains( elem.ownerDocument, elem ) || + elem.getRootNode( composed ) === elem.ownerDocument; + }; + } +var isHiddenWithinTree = function( elem, el ) { + + // isHiddenWithinTree might be called from jQuery#filter function; + // in that case, element will be second argument + elem = el || elem; + + // Inline style trumps all + return elem.style.display === "none" || + elem.style.display === "" && + + // Otherwise, check computed style + // Support: Firefox <=43 - 45 + // Disconnected elements can have computed display: none, so first confirm that elem is + // in the document. + isAttached( elem ) && + + jQuery.css( elem, "display" ) === "none"; + }; + + + +function adjustCSS( elem, prop, valueParts, tween ) { + var adjusted, scale, + maxIterations = 20, + currentValue = tween ? + function() { + return tween.cur(); + } : + function() { + return jQuery.css( elem, prop, "" ); + }, + initial = currentValue(), + unit = valueParts && valueParts[ 3 ] || ( jQuery.cssNumber[ prop ] ? "" : "px" ), + + // Starting value computation is required for potential unit mismatches + initialInUnit = elem.nodeType && + ( jQuery.cssNumber[ prop ] || unit !== "px" && +initial ) && + rcssNum.exec( jQuery.css( elem, prop ) ); + + if ( initialInUnit && initialInUnit[ 3 ] !== unit ) { + + // Support: Firefox <=54 + // Halve the iteration target value to prevent interference from CSS upper bounds (gh-2144) + initial = initial / 2; + + // Trust units reported by jQuery.css + unit = unit || initialInUnit[ 3 ]; + + // Iteratively approximate from a nonzero starting point + initialInUnit = +initial || 1; + + while ( maxIterations-- ) { + + // Evaluate and update our best guess (doubling guesses that zero out). + // Finish if the scale equals or crosses 1 (making the old*new product non-positive). + jQuery.style( elem, prop, initialInUnit + unit ); + if ( ( 1 - scale ) * ( 1 - ( scale = currentValue() / initial || 0.5 ) ) <= 0 ) { + maxIterations = 0; + } + initialInUnit = initialInUnit / scale; + + } + + initialInUnit = initialInUnit * 2; + jQuery.style( elem, prop, initialInUnit + unit ); + + // Make sure we update the tween properties later on + valueParts = valueParts || []; + } + + if ( valueParts ) { + initialInUnit = +initialInUnit || +initial || 0; + + // Apply relative offset (+=/-=) if specified + adjusted = valueParts[ 1 ] ? + initialInUnit + ( valueParts[ 1 ] + 1 ) * valueParts[ 2 ] : + +valueParts[ 2 ]; + if ( tween ) { + tween.unit = unit; + tween.start = initialInUnit; + tween.end = adjusted; + } + } + return adjusted; +} + + +var defaultDisplayMap = {}; + +function getDefaultDisplay( elem ) { + var temp, + doc = elem.ownerDocument, + nodeName = elem.nodeName, + display = defaultDisplayMap[ nodeName ]; + + if ( display ) { + return display; + } + + temp = doc.body.appendChild( doc.createElement( nodeName ) ); + display = jQuery.css( temp, "display" ); + + temp.parentNode.removeChild( temp ); + + if ( display === "none" ) { + display = "block"; + } + defaultDisplayMap[ nodeName ] = display; + + return display; +} + +function showHide( elements, show ) { + var display, elem, + values = [], + index = 0, + length = elements.length; + + // Determine new display value for elements that need to change + for ( ; index < length; index++ ) { + elem = elements[ index ]; + if ( !elem.style ) { + continue; + } + + display = elem.style.display; + if ( show ) { + + // Since we force visibility upon cascade-hidden elements, an immediate (and slow) + // check is required in this first loop unless we have a nonempty display value (either + // inline or about-to-be-restored) + if ( display === "none" ) { + values[ index ] = dataPriv.get( elem, "display" ) || null; + if ( !values[ index ] ) { + elem.style.display = ""; + } + } + if ( elem.style.display === "" && isHiddenWithinTree( elem ) ) { + values[ index ] = getDefaultDisplay( elem ); + } + } else { + if ( display !== "none" ) { + values[ index ] = "none"; + + // Remember what we're overwriting + dataPriv.set( elem, "display", display ); + } + } + } + + // Set the display of the elements in a second loop to avoid constant reflow + for ( index = 0; index < length; index++ ) { + if ( values[ index ] != null ) { + elements[ index ].style.display = values[ index ]; + } + } + + return elements; +} + +jQuery.fn.extend( { + show: function() { + return showHide( this, true ); + }, + hide: function() { + return showHide( this ); + }, + toggle: function( state ) { + if ( typeof state === "boolean" ) { + return state ? this.show() : this.hide(); + } + + return this.each( function() { + if ( isHiddenWithinTree( this ) ) { + jQuery( this ).show(); + } else { + jQuery( this ).hide(); + } + } ); + } +} ); +var rcheckableType = ( /^(?:checkbox|radio)$/i ); + +var rtagName = ( /<([a-z][^\/\0>\x20\t\r\n\f]*)/i ); + +var rscriptType = ( /^$|^module$|\/(?:java|ecma)script/i ); + + + +( function() { + var fragment = document.createDocumentFragment(), + div = fragment.appendChild( document.createElement( "div" ) ), + input = document.createElement( "input" ); + + // Support: Android 4.0 - 4.3 only + // Check state lost if the name is set (#11217) + // Support: Windows Web Apps (WWA) + // `name` and `type` must use .setAttribute for WWA (#14901) + input.setAttribute( "type", "radio" ); + input.setAttribute( "checked", "checked" ); + input.setAttribute( "name", "t" ); + + div.appendChild( input ); + + // Support: Android <=4.1 only + // Older WebKit doesn't clone checked state correctly in fragments + support.checkClone = div.cloneNode( true ).cloneNode( true ).lastChild.checked; + + // Support: IE <=11 only + // Make sure textarea (and checkbox) defaultValue is properly cloned + div.innerHTML = ""; + support.noCloneChecked = !!div.cloneNode( true ).lastChild.defaultValue; + + // Support: IE <=9 only + // IE <=9 replaces "; + support.option = !!div.lastChild; +} )(); + + +// We have to close these tags to support XHTML (#13200) +var wrapMap = { + + // XHTML parsers do not magically insert elements in the + // same way that tag soup parsers do. So we cannot shorten + // this by omitting or other required elements. + thead: [ 1, "", "
" ], + col: [ 2, "", "
" ], + tr: [ 2, "", "
" ], + td: [ 3, "", "
" ], + + _default: [ 0, "", "" ] +}; + +wrapMap.tbody = wrapMap.tfoot = wrapMap.colgroup = wrapMap.caption = wrapMap.thead; +wrapMap.th = wrapMap.td; + +// Support: IE <=9 only +if ( !support.option ) { + wrapMap.optgroup = wrapMap.option = [ 1, "" ]; +} + + +function getAll( context, tag ) { + + // Support: IE <=9 - 11 only + // Use typeof to avoid zero-argument method invocation on host objects (#15151) + var ret; + + if ( typeof context.getElementsByTagName !== "undefined" ) { + ret = context.getElementsByTagName( tag || "*" ); + + } else if ( typeof context.querySelectorAll !== "undefined" ) { + ret = context.querySelectorAll( tag || "*" ); + + } else { + ret = []; + } + + if ( tag === undefined || tag && nodeName( context, tag ) ) { + return jQuery.merge( [ context ], ret ); + } + + return ret; +} + + +// Mark scripts as having already been evaluated +function setGlobalEval( elems, refElements ) { + var i = 0, + l = elems.length; + + for ( ; i < l; i++ ) { + dataPriv.set( + elems[ i ], + "globalEval", + !refElements || dataPriv.get( refElements[ i ], "globalEval" ) + ); + } +} + + +var rhtml = /<|&#?\w+;/; + +function buildFragment( elems, context, scripts, selection, ignored ) { + var elem, tmp, tag, wrap, attached, j, + fragment = context.createDocumentFragment(), + nodes = [], + i = 0, + l = elems.length; + + for ( ; i < l; i++ ) { + elem = elems[ i ]; + + if ( elem || elem === 0 ) { + + // Add nodes directly + if ( toType( elem ) === "object" ) { + + // Support: Android <=4.0 only, PhantomJS 1 only + // push.apply(_, arraylike) throws on ancient WebKit + jQuery.merge( nodes, elem.nodeType ? [ elem ] : elem ); + + // Convert non-html into a text node + } else if ( !rhtml.test( elem ) ) { + nodes.push( context.createTextNode( elem ) ); + + // Convert html into DOM nodes + } else { + tmp = tmp || fragment.appendChild( context.createElement( "div" ) ); + + // Deserialize a standard representation + tag = ( rtagName.exec( elem ) || [ "", "" ] )[ 1 ].toLowerCase(); + wrap = wrapMap[ tag ] || wrapMap._default; + tmp.innerHTML = wrap[ 1 ] + jQuery.htmlPrefilter( elem ) + wrap[ 2 ]; + + // Descend through wrappers to the right content + j = wrap[ 0 ]; + while ( j-- ) { + tmp = tmp.lastChild; + } + + // Support: Android <=4.0 only, PhantomJS 1 only + // push.apply(_, arraylike) throws on ancient WebKit + jQuery.merge( nodes, tmp.childNodes ); + + // Remember the top-level container + tmp = fragment.firstChild; + + // Ensure the created nodes are orphaned (#12392) + tmp.textContent = ""; + } + } + } + + // Remove wrapper from fragment + fragment.textContent = ""; + + i = 0; + while ( ( elem = nodes[ i++ ] ) ) { + + // Skip elements already in the context collection (trac-4087) + if ( selection && jQuery.inArray( elem, selection ) > -1 ) { + if ( ignored ) { + ignored.push( elem ); + } + continue; + } + + attached = isAttached( elem ); + + // Append to fragment + tmp = getAll( fragment.appendChild( elem ), "script" ); + + // Preserve script evaluation history + if ( attached ) { + setGlobalEval( tmp ); + } + + // Capture executables + if ( scripts ) { + j = 0; + while ( ( elem = tmp[ j++ ] ) ) { + if ( rscriptType.test( elem.type || "" ) ) { + scripts.push( elem ); + } + } + } + } + + return fragment; +} + + +var + rkeyEvent = /^key/, + rmouseEvent = /^(?:mouse|pointer|contextmenu|drag|drop)|click/, + rtypenamespace = /^([^.]*)(?:\.(.+)|)/; + +function returnTrue() { + return true; +} + +function returnFalse() { + return false; +} + +// Support: IE <=9 - 11+ +// focus() and blur() are asynchronous, except when they are no-op. +// So expect focus to be synchronous when the element is already active, +// and blur to be synchronous when the element is not already active. +// (focus and blur are always synchronous in other supported browsers, +// this just defines when we can count on it). +function expectSync( elem, type ) { + return ( elem === safeActiveElement() ) === ( type === "focus" ); +} + +// Support: IE <=9 only +// Accessing document.activeElement can throw unexpectedly +// https://bugs.jquery.com/ticket/13393 +function safeActiveElement() { + try { + return document.activeElement; + } catch ( err ) { } +} + +function on( elem, types, selector, data, fn, one ) { + var origFn, type; + + // Types can be a map of types/handlers + if ( typeof types === "object" ) { + + // ( types-Object, selector, data ) + if ( typeof selector !== "string" ) { + + // ( types-Object, data ) + data = data || selector; + selector = undefined; + } + for ( type in types ) { + on( elem, type, selector, data, types[ type ], one ); + } + return elem; + } + + if ( data == null && fn == null ) { + + // ( types, fn ) + fn = selector; + data = selector = undefined; + } else if ( fn == null ) { + if ( typeof selector === "string" ) { + + // ( types, selector, fn ) + fn = data; + data = undefined; + } else { + + // ( types, data, fn ) + fn = data; + data = selector; + selector = undefined; + } + } + if ( fn === false ) { + fn = returnFalse; + } else if ( !fn ) { + return elem; + } + + if ( one === 1 ) { + origFn = fn; + fn = function( event ) { + + // Can use an empty set, since event contains the info + jQuery().off( event ); + return origFn.apply( this, arguments ); + }; + + // Use same guid so caller can remove using origFn + fn.guid = origFn.guid || ( origFn.guid = jQuery.guid++ ); + } + return elem.each( function() { + jQuery.event.add( this, types, fn, data, selector ); + } ); +} + +/* + * Helper functions for managing events -- not part of the public interface. + * Props to Dean Edwards' addEvent library for many of the ideas. + */ +jQuery.event = { + + global: {}, + + add: function( elem, types, handler, data, selector ) { + + var handleObjIn, eventHandle, tmp, + events, t, handleObj, + special, handlers, type, namespaces, origType, + elemData = dataPriv.get( elem ); + + // Only attach events to objects that accept data + if ( !acceptData( elem ) ) { + return; + } + + // Caller can pass in an object of custom data in lieu of the handler + if ( handler.handler ) { + handleObjIn = handler; + handler = handleObjIn.handler; + selector = handleObjIn.selector; + } + + // Ensure that invalid selectors throw exceptions at attach time + // Evaluate against documentElement in case elem is a non-element node (e.g., document) + if ( selector ) { + jQuery.find.matchesSelector( documentElement, selector ); + } + + // Make sure that the handler has a unique ID, used to find/remove it later + if ( !handler.guid ) { + handler.guid = jQuery.guid++; + } + + // Init the element's event structure and main handler, if this is the first + if ( !( events = elemData.events ) ) { + events = elemData.events = Object.create( null ); + } + if ( !( eventHandle = elemData.handle ) ) { + eventHandle = elemData.handle = function( e ) { + + // Discard the second event of a jQuery.event.trigger() and + // when an event is called after a page has unloaded + return typeof jQuery !== "undefined" && jQuery.event.triggered !== e.type ? + jQuery.event.dispatch.apply( elem, arguments ) : undefined; + }; + } + + // Handle multiple events separated by a space + types = ( types || "" ).match( rnothtmlwhite ) || [ "" ]; + t = types.length; + while ( t-- ) { + tmp = rtypenamespace.exec( types[ t ] ) || []; + type = origType = tmp[ 1 ]; + namespaces = ( tmp[ 2 ] || "" ).split( "." ).sort(); + + // There *must* be a type, no attaching namespace-only handlers + if ( !type ) { + continue; + } + + // If event changes its type, use the special event handlers for the changed type + special = jQuery.event.special[ type ] || {}; + + // If selector defined, determine special event api type, otherwise given type + type = ( selector ? special.delegateType : special.bindType ) || type; + + // Update special based on newly reset type + special = jQuery.event.special[ type ] || {}; + + // handleObj is passed to all event handlers + handleObj = jQuery.extend( { + type: type, + origType: origType, + data: data, + handler: handler, + guid: handler.guid, + selector: selector, + needsContext: selector && jQuery.expr.match.needsContext.test( selector ), + namespace: namespaces.join( "." ) + }, handleObjIn ); + + // Init the event handler queue if we're the first + if ( !( handlers = events[ type ] ) ) { + handlers = events[ type ] = []; + handlers.delegateCount = 0; + + // Only use addEventListener if the special events handler returns false + if ( !special.setup || + special.setup.call( elem, data, namespaces, eventHandle ) === false ) { + + if ( elem.addEventListener ) { + elem.addEventListener( type, eventHandle ); + } + } + } + + if ( special.add ) { + special.add.call( elem, handleObj ); + + if ( !handleObj.handler.guid ) { + handleObj.handler.guid = handler.guid; + } + } + + // Add to the element's handler list, delegates in front + if ( selector ) { + handlers.splice( handlers.delegateCount++, 0, handleObj ); + } else { + handlers.push( handleObj ); + } + + // Keep track of which events have ever been used, for event optimization + jQuery.event.global[ type ] = true; + } + + }, + + // Detach an event or set of events from an element + remove: function( elem, types, handler, selector, mappedTypes ) { + + var j, origCount, tmp, + events, t, handleObj, + special, handlers, type, namespaces, origType, + elemData = dataPriv.hasData( elem ) && dataPriv.get( elem ); + + if ( !elemData || !( events = elemData.events ) ) { + return; + } + + // Once for each type.namespace in types; type may be omitted + types = ( types || "" ).match( rnothtmlwhite ) || [ "" ]; + t = types.length; + while ( t-- ) { + tmp = rtypenamespace.exec( types[ t ] ) || []; + type = origType = tmp[ 1 ]; + namespaces = ( tmp[ 2 ] || "" ).split( "." ).sort(); + + // Unbind all events (on this namespace, if provided) for the element + if ( !type ) { + for ( type in events ) { + jQuery.event.remove( elem, type + types[ t ], handler, selector, true ); + } + continue; + } + + special = jQuery.event.special[ type ] || {}; + type = ( selector ? special.delegateType : special.bindType ) || type; + handlers = events[ type ] || []; + tmp = tmp[ 2 ] && + new RegExp( "(^|\\.)" + namespaces.join( "\\.(?:.*\\.|)" ) + "(\\.|$)" ); + + // Remove matching events + origCount = j = handlers.length; + while ( j-- ) { + handleObj = handlers[ j ]; + + if ( ( mappedTypes || origType === handleObj.origType ) && + ( !handler || handler.guid === handleObj.guid ) && + ( !tmp || tmp.test( handleObj.namespace ) ) && + ( !selector || selector === handleObj.selector || + selector === "**" && handleObj.selector ) ) { + handlers.splice( j, 1 ); + + if ( handleObj.selector ) { + handlers.delegateCount--; + } + if ( special.remove ) { + special.remove.call( elem, handleObj ); + } + } + } + + // Remove generic event handler if we removed something and no more handlers exist + // (avoids potential for endless recursion during removal of special event handlers) + if ( origCount && !handlers.length ) { + if ( !special.teardown || + special.teardown.call( elem, namespaces, elemData.handle ) === false ) { + + jQuery.removeEvent( elem, type, elemData.handle ); + } + + delete events[ type ]; + } + } + + // Remove data and the expando if it's no longer used + if ( jQuery.isEmptyObject( events ) ) { + dataPriv.remove( elem, "handle events" ); + } + }, + + dispatch: function( nativeEvent ) { + + var i, j, ret, matched, handleObj, handlerQueue, + args = new Array( arguments.length ), + + // Make a writable jQuery.Event from the native event object + event = jQuery.event.fix( nativeEvent ), + + handlers = ( + dataPriv.get( this, "events" ) || Object.create( null ) + )[ event.type ] || [], + special = jQuery.event.special[ event.type ] || {}; + + // Use the fix-ed jQuery.Event rather than the (read-only) native event + args[ 0 ] = event; + + for ( i = 1; i < arguments.length; i++ ) { + args[ i ] = arguments[ i ]; + } + + event.delegateTarget = this; + + // Call the preDispatch hook for the mapped type, and let it bail if desired + if ( special.preDispatch && special.preDispatch.call( this, event ) === false ) { + return; + } + + // Determine handlers + handlerQueue = jQuery.event.handlers.call( this, event, handlers ); + + // Run delegates first; they may want to stop propagation beneath us + i = 0; + while ( ( matched = handlerQueue[ i++ ] ) && !event.isPropagationStopped() ) { + event.currentTarget = matched.elem; + + j = 0; + while ( ( handleObj = matched.handlers[ j++ ] ) && + !event.isImmediatePropagationStopped() ) { + + // If the event is namespaced, then each handler is only invoked if it is + // specially universal or its namespaces are a superset of the event's. + if ( !event.rnamespace || handleObj.namespace === false || + event.rnamespace.test( handleObj.namespace ) ) { + + event.handleObj = handleObj; + event.data = handleObj.data; + + ret = ( ( jQuery.event.special[ handleObj.origType ] || {} ).handle || + handleObj.handler ).apply( matched.elem, args ); + + if ( ret !== undefined ) { + if ( ( event.result = ret ) === false ) { + event.preventDefault(); + event.stopPropagation(); + } + } + } + } + } + + // Call the postDispatch hook for the mapped type + if ( special.postDispatch ) { + special.postDispatch.call( this, event ); + } + + return event.result; + }, + + handlers: function( event, handlers ) { + var i, handleObj, sel, matchedHandlers, matchedSelectors, + handlerQueue = [], + delegateCount = handlers.delegateCount, + cur = event.target; + + // Find delegate handlers + if ( delegateCount && + + // Support: IE <=9 + // Black-hole SVG instance trees (trac-13180) + cur.nodeType && + + // Support: Firefox <=42 + // Suppress spec-violating clicks indicating a non-primary pointer button (trac-3861) + // https://www.w3.org/TR/DOM-Level-3-Events/#event-type-click + // Support: IE 11 only + // ...but not arrow key "clicks" of radio inputs, which can have `button` -1 (gh-2343) + !( event.type === "click" && event.button >= 1 ) ) { + + for ( ; cur !== this; cur = cur.parentNode || this ) { + + // Don't check non-elements (#13208) + // Don't process clicks on disabled elements (#6911, #8165, #11382, #11764) + if ( cur.nodeType === 1 && !( event.type === "click" && cur.disabled === true ) ) { + matchedHandlers = []; + matchedSelectors = {}; + for ( i = 0; i < delegateCount; i++ ) { + handleObj = handlers[ i ]; + + // Don't conflict with Object.prototype properties (#13203) + sel = handleObj.selector + " "; + + if ( matchedSelectors[ sel ] === undefined ) { + matchedSelectors[ sel ] = handleObj.needsContext ? + jQuery( sel, this ).index( cur ) > -1 : + jQuery.find( sel, this, null, [ cur ] ).length; + } + if ( matchedSelectors[ sel ] ) { + matchedHandlers.push( handleObj ); + } + } + if ( matchedHandlers.length ) { + handlerQueue.push( { elem: cur, handlers: matchedHandlers } ); + } + } + } + } + + // Add the remaining (directly-bound) handlers + cur = this; + if ( delegateCount < handlers.length ) { + handlerQueue.push( { elem: cur, handlers: handlers.slice( delegateCount ) } ); + } + + return handlerQueue; + }, + + addProp: function( name, hook ) { + Object.defineProperty( jQuery.Event.prototype, name, { + enumerable: true, + configurable: true, + + get: isFunction( hook ) ? + function() { + if ( this.originalEvent ) { + return hook( this.originalEvent ); + } + } : + function() { + if ( this.originalEvent ) { + return this.originalEvent[ name ]; + } + }, + + set: function( value ) { + Object.defineProperty( this, name, { + enumerable: true, + configurable: true, + writable: true, + value: value + } ); + } + } ); + }, + + fix: function( originalEvent ) { + return originalEvent[ jQuery.expando ] ? + originalEvent : + new jQuery.Event( originalEvent ); + }, + + special: { + load: { + + // Prevent triggered image.load events from bubbling to window.load + noBubble: true + }, + click: { + + // Utilize native event to ensure correct state for checkable inputs + setup: function( data ) { + + // For mutual compressibility with _default, replace `this` access with a local var. + // `|| data` is dead code meant only to preserve the variable through minification. + var el = this || data; + + // Claim the first handler + if ( rcheckableType.test( el.type ) && + el.click && nodeName( el, "input" ) ) { + + // dataPriv.set( el, "click", ... ) + leverageNative( el, "click", returnTrue ); + } + + // Return false to allow normal processing in the caller + return false; + }, + trigger: function( data ) { + + // For mutual compressibility with _default, replace `this` access with a local var. + // `|| data` is dead code meant only to preserve the variable through minification. + var el = this || data; + + // Force setup before triggering a click + if ( rcheckableType.test( el.type ) && + el.click && nodeName( el, "input" ) ) { + + leverageNative( el, "click" ); + } + + // Return non-false to allow normal event-path propagation + return true; + }, + + // For cross-browser consistency, suppress native .click() on links + // Also prevent it if we're currently inside a leveraged native-event stack + _default: function( event ) { + var target = event.target; + return rcheckableType.test( target.type ) && + target.click && nodeName( target, "input" ) && + dataPriv.get( target, "click" ) || + nodeName( target, "a" ); + } + }, + + beforeunload: { + postDispatch: function( event ) { + + // Support: Firefox 20+ + // Firefox doesn't alert if the returnValue field is not set. + if ( event.result !== undefined && event.originalEvent ) { + event.originalEvent.returnValue = event.result; + } + } + } + } +}; + +// Ensure the presence of an event listener that handles manually-triggered +// synthetic events by interrupting progress until reinvoked in response to +// *native* events that it fires directly, ensuring that state changes have +// already occurred before other listeners are invoked. +function leverageNative( el, type, expectSync ) { + + // Missing expectSync indicates a trigger call, which must force setup through jQuery.event.add + if ( !expectSync ) { + if ( dataPriv.get( el, type ) === undefined ) { + jQuery.event.add( el, type, returnTrue ); + } + return; + } + + // Register the controller as a special universal handler for all event namespaces + dataPriv.set( el, type, false ); + jQuery.event.add( el, type, { + namespace: false, + handler: function( event ) { + var notAsync, result, + saved = dataPriv.get( this, type ); + + if ( ( event.isTrigger & 1 ) && this[ type ] ) { + + // Interrupt processing of the outer synthetic .trigger()ed event + // Saved data should be false in such cases, but might be a leftover capture object + // from an async native handler (gh-4350) + if ( !saved.length ) { + + // Store arguments for use when handling the inner native event + // There will always be at least one argument (an event object), so this array + // will not be confused with a leftover capture object. + saved = slice.call( arguments ); + dataPriv.set( this, type, saved ); + + // Trigger the native event and capture its result + // Support: IE <=9 - 11+ + // focus() and blur() are asynchronous + notAsync = expectSync( this, type ); + this[ type ](); + result = dataPriv.get( this, type ); + if ( saved !== result || notAsync ) { + dataPriv.set( this, type, false ); + } else { + result = {}; + } + if ( saved !== result ) { + + // Cancel the outer synthetic event + event.stopImmediatePropagation(); + event.preventDefault(); + return result.value; + } + + // If this is an inner synthetic event for an event with a bubbling surrogate + // (focus or blur), assume that the surrogate already propagated from triggering the + // native event and prevent that from happening again here. + // This technically gets the ordering wrong w.r.t. to `.trigger()` (in which the + // bubbling surrogate propagates *after* the non-bubbling base), but that seems + // less bad than duplication. + } else if ( ( jQuery.event.special[ type ] || {} ).delegateType ) { + event.stopPropagation(); + } + + // If this is a native event triggered above, everything is now in order + // Fire an inner synthetic event with the original arguments + } else if ( saved.length ) { + + // ...and capture the result + dataPriv.set( this, type, { + value: jQuery.event.trigger( + + // Support: IE <=9 - 11+ + // Extend with the prototype to reset the above stopImmediatePropagation() + jQuery.extend( saved[ 0 ], jQuery.Event.prototype ), + saved.slice( 1 ), + this + ) + } ); + + // Abort handling of the native event + event.stopImmediatePropagation(); + } + } + } ); +} + +jQuery.removeEvent = function( elem, type, handle ) { + + // This "if" is needed for plain objects + if ( elem.removeEventListener ) { + elem.removeEventListener( type, handle ); + } +}; + +jQuery.Event = function( src, props ) { + + // Allow instantiation without the 'new' keyword + if ( !( this instanceof jQuery.Event ) ) { + return new jQuery.Event( src, props ); + } + + // Event object + if ( src && src.type ) { + this.originalEvent = src; + this.type = src.type; + + // Events bubbling up the document may have been marked as prevented + // by a handler lower down the tree; reflect the correct value. + this.isDefaultPrevented = src.defaultPrevented || + src.defaultPrevented === undefined && + + // Support: Android <=2.3 only + src.returnValue === false ? + returnTrue : + returnFalse; + + // Create target properties + // Support: Safari <=6 - 7 only + // Target should not be a text node (#504, #13143) + this.target = ( src.target && src.target.nodeType === 3 ) ? + src.target.parentNode : + src.target; + + this.currentTarget = src.currentTarget; + this.relatedTarget = src.relatedTarget; + + // Event type + } else { + this.type = src; + } + + // Put explicitly provided properties onto the event object + if ( props ) { + jQuery.extend( this, props ); + } + + // Create a timestamp if incoming event doesn't have one + this.timeStamp = src && src.timeStamp || Date.now(); + + // Mark it as fixed + this[ jQuery.expando ] = true; +}; + +// jQuery.Event is based on DOM3 Events as specified by the ECMAScript Language Binding +// https://www.w3.org/TR/2003/WD-DOM-Level-3-Events-20030331/ecma-script-binding.html +jQuery.Event.prototype = { + constructor: jQuery.Event, + isDefaultPrevented: returnFalse, + isPropagationStopped: returnFalse, + isImmediatePropagationStopped: returnFalse, + isSimulated: false, + + preventDefault: function() { + var e = this.originalEvent; + + this.isDefaultPrevented = returnTrue; + + if ( e && !this.isSimulated ) { + e.preventDefault(); + } + }, + stopPropagation: function() { + var e = this.originalEvent; + + this.isPropagationStopped = returnTrue; + + if ( e && !this.isSimulated ) { + e.stopPropagation(); + } + }, + stopImmediatePropagation: function() { + var e = this.originalEvent; + + this.isImmediatePropagationStopped = returnTrue; + + if ( e && !this.isSimulated ) { + e.stopImmediatePropagation(); + } + + this.stopPropagation(); + } +}; + +// Includes all common event props including KeyEvent and MouseEvent specific props +jQuery.each( { + altKey: true, + bubbles: true, + cancelable: true, + changedTouches: true, + ctrlKey: true, + detail: true, + eventPhase: true, + metaKey: true, + pageX: true, + pageY: true, + shiftKey: true, + view: true, + "char": true, + code: true, + charCode: true, + key: true, + keyCode: true, + button: true, + buttons: true, + clientX: true, + clientY: true, + offsetX: true, + offsetY: true, + pointerId: true, + pointerType: true, + screenX: true, + screenY: true, + targetTouches: true, + toElement: true, + touches: true, + + which: function( event ) { + var button = event.button; + + // Add which for key events + if ( event.which == null && rkeyEvent.test( event.type ) ) { + return event.charCode != null ? event.charCode : event.keyCode; + } + + // Add which for click: 1 === left; 2 === middle; 3 === right + if ( !event.which && button !== undefined && rmouseEvent.test( event.type ) ) { + if ( button & 1 ) { + return 1; + } + + if ( button & 2 ) { + return 3; + } + + if ( button & 4 ) { + return 2; + } + + return 0; + } + + return event.which; + } +}, jQuery.event.addProp ); + +jQuery.each( { focus: "focusin", blur: "focusout" }, function( type, delegateType ) { + jQuery.event.special[ type ] = { + + // Utilize native event if possible so blur/focus sequence is correct + setup: function() { + + // Claim the first handler + // dataPriv.set( this, "focus", ... ) + // dataPriv.set( this, "blur", ... ) + leverageNative( this, type, expectSync ); + + // Return false to allow normal processing in the caller + return false; + }, + trigger: function() { + + // Force setup before trigger + leverageNative( this, type ); + + // Return non-false to allow normal event-path propagation + return true; + }, + + delegateType: delegateType + }; +} ); + +// Create mouseenter/leave events using mouseover/out and event-time checks +// so that event delegation works in jQuery. +// Do the same for pointerenter/pointerleave and pointerover/pointerout +// +// Support: Safari 7 only +// Safari sends mouseenter too often; see: +// https://bugs.chromium.org/p/chromium/issues/detail?id=470258 +// for the description of the bug (it existed in older Chrome versions as well). +jQuery.each( { + mouseenter: "mouseover", + mouseleave: "mouseout", + pointerenter: "pointerover", + pointerleave: "pointerout" +}, function( orig, fix ) { + jQuery.event.special[ orig ] = { + delegateType: fix, + bindType: fix, + + handle: function( event ) { + var ret, + target = this, + related = event.relatedTarget, + handleObj = event.handleObj; + + // For mouseenter/leave call the handler if related is outside the target. + // NB: No relatedTarget if the mouse left/entered the browser window + if ( !related || ( related !== target && !jQuery.contains( target, related ) ) ) { + event.type = handleObj.origType; + ret = handleObj.handler.apply( this, arguments ); + event.type = fix; + } + return ret; + } + }; +} ); + +jQuery.fn.extend( { + + on: function( types, selector, data, fn ) { + return on( this, types, selector, data, fn ); + }, + one: function( types, selector, data, fn ) { + return on( this, types, selector, data, fn, 1 ); + }, + off: function( types, selector, fn ) { + var handleObj, type; + if ( types && types.preventDefault && types.handleObj ) { + + // ( event ) dispatched jQuery.Event + handleObj = types.handleObj; + jQuery( types.delegateTarget ).off( + handleObj.namespace ? + handleObj.origType + "." + handleObj.namespace : + handleObj.origType, + handleObj.selector, + handleObj.handler + ); + return this; + } + if ( typeof types === "object" ) { + + // ( types-object [, selector] ) + for ( type in types ) { + this.off( type, selector, types[ type ] ); + } + return this; + } + if ( selector === false || typeof selector === "function" ) { + + // ( types [, fn] ) + fn = selector; + selector = undefined; + } + if ( fn === false ) { + fn = returnFalse; + } + return this.each( function() { + jQuery.event.remove( this, types, fn, selector ); + } ); + } +} ); + + +var + + // Support: IE <=10 - 11, Edge 12 - 13 only + // In IE/Edge using regex groups here causes severe slowdowns. + // See https://connect.microsoft.com/IE/feedback/details/1736512/ + rnoInnerhtml = /\s*$/g; + +// Prefer a tbody over its parent table for containing new rows +function manipulationTarget( elem, content ) { + if ( nodeName( elem, "table" ) && + nodeName( content.nodeType !== 11 ? content : content.firstChild, "tr" ) ) { + + return jQuery( elem ).children( "tbody" )[ 0 ] || elem; + } + + return elem; +} + +// Replace/restore the type attribute of script elements for safe DOM manipulation +function disableScript( elem ) { + elem.type = ( elem.getAttribute( "type" ) !== null ) + "/" + elem.type; + return elem; +} +function restoreScript( elem ) { + if ( ( elem.type || "" ).slice( 0, 5 ) === "true/" ) { + elem.type = elem.type.slice( 5 ); + } else { + elem.removeAttribute( "type" ); + } + + return elem; +} + +function cloneCopyEvent( src, dest ) { + var i, l, type, pdataOld, udataOld, udataCur, events; + + if ( dest.nodeType !== 1 ) { + return; + } + + // 1. Copy private data: events, handlers, etc. + if ( dataPriv.hasData( src ) ) { + pdataOld = dataPriv.get( src ); + events = pdataOld.events; + + if ( events ) { + dataPriv.remove( dest, "handle events" ); + + for ( type in events ) { + for ( i = 0, l = events[ type ].length; i < l; i++ ) { + jQuery.event.add( dest, type, events[ type ][ i ] ); + } + } + } + } + + // 2. Copy user data + if ( dataUser.hasData( src ) ) { + udataOld = dataUser.access( src ); + udataCur = jQuery.extend( {}, udataOld ); + + dataUser.set( dest, udataCur ); + } +} + +// Fix IE bugs, see support tests +function fixInput( src, dest ) { + var nodeName = dest.nodeName.toLowerCase(); + + // Fails to persist the checked state of a cloned checkbox or radio button. + if ( nodeName === "input" && rcheckableType.test( src.type ) ) { + dest.checked = src.checked; + + // Fails to return the selected option to the default selected state when cloning options + } else if ( nodeName === "input" || nodeName === "textarea" ) { + dest.defaultValue = src.defaultValue; + } +} + +function domManip( collection, args, callback, ignored ) { + + // Flatten any nested arrays + args = flat( args ); + + var fragment, first, scripts, hasScripts, node, doc, + i = 0, + l = collection.length, + iNoClone = l - 1, + value = args[ 0 ], + valueIsFunction = isFunction( value ); + + // We can't cloneNode fragments that contain checked, in WebKit + if ( valueIsFunction || + ( l > 1 && typeof value === "string" && + !support.checkClone && rchecked.test( value ) ) ) { + return collection.each( function( index ) { + var self = collection.eq( index ); + if ( valueIsFunction ) { + args[ 0 ] = value.call( this, index, self.html() ); + } + domManip( self, args, callback, ignored ); + } ); + } + + if ( l ) { + fragment = buildFragment( args, collection[ 0 ].ownerDocument, false, collection, ignored ); + first = fragment.firstChild; + + if ( fragment.childNodes.length === 1 ) { + fragment = first; + } + + // Require either new content or an interest in ignored elements to invoke the callback + if ( first || ignored ) { + scripts = jQuery.map( getAll( fragment, "script" ), disableScript ); + hasScripts = scripts.length; + + // Use the original fragment for the last item + // instead of the first because it can end up + // being emptied incorrectly in certain situations (#8070). + for ( ; i < l; i++ ) { + node = fragment; + + if ( i !== iNoClone ) { + node = jQuery.clone( node, true, true ); + + // Keep references to cloned scripts for later restoration + if ( hasScripts ) { + + // Support: Android <=4.0 only, PhantomJS 1 only + // push.apply(_, arraylike) throws on ancient WebKit + jQuery.merge( scripts, getAll( node, "script" ) ); + } + } + + callback.call( collection[ i ], node, i ); + } + + if ( hasScripts ) { + doc = scripts[ scripts.length - 1 ].ownerDocument; + + // Reenable scripts + jQuery.map( scripts, restoreScript ); + + // Evaluate executable scripts on first document insertion + for ( i = 0; i < hasScripts; i++ ) { + node = scripts[ i ]; + if ( rscriptType.test( node.type || "" ) && + !dataPriv.access( node, "globalEval" ) && + jQuery.contains( doc, node ) ) { + + if ( node.src && ( node.type || "" ).toLowerCase() !== "module" ) { + + // Optional AJAX dependency, but won't run scripts if not present + if ( jQuery._evalUrl && !node.noModule ) { + jQuery._evalUrl( node.src, { + nonce: node.nonce || node.getAttribute( "nonce" ) + }, doc ); + } + } else { + DOMEval( node.textContent.replace( rcleanScript, "" ), node, doc ); + } + } + } + } + } + } + + return collection; +} + +function remove( elem, selector, keepData ) { + var node, + nodes = selector ? jQuery.filter( selector, elem ) : elem, + i = 0; + + for ( ; ( node = nodes[ i ] ) != null; i++ ) { + if ( !keepData && node.nodeType === 1 ) { + jQuery.cleanData( getAll( node ) ); + } + + if ( node.parentNode ) { + if ( keepData && isAttached( node ) ) { + setGlobalEval( getAll( node, "script" ) ); + } + node.parentNode.removeChild( node ); + } + } + + return elem; +} + +jQuery.extend( { + htmlPrefilter: function( html ) { + return html; + }, + + clone: function( elem, dataAndEvents, deepDataAndEvents ) { + var i, l, srcElements, destElements, + clone = elem.cloneNode( true ), + inPage = isAttached( elem ); + + // Fix IE cloning issues + if ( !support.noCloneChecked && ( elem.nodeType === 1 || elem.nodeType === 11 ) && + !jQuery.isXMLDoc( elem ) ) { + + // We eschew Sizzle here for performance reasons: https://jsperf.com/getall-vs-sizzle/2 + destElements = getAll( clone ); + srcElements = getAll( elem ); + + for ( i = 0, l = srcElements.length; i < l; i++ ) { + fixInput( srcElements[ i ], destElements[ i ] ); + } + } + + // Copy the events from the original to the clone + if ( dataAndEvents ) { + if ( deepDataAndEvents ) { + srcElements = srcElements || getAll( elem ); + destElements = destElements || getAll( clone ); + + for ( i = 0, l = srcElements.length; i < l; i++ ) { + cloneCopyEvent( srcElements[ i ], destElements[ i ] ); + } + } else { + cloneCopyEvent( elem, clone ); + } + } + + // Preserve script evaluation history + destElements = getAll( clone, "script" ); + if ( destElements.length > 0 ) { + setGlobalEval( destElements, !inPage && getAll( elem, "script" ) ); + } + + // Return the cloned set + return clone; + }, + + cleanData: function( elems ) { + var data, elem, type, + special = jQuery.event.special, + i = 0; + + for ( ; ( elem = elems[ i ] ) !== undefined; i++ ) { + if ( acceptData( elem ) ) { + if ( ( data = elem[ dataPriv.expando ] ) ) { + if ( data.events ) { + for ( type in data.events ) { + if ( special[ type ] ) { + jQuery.event.remove( elem, type ); + + // This is a shortcut to avoid jQuery.event.remove's overhead + } else { + jQuery.removeEvent( elem, type, data.handle ); + } + } + } + + // Support: Chrome <=35 - 45+ + // Assign undefined instead of using delete, see Data#remove + elem[ dataPriv.expando ] = undefined; + } + if ( elem[ dataUser.expando ] ) { + + // Support: Chrome <=35 - 45+ + // Assign undefined instead of using delete, see Data#remove + elem[ dataUser.expando ] = undefined; + } + } + } + } +} ); + +jQuery.fn.extend( { + detach: function( selector ) { + return remove( this, selector, true ); + }, + + remove: function( selector ) { + return remove( this, selector ); + }, + + text: function( value ) { + return access( this, function( value ) { + return value === undefined ? + jQuery.text( this ) : + this.empty().each( function() { + if ( this.nodeType === 1 || this.nodeType === 11 || this.nodeType === 9 ) { + this.textContent = value; + } + } ); + }, null, value, arguments.length ); + }, + + append: function() { + return domManip( this, arguments, function( elem ) { + if ( this.nodeType === 1 || this.nodeType === 11 || this.nodeType === 9 ) { + var target = manipulationTarget( this, elem ); + target.appendChild( elem ); + } + } ); + }, + + prepend: function() { + return domManip( this, arguments, function( elem ) { + if ( this.nodeType === 1 || this.nodeType === 11 || this.nodeType === 9 ) { + var target = manipulationTarget( this, elem ); + target.insertBefore( elem, target.firstChild ); + } + } ); + }, + + before: function() { + return domManip( this, arguments, function( elem ) { + if ( this.parentNode ) { + this.parentNode.insertBefore( elem, this ); + } + } ); + }, + + after: function() { + return domManip( this, arguments, function( elem ) { + if ( this.parentNode ) { + this.parentNode.insertBefore( elem, this.nextSibling ); + } + } ); + }, + + empty: function() { + var elem, + i = 0; + + for ( ; ( elem = this[ i ] ) != null; i++ ) { + if ( elem.nodeType === 1 ) { + + // Prevent memory leaks + jQuery.cleanData( getAll( elem, false ) ); + + // Remove any remaining nodes + elem.textContent = ""; + } + } + + return this; + }, + + clone: function( dataAndEvents, deepDataAndEvents ) { + dataAndEvents = dataAndEvents == null ? false : dataAndEvents; + deepDataAndEvents = deepDataAndEvents == null ? dataAndEvents : deepDataAndEvents; + + return this.map( function() { + return jQuery.clone( this, dataAndEvents, deepDataAndEvents ); + } ); + }, + + html: function( value ) { + return access( this, function( value ) { + var elem = this[ 0 ] || {}, + i = 0, + l = this.length; + + if ( value === undefined && elem.nodeType === 1 ) { + return elem.innerHTML; + } + + // See if we can take a shortcut and just use innerHTML + if ( typeof value === "string" && !rnoInnerhtml.test( value ) && + !wrapMap[ ( rtagName.exec( value ) || [ "", "" ] )[ 1 ].toLowerCase() ] ) { + + value = jQuery.htmlPrefilter( value ); + + try { + for ( ; i < l; i++ ) { + elem = this[ i ] || {}; + + // Remove element nodes and prevent memory leaks + if ( elem.nodeType === 1 ) { + jQuery.cleanData( getAll( elem, false ) ); + elem.innerHTML = value; + } + } + + elem = 0; + + // If using innerHTML throws an exception, use the fallback method + } catch ( e ) {} + } + + if ( elem ) { + this.empty().append( value ); + } + }, null, value, arguments.length ); + }, + + replaceWith: function() { + var ignored = []; + + // Make the changes, replacing each non-ignored context element with the new content + return domManip( this, arguments, function( elem ) { + var parent = this.parentNode; + + if ( jQuery.inArray( this, ignored ) < 0 ) { + jQuery.cleanData( getAll( this ) ); + if ( parent ) { + parent.replaceChild( elem, this ); + } + } + + // Force callback invocation + }, ignored ); + } +} ); + +jQuery.each( { + appendTo: "append", + prependTo: "prepend", + insertBefore: "before", + insertAfter: "after", + replaceAll: "replaceWith" +}, function( name, original ) { + jQuery.fn[ name ] = function( selector ) { + var elems, + ret = [], + insert = jQuery( selector ), + last = insert.length - 1, + i = 0; + + for ( ; i <= last; i++ ) { + elems = i === last ? this : this.clone( true ); + jQuery( insert[ i ] )[ original ]( elems ); + + // Support: Android <=4.0 only, PhantomJS 1 only + // .get() because push.apply(_, arraylike) throws on ancient WebKit + push.apply( ret, elems.get() ); + } + + return this.pushStack( ret ); + }; +} ); +var rnumnonpx = new RegExp( "^(" + pnum + ")(?!px)[a-z%]+$", "i" ); + +var getStyles = function( elem ) { + + // Support: IE <=11 only, Firefox <=30 (#15098, #14150) + // IE throws on elements created in popups + // FF meanwhile throws on frame elements through "defaultView.getComputedStyle" + var view = elem.ownerDocument.defaultView; + + if ( !view || !view.opener ) { + view = window; + } + + return view.getComputedStyle( elem ); + }; + +var swap = function( elem, options, callback ) { + var ret, name, + old = {}; + + // Remember the old values, and insert the new ones + for ( name in options ) { + old[ name ] = elem.style[ name ]; + elem.style[ name ] = options[ name ]; + } + + ret = callback.call( elem ); + + // Revert the old values + for ( name in options ) { + elem.style[ name ] = old[ name ]; + } + + return ret; +}; + + +var rboxStyle = new RegExp( cssExpand.join( "|" ), "i" ); + + + +( function() { + + // Executing both pixelPosition & boxSizingReliable tests require only one layout + // so they're executed at the same time to save the second computation. + function computeStyleTests() { + + // This is a singleton, we need to execute it only once + if ( !div ) { + return; + } + + container.style.cssText = "position:absolute;left:-11111px;width:60px;" + + "margin-top:1px;padding:0;border:0"; + div.style.cssText = + "position:relative;display:block;box-sizing:border-box;overflow:scroll;" + + "margin:auto;border:1px;padding:1px;" + + "width:60%;top:1%"; + documentElement.appendChild( container ).appendChild( div ); + + var divStyle = window.getComputedStyle( div ); + pixelPositionVal = divStyle.top !== "1%"; + + // Support: Android 4.0 - 4.3 only, Firefox <=3 - 44 + reliableMarginLeftVal = roundPixelMeasures( divStyle.marginLeft ) === 12; + + // Support: Android 4.0 - 4.3 only, Safari <=9.1 - 10.1, iOS <=7.0 - 9.3 + // Some styles come back with percentage values, even though they shouldn't + div.style.right = "60%"; + pixelBoxStylesVal = roundPixelMeasures( divStyle.right ) === 36; + + // Support: IE 9 - 11 only + // Detect misreporting of content dimensions for box-sizing:border-box elements + boxSizingReliableVal = roundPixelMeasures( divStyle.width ) === 36; + + // Support: IE 9 only + // Detect overflow:scroll screwiness (gh-3699) + // Support: Chrome <=64 + // Don't get tricked when zoom affects offsetWidth (gh-4029) + div.style.position = "absolute"; + scrollboxSizeVal = roundPixelMeasures( div.offsetWidth / 3 ) === 12; + + documentElement.removeChild( container ); + + // Nullify the div so it wouldn't be stored in the memory and + // it will also be a sign that checks already performed + div = null; + } + + function roundPixelMeasures( measure ) { + return Math.round( parseFloat( measure ) ); + } + + var pixelPositionVal, boxSizingReliableVal, scrollboxSizeVal, pixelBoxStylesVal, + reliableTrDimensionsVal, reliableMarginLeftVal, + container = document.createElement( "div" ), + div = document.createElement( "div" ); + + // Finish early in limited (non-browser) environments + if ( !div.style ) { + return; + } + + // Support: IE <=9 - 11 only + // Style of cloned element affects source element cloned (#8908) + div.style.backgroundClip = "content-box"; + div.cloneNode( true ).style.backgroundClip = ""; + support.clearCloneStyle = div.style.backgroundClip === "content-box"; + + jQuery.extend( support, { + boxSizingReliable: function() { + computeStyleTests(); + return boxSizingReliableVal; + }, + pixelBoxStyles: function() { + computeStyleTests(); + return pixelBoxStylesVal; + }, + pixelPosition: function() { + computeStyleTests(); + return pixelPositionVal; + }, + reliableMarginLeft: function() { + computeStyleTests(); + return reliableMarginLeftVal; + }, + scrollboxSize: function() { + computeStyleTests(); + return scrollboxSizeVal; + }, + + // Support: IE 9 - 11+, Edge 15 - 18+ + // IE/Edge misreport `getComputedStyle` of table rows with width/height + // set in CSS while `offset*` properties report correct values. + // Behavior in IE 9 is more subtle than in newer versions & it passes + // some versions of this test; make sure not to make it pass there! + reliableTrDimensions: function() { + var table, tr, trChild, trStyle; + if ( reliableTrDimensionsVal == null ) { + table = document.createElement( "table" ); + tr = document.createElement( "tr" ); + trChild = document.createElement( "div" ); + + table.style.cssText = "position:absolute;left:-11111px"; + tr.style.height = "1px"; + trChild.style.height = "9px"; + + documentElement + .appendChild( table ) + .appendChild( tr ) + .appendChild( trChild ); + + trStyle = window.getComputedStyle( tr ); + reliableTrDimensionsVal = parseInt( trStyle.height ) > 3; + + documentElement.removeChild( table ); + } + return reliableTrDimensionsVal; + } + } ); +} )(); + + +function curCSS( elem, name, computed ) { + var width, minWidth, maxWidth, ret, + + // Support: Firefox 51+ + // Retrieving style before computed somehow + // fixes an issue with getting wrong values + // on detached elements + style = elem.style; + + computed = computed || getStyles( elem ); + + // getPropertyValue is needed for: + // .css('filter') (IE 9 only, #12537) + // .css('--customProperty) (#3144) + if ( computed ) { + ret = computed.getPropertyValue( name ) || computed[ name ]; + + if ( ret === "" && !isAttached( elem ) ) { + ret = jQuery.style( elem, name ); + } + + // A tribute to the "awesome hack by Dean Edwards" + // Android Browser returns percentage for some values, + // but width seems to be reliably pixels. + // This is against the CSSOM draft spec: + // https://drafts.csswg.org/cssom/#resolved-values + if ( !support.pixelBoxStyles() && rnumnonpx.test( ret ) && rboxStyle.test( name ) ) { + + // Remember the original values + width = style.width; + minWidth = style.minWidth; + maxWidth = style.maxWidth; + + // Put in the new values to get a computed value out + style.minWidth = style.maxWidth = style.width = ret; + ret = computed.width; + + // Revert the changed values + style.width = width; + style.minWidth = minWidth; + style.maxWidth = maxWidth; + } + } + + return ret !== undefined ? + + // Support: IE <=9 - 11 only + // IE returns zIndex value as an integer. + ret + "" : + ret; +} + + +function addGetHookIf( conditionFn, hookFn ) { + + // Define the hook, we'll check on the first run if it's really needed. + return { + get: function() { + if ( conditionFn() ) { + + // Hook not needed (or it's not possible to use it due + // to missing dependency), remove it. + delete this.get; + return; + } + + // Hook needed; redefine it so that the support test is not executed again. + return ( this.get = hookFn ).apply( this, arguments ); + } + }; +} + + +var cssPrefixes = [ "Webkit", "Moz", "ms" ], + emptyStyle = document.createElement( "div" ).style, + vendorProps = {}; + +// Return a vendor-prefixed property or undefined +function vendorPropName( name ) { + + // Check for vendor prefixed names + var capName = name[ 0 ].toUpperCase() + name.slice( 1 ), + i = cssPrefixes.length; + + while ( i-- ) { + name = cssPrefixes[ i ] + capName; + if ( name in emptyStyle ) { + return name; + } + } +} + +// Return a potentially-mapped jQuery.cssProps or vendor prefixed property +function finalPropName( name ) { + var final = jQuery.cssProps[ name ] || vendorProps[ name ]; + + if ( final ) { + return final; + } + if ( name in emptyStyle ) { + return name; + } + return vendorProps[ name ] = vendorPropName( name ) || name; +} + + +var + + // Swappable if display is none or starts with table + // except "table", "table-cell", or "table-caption" + // See here for display values: https://developer.mozilla.org/en-US/docs/CSS/display + rdisplayswap = /^(none|table(?!-c[ea]).+)/, + rcustomProp = /^--/, + cssShow = { position: "absolute", visibility: "hidden", display: "block" }, + cssNormalTransform = { + letterSpacing: "0", + fontWeight: "400" + }; + +function setPositiveNumber( _elem, value, subtract ) { + + // Any relative (+/-) values have already been + // normalized at this point + var matches = rcssNum.exec( value ); + return matches ? + + // Guard against undefined "subtract", e.g., when used as in cssHooks + Math.max( 0, matches[ 2 ] - ( subtract || 0 ) ) + ( matches[ 3 ] || "px" ) : + value; +} + +function boxModelAdjustment( elem, dimension, box, isBorderBox, styles, computedVal ) { + var i = dimension === "width" ? 1 : 0, + extra = 0, + delta = 0; + + // Adjustment may not be necessary + if ( box === ( isBorderBox ? "border" : "content" ) ) { + return 0; + } + + for ( ; i < 4; i += 2 ) { + + // Both box models exclude margin + if ( box === "margin" ) { + delta += jQuery.css( elem, box + cssExpand[ i ], true, styles ); + } + + // If we get here with a content-box, we're seeking "padding" or "border" or "margin" + if ( !isBorderBox ) { + + // Add padding + delta += jQuery.css( elem, "padding" + cssExpand[ i ], true, styles ); + + // For "border" or "margin", add border + if ( box !== "padding" ) { + delta += jQuery.css( elem, "border" + cssExpand[ i ] + "Width", true, styles ); + + // But still keep track of it otherwise + } else { + extra += jQuery.css( elem, "border" + cssExpand[ i ] + "Width", true, styles ); + } + + // If we get here with a border-box (content + padding + border), we're seeking "content" or + // "padding" or "margin" + } else { + + // For "content", subtract padding + if ( box === "content" ) { + delta -= jQuery.css( elem, "padding" + cssExpand[ i ], true, styles ); + } + + // For "content" or "padding", subtract border + if ( box !== "margin" ) { + delta -= jQuery.css( elem, "border" + cssExpand[ i ] + "Width", true, styles ); + } + } + } + + // Account for positive content-box scroll gutter when requested by providing computedVal + if ( !isBorderBox && computedVal >= 0 ) { + + // offsetWidth/offsetHeight is a rounded sum of content, padding, scroll gutter, and border + // Assuming integer scroll gutter, subtract the rest and round down + delta += Math.max( 0, Math.ceil( + elem[ "offset" + dimension[ 0 ].toUpperCase() + dimension.slice( 1 ) ] - + computedVal - + delta - + extra - + 0.5 + + // If offsetWidth/offsetHeight is unknown, then we can't determine content-box scroll gutter + // Use an explicit zero to avoid NaN (gh-3964) + ) ) || 0; + } + + return delta; +} + +function getWidthOrHeight( elem, dimension, extra ) { + + // Start with computed style + var styles = getStyles( elem ), + + // To avoid forcing a reflow, only fetch boxSizing if we need it (gh-4322). + // Fake content-box until we know it's needed to know the true value. + boxSizingNeeded = !support.boxSizingReliable() || extra, + isBorderBox = boxSizingNeeded && + jQuery.css( elem, "boxSizing", false, styles ) === "border-box", + valueIsBorderBox = isBorderBox, + + val = curCSS( elem, dimension, styles ), + offsetProp = "offset" + dimension[ 0 ].toUpperCase() + dimension.slice( 1 ); + + // Support: Firefox <=54 + // Return a confounding non-pixel value or feign ignorance, as appropriate. + if ( rnumnonpx.test( val ) ) { + if ( !extra ) { + return val; + } + val = "auto"; + } + + + // Support: IE 9 - 11 only + // Use offsetWidth/offsetHeight for when box sizing is unreliable. + // In those cases, the computed value can be trusted to be border-box. + if ( ( !support.boxSizingReliable() && isBorderBox || + + // Support: IE 10 - 11+, Edge 15 - 18+ + // IE/Edge misreport `getComputedStyle` of table rows with width/height + // set in CSS while `offset*` properties report correct values. + // Interestingly, in some cases IE 9 doesn't suffer from this issue. + !support.reliableTrDimensions() && nodeName( elem, "tr" ) || + + // Fall back to offsetWidth/offsetHeight when value is "auto" + // This happens for inline elements with no explicit setting (gh-3571) + val === "auto" || + + // Support: Android <=4.1 - 4.3 only + // Also use offsetWidth/offsetHeight for misreported inline dimensions (gh-3602) + !parseFloat( val ) && jQuery.css( elem, "display", false, styles ) === "inline" ) && + + // Make sure the element is visible & connected + elem.getClientRects().length ) { + + isBorderBox = jQuery.css( elem, "boxSizing", false, styles ) === "border-box"; + + // Where available, offsetWidth/offsetHeight approximate border box dimensions. + // Where not available (e.g., SVG), assume unreliable box-sizing and interpret the + // retrieved value as a content box dimension. + valueIsBorderBox = offsetProp in elem; + if ( valueIsBorderBox ) { + val = elem[ offsetProp ]; + } + } + + // Normalize "" and auto + val = parseFloat( val ) || 0; + + // Adjust for the element's box model + return ( val + + boxModelAdjustment( + elem, + dimension, + extra || ( isBorderBox ? "border" : "content" ), + valueIsBorderBox, + styles, + + // Provide the current computed size to request scroll gutter calculation (gh-3589) + val + ) + ) + "px"; +} + +jQuery.extend( { + + // Add in style property hooks for overriding the default + // behavior of getting and setting a style property + cssHooks: { + opacity: { + get: function( elem, computed ) { + if ( computed ) { + + // We should always get a number back from opacity + var ret = curCSS( elem, "opacity" ); + return ret === "" ? "1" : ret; + } + } + } + }, + + // Don't automatically add "px" to these possibly-unitless properties + cssNumber: { + "animationIterationCount": true, + "columnCount": true, + "fillOpacity": true, + "flexGrow": true, + "flexShrink": true, + "fontWeight": true, + "gridArea": true, + "gridColumn": true, + "gridColumnEnd": true, + "gridColumnStart": true, + "gridRow": true, + "gridRowEnd": true, + "gridRowStart": true, + "lineHeight": true, + "opacity": true, + "order": true, + "orphans": true, + "widows": true, + "zIndex": true, + "zoom": true + }, + + // Add in properties whose names you wish to fix before + // setting or getting the value + cssProps: {}, + + // Get and set the style property on a DOM Node + style: function( elem, name, value, extra ) { + + // Don't set styles on text and comment nodes + if ( !elem || elem.nodeType === 3 || elem.nodeType === 8 || !elem.style ) { + return; + } + + // Make sure that we're working with the right name + var ret, type, hooks, + origName = camelCase( name ), + isCustomProp = rcustomProp.test( name ), + style = elem.style; + + // Make sure that we're working with the right name. We don't + // want to query the value if it is a CSS custom property + // since they are user-defined. + if ( !isCustomProp ) { + name = finalPropName( origName ); + } + + // Gets hook for the prefixed version, then unprefixed version + hooks = jQuery.cssHooks[ name ] || jQuery.cssHooks[ origName ]; + + // Check if we're setting a value + if ( value !== undefined ) { + type = typeof value; + + // Convert "+=" or "-=" to relative numbers (#7345) + if ( type === "string" && ( ret = rcssNum.exec( value ) ) && ret[ 1 ] ) { + value = adjustCSS( elem, name, ret ); + + // Fixes bug #9237 + type = "number"; + } + + // Make sure that null and NaN values aren't set (#7116) + if ( value == null || value !== value ) { + return; + } + + // If a number was passed in, add the unit (except for certain CSS properties) + // The isCustomProp check can be removed in jQuery 4.0 when we only auto-append + // "px" to a few hardcoded values. + if ( type === "number" && !isCustomProp ) { + value += ret && ret[ 3 ] || ( jQuery.cssNumber[ origName ] ? "" : "px" ); + } + + // background-* props affect original clone's values + if ( !support.clearCloneStyle && value === "" && name.indexOf( "background" ) === 0 ) { + style[ name ] = "inherit"; + } + + // If a hook was provided, use that value, otherwise just set the specified value + if ( !hooks || !( "set" in hooks ) || + ( value = hooks.set( elem, value, extra ) ) !== undefined ) { + + if ( isCustomProp ) { + style.setProperty( name, value ); + } else { + style[ name ] = value; + } + } + + } else { + + // If a hook was provided get the non-computed value from there + if ( hooks && "get" in hooks && + ( ret = hooks.get( elem, false, extra ) ) !== undefined ) { + + return ret; + } + + // Otherwise just get the value from the style object + return style[ name ]; + } + }, + + css: function( elem, name, extra, styles ) { + var val, num, hooks, + origName = camelCase( name ), + isCustomProp = rcustomProp.test( name ); + + // Make sure that we're working with the right name. We don't + // want to modify the value if it is a CSS custom property + // since they are user-defined. + if ( !isCustomProp ) { + name = finalPropName( origName ); + } + + // Try prefixed name followed by the unprefixed name + hooks = jQuery.cssHooks[ name ] || jQuery.cssHooks[ origName ]; + + // If a hook was provided get the computed value from there + if ( hooks && "get" in hooks ) { + val = hooks.get( elem, true, extra ); + } + + // Otherwise, if a way to get the computed value exists, use that + if ( val === undefined ) { + val = curCSS( elem, name, styles ); + } + + // Convert "normal" to computed value + if ( val === "normal" && name in cssNormalTransform ) { + val = cssNormalTransform[ name ]; + } + + // Make numeric if forced or a qualifier was provided and val looks numeric + if ( extra === "" || extra ) { + num = parseFloat( val ); + return extra === true || isFinite( num ) ? num || 0 : val; + } + + return val; + } +} ); + +jQuery.each( [ "height", "width" ], function( _i, dimension ) { + jQuery.cssHooks[ dimension ] = { + get: function( elem, computed, extra ) { + if ( computed ) { + + // Certain elements can have dimension info if we invisibly show them + // but it must have a current display style that would benefit + return rdisplayswap.test( jQuery.css( elem, "display" ) ) && + + // Support: Safari 8+ + // Table columns in Safari have non-zero offsetWidth & zero + // getBoundingClientRect().width unless display is changed. + // Support: IE <=11 only + // Running getBoundingClientRect on a disconnected node + // in IE throws an error. + ( !elem.getClientRects().length || !elem.getBoundingClientRect().width ) ? + swap( elem, cssShow, function() { + return getWidthOrHeight( elem, dimension, extra ); + } ) : + getWidthOrHeight( elem, dimension, extra ); + } + }, + + set: function( elem, value, extra ) { + var matches, + styles = getStyles( elem ), + + // Only read styles.position if the test has a chance to fail + // to avoid forcing a reflow. + scrollboxSizeBuggy = !support.scrollboxSize() && + styles.position === "absolute", + + // To avoid forcing a reflow, only fetch boxSizing if we need it (gh-3991) + boxSizingNeeded = scrollboxSizeBuggy || extra, + isBorderBox = boxSizingNeeded && + jQuery.css( elem, "boxSizing", false, styles ) === "border-box", + subtract = extra ? + boxModelAdjustment( + elem, + dimension, + extra, + isBorderBox, + styles + ) : + 0; + + // Account for unreliable border-box dimensions by comparing offset* to computed and + // faking a content-box to get border and padding (gh-3699) + if ( isBorderBox && scrollboxSizeBuggy ) { + subtract -= Math.ceil( + elem[ "offset" + dimension[ 0 ].toUpperCase() + dimension.slice( 1 ) ] - + parseFloat( styles[ dimension ] ) - + boxModelAdjustment( elem, dimension, "border", false, styles ) - + 0.5 + ); + } + + // Convert to pixels if value adjustment is needed + if ( subtract && ( matches = rcssNum.exec( value ) ) && + ( matches[ 3 ] || "px" ) !== "px" ) { + + elem.style[ dimension ] = value; + value = jQuery.css( elem, dimension ); + } + + return setPositiveNumber( elem, value, subtract ); + } + }; +} ); + +jQuery.cssHooks.marginLeft = addGetHookIf( support.reliableMarginLeft, + function( elem, computed ) { + if ( computed ) { + return ( parseFloat( curCSS( elem, "marginLeft" ) ) || + elem.getBoundingClientRect().left - + swap( elem, { marginLeft: 0 }, function() { + return elem.getBoundingClientRect().left; + } ) + ) + "px"; + } + } +); + +// These hooks are used by animate to expand properties +jQuery.each( { + margin: "", + padding: "", + border: "Width" +}, function( prefix, suffix ) { + jQuery.cssHooks[ prefix + suffix ] = { + expand: function( value ) { + var i = 0, + expanded = {}, + + // Assumes a single number if not a string + parts = typeof value === "string" ? value.split( " " ) : [ value ]; + + for ( ; i < 4; i++ ) { + expanded[ prefix + cssExpand[ i ] + suffix ] = + parts[ i ] || parts[ i - 2 ] || parts[ 0 ]; + } + + return expanded; + } + }; + + if ( prefix !== "margin" ) { + jQuery.cssHooks[ prefix + suffix ].set = setPositiveNumber; + } +} ); + +jQuery.fn.extend( { + css: function( name, value ) { + return access( this, function( elem, name, value ) { + var styles, len, + map = {}, + i = 0; + + if ( Array.isArray( name ) ) { + styles = getStyles( elem ); + len = name.length; + + for ( ; i < len; i++ ) { + map[ name[ i ] ] = jQuery.css( elem, name[ i ], false, styles ); + } + + return map; + } + + return value !== undefined ? + jQuery.style( elem, name, value ) : + jQuery.css( elem, name ); + }, name, value, arguments.length > 1 ); + } +} ); + + +function Tween( elem, options, prop, end, easing ) { + return new Tween.prototype.init( elem, options, prop, end, easing ); +} +jQuery.Tween = Tween; + +Tween.prototype = { + constructor: Tween, + init: function( elem, options, prop, end, easing, unit ) { + this.elem = elem; + this.prop = prop; + this.easing = easing || jQuery.easing._default; + this.options = options; + this.start = this.now = this.cur(); + this.end = end; + this.unit = unit || ( jQuery.cssNumber[ prop ] ? "" : "px" ); + }, + cur: function() { + var hooks = Tween.propHooks[ this.prop ]; + + return hooks && hooks.get ? + hooks.get( this ) : + Tween.propHooks._default.get( this ); + }, + run: function( percent ) { + var eased, + hooks = Tween.propHooks[ this.prop ]; + + if ( this.options.duration ) { + this.pos = eased = jQuery.easing[ this.easing ]( + percent, this.options.duration * percent, 0, 1, this.options.duration + ); + } else { + this.pos = eased = percent; + } + this.now = ( this.end - this.start ) * eased + this.start; + + if ( this.options.step ) { + this.options.step.call( this.elem, this.now, this ); + } + + if ( hooks && hooks.set ) { + hooks.set( this ); + } else { + Tween.propHooks._default.set( this ); + } + return this; + } +}; + +Tween.prototype.init.prototype = Tween.prototype; + +Tween.propHooks = { + _default: { + get: function( tween ) { + var result; + + // Use a property on the element directly when it is not a DOM element, + // or when there is no matching style property that exists. + if ( tween.elem.nodeType !== 1 || + tween.elem[ tween.prop ] != null && tween.elem.style[ tween.prop ] == null ) { + return tween.elem[ tween.prop ]; + } + + // Passing an empty string as a 3rd parameter to .css will automatically + // attempt a parseFloat and fallback to a string if the parse fails. + // Simple values such as "10px" are parsed to Float; + // complex values such as "rotate(1rad)" are returned as-is. + result = jQuery.css( tween.elem, tween.prop, "" ); + + // Empty strings, null, undefined and "auto" are converted to 0. + return !result || result === "auto" ? 0 : result; + }, + set: function( tween ) { + + // Use step hook for back compat. + // Use cssHook if its there. + // Use .style if available and use plain properties where available. + if ( jQuery.fx.step[ tween.prop ] ) { + jQuery.fx.step[ tween.prop ]( tween ); + } else if ( tween.elem.nodeType === 1 && ( + jQuery.cssHooks[ tween.prop ] || + tween.elem.style[ finalPropName( tween.prop ) ] != null ) ) { + jQuery.style( tween.elem, tween.prop, tween.now + tween.unit ); + } else { + tween.elem[ tween.prop ] = tween.now; + } + } + } +}; + +// Support: IE <=9 only +// Panic based approach to setting things on disconnected nodes +Tween.propHooks.scrollTop = Tween.propHooks.scrollLeft = { + set: function( tween ) { + if ( tween.elem.nodeType && tween.elem.parentNode ) { + tween.elem[ tween.prop ] = tween.now; + } + } +}; + +jQuery.easing = { + linear: function( p ) { + return p; + }, + swing: function( p ) { + return 0.5 - Math.cos( p * Math.PI ) / 2; + }, + _default: "swing" +}; + +jQuery.fx = Tween.prototype.init; + +// Back compat <1.8 extension point +jQuery.fx.step = {}; + + + + +var + fxNow, inProgress, + rfxtypes = /^(?:toggle|show|hide)$/, + rrun = /queueHooks$/; + +function schedule() { + if ( inProgress ) { + if ( document.hidden === false && window.requestAnimationFrame ) { + window.requestAnimationFrame( schedule ); + } else { + window.setTimeout( schedule, jQuery.fx.interval ); + } + + jQuery.fx.tick(); + } +} + +// Animations created synchronously will run synchronously +function createFxNow() { + window.setTimeout( function() { + fxNow = undefined; + } ); + return ( fxNow = Date.now() ); +} + +// Generate parameters to create a standard animation +function genFx( type, includeWidth ) { + var which, + i = 0, + attrs = { height: type }; + + // If we include width, step value is 1 to do all cssExpand values, + // otherwise step value is 2 to skip over Left and Right + includeWidth = includeWidth ? 1 : 0; + for ( ; i < 4; i += 2 - includeWidth ) { + which = cssExpand[ i ]; + attrs[ "margin" + which ] = attrs[ "padding" + which ] = type; + } + + if ( includeWidth ) { + attrs.opacity = attrs.width = type; + } + + return attrs; +} + +function createTween( value, prop, animation ) { + var tween, + collection = ( Animation.tweeners[ prop ] || [] ).concat( Animation.tweeners[ "*" ] ), + index = 0, + length = collection.length; + for ( ; index < length; index++ ) { + if ( ( tween = collection[ index ].call( animation, prop, value ) ) ) { + + // We're done with this property + return tween; + } + } +} + +function defaultPrefilter( elem, props, opts ) { + var prop, value, toggle, hooks, oldfire, propTween, restoreDisplay, display, + isBox = "width" in props || "height" in props, + anim = this, + orig = {}, + style = elem.style, + hidden = elem.nodeType && isHiddenWithinTree( elem ), + dataShow = dataPriv.get( elem, "fxshow" ); + + // Queue-skipping animations hijack the fx hooks + if ( !opts.queue ) { + hooks = jQuery._queueHooks( elem, "fx" ); + if ( hooks.unqueued == null ) { + hooks.unqueued = 0; + oldfire = hooks.empty.fire; + hooks.empty.fire = function() { + if ( !hooks.unqueued ) { + oldfire(); + } + }; + } + hooks.unqueued++; + + anim.always( function() { + + // Ensure the complete handler is called before this completes + anim.always( function() { + hooks.unqueued--; + if ( !jQuery.queue( elem, "fx" ).length ) { + hooks.empty.fire(); + } + } ); + } ); + } + + // Detect show/hide animations + for ( prop in props ) { + value = props[ prop ]; + if ( rfxtypes.test( value ) ) { + delete props[ prop ]; + toggle = toggle || value === "toggle"; + if ( value === ( hidden ? "hide" : "show" ) ) { + + // Pretend to be hidden if this is a "show" and + // there is still data from a stopped show/hide + if ( value === "show" && dataShow && dataShow[ prop ] !== undefined ) { + hidden = true; + + // Ignore all other no-op show/hide data + } else { + continue; + } + } + orig[ prop ] = dataShow && dataShow[ prop ] || jQuery.style( elem, prop ); + } + } + + // Bail out if this is a no-op like .hide().hide() + propTween = !jQuery.isEmptyObject( props ); + if ( !propTween && jQuery.isEmptyObject( orig ) ) { + return; + } + + // Restrict "overflow" and "display" styles during box animations + if ( isBox && elem.nodeType === 1 ) { + + // Support: IE <=9 - 11, Edge 12 - 15 + // Record all 3 overflow attributes because IE does not infer the shorthand + // from identically-valued overflowX and overflowY and Edge just mirrors + // the overflowX value there. + opts.overflow = [ style.overflow, style.overflowX, style.overflowY ]; + + // Identify a display type, preferring old show/hide data over the CSS cascade + restoreDisplay = dataShow && dataShow.display; + if ( restoreDisplay == null ) { + restoreDisplay = dataPriv.get( elem, "display" ); + } + display = jQuery.css( elem, "display" ); + if ( display === "none" ) { + if ( restoreDisplay ) { + display = restoreDisplay; + } else { + + // Get nonempty value(s) by temporarily forcing visibility + showHide( [ elem ], true ); + restoreDisplay = elem.style.display || restoreDisplay; + display = jQuery.css( elem, "display" ); + showHide( [ elem ] ); + } + } + + // Animate inline elements as inline-block + if ( display === "inline" || display === "inline-block" && restoreDisplay != null ) { + if ( jQuery.css( elem, "float" ) === "none" ) { + + // Restore the original display value at the end of pure show/hide animations + if ( !propTween ) { + anim.done( function() { + style.display = restoreDisplay; + } ); + if ( restoreDisplay == null ) { + display = style.display; + restoreDisplay = display === "none" ? "" : display; + } + } + style.display = "inline-block"; + } + } + } + + if ( opts.overflow ) { + style.overflow = "hidden"; + anim.always( function() { + style.overflow = opts.overflow[ 0 ]; + style.overflowX = opts.overflow[ 1 ]; + style.overflowY = opts.overflow[ 2 ]; + } ); + } + + // Implement show/hide animations + propTween = false; + for ( prop in orig ) { + + // General show/hide setup for this element animation + if ( !propTween ) { + if ( dataShow ) { + if ( "hidden" in dataShow ) { + hidden = dataShow.hidden; + } + } else { + dataShow = dataPriv.access( elem, "fxshow", { display: restoreDisplay } ); + } + + // Store hidden/visible for toggle so `.stop().toggle()` "reverses" + if ( toggle ) { + dataShow.hidden = !hidden; + } + + // Show elements before animating them + if ( hidden ) { + showHide( [ elem ], true ); + } + + /* eslint-disable no-loop-func */ + + anim.done( function() { + + /* eslint-enable no-loop-func */ + + // The final step of a "hide" animation is actually hiding the element + if ( !hidden ) { + showHide( [ elem ] ); + } + dataPriv.remove( elem, "fxshow" ); + for ( prop in orig ) { + jQuery.style( elem, prop, orig[ prop ] ); + } + } ); + } + + // Per-property setup + propTween = createTween( hidden ? dataShow[ prop ] : 0, prop, anim ); + if ( !( prop in dataShow ) ) { + dataShow[ prop ] = propTween.start; + if ( hidden ) { + propTween.end = propTween.start; + propTween.start = 0; + } + } + } +} + +function propFilter( props, specialEasing ) { + var index, name, easing, value, hooks; + + // camelCase, specialEasing and expand cssHook pass + for ( index in props ) { + name = camelCase( index ); + easing = specialEasing[ name ]; + value = props[ index ]; + if ( Array.isArray( value ) ) { + easing = value[ 1 ]; + value = props[ index ] = value[ 0 ]; + } + + if ( index !== name ) { + props[ name ] = value; + delete props[ index ]; + } + + hooks = jQuery.cssHooks[ name ]; + if ( hooks && "expand" in hooks ) { + value = hooks.expand( value ); + delete props[ name ]; + + // Not quite $.extend, this won't overwrite existing keys. + // Reusing 'index' because we have the correct "name" + for ( index in value ) { + if ( !( index in props ) ) { + props[ index ] = value[ index ]; + specialEasing[ index ] = easing; + } + } + } else { + specialEasing[ name ] = easing; + } + } +} + +function Animation( elem, properties, options ) { + var result, + stopped, + index = 0, + length = Animation.prefilters.length, + deferred = jQuery.Deferred().always( function() { + + // Don't match elem in the :animated selector + delete tick.elem; + } ), + tick = function() { + if ( stopped ) { + return false; + } + var currentTime = fxNow || createFxNow(), + remaining = Math.max( 0, animation.startTime + animation.duration - currentTime ), + + // Support: Android 2.3 only + // Archaic crash bug won't allow us to use `1 - ( 0.5 || 0 )` (#12497) + temp = remaining / animation.duration || 0, + percent = 1 - temp, + index = 0, + length = animation.tweens.length; + + for ( ; index < length; index++ ) { + animation.tweens[ index ].run( percent ); + } + + deferred.notifyWith( elem, [ animation, percent, remaining ] ); + + // If there's more to do, yield + if ( percent < 1 && length ) { + return remaining; + } + + // If this was an empty animation, synthesize a final progress notification + if ( !length ) { + deferred.notifyWith( elem, [ animation, 1, 0 ] ); + } + + // Resolve the animation and report its conclusion + deferred.resolveWith( elem, [ animation ] ); + return false; + }, + animation = deferred.promise( { + elem: elem, + props: jQuery.extend( {}, properties ), + opts: jQuery.extend( true, { + specialEasing: {}, + easing: jQuery.easing._default + }, options ), + originalProperties: properties, + originalOptions: options, + startTime: fxNow || createFxNow(), + duration: options.duration, + tweens: [], + createTween: function( prop, end ) { + var tween = jQuery.Tween( elem, animation.opts, prop, end, + animation.opts.specialEasing[ prop ] || animation.opts.easing ); + animation.tweens.push( tween ); + return tween; + }, + stop: function( gotoEnd ) { + var index = 0, + + // If we are going to the end, we want to run all the tweens + // otherwise we skip this part + length = gotoEnd ? animation.tweens.length : 0; + if ( stopped ) { + return this; + } + stopped = true; + for ( ; index < length; index++ ) { + animation.tweens[ index ].run( 1 ); + } + + // Resolve when we played the last frame; otherwise, reject + if ( gotoEnd ) { + deferred.notifyWith( elem, [ animation, 1, 0 ] ); + deferred.resolveWith( elem, [ animation, gotoEnd ] ); + } else { + deferred.rejectWith( elem, [ animation, gotoEnd ] ); + } + return this; + } + } ), + props = animation.props; + + propFilter( props, animation.opts.specialEasing ); + + for ( ; index < length; index++ ) { + result = Animation.prefilters[ index ].call( animation, elem, props, animation.opts ); + if ( result ) { + if ( isFunction( result.stop ) ) { + jQuery._queueHooks( animation.elem, animation.opts.queue ).stop = + result.stop.bind( result ); + } + return result; + } + } + + jQuery.map( props, createTween, animation ); + + if ( isFunction( animation.opts.start ) ) { + animation.opts.start.call( elem, animation ); + } + + // Attach callbacks from options + animation + .progress( animation.opts.progress ) + .done( animation.opts.done, animation.opts.complete ) + .fail( animation.opts.fail ) + .always( animation.opts.always ); + + jQuery.fx.timer( + jQuery.extend( tick, { + elem: elem, + anim: animation, + queue: animation.opts.queue + } ) + ); + + return animation; +} + +jQuery.Animation = jQuery.extend( Animation, { + + tweeners: { + "*": [ function( prop, value ) { + var tween = this.createTween( prop, value ); + adjustCSS( tween.elem, prop, rcssNum.exec( value ), tween ); + return tween; + } ] + }, + + tweener: function( props, callback ) { + if ( isFunction( props ) ) { + callback = props; + props = [ "*" ]; + } else { + props = props.match( rnothtmlwhite ); + } + + var prop, + index = 0, + length = props.length; + + for ( ; index < length; index++ ) { + prop = props[ index ]; + Animation.tweeners[ prop ] = Animation.tweeners[ prop ] || []; + Animation.tweeners[ prop ].unshift( callback ); + } + }, + + prefilters: [ defaultPrefilter ], + + prefilter: function( callback, prepend ) { + if ( prepend ) { + Animation.prefilters.unshift( callback ); + } else { + Animation.prefilters.push( callback ); + } + } +} ); + +jQuery.speed = function( speed, easing, fn ) { + var opt = speed && typeof speed === "object" ? jQuery.extend( {}, speed ) : { + complete: fn || !fn && easing || + isFunction( speed ) && speed, + duration: speed, + easing: fn && easing || easing && !isFunction( easing ) && easing + }; + + // Go to the end state if fx are off + if ( jQuery.fx.off ) { + opt.duration = 0; + + } else { + if ( typeof opt.duration !== "number" ) { + if ( opt.duration in jQuery.fx.speeds ) { + opt.duration = jQuery.fx.speeds[ opt.duration ]; + + } else { + opt.duration = jQuery.fx.speeds._default; + } + } + } + + // Normalize opt.queue - true/undefined/null -> "fx" + if ( opt.queue == null || opt.queue === true ) { + opt.queue = "fx"; + } + + // Queueing + opt.old = opt.complete; + + opt.complete = function() { + if ( isFunction( opt.old ) ) { + opt.old.call( this ); + } + + if ( opt.queue ) { + jQuery.dequeue( this, opt.queue ); + } + }; + + return opt; +}; + +jQuery.fn.extend( { + fadeTo: function( speed, to, easing, callback ) { + + // Show any hidden elements after setting opacity to 0 + return this.filter( isHiddenWithinTree ).css( "opacity", 0 ).show() + + // Animate to the value specified + .end().animate( { opacity: to }, speed, easing, callback ); + }, + animate: function( prop, speed, easing, callback ) { + var empty = jQuery.isEmptyObject( prop ), + optall = jQuery.speed( speed, easing, callback ), + doAnimation = function() { + + // Operate on a copy of prop so per-property easing won't be lost + var anim = Animation( this, jQuery.extend( {}, prop ), optall ); + + // Empty animations, or finishing resolves immediately + if ( empty || dataPriv.get( this, "finish" ) ) { + anim.stop( true ); + } + }; + doAnimation.finish = doAnimation; + + return empty || optall.queue === false ? + this.each( doAnimation ) : + this.queue( optall.queue, doAnimation ); + }, + stop: function( type, clearQueue, gotoEnd ) { + var stopQueue = function( hooks ) { + var stop = hooks.stop; + delete hooks.stop; + stop( gotoEnd ); + }; + + if ( typeof type !== "string" ) { + gotoEnd = clearQueue; + clearQueue = type; + type = undefined; + } + if ( clearQueue ) { + this.queue( type || "fx", [] ); + } + + return this.each( function() { + var dequeue = true, + index = type != null && type + "queueHooks", + timers = jQuery.timers, + data = dataPriv.get( this ); + + if ( index ) { + if ( data[ index ] && data[ index ].stop ) { + stopQueue( data[ index ] ); + } + } else { + for ( index in data ) { + if ( data[ index ] && data[ index ].stop && rrun.test( index ) ) { + stopQueue( data[ index ] ); + } + } + } + + for ( index = timers.length; index--; ) { + if ( timers[ index ].elem === this && + ( type == null || timers[ index ].queue === type ) ) { + + timers[ index ].anim.stop( gotoEnd ); + dequeue = false; + timers.splice( index, 1 ); + } + } + + // Start the next in the queue if the last step wasn't forced. + // Timers currently will call their complete callbacks, which + // will dequeue but only if they were gotoEnd. + if ( dequeue || !gotoEnd ) { + jQuery.dequeue( this, type ); + } + } ); + }, + finish: function( type ) { + if ( type !== false ) { + type = type || "fx"; + } + return this.each( function() { + var index, + data = dataPriv.get( this ), + queue = data[ type + "queue" ], + hooks = data[ type + "queueHooks" ], + timers = jQuery.timers, + length = queue ? queue.length : 0; + + // Enable finishing flag on private data + data.finish = true; + + // Empty the queue first + jQuery.queue( this, type, [] ); + + if ( hooks && hooks.stop ) { + hooks.stop.call( this, true ); + } + + // Look for any active animations, and finish them + for ( index = timers.length; index--; ) { + if ( timers[ index ].elem === this && timers[ index ].queue === type ) { + timers[ index ].anim.stop( true ); + timers.splice( index, 1 ); + } + } + + // Look for any animations in the old queue and finish them + for ( index = 0; index < length; index++ ) { + if ( queue[ index ] && queue[ index ].finish ) { + queue[ index ].finish.call( this ); + } + } + + // Turn off finishing flag + delete data.finish; + } ); + } +} ); + +jQuery.each( [ "toggle", "show", "hide" ], function( _i, name ) { + var cssFn = jQuery.fn[ name ]; + jQuery.fn[ name ] = function( speed, easing, callback ) { + return speed == null || typeof speed === "boolean" ? + cssFn.apply( this, arguments ) : + this.animate( genFx( name, true ), speed, easing, callback ); + }; +} ); + +// Generate shortcuts for custom animations +jQuery.each( { + slideDown: genFx( "show" ), + slideUp: genFx( "hide" ), + slideToggle: genFx( "toggle" ), + fadeIn: { opacity: "show" }, + fadeOut: { opacity: "hide" }, + fadeToggle: { opacity: "toggle" } +}, function( name, props ) { + jQuery.fn[ name ] = function( speed, easing, callback ) { + return this.animate( props, speed, easing, callback ); + }; +} ); + +jQuery.timers = []; +jQuery.fx.tick = function() { + var timer, + i = 0, + timers = jQuery.timers; + + fxNow = Date.now(); + + for ( ; i < timers.length; i++ ) { + timer = timers[ i ]; + + // Run the timer and safely remove it when done (allowing for external removal) + if ( !timer() && timers[ i ] === timer ) { + timers.splice( i--, 1 ); + } + } + + if ( !timers.length ) { + jQuery.fx.stop(); + } + fxNow = undefined; +}; + +jQuery.fx.timer = function( timer ) { + jQuery.timers.push( timer ); + jQuery.fx.start(); +}; + +jQuery.fx.interval = 13; +jQuery.fx.start = function() { + if ( inProgress ) { + return; + } + + inProgress = true; + schedule(); +}; + +jQuery.fx.stop = function() { + inProgress = null; +}; + +jQuery.fx.speeds = { + slow: 600, + fast: 200, + + // Default speed + _default: 400 +}; + + +// Based off of the plugin by Clint Helfers, with permission. +// https://web.archive.org/web/20100324014747/http://blindsignals.com/index.php/2009/07/jquery-delay/ +jQuery.fn.delay = function( time, type ) { + time = jQuery.fx ? jQuery.fx.speeds[ time ] || time : time; + type = type || "fx"; + + return this.queue( type, function( next, hooks ) { + var timeout = window.setTimeout( next, time ); + hooks.stop = function() { + window.clearTimeout( timeout ); + }; + } ); +}; + + +( function() { + var input = document.createElement( "input" ), + select = document.createElement( "select" ), + opt = select.appendChild( document.createElement( "option" ) ); + + input.type = "checkbox"; + + // Support: Android <=4.3 only + // Default value for a checkbox should be "on" + support.checkOn = input.value !== ""; + + // Support: IE <=11 only + // Must access selectedIndex to make default options select + support.optSelected = opt.selected; + + // Support: IE <=11 only + // An input loses its value after becoming a radio + input = document.createElement( "input" ); + input.value = "t"; + input.type = "radio"; + support.radioValue = input.value === "t"; +} )(); + + +var boolHook, + attrHandle = jQuery.expr.attrHandle; + +jQuery.fn.extend( { + attr: function( name, value ) { + return access( this, jQuery.attr, name, value, arguments.length > 1 ); + }, + + removeAttr: function( name ) { + return this.each( function() { + jQuery.removeAttr( this, name ); + } ); + } +} ); + +jQuery.extend( { + attr: function( elem, name, value ) { + var ret, hooks, + nType = elem.nodeType; + + // Don't get/set attributes on text, comment and attribute nodes + if ( nType === 3 || nType === 8 || nType === 2 ) { + return; + } + + // Fallback to prop when attributes are not supported + if ( typeof elem.getAttribute === "undefined" ) { + return jQuery.prop( elem, name, value ); + } + + // Attribute hooks are determined by the lowercase version + // Grab necessary hook if one is defined + if ( nType !== 1 || !jQuery.isXMLDoc( elem ) ) { + hooks = jQuery.attrHooks[ name.toLowerCase() ] || + ( jQuery.expr.match.bool.test( name ) ? boolHook : undefined ); + } + + if ( value !== undefined ) { + if ( value === null ) { + jQuery.removeAttr( elem, name ); + return; + } + + if ( hooks && "set" in hooks && + ( ret = hooks.set( elem, value, name ) ) !== undefined ) { + return ret; + } + + elem.setAttribute( name, value + "" ); + return value; + } + + if ( hooks && "get" in hooks && ( ret = hooks.get( elem, name ) ) !== null ) { + return ret; + } + + ret = jQuery.find.attr( elem, name ); + + // Non-existent attributes return null, we normalize to undefined + return ret == null ? undefined : ret; + }, + + attrHooks: { + type: { + set: function( elem, value ) { + if ( !support.radioValue && value === "radio" && + nodeName( elem, "input" ) ) { + var val = elem.value; + elem.setAttribute( "type", value ); + if ( val ) { + elem.value = val; + } + return value; + } + } + } + }, + + removeAttr: function( elem, value ) { + var name, + i = 0, + + // Attribute names can contain non-HTML whitespace characters + // https://html.spec.whatwg.org/multipage/syntax.html#attributes-2 + attrNames = value && value.match( rnothtmlwhite ); + + if ( attrNames && elem.nodeType === 1 ) { + while ( ( name = attrNames[ i++ ] ) ) { + elem.removeAttribute( name ); + } + } + } +} ); + +// Hooks for boolean attributes +boolHook = { + set: function( elem, value, name ) { + if ( value === false ) { + + // Remove boolean attributes when set to false + jQuery.removeAttr( elem, name ); + } else { + elem.setAttribute( name, name ); + } + return name; + } +}; + +jQuery.each( jQuery.expr.match.bool.source.match( /\w+/g ), function( _i, name ) { + var getter = attrHandle[ name ] || jQuery.find.attr; + + attrHandle[ name ] = function( elem, name, isXML ) { + var ret, handle, + lowercaseName = name.toLowerCase(); + + if ( !isXML ) { + + // Avoid an infinite loop by temporarily removing this function from the getter + handle = attrHandle[ lowercaseName ]; + attrHandle[ lowercaseName ] = ret; + ret = getter( elem, name, isXML ) != null ? + lowercaseName : + null; + attrHandle[ lowercaseName ] = handle; + } + return ret; + }; +} ); + + + + +var rfocusable = /^(?:input|select|textarea|button)$/i, + rclickable = /^(?:a|area)$/i; + +jQuery.fn.extend( { + prop: function( name, value ) { + return access( this, jQuery.prop, name, value, arguments.length > 1 ); + }, + + removeProp: function( name ) { + return this.each( function() { + delete this[ jQuery.propFix[ name ] || name ]; + } ); + } +} ); + +jQuery.extend( { + prop: function( elem, name, value ) { + var ret, hooks, + nType = elem.nodeType; + + // Don't get/set properties on text, comment and attribute nodes + if ( nType === 3 || nType === 8 || nType === 2 ) { + return; + } + + if ( nType !== 1 || !jQuery.isXMLDoc( elem ) ) { + + // Fix name and attach hooks + name = jQuery.propFix[ name ] || name; + hooks = jQuery.propHooks[ name ]; + } + + if ( value !== undefined ) { + if ( hooks && "set" in hooks && + ( ret = hooks.set( elem, value, name ) ) !== undefined ) { + return ret; + } + + return ( elem[ name ] = value ); + } + + if ( hooks && "get" in hooks && ( ret = hooks.get( elem, name ) ) !== null ) { + return ret; + } + + return elem[ name ]; + }, + + propHooks: { + tabIndex: { + get: function( elem ) { + + // Support: IE <=9 - 11 only + // elem.tabIndex doesn't always return the + // correct value when it hasn't been explicitly set + // https://web.archive.org/web/20141116233347/http://fluidproject.org/blog/2008/01/09/getting-setting-and-removing-tabindex-values-with-javascript/ + // Use proper attribute retrieval(#12072) + var tabindex = jQuery.find.attr( elem, "tabindex" ); + + if ( tabindex ) { + return parseInt( tabindex, 10 ); + } + + if ( + rfocusable.test( elem.nodeName ) || + rclickable.test( elem.nodeName ) && + elem.href + ) { + return 0; + } + + return -1; + } + } + }, + + propFix: { + "for": "htmlFor", + "class": "className" + } +} ); + +// Support: IE <=11 only +// Accessing the selectedIndex property +// forces the browser to respect setting selected +// on the option +// The getter ensures a default option is selected +// when in an optgroup +// eslint rule "no-unused-expressions" is disabled for this code +// since it considers such accessions noop +if ( !support.optSelected ) { + jQuery.propHooks.selected = { + get: function( elem ) { + + /* eslint no-unused-expressions: "off" */ + + var parent = elem.parentNode; + if ( parent && parent.parentNode ) { + parent.parentNode.selectedIndex; + } + return null; + }, + set: function( elem ) { + + /* eslint no-unused-expressions: "off" */ + + var parent = elem.parentNode; + if ( parent ) { + parent.selectedIndex; + + if ( parent.parentNode ) { + parent.parentNode.selectedIndex; + } + } + } + }; +} + +jQuery.each( [ + "tabIndex", + "readOnly", + "maxLength", + "cellSpacing", + "cellPadding", + "rowSpan", + "colSpan", + "useMap", + "frameBorder", + "contentEditable" +], function() { + jQuery.propFix[ this.toLowerCase() ] = this; +} ); + + + + + // Strip and collapse whitespace according to HTML spec + // https://infra.spec.whatwg.org/#strip-and-collapse-ascii-whitespace + function stripAndCollapse( value ) { + var tokens = value.match( rnothtmlwhite ) || []; + return tokens.join( " " ); + } + + +function getClass( elem ) { + return elem.getAttribute && elem.getAttribute( "class" ) || ""; +} + +function classesToArray( value ) { + if ( Array.isArray( value ) ) { + return value; + } + if ( typeof value === "string" ) { + return value.match( rnothtmlwhite ) || []; + } + return []; +} + +jQuery.fn.extend( { + addClass: function( value ) { + var classes, elem, cur, curValue, clazz, j, finalValue, + i = 0; + + if ( isFunction( value ) ) { + return this.each( function( j ) { + jQuery( this ).addClass( value.call( this, j, getClass( this ) ) ); + } ); + } + + classes = classesToArray( value ); + + if ( classes.length ) { + while ( ( elem = this[ i++ ] ) ) { + curValue = getClass( elem ); + cur = elem.nodeType === 1 && ( " " + stripAndCollapse( curValue ) + " " ); + + if ( cur ) { + j = 0; + while ( ( clazz = classes[ j++ ] ) ) { + if ( cur.indexOf( " " + clazz + " " ) < 0 ) { + cur += clazz + " "; + } + } + + // Only assign if different to avoid unneeded rendering. + finalValue = stripAndCollapse( cur ); + if ( curValue !== finalValue ) { + elem.setAttribute( "class", finalValue ); + } + } + } + } + + return this; + }, + + removeClass: function( value ) { + var classes, elem, cur, curValue, clazz, j, finalValue, + i = 0; + + if ( isFunction( value ) ) { + return this.each( function( j ) { + jQuery( this ).removeClass( value.call( this, j, getClass( this ) ) ); + } ); + } + + if ( !arguments.length ) { + return this.attr( "class", "" ); + } + + classes = classesToArray( value ); + + if ( classes.length ) { + while ( ( elem = this[ i++ ] ) ) { + curValue = getClass( elem ); + + // This expression is here for better compressibility (see addClass) + cur = elem.nodeType === 1 && ( " " + stripAndCollapse( curValue ) + " " ); + + if ( cur ) { + j = 0; + while ( ( clazz = classes[ j++ ] ) ) { + + // Remove *all* instances + while ( cur.indexOf( " " + clazz + " " ) > -1 ) { + cur = cur.replace( " " + clazz + " ", " " ); + } + } + + // Only assign if different to avoid unneeded rendering. + finalValue = stripAndCollapse( cur ); + if ( curValue !== finalValue ) { + elem.setAttribute( "class", finalValue ); + } + } + } + } + + return this; + }, + + toggleClass: function( value, stateVal ) { + var type = typeof value, + isValidValue = type === "string" || Array.isArray( value ); + + if ( typeof stateVal === "boolean" && isValidValue ) { + return stateVal ? this.addClass( value ) : this.removeClass( value ); + } + + if ( isFunction( value ) ) { + return this.each( function( i ) { + jQuery( this ).toggleClass( + value.call( this, i, getClass( this ), stateVal ), + stateVal + ); + } ); + } + + return this.each( function() { + var className, i, self, classNames; + + if ( isValidValue ) { + + // Toggle individual class names + i = 0; + self = jQuery( this ); + classNames = classesToArray( value ); + + while ( ( className = classNames[ i++ ] ) ) { + + // Check each className given, space separated list + if ( self.hasClass( className ) ) { + self.removeClass( className ); + } else { + self.addClass( className ); + } + } + + // Toggle whole class name + } else if ( value === undefined || type === "boolean" ) { + className = getClass( this ); + if ( className ) { + + // Store className if set + dataPriv.set( this, "__className__", className ); + } + + // If the element has a class name or if we're passed `false`, + // then remove the whole classname (if there was one, the above saved it). + // Otherwise bring back whatever was previously saved (if anything), + // falling back to the empty string if nothing was stored. + if ( this.setAttribute ) { + this.setAttribute( "class", + className || value === false ? + "" : + dataPriv.get( this, "__className__" ) || "" + ); + } + } + } ); + }, + + hasClass: function( selector ) { + var className, elem, + i = 0; + + className = " " + selector + " "; + while ( ( elem = this[ i++ ] ) ) { + if ( elem.nodeType === 1 && + ( " " + stripAndCollapse( getClass( elem ) ) + " " ).indexOf( className ) > -1 ) { + return true; + } + } + + return false; + } +} ); + + + + +var rreturn = /\r/g; + +jQuery.fn.extend( { + val: function( value ) { + var hooks, ret, valueIsFunction, + elem = this[ 0 ]; + + if ( !arguments.length ) { + if ( elem ) { + hooks = jQuery.valHooks[ elem.type ] || + jQuery.valHooks[ elem.nodeName.toLowerCase() ]; + + if ( hooks && + "get" in hooks && + ( ret = hooks.get( elem, "value" ) ) !== undefined + ) { + return ret; + } + + ret = elem.value; + + // Handle most common string cases + if ( typeof ret === "string" ) { + return ret.replace( rreturn, "" ); + } + + // Handle cases where value is null/undef or number + return ret == null ? "" : ret; + } + + return; + } + + valueIsFunction = isFunction( value ); + + return this.each( function( i ) { + var val; + + if ( this.nodeType !== 1 ) { + return; + } + + if ( valueIsFunction ) { + val = value.call( this, i, jQuery( this ).val() ); + } else { + val = value; + } + + // Treat null/undefined as ""; convert numbers to string + if ( val == null ) { + val = ""; + + } else if ( typeof val === "number" ) { + val += ""; + + } else if ( Array.isArray( val ) ) { + val = jQuery.map( val, function( value ) { + return value == null ? "" : value + ""; + } ); + } + + hooks = jQuery.valHooks[ this.type ] || jQuery.valHooks[ this.nodeName.toLowerCase() ]; + + // If set returns undefined, fall back to normal setting + if ( !hooks || !( "set" in hooks ) || hooks.set( this, val, "value" ) === undefined ) { + this.value = val; + } + } ); + } +} ); + +jQuery.extend( { + valHooks: { + option: { + get: function( elem ) { + + var val = jQuery.find.attr( elem, "value" ); + return val != null ? + val : + + // Support: IE <=10 - 11 only + // option.text throws exceptions (#14686, #14858) + // Strip and collapse whitespace + // https://html.spec.whatwg.org/#strip-and-collapse-whitespace + stripAndCollapse( jQuery.text( elem ) ); + } + }, + select: { + get: function( elem ) { + var value, option, i, + options = elem.options, + index = elem.selectedIndex, + one = elem.type === "select-one", + values = one ? null : [], + max = one ? index + 1 : options.length; + + if ( index < 0 ) { + i = max; + + } else { + i = one ? index : 0; + } + + // Loop through all the selected options + for ( ; i < max; i++ ) { + option = options[ i ]; + + // Support: IE <=9 only + // IE8-9 doesn't update selected after form reset (#2551) + if ( ( option.selected || i === index ) && + + // Don't return options that are disabled or in a disabled optgroup + !option.disabled && + ( !option.parentNode.disabled || + !nodeName( option.parentNode, "optgroup" ) ) ) { + + // Get the specific value for the option + value = jQuery( option ).val(); + + // We don't need an array for one selects + if ( one ) { + return value; + } + + // Multi-Selects return an array + values.push( value ); + } + } + + return values; + }, + + set: function( elem, value ) { + var optionSet, option, + options = elem.options, + values = jQuery.makeArray( value ), + i = options.length; + + while ( i-- ) { + option = options[ i ]; + + /* eslint-disable no-cond-assign */ + + if ( option.selected = + jQuery.inArray( jQuery.valHooks.option.get( option ), values ) > -1 + ) { + optionSet = true; + } + + /* eslint-enable no-cond-assign */ + } + + // Force browsers to behave consistently when non-matching value is set + if ( !optionSet ) { + elem.selectedIndex = -1; + } + return values; + } + } + } +} ); + +// Radios and checkboxes getter/setter +jQuery.each( [ "radio", "checkbox" ], function() { + jQuery.valHooks[ this ] = { + set: function( elem, value ) { + if ( Array.isArray( value ) ) { + return ( elem.checked = jQuery.inArray( jQuery( elem ).val(), value ) > -1 ); + } + } + }; + if ( !support.checkOn ) { + jQuery.valHooks[ this ].get = function( elem ) { + return elem.getAttribute( "value" ) === null ? "on" : elem.value; + }; + } +} ); + + + + +// Return jQuery for attributes-only inclusion + + +support.focusin = "onfocusin" in window; + + +var rfocusMorph = /^(?:focusinfocus|focusoutblur)$/, + stopPropagationCallback = function( e ) { + e.stopPropagation(); + }; + +jQuery.extend( jQuery.event, { + + trigger: function( event, data, elem, onlyHandlers ) { + + var i, cur, tmp, bubbleType, ontype, handle, special, lastElement, + eventPath = [ elem || document ], + type = hasOwn.call( event, "type" ) ? event.type : event, + namespaces = hasOwn.call( event, "namespace" ) ? event.namespace.split( "." ) : []; + + cur = lastElement = tmp = elem = elem || document; + + // Don't do events on text and comment nodes + if ( elem.nodeType === 3 || elem.nodeType === 8 ) { + return; + } + + // focus/blur morphs to focusin/out; ensure we're not firing them right now + if ( rfocusMorph.test( type + jQuery.event.triggered ) ) { + return; + } + + if ( type.indexOf( "." ) > -1 ) { + + // Namespaced trigger; create a regexp to match event type in handle() + namespaces = type.split( "." ); + type = namespaces.shift(); + namespaces.sort(); + } + ontype = type.indexOf( ":" ) < 0 && "on" + type; + + // Caller can pass in a jQuery.Event object, Object, or just an event type string + event = event[ jQuery.expando ] ? + event : + new jQuery.Event( type, typeof event === "object" && event ); + + // Trigger bitmask: & 1 for native handlers; & 2 for jQuery (always true) + event.isTrigger = onlyHandlers ? 2 : 3; + event.namespace = namespaces.join( "." ); + event.rnamespace = event.namespace ? + new RegExp( "(^|\\.)" + namespaces.join( "\\.(?:.*\\.|)" ) + "(\\.|$)" ) : + null; + + // Clean up the event in case it is being reused + event.result = undefined; + if ( !event.target ) { + event.target = elem; + } + + // Clone any incoming data and prepend the event, creating the handler arg list + data = data == null ? + [ event ] : + jQuery.makeArray( data, [ event ] ); + + // Allow special events to draw outside the lines + special = jQuery.event.special[ type ] || {}; + if ( !onlyHandlers && special.trigger && special.trigger.apply( elem, data ) === false ) { + return; + } + + // Determine event propagation path in advance, per W3C events spec (#9951) + // Bubble up to document, then to window; watch for a global ownerDocument var (#9724) + if ( !onlyHandlers && !special.noBubble && !isWindow( elem ) ) { + + bubbleType = special.delegateType || type; + if ( !rfocusMorph.test( bubbleType + type ) ) { + cur = cur.parentNode; + } + for ( ; cur; cur = cur.parentNode ) { + eventPath.push( cur ); + tmp = cur; + } + + // Only add window if we got to document (e.g., not plain obj or detached DOM) + if ( tmp === ( elem.ownerDocument || document ) ) { + eventPath.push( tmp.defaultView || tmp.parentWindow || window ); + } + } + + // Fire handlers on the event path + i = 0; + while ( ( cur = eventPath[ i++ ] ) && !event.isPropagationStopped() ) { + lastElement = cur; + event.type = i > 1 ? + bubbleType : + special.bindType || type; + + // jQuery handler + handle = ( + dataPriv.get( cur, "events" ) || Object.create( null ) + )[ event.type ] && + dataPriv.get( cur, "handle" ); + if ( handle ) { + handle.apply( cur, data ); + } + + // Native handler + handle = ontype && cur[ ontype ]; + if ( handle && handle.apply && acceptData( cur ) ) { + event.result = handle.apply( cur, data ); + if ( event.result === false ) { + event.preventDefault(); + } + } + } + event.type = type; + + // If nobody prevented the default action, do it now + if ( !onlyHandlers && !event.isDefaultPrevented() ) { + + if ( ( !special._default || + special._default.apply( eventPath.pop(), data ) === false ) && + acceptData( elem ) ) { + + // Call a native DOM method on the target with the same name as the event. + // Don't do default actions on window, that's where global variables be (#6170) + if ( ontype && isFunction( elem[ type ] ) && !isWindow( elem ) ) { + + // Don't re-trigger an onFOO event when we call its FOO() method + tmp = elem[ ontype ]; + + if ( tmp ) { + elem[ ontype ] = null; + } + + // Prevent re-triggering of the same event, since we already bubbled it above + jQuery.event.triggered = type; + + if ( event.isPropagationStopped() ) { + lastElement.addEventListener( type, stopPropagationCallback ); + } + + elem[ type ](); + + if ( event.isPropagationStopped() ) { + lastElement.removeEventListener( type, stopPropagationCallback ); + } + + jQuery.event.triggered = undefined; + + if ( tmp ) { + elem[ ontype ] = tmp; + } + } + } + } + + return event.result; + }, + + // Piggyback on a donor event to simulate a different one + // Used only for `focus(in | out)` events + simulate: function( type, elem, event ) { + var e = jQuery.extend( + new jQuery.Event(), + event, + { + type: type, + isSimulated: true + } + ); + + jQuery.event.trigger( e, null, elem ); + } + +} ); + +jQuery.fn.extend( { + + trigger: function( type, data ) { + return this.each( function() { + jQuery.event.trigger( type, data, this ); + } ); + }, + triggerHandler: function( type, data ) { + var elem = this[ 0 ]; + if ( elem ) { + return jQuery.event.trigger( type, data, elem, true ); + } + } +} ); + + +// Support: Firefox <=44 +// Firefox doesn't have focus(in | out) events +// Related ticket - https://bugzilla.mozilla.org/show_bug.cgi?id=687787 +// +// Support: Chrome <=48 - 49, Safari <=9.0 - 9.1 +// focus(in | out) events fire after focus & blur events, +// which is spec violation - http://www.w3.org/TR/DOM-Level-3-Events/#events-focusevent-event-order +// Related ticket - https://bugs.chromium.org/p/chromium/issues/detail?id=449857 +if ( !support.focusin ) { + jQuery.each( { focus: "focusin", blur: "focusout" }, function( orig, fix ) { + + // Attach a single capturing handler on the document while someone wants focusin/focusout + var handler = function( event ) { + jQuery.event.simulate( fix, event.target, jQuery.event.fix( event ) ); + }; + + jQuery.event.special[ fix ] = { + setup: function() { + + // Handle: regular nodes (via `this.ownerDocument`), window + // (via `this.document`) & document (via `this`). + var doc = this.ownerDocument || this.document || this, + attaches = dataPriv.access( doc, fix ); + + if ( !attaches ) { + doc.addEventListener( orig, handler, true ); + } + dataPriv.access( doc, fix, ( attaches || 0 ) + 1 ); + }, + teardown: function() { + var doc = this.ownerDocument || this.document || this, + attaches = dataPriv.access( doc, fix ) - 1; + + if ( !attaches ) { + doc.removeEventListener( orig, handler, true ); + dataPriv.remove( doc, fix ); + + } else { + dataPriv.access( doc, fix, attaches ); + } + } + }; + } ); +} +var location = window.location; + +var nonce = { guid: Date.now() }; + +var rquery = ( /\?/ ); + + + +// Cross-browser xml parsing +jQuery.parseXML = function( data ) { + var xml; + if ( !data || typeof data !== "string" ) { + return null; + } + + // Support: IE 9 - 11 only + // IE throws on parseFromString with invalid input. + try { + xml = ( new window.DOMParser() ).parseFromString( data, "text/xml" ); + } catch ( e ) { + xml = undefined; + } + + if ( !xml || xml.getElementsByTagName( "parsererror" ).length ) { + jQuery.error( "Invalid XML: " + data ); + } + return xml; +}; + + +var + rbracket = /\[\]$/, + rCRLF = /\r?\n/g, + rsubmitterTypes = /^(?:submit|button|image|reset|file)$/i, + rsubmittable = /^(?:input|select|textarea|keygen)/i; + +function buildParams( prefix, obj, traditional, add ) { + var name; + + if ( Array.isArray( obj ) ) { + + // Serialize array item. + jQuery.each( obj, function( i, v ) { + if ( traditional || rbracket.test( prefix ) ) { + + // Treat each array item as a scalar. + add( prefix, v ); + + } else { + + // Item is non-scalar (array or object), encode its numeric index. + buildParams( + prefix + "[" + ( typeof v === "object" && v != null ? i : "" ) + "]", + v, + traditional, + add + ); + } + } ); + + } else if ( !traditional && toType( obj ) === "object" ) { + + // Serialize object item. + for ( name in obj ) { + buildParams( prefix + "[" + name + "]", obj[ name ], traditional, add ); + } + + } else { + + // Serialize scalar item. + add( prefix, obj ); + } +} + +// Serialize an array of form elements or a set of +// key/values into a query string +jQuery.param = function( a, traditional ) { + var prefix, + s = [], + add = function( key, valueOrFunction ) { + + // If value is a function, invoke it and use its return value + var value = isFunction( valueOrFunction ) ? + valueOrFunction() : + valueOrFunction; + + s[ s.length ] = encodeURIComponent( key ) + "=" + + encodeURIComponent( value == null ? "" : value ); + }; + + if ( a == null ) { + return ""; + } + + // If an array was passed in, assume that it is an array of form elements. + if ( Array.isArray( a ) || ( a.jquery && !jQuery.isPlainObject( a ) ) ) { + + // Serialize the form elements + jQuery.each( a, function() { + add( this.name, this.value ); + } ); + + } else { + + // If traditional, encode the "old" way (the way 1.3.2 or older + // did it), otherwise encode params recursively. + for ( prefix in a ) { + buildParams( prefix, a[ prefix ], traditional, add ); + } + } + + // Return the resulting serialization + return s.join( "&" ); +}; + +jQuery.fn.extend( { + serialize: function() { + return jQuery.param( this.serializeArray() ); + }, + serializeArray: function() { + return this.map( function() { + + // Can add propHook for "elements" to filter or add form elements + var elements = jQuery.prop( this, "elements" ); + return elements ? jQuery.makeArray( elements ) : this; + } ) + .filter( function() { + var type = this.type; + + // Use .is( ":disabled" ) so that fieldset[disabled] works + return this.name && !jQuery( this ).is( ":disabled" ) && + rsubmittable.test( this.nodeName ) && !rsubmitterTypes.test( type ) && + ( this.checked || !rcheckableType.test( type ) ); + } ) + .map( function( _i, elem ) { + var val = jQuery( this ).val(); + + if ( val == null ) { + return null; + } + + if ( Array.isArray( val ) ) { + return jQuery.map( val, function( val ) { + return { name: elem.name, value: val.replace( rCRLF, "\r\n" ) }; + } ); + } + + return { name: elem.name, value: val.replace( rCRLF, "\r\n" ) }; + } ).get(); + } +} ); + + +var + r20 = /%20/g, + rhash = /#.*$/, + rantiCache = /([?&])_=[^&]*/, + rheaders = /^(.*?):[ \t]*([^\r\n]*)$/mg, + + // #7653, #8125, #8152: local protocol detection + rlocalProtocol = /^(?:about|app|app-storage|.+-extension|file|res|widget):$/, + rnoContent = /^(?:GET|HEAD)$/, + rprotocol = /^\/\//, + + /* Prefilters + * 1) They are useful to introduce custom dataTypes (see ajax/jsonp.js for an example) + * 2) These are called: + * - BEFORE asking for a transport + * - AFTER param serialization (s.data is a string if s.processData is true) + * 3) key is the dataType + * 4) the catchall symbol "*" can be used + * 5) execution will start with transport dataType and THEN continue down to "*" if needed + */ + prefilters = {}, + + /* Transports bindings + * 1) key is the dataType + * 2) the catchall symbol "*" can be used + * 3) selection will start with transport dataType and THEN go to "*" if needed + */ + transports = {}, + + // Avoid comment-prolog char sequence (#10098); must appease lint and evade compression + allTypes = "*/".concat( "*" ), + + // Anchor tag for parsing the document origin + originAnchor = document.createElement( "a" ); + originAnchor.href = location.href; + +// Base "constructor" for jQuery.ajaxPrefilter and jQuery.ajaxTransport +function addToPrefiltersOrTransports( structure ) { + + // dataTypeExpression is optional and defaults to "*" + return function( dataTypeExpression, func ) { + + if ( typeof dataTypeExpression !== "string" ) { + func = dataTypeExpression; + dataTypeExpression = "*"; + } + + var dataType, + i = 0, + dataTypes = dataTypeExpression.toLowerCase().match( rnothtmlwhite ) || []; + + if ( isFunction( func ) ) { + + // For each dataType in the dataTypeExpression + while ( ( dataType = dataTypes[ i++ ] ) ) { + + // Prepend if requested + if ( dataType[ 0 ] === "+" ) { + dataType = dataType.slice( 1 ) || "*"; + ( structure[ dataType ] = structure[ dataType ] || [] ).unshift( func ); + + // Otherwise append + } else { + ( structure[ dataType ] = structure[ dataType ] || [] ).push( func ); + } + } + } + }; +} + +// Base inspection function for prefilters and transports +function inspectPrefiltersOrTransports( structure, options, originalOptions, jqXHR ) { + + var inspected = {}, + seekingTransport = ( structure === transports ); + + function inspect( dataType ) { + var selected; + inspected[ dataType ] = true; + jQuery.each( structure[ dataType ] || [], function( _, prefilterOrFactory ) { + var dataTypeOrTransport = prefilterOrFactory( options, originalOptions, jqXHR ); + if ( typeof dataTypeOrTransport === "string" && + !seekingTransport && !inspected[ dataTypeOrTransport ] ) { + + options.dataTypes.unshift( dataTypeOrTransport ); + inspect( dataTypeOrTransport ); + return false; + } else if ( seekingTransport ) { + return !( selected = dataTypeOrTransport ); + } + } ); + return selected; + } + + return inspect( options.dataTypes[ 0 ] ) || !inspected[ "*" ] && inspect( "*" ); +} + +// A special extend for ajax options +// that takes "flat" options (not to be deep extended) +// Fixes #9887 +function ajaxExtend( target, src ) { + var key, deep, + flatOptions = jQuery.ajaxSettings.flatOptions || {}; + + for ( key in src ) { + if ( src[ key ] !== undefined ) { + ( flatOptions[ key ] ? target : ( deep || ( deep = {} ) ) )[ key ] = src[ key ]; + } + } + if ( deep ) { + jQuery.extend( true, target, deep ); + } + + return target; +} + +/* Handles responses to an ajax request: + * - finds the right dataType (mediates between content-type and expected dataType) + * - returns the corresponding response + */ +function ajaxHandleResponses( s, jqXHR, responses ) { + + var ct, type, finalDataType, firstDataType, + contents = s.contents, + dataTypes = s.dataTypes; + + // Remove auto dataType and get content-type in the process + while ( dataTypes[ 0 ] === "*" ) { + dataTypes.shift(); + if ( ct === undefined ) { + ct = s.mimeType || jqXHR.getResponseHeader( "Content-Type" ); + } + } + + // Check if we're dealing with a known content-type + if ( ct ) { + for ( type in contents ) { + if ( contents[ type ] && contents[ type ].test( ct ) ) { + dataTypes.unshift( type ); + break; + } + } + } + + // Check to see if we have a response for the expected dataType + if ( dataTypes[ 0 ] in responses ) { + finalDataType = dataTypes[ 0 ]; + } else { + + // Try convertible dataTypes + for ( type in responses ) { + if ( !dataTypes[ 0 ] || s.converters[ type + " " + dataTypes[ 0 ] ] ) { + finalDataType = type; + break; + } + if ( !firstDataType ) { + firstDataType = type; + } + } + + // Or just use first one + finalDataType = finalDataType || firstDataType; + } + + // If we found a dataType + // We add the dataType to the list if needed + // and return the corresponding response + if ( finalDataType ) { + if ( finalDataType !== dataTypes[ 0 ] ) { + dataTypes.unshift( finalDataType ); + } + return responses[ finalDataType ]; + } +} + +/* Chain conversions given the request and the original response + * Also sets the responseXXX fields on the jqXHR instance + */ +function ajaxConvert( s, response, jqXHR, isSuccess ) { + var conv2, current, conv, tmp, prev, + converters = {}, + + // Work with a copy of dataTypes in case we need to modify it for conversion + dataTypes = s.dataTypes.slice(); + + // Create converters map with lowercased keys + if ( dataTypes[ 1 ] ) { + for ( conv in s.converters ) { + converters[ conv.toLowerCase() ] = s.converters[ conv ]; + } + } + + current = dataTypes.shift(); + + // Convert to each sequential dataType + while ( current ) { + + if ( s.responseFields[ current ] ) { + jqXHR[ s.responseFields[ current ] ] = response; + } + + // Apply the dataFilter if provided + if ( !prev && isSuccess && s.dataFilter ) { + response = s.dataFilter( response, s.dataType ); + } + + prev = current; + current = dataTypes.shift(); + + if ( current ) { + + // There's only work to do if current dataType is non-auto + if ( current === "*" ) { + + current = prev; + + // Convert response if prev dataType is non-auto and differs from current + } else if ( prev !== "*" && prev !== current ) { + + // Seek a direct converter + conv = converters[ prev + " " + current ] || converters[ "* " + current ]; + + // If none found, seek a pair + if ( !conv ) { + for ( conv2 in converters ) { + + // If conv2 outputs current + tmp = conv2.split( " " ); + if ( tmp[ 1 ] === current ) { + + // If prev can be converted to accepted input + conv = converters[ prev + " " + tmp[ 0 ] ] || + converters[ "* " + tmp[ 0 ] ]; + if ( conv ) { + + // Condense equivalence converters + if ( conv === true ) { + conv = converters[ conv2 ]; + + // Otherwise, insert the intermediate dataType + } else if ( converters[ conv2 ] !== true ) { + current = tmp[ 0 ]; + dataTypes.unshift( tmp[ 1 ] ); + } + break; + } + } + } + } + + // Apply converter (if not an equivalence) + if ( conv !== true ) { + + // Unless errors are allowed to bubble, catch and return them + if ( conv && s.throws ) { + response = conv( response ); + } else { + try { + response = conv( response ); + } catch ( e ) { + return { + state: "parsererror", + error: conv ? e : "No conversion from " + prev + " to " + current + }; + } + } + } + } + } + } + + return { state: "success", data: response }; +} + +jQuery.extend( { + + // Counter for holding the number of active queries + active: 0, + + // Last-Modified header cache for next request + lastModified: {}, + etag: {}, + + ajaxSettings: { + url: location.href, + type: "GET", + isLocal: rlocalProtocol.test( location.protocol ), + global: true, + processData: true, + async: true, + contentType: "application/x-www-form-urlencoded; charset=UTF-8", + + /* + timeout: 0, + data: null, + dataType: null, + username: null, + password: null, + cache: null, + throws: false, + traditional: false, + headers: {}, + */ + + accepts: { + "*": allTypes, + text: "text/plain", + html: "text/html", + xml: "application/xml, text/xml", + json: "application/json, text/javascript" + }, + + contents: { + xml: /\bxml\b/, + html: /\bhtml/, + json: /\bjson\b/ + }, + + responseFields: { + xml: "responseXML", + text: "responseText", + json: "responseJSON" + }, + + // Data converters + // Keys separate source (or catchall "*") and destination types with a single space + converters: { + + // Convert anything to text + "* text": String, + + // Text to html (true = no transformation) + "text html": true, + + // Evaluate text as a json expression + "text json": JSON.parse, + + // Parse text as xml + "text xml": jQuery.parseXML + }, + + // For options that shouldn't be deep extended: + // you can add your own custom options here if + // and when you create one that shouldn't be + // deep extended (see ajaxExtend) + flatOptions: { + url: true, + context: true + } + }, + + // Creates a full fledged settings object into target + // with both ajaxSettings and settings fields. + // If target is omitted, writes into ajaxSettings. + ajaxSetup: function( target, settings ) { + return settings ? + + // Building a settings object + ajaxExtend( ajaxExtend( target, jQuery.ajaxSettings ), settings ) : + + // Extending ajaxSettings + ajaxExtend( jQuery.ajaxSettings, target ); + }, + + ajaxPrefilter: addToPrefiltersOrTransports( prefilters ), + ajaxTransport: addToPrefiltersOrTransports( transports ), + + // Main method + ajax: function( url, options ) { + + // If url is an object, simulate pre-1.5 signature + if ( typeof url === "object" ) { + options = url; + url = undefined; + } + + // Force options to be an object + options = options || {}; + + var transport, + + // URL without anti-cache param + cacheURL, + + // Response headers + responseHeadersString, + responseHeaders, + + // timeout handle + timeoutTimer, + + // Url cleanup var + urlAnchor, + + // Request state (becomes false upon send and true upon completion) + completed, + + // To know if global events are to be dispatched + fireGlobals, + + // Loop variable + i, + + // uncached part of the url + uncached, + + // Create the final options object + s = jQuery.ajaxSetup( {}, options ), + + // Callbacks context + callbackContext = s.context || s, + + // Context for global events is callbackContext if it is a DOM node or jQuery collection + globalEventContext = s.context && + ( callbackContext.nodeType || callbackContext.jquery ) ? + jQuery( callbackContext ) : + jQuery.event, + + // Deferreds + deferred = jQuery.Deferred(), + completeDeferred = jQuery.Callbacks( "once memory" ), + + // Status-dependent callbacks + statusCode = s.statusCode || {}, + + // Headers (they are sent all at once) + requestHeaders = {}, + requestHeadersNames = {}, + + // Default abort message + strAbort = "canceled", + + // Fake xhr + jqXHR = { + readyState: 0, + + // Builds headers hashtable if needed + getResponseHeader: function( key ) { + var match; + if ( completed ) { + if ( !responseHeaders ) { + responseHeaders = {}; + while ( ( match = rheaders.exec( responseHeadersString ) ) ) { + responseHeaders[ match[ 1 ].toLowerCase() + " " ] = + ( responseHeaders[ match[ 1 ].toLowerCase() + " " ] || [] ) + .concat( match[ 2 ] ); + } + } + match = responseHeaders[ key.toLowerCase() + " " ]; + } + return match == null ? null : match.join( ", " ); + }, + + // Raw string + getAllResponseHeaders: function() { + return completed ? responseHeadersString : null; + }, + + // Caches the header + setRequestHeader: function( name, value ) { + if ( completed == null ) { + name = requestHeadersNames[ name.toLowerCase() ] = + requestHeadersNames[ name.toLowerCase() ] || name; + requestHeaders[ name ] = value; + } + return this; + }, + + // Overrides response content-type header + overrideMimeType: function( type ) { + if ( completed == null ) { + s.mimeType = type; + } + return this; + }, + + // Status-dependent callbacks + statusCode: function( map ) { + var code; + if ( map ) { + if ( completed ) { + + // Execute the appropriate callbacks + jqXHR.always( map[ jqXHR.status ] ); + } else { + + // Lazy-add the new callbacks in a way that preserves old ones + for ( code in map ) { + statusCode[ code ] = [ statusCode[ code ], map[ code ] ]; + } + } + } + return this; + }, + + // Cancel the request + abort: function( statusText ) { + var finalText = statusText || strAbort; + if ( transport ) { + transport.abort( finalText ); + } + done( 0, finalText ); + return this; + } + }; + + // Attach deferreds + deferred.promise( jqXHR ); + + // Add protocol if not provided (prefilters might expect it) + // Handle falsy url in the settings object (#10093: consistency with old signature) + // We also use the url parameter if available + s.url = ( ( url || s.url || location.href ) + "" ) + .replace( rprotocol, location.protocol + "//" ); + + // Alias method option to type as per ticket #12004 + s.type = options.method || options.type || s.method || s.type; + + // Extract dataTypes list + s.dataTypes = ( s.dataType || "*" ).toLowerCase().match( rnothtmlwhite ) || [ "" ]; + + // A cross-domain request is in order when the origin doesn't match the current origin. + if ( s.crossDomain == null ) { + urlAnchor = document.createElement( "a" ); + + // Support: IE <=8 - 11, Edge 12 - 15 + // IE throws exception on accessing the href property if url is malformed, + // e.g. http://example.com:80x/ + try { + urlAnchor.href = s.url; + + // Support: IE <=8 - 11 only + // Anchor's host property isn't correctly set when s.url is relative + urlAnchor.href = urlAnchor.href; + s.crossDomain = originAnchor.protocol + "//" + originAnchor.host !== + urlAnchor.protocol + "//" + urlAnchor.host; + } catch ( e ) { + + // If there is an error parsing the URL, assume it is crossDomain, + // it can be rejected by the transport if it is invalid + s.crossDomain = true; + } + } + + // Convert data if not already a string + if ( s.data && s.processData && typeof s.data !== "string" ) { + s.data = jQuery.param( s.data, s.traditional ); + } + + // Apply prefilters + inspectPrefiltersOrTransports( prefilters, s, options, jqXHR ); + + // If request was aborted inside a prefilter, stop there + if ( completed ) { + return jqXHR; + } + + // We can fire global events as of now if asked to + // Don't fire events if jQuery.event is undefined in an AMD-usage scenario (#15118) + fireGlobals = jQuery.event && s.global; + + // Watch for a new set of requests + if ( fireGlobals && jQuery.active++ === 0 ) { + jQuery.event.trigger( "ajaxStart" ); + } + + // Uppercase the type + s.type = s.type.toUpperCase(); + + // Determine if request has content + s.hasContent = !rnoContent.test( s.type ); + + // Save the URL in case we're toying with the If-Modified-Since + // and/or If-None-Match header later on + // Remove hash to simplify url manipulation + cacheURL = s.url.replace( rhash, "" ); + + // More options handling for requests with no content + if ( !s.hasContent ) { + + // Remember the hash so we can put it back + uncached = s.url.slice( cacheURL.length ); + + // If data is available and should be processed, append data to url + if ( s.data && ( s.processData || typeof s.data === "string" ) ) { + cacheURL += ( rquery.test( cacheURL ) ? "&" : "?" ) + s.data; + + // #9682: remove data so that it's not used in an eventual retry + delete s.data; + } + + // Add or update anti-cache param if needed + if ( s.cache === false ) { + cacheURL = cacheURL.replace( rantiCache, "$1" ); + uncached = ( rquery.test( cacheURL ) ? "&" : "?" ) + "_=" + ( nonce.guid++ ) + + uncached; + } + + // Put hash and anti-cache on the URL that will be requested (gh-1732) + s.url = cacheURL + uncached; + + // Change '%20' to '+' if this is encoded form body content (gh-2658) + } else if ( s.data && s.processData && + ( s.contentType || "" ).indexOf( "application/x-www-form-urlencoded" ) === 0 ) { + s.data = s.data.replace( r20, "+" ); + } + + // Set the If-Modified-Since and/or If-None-Match header, if in ifModified mode. + if ( s.ifModified ) { + if ( jQuery.lastModified[ cacheURL ] ) { + jqXHR.setRequestHeader( "If-Modified-Since", jQuery.lastModified[ cacheURL ] ); + } + if ( jQuery.etag[ cacheURL ] ) { + jqXHR.setRequestHeader( "If-None-Match", jQuery.etag[ cacheURL ] ); + } + } + + // Set the correct header, if data is being sent + if ( s.data && s.hasContent && s.contentType !== false || options.contentType ) { + jqXHR.setRequestHeader( "Content-Type", s.contentType ); + } + + // Set the Accepts header for the server, depending on the dataType + jqXHR.setRequestHeader( + "Accept", + s.dataTypes[ 0 ] && s.accepts[ s.dataTypes[ 0 ] ] ? + s.accepts[ s.dataTypes[ 0 ] ] + + ( s.dataTypes[ 0 ] !== "*" ? ", " + allTypes + "; q=0.01" : "" ) : + s.accepts[ "*" ] + ); + + // Check for headers option + for ( i in s.headers ) { + jqXHR.setRequestHeader( i, s.headers[ i ] ); + } + + // Allow custom headers/mimetypes and early abort + if ( s.beforeSend && + ( s.beforeSend.call( callbackContext, jqXHR, s ) === false || completed ) ) { + + // Abort if not done already and return + return jqXHR.abort(); + } + + // Aborting is no longer a cancellation + strAbort = "abort"; + + // Install callbacks on deferreds + completeDeferred.add( s.complete ); + jqXHR.done( s.success ); + jqXHR.fail( s.error ); + + // Get transport + transport = inspectPrefiltersOrTransports( transports, s, options, jqXHR ); + + // If no transport, we auto-abort + if ( !transport ) { + done( -1, "No Transport" ); + } else { + jqXHR.readyState = 1; + + // Send global event + if ( fireGlobals ) { + globalEventContext.trigger( "ajaxSend", [ jqXHR, s ] ); + } + + // If request was aborted inside ajaxSend, stop there + if ( completed ) { + return jqXHR; + } + + // Timeout + if ( s.async && s.timeout > 0 ) { + timeoutTimer = window.setTimeout( function() { + jqXHR.abort( "timeout" ); + }, s.timeout ); + } + + try { + completed = false; + transport.send( requestHeaders, done ); + } catch ( e ) { + + // Rethrow post-completion exceptions + if ( completed ) { + throw e; + } + + // Propagate others as results + done( -1, e ); + } + } + + // Callback for when everything is done + function done( status, nativeStatusText, responses, headers ) { + var isSuccess, success, error, response, modified, + statusText = nativeStatusText; + + // Ignore repeat invocations + if ( completed ) { + return; + } + + completed = true; + + // Clear timeout if it exists + if ( timeoutTimer ) { + window.clearTimeout( timeoutTimer ); + } + + // Dereference transport for early garbage collection + // (no matter how long the jqXHR object will be used) + transport = undefined; + + // Cache response headers + responseHeadersString = headers || ""; + + // Set readyState + jqXHR.readyState = status > 0 ? 4 : 0; + + // Determine if successful + isSuccess = status >= 200 && status < 300 || status === 304; + + // Get response data + if ( responses ) { + response = ajaxHandleResponses( s, jqXHR, responses ); + } + + // Use a noop converter for missing script + if ( !isSuccess && jQuery.inArray( "script", s.dataTypes ) > -1 ) { + s.converters[ "text script" ] = function() {}; + } + + // Convert no matter what (that way responseXXX fields are always set) + response = ajaxConvert( s, response, jqXHR, isSuccess ); + + // If successful, handle type chaining + if ( isSuccess ) { + + // Set the If-Modified-Since and/or If-None-Match header, if in ifModified mode. + if ( s.ifModified ) { + modified = jqXHR.getResponseHeader( "Last-Modified" ); + if ( modified ) { + jQuery.lastModified[ cacheURL ] = modified; + } + modified = jqXHR.getResponseHeader( "etag" ); + if ( modified ) { + jQuery.etag[ cacheURL ] = modified; + } + } + + // if no content + if ( status === 204 || s.type === "HEAD" ) { + statusText = "nocontent"; + + // if not modified + } else if ( status === 304 ) { + statusText = "notmodified"; + + // If we have data, let's convert it + } else { + statusText = response.state; + success = response.data; + error = response.error; + isSuccess = !error; + } + } else { + + // Extract error from statusText and normalize for non-aborts + error = statusText; + if ( status || !statusText ) { + statusText = "error"; + if ( status < 0 ) { + status = 0; + } + } + } + + // Set data for the fake xhr object + jqXHR.status = status; + jqXHR.statusText = ( nativeStatusText || statusText ) + ""; + + // Success/Error + if ( isSuccess ) { + deferred.resolveWith( callbackContext, [ success, statusText, jqXHR ] ); + } else { + deferred.rejectWith( callbackContext, [ jqXHR, statusText, error ] ); + } + + // Status-dependent callbacks + jqXHR.statusCode( statusCode ); + statusCode = undefined; + + if ( fireGlobals ) { + globalEventContext.trigger( isSuccess ? "ajaxSuccess" : "ajaxError", + [ jqXHR, s, isSuccess ? success : error ] ); + } + + // Complete + completeDeferred.fireWith( callbackContext, [ jqXHR, statusText ] ); + + if ( fireGlobals ) { + globalEventContext.trigger( "ajaxComplete", [ jqXHR, s ] ); + + // Handle the global AJAX counter + if ( !( --jQuery.active ) ) { + jQuery.event.trigger( "ajaxStop" ); + } + } + } + + return jqXHR; + }, + + getJSON: function( url, data, callback ) { + return jQuery.get( url, data, callback, "json" ); + }, + + getScript: function( url, callback ) { + return jQuery.get( url, undefined, callback, "script" ); + } +} ); + +jQuery.each( [ "get", "post" ], function( _i, method ) { + jQuery[ method ] = function( url, data, callback, type ) { + + // Shift arguments if data argument was omitted + if ( isFunction( data ) ) { + type = type || callback; + callback = data; + data = undefined; + } + + // The url can be an options object (which then must have .url) + return jQuery.ajax( jQuery.extend( { + url: url, + type: method, + dataType: type, + data: data, + success: callback + }, jQuery.isPlainObject( url ) && url ) ); + }; +} ); + +jQuery.ajaxPrefilter( function( s ) { + var i; + for ( i in s.headers ) { + if ( i.toLowerCase() === "content-type" ) { + s.contentType = s.headers[ i ] || ""; + } + } +} ); + + +jQuery._evalUrl = function( url, options, doc ) { + return jQuery.ajax( { + url: url, + + // Make this explicit, since user can override this through ajaxSetup (#11264) + type: "GET", + dataType: "script", + cache: true, + async: false, + global: false, + + // Only evaluate the response if it is successful (gh-4126) + // dataFilter is not invoked for failure responses, so using it instead + // of the default converter is kludgy but it works. + converters: { + "text script": function() {} + }, + dataFilter: function( response ) { + jQuery.globalEval( response, options, doc ); + } + } ); +}; + + +jQuery.fn.extend( { + wrapAll: function( html ) { + var wrap; + + if ( this[ 0 ] ) { + if ( isFunction( html ) ) { + html = html.call( this[ 0 ] ); + } + + // The elements to wrap the target around + wrap = jQuery( html, this[ 0 ].ownerDocument ).eq( 0 ).clone( true ); + + if ( this[ 0 ].parentNode ) { + wrap.insertBefore( this[ 0 ] ); + } + + wrap.map( function() { + var elem = this; + + while ( elem.firstElementChild ) { + elem = elem.firstElementChild; + } + + return elem; + } ).append( this ); + } + + return this; + }, + + wrapInner: function( html ) { + if ( isFunction( html ) ) { + return this.each( function( i ) { + jQuery( this ).wrapInner( html.call( this, i ) ); + } ); + } + + return this.each( function() { + var self = jQuery( this ), + contents = self.contents(); + + if ( contents.length ) { + contents.wrapAll( html ); + + } else { + self.append( html ); + } + } ); + }, + + wrap: function( html ) { + var htmlIsFunction = isFunction( html ); + + return this.each( function( i ) { + jQuery( this ).wrapAll( htmlIsFunction ? html.call( this, i ) : html ); + } ); + }, + + unwrap: function( selector ) { + this.parent( selector ).not( "body" ).each( function() { + jQuery( this ).replaceWith( this.childNodes ); + } ); + return this; + } +} ); + + +jQuery.expr.pseudos.hidden = function( elem ) { + return !jQuery.expr.pseudos.visible( elem ); +}; +jQuery.expr.pseudos.visible = function( elem ) { + return !!( elem.offsetWidth || elem.offsetHeight || elem.getClientRects().length ); +}; + + + + +jQuery.ajaxSettings.xhr = function() { + try { + return new window.XMLHttpRequest(); + } catch ( e ) {} +}; + +var xhrSuccessStatus = { + + // File protocol always yields status code 0, assume 200 + 0: 200, + + // Support: IE <=9 only + // #1450: sometimes IE returns 1223 when it should be 204 + 1223: 204 + }, + xhrSupported = jQuery.ajaxSettings.xhr(); + +support.cors = !!xhrSupported && ( "withCredentials" in xhrSupported ); +support.ajax = xhrSupported = !!xhrSupported; + +jQuery.ajaxTransport( function( options ) { + var callback, errorCallback; + + // Cross domain only allowed if supported through XMLHttpRequest + if ( support.cors || xhrSupported && !options.crossDomain ) { + return { + send: function( headers, complete ) { + var i, + xhr = options.xhr(); + + xhr.open( + options.type, + options.url, + options.async, + options.username, + options.password + ); + + // Apply custom fields if provided + if ( options.xhrFields ) { + for ( i in options.xhrFields ) { + xhr[ i ] = options.xhrFields[ i ]; + } + } + + // Override mime type if needed + if ( options.mimeType && xhr.overrideMimeType ) { + xhr.overrideMimeType( options.mimeType ); + } + + // X-Requested-With header + // For cross-domain requests, seeing as conditions for a preflight are + // akin to a jigsaw puzzle, we simply never set it to be sure. + // (it can always be set on a per-request basis or even using ajaxSetup) + // For same-domain requests, won't change header if already provided. + if ( !options.crossDomain && !headers[ "X-Requested-With" ] ) { + headers[ "X-Requested-With" ] = "XMLHttpRequest"; + } + + // Set headers + for ( i in headers ) { + xhr.setRequestHeader( i, headers[ i ] ); + } + + // Callback + callback = function( type ) { + return function() { + if ( callback ) { + callback = errorCallback = xhr.onload = + xhr.onerror = xhr.onabort = xhr.ontimeout = + xhr.onreadystatechange = null; + + if ( type === "abort" ) { + xhr.abort(); + } else if ( type === "error" ) { + + // Support: IE <=9 only + // On a manual native abort, IE9 throws + // errors on any property access that is not readyState + if ( typeof xhr.status !== "number" ) { + complete( 0, "error" ); + } else { + complete( + + // File: protocol always yields status 0; see #8605, #14207 + xhr.status, + xhr.statusText + ); + } + } else { + complete( + xhrSuccessStatus[ xhr.status ] || xhr.status, + xhr.statusText, + + // Support: IE <=9 only + // IE9 has no XHR2 but throws on binary (trac-11426) + // For XHR2 non-text, let the caller handle it (gh-2498) + ( xhr.responseType || "text" ) !== "text" || + typeof xhr.responseText !== "string" ? + { binary: xhr.response } : + { text: xhr.responseText }, + xhr.getAllResponseHeaders() + ); + } + } + }; + }; + + // Listen to events + xhr.onload = callback(); + errorCallback = xhr.onerror = xhr.ontimeout = callback( "error" ); + + // Support: IE 9 only + // Use onreadystatechange to replace onabort + // to handle uncaught aborts + if ( xhr.onabort !== undefined ) { + xhr.onabort = errorCallback; + } else { + xhr.onreadystatechange = function() { + + // Check readyState before timeout as it changes + if ( xhr.readyState === 4 ) { + + // Allow onerror to be called first, + // but that will not handle a native abort + // Also, save errorCallback to a variable + // as xhr.onerror cannot be accessed + window.setTimeout( function() { + if ( callback ) { + errorCallback(); + } + } ); + } + }; + } + + // Create the abort callback + callback = callback( "abort" ); + + try { + + // Do send the request (this may raise an exception) + xhr.send( options.hasContent && options.data || null ); + } catch ( e ) { + + // #14683: Only rethrow if this hasn't been notified as an error yet + if ( callback ) { + throw e; + } + } + }, + + abort: function() { + if ( callback ) { + callback(); + } + } + }; + } +} ); + + + + +// Prevent auto-execution of scripts when no explicit dataType was provided (See gh-2432) +jQuery.ajaxPrefilter( function( s ) { + if ( s.crossDomain ) { + s.contents.script = false; + } +} ); + +// Install script dataType +jQuery.ajaxSetup( { + accepts: { + script: "text/javascript, application/javascript, " + + "application/ecmascript, application/x-ecmascript" + }, + contents: { + script: /\b(?:java|ecma)script\b/ + }, + converters: { + "text script": function( text ) { + jQuery.globalEval( text ); + return text; + } + } +} ); + +// Handle cache's special case and crossDomain +jQuery.ajaxPrefilter( "script", function( s ) { + if ( s.cache === undefined ) { + s.cache = false; + } + if ( s.crossDomain ) { + s.type = "GET"; + } +} ); + +// Bind script tag hack transport +jQuery.ajaxTransport( "script", function( s ) { + + // This transport only deals with cross domain or forced-by-attrs requests + if ( s.crossDomain || s.scriptAttrs ) { + var script, callback; + return { + send: function( _, complete ) { + script = jQuery( " + + + + + + + + + + + + + +
+ + +
+ +
+
+
+
    +
  • + +
  • +
  • +
+
+
+
+
+ +
+

Robot Arms

+
+ +
+
+ + +
+
+
+ +
+ +
+

© Copyright 2023, Jesse Haviland and Peter Corke. + Last updated on 11-May-2024. +

+
+ + + +
+
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/arm_backend.html b/arm_backend.html new file mode 100644 index 00000000..34fee22a --- /dev/null +++ b/arm_backend.html @@ -0,0 +1,137 @@ + + + + + + + + + + + Backends — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+
    +
  • + +
  • +
  • +
+
+
+
+
+ +
+

Backends

+

The Robotics Toolbox supports a number of backends, through a standard API, to communicate with simulators or +physical robots.

+ +
+ + +
+
+
+ +
+ +
+

© Copyright 2023, Jesse Haviland and Peter Corke. + Last updated on 11-May-2024. +

+
+ + + +
+
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/arm_backend_pyplot.html b/arm_backend_pyplot.html new file mode 100644 index 00000000..8f666416 --- /dev/null +++ b/arm_backend_pyplot.html @@ -0,0 +1,351 @@ + + + + + + + + + + + PyPlot (matplotlib) — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+
    +
  • + +
  • +
  • +
+
+
+
+
+ +
+

PyPlot (matplotlib)

+
+
+class roboticstoolbox.backends.PyPlot.PyPlot[source]
+

Bases: Connector

+

Graphical backend using matplotlib

+

matplotlib is a common and highly portable graphics library for Python, +but has relatively limited 3D capability.

+

Example:

+
1import roboticstoolbox as rtb
+2
+3robot = rtb.models.DH.Panda()  # create a robot
+4
+5pyplot = rtb.backends.PyPlot()  # create a PyPlot backend
+6pyplot.add(robot)              # add the robot to the backend
+7robot.q = robot.qz             # set the robot configuration
+8pyplot.step()                  # update the backend and graphical view
+
+
+
+

Note

+

PyPlot is the default backend, and robot.plot(q) effectively +performs lines 7-8 above.

+
+
+
+launch(name=None, fig=None, limits=None, **kwargs)[source]
+

Launch a graphical interface

+

`env = launch()` creates a blank 3D matplotlib figure and returns +a reference to the backend.

+
+ +
+
+step(dt=0.05)[source]
+

Update the graphical scene

+
+
Parameters:
+

dt (int, optional) – time step in seconds, defaults to 50 (0.05 s)

+
+
+

env.step(args) triggers an update of the 3D scene in the matplotlib +window referenced by env.

+
+

Note

+
    +
  • Each robot in the scene is updated based on +their control type (position, velocity, acceleration, or torque).

  • +
  • Upon acting, the other three of the four control types will be +updated in the internal state of the robot object.

  • +
  • The control type is defined by the robot object, and not all +robot objects support all control types.

  • +
  • Execution is blocked for the specified interval

  • +
+
+
+ +
+
+reset()[source]
+

Reset the graphical scene

+

env.reset() triggers a reset of the 3D scene in the matplotlib +window referenced by env. It is restored to the original state +defined by launch().

+
+ +
+
+restart()[source]
+

Restart the graphics display

+

env.restart() triggers a restart of the matplotlib view referenced +by env. It is closed and relaunched to the original state defined +by launch().

+
+ +
+
+close()[source]
+

env.close() gracefully closes the matplotlib window +referenced by env.

+
+ +
+
+add(ob, readonly=False, display=True, jointaxes=True, jointlabels=False, eeframe=True, shadow=True, name=True, options=None)[source]
+

Add a robot to the graphical scene

+
+
Parameters:
+
    +
  • ob (DHRobot or EllipsePlot) – The object to add to the plot (robot or ellipse)

  • +
  • readonly (bool, optional) – Do not update the state of the object +(i.e. display not simulate), defaults to False

  • +
  • display (bool, optional) – Display the object, defaults to True

  • +
  • jointaxes (bool, optional) – Show the joint axes of the robot with arrows, +defaults to True

  • +
  • eeframe (bool, optional) – Show the end-effector frame of the robot, +defaults to True

  • +
  • shadow (bool, optional) – Display a shadow of the robot on the x-y gound plane, +defaults to True

  • +
  • name (bool, optional) – Display the name of the robot, defaults to True

  • +
+
+
+
+
id = env.add(robot) adds the robot to the graphical

environment.

+
+
+
+

Note

+
    +
  • robot must be of an appropriate class.

  • +
  • Adds the robot object to a list of robots which will be updated +when the step() method is called.

  • +
+
+
+ +
+
+remove(id)[source]
+

Remove a robot or shape from the graphical scene

+
+
Parameters:
+
    +
  • id (class:DHRobot, +class:roboticstoolbox.backends.VPython.graphics_robot.GraphicalRobot) – The id of the robot to remove. Can be either the DHLink or +GraphicalRobot

  • +
  • fig_num (int, optional) – The canvas index to delete the robot from, defaults to +the initial one

  • +
+
+
Raises:
+
    +
  • ValueError – Figure number must be between 0 and total number +of canvases

  • +
  • TypeError – Input must be a DHLink or GraphicalRobot

  • +
+
+
+
+
env.remove(robot) removes the robot from the graphical

environment.

+
+
+
+ +
+
+hold()[source]
+

hold() keeps the plot open i.e. stops the plot from closing once +the main script has finished.

+
+ +
+
+getframe()[source]
+
+ +
+ +
+
+class roboticstoolbox.backends.PyPlot.PyPlot2[source]
+

Bases: Connector

+
+
+launch(name=None, limits=None, **kwargs)[source]
+

env = launch() launchs a blank 2D matplotlib figure

+
+ +
+
+step(dt=0.05)[source]
+

state = step(args) triggers the external program to make a time step +of defined time updating the state of the environment as defined by +the robot’s actions.

+

The will go through each robot in the list and make them act based on +their control type (position, velocity, acceleration, or torque). Upon +acting, the other three of the four control types will be updated in +the internal state of the robot object. The control type is defined +by the robot object, and not all robot objects support all control +types.

+
+ +
+
+reset()[source]
+

state = reset() triggers the external program to reset to the +original state defined by launch

+
+ +
+
+restart()[source]
+

state = restart() triggers the external program to close and relaunch +to thestate defined by launch

+
+ +
+
+close()[source]
+

close() closes the plot

+
+ +
+
+add(ob, readonly=False, display=True, eeframe=True, name=False, **kwargs)[source]
+

id = add(robot) adds the robot to the external environment. robot must +be of an appropriate class. This adds a robot object to a list of +robots which will act upon the step() method being called.

+
+ +
+
+remove()[source]
+

id = remove(robot) removes the robot to the external environment.

+
+ +
+
+hold()[source]
+

hold() keeps the plot open i.e. stops the plot from closing once +the main script has finished.

+
+ +
+ +
+ + +
+
+
+ +
+ +
+

© Copyright 2023, Jesse Haviland and Peter Corke. + Last updated on 11-May-2024. +

+
+ + + +
+
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/arm_backend_swift.html b/arm_backend_swift.html new file mode 100644 index 00000000..aef60ef5 --- /dev/null +++ b/arm_backend_swift.html @@ -0,0 +1,139 @@ + + + + + + + Swift — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+ +
+
+
+ + + + + + + \ No newline at end of file diff --git a/arm_backend_vpython.html b/arm_backend_vpython.html new file mode 100644 index 00000000..e7cf4e07 --- /dev/null +++ b/arm_backend_vpython.html @@ -0,0 +1,139 @@ + + + + + + + VPython — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+ +
+
+
+ + + + + + + \ No newline at end of file diff --git a/arm_dh.html b/arm_dh.html new file mode 100644 index 00000000..0661d3dd --- /dev/null +++ b/arm_dh.html @@ -0,0 +1,10283 @@ + + + + + + + + + + + Denavit-Hartenberg models — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

Denavit-Hartenberg models

+

Code author: Jesse Haviland

+

This class is used to model robots defined in terms of standard or modified +Denavit-Hartenberg notation.

+
+

Note

+

These classes provide similar functionality and notation to MATLAB Toolbox SerialLink and +Link classes.

+
+
+

DHRobot

+
Inheritance diagram of roboticstoolbox.DHRobot
+ + + +

The various Models all subclass this class.

+
+
+class roboticstoolbox.robot.DHRobot.DHRobot(links, meshdir=None, **kwargs)[source]
+

Bases: Robot

+

Class for robots defined using Denavit-Hartenberg notation

+
+
Parameters:
+
    +
  • L (list(n)) – List of links which define the robot

  • +
  • name (str) – Name of the robot

  • +
  • manufacturer (str) – Manufacturer of the robot

  • +
  • base (SE3) – Location of the base

  • +
  • tool (SE3) – Location of the tool

  • +
  • gravity (ndarray(3)) – Gravitational acceleration vector

  • +
+
+
+

A concrete superclass for arm type robots defined using Denavit-Hartenberg +notation, that represents a serial-link arm-type robot. Each link and +joint in the chain is described by a DHLink-class object using +Denavit-Hartenberg parameters (standard or modified).

+
+

Note

+

Link subclass elements passed in must be all standard, or all +modified, DH parameters.

+
+
+
Reference:
+
    +
  • Robotics, Vision & Control in Python, 3e, P. Corke, Springer 2023, Chap 7-9.

  • +
  • Robot, Modeling & Control, +M.Spong, S. Hutchinson & M. Vidyasagar, Wiley 2006.

  • +
+
+
+
+
+__str__()[source]
+

Pretty prints the DH Model of the robot. Will output angles in degrees

+
+
Returns:
+

Pretty print of the robot model

+
+
Return type:
+

str

+
+
+
+ +
+
+property mdh
+

Modified Denavit-Hartenberg status

+
+
Returns:
+

whether robot is defined using modified Denavit-Hartenberg +notation

+
+
Return type:
+

bool

+
+
+

Example:

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> puma.mdh
+0
+>>> panda = rtb.models.DH.Panda()
+>>> panda.mdh
+1
+
+
+
+ +
+
+property d
+

Link offset values

+
+
Returns:
+

List of link offset values \(d_j\).

+
+
Return type:
+

ndarray(n,)

+
+
+

The following are equivalent:

+
robot.links[j].d
+robot.d[j]
+
+
+

Example:

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.d
+[0.6718299999999999, 0, 0.15005, 0.4318, 0, 0]
+
+
+
+ +
+
+property a
+

Link length values

+
+
Returns:
+

List of link length values \(a_j\).

+
+
Return type:
+

ndarray(n,)

+
+
+

The following are equivalent:

+
robot.links[j].a
+robot.a[j]
+
+
+

Example:

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.a
+[0, 0.4318, 0.0203, 0, 0, 0]
+
+
+
+ +
+
+property theta
+

Joint angle values

+
+
Returns:
+

List of joint angle values \(\theta_j\).

+
+
Return type:
+

ndarray(n,)

+
+
+

The following are equivalent:

+
robot.links[j].theta
+robot.theta[j]
+
+
+

Example:

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.theta
+[0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
+
+
+
+ +
+
+property alpha
+

Link twist values

+
+
Returns:
+

List of link twist values \(\alpha_j\).

+
+
Return type:
+

ndarray(n,)

+
+
+

The following are equivalent:

+
robot.links[j].alpha
+robot.alpha[j]
+
+
+

Example:

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.alpha
+[1.5707963267948966, 0.0, -1.5707963267948966, 1.5707963267948966, -1.5707963267948966, 0.0]
+
+
+
+ +
+
+property r
+

Link centre of mass values

+
+
Returns:
+

Array of link centre of mass values \(r_j\).

+
+
Return type:
+

ndarray(3,n)

+
+
+

Example:

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.r
+array([[ 0.    , -0.3638, -0.0203,  0.    ,  0.    ,  0.    ],
+       [ 0.    ,  0.006 , -0.0141,  0.019 ,  0.    ,  0.    ],
+       [ 0.    ,  0.2275,  0.07  ,  0.    ,  0.    ,  0.032 ]])
+
+
+
+ +
+
+property offset
+

Joint offset values

+
+
Returns:
+

List of joint offset values \(\bar{q}_j\).

+
+
Return type:
+

ndarray(n,)

+
+
+

Example:

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.offset
+[0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
+
+
+
+ +
+
+property reach
+

Reach of the robot

+
+
Returns:
+

Maximum reach of the robot

+
+
Return type:
+

float

+
+
+

A conservative estimate of the reach of the robot. It is computed as +\(\sum_j |a_j| + |d_j|\) where \(d_j\) is taken as the maximum +joint coordinate (qlim) if the joint is primsmatic.

+
+

Note

+
    +
  • This is the length sum referred to in Craig’s book

  • +
  • Probably an overestimate of the actual reach

  • +
  • Used by numerical inverse kinematics to scale translational +error.

  • +
  • For a prismatic joint, uses qlim if it is set

  • +
+
+
+

Warning

+

Computed on the first access. If kinematic parameters +subsequently change this will not be reflected.

+
+
+ +
+
+property nbranches
+

Number of branches

+
+
Returns:
+

number of branches in the robot’s kinematic tree

+
+
Return type:
+

int

+
+
+

Number of branches in this robot.

+

Example:

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Panda()
+>>> robot.nbranches
+1
+
+
+
+
Seealso:
+

n(), nlinks()

+
+
+
+ +
+
+A(j, q=None)[source]
+

Link forward kinematics

+
+
Parameters:
+
    +
  • j (int, 2-tuple) – Joints to compute link transform for

  • +
  • q (float ndarray(1,n)) – The joint configuration of the robot (Optional, +if not supplied will use the stored q values)

  • +
+
+
Return T:
+

The transform between link frames

+
+
Rtype T:
+

SE3

+
+
+
    +
  • robot.A(j, q) transform between link frames {0} and {j}. q +is a vector (n) of joint variables.

  • +
  • robot.A([j1, j2], q) as above between link frames {j1} and {j2}.

  • +
  • robot.A(j) as above except uses the stored q value of the +robot object.

  • +
+
+

Note

+

Base and tool transforms are not applied.

+
+
+ +
+
+islimit(q=None)[source]
+

Joint limit test

+
+
Parameters:
+

q (ndarray(n) – The joint configuration of the robot (Optional, +if not supplied will use the stored q values)

+
+
Return v:
+

is a vector of boolean values, one per joint, False if +q[j] is within the joint limits, else True

+
+
Rtype v:
+

bool list

+
+
+
    +
  • robot.islimit(q) is a list of boolean values indicating if the +joint configuration q is in violation of the joint limits.

  • +
  • robot.jointlimit() as above except uses the stored q value of the +robot object.

  • +
+

Example:

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.islimit([0, 0, -4, 4, 0, 0])
+[False, False, True, False, False, False]
+
+
+
+ +
+
+isspherical()[source]
+

Test for spherical wrist

+
+
Returns:
+

True if spherical wrist

+
+
Return type:
+

bool

+
+
+

Tests if the robot has a spherical wrist, that is, the last 3 axes are +revolute and their axes intersect at a point.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.isspherical()
+True
+
+
+
+ +
+
+dhunique()[source]
+

Print the unique DH parameters

+

Print a table showing all the non-zero DH parameters, and their +values. This is the minimum set of kinematic parameters required +to describe the robot.

+

Example:

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> puma.dhunique()
+┌──────┬────────┐
+│param │ value  │
+├──────┼────────┤
+│   d0 │ 0.6718 │
+│   ⍺0 │ 1.571  │
+│   a1 │ 0.4318 │
+│   d2 │ 0.15   │
+│   a2 │ 0.0203 │
+│   ⍺2 │ -1.571 │
+│   d3 │ 0.4318 │
+│   ⍺3 │ 1.571  │
+│   ⍺4 │ -1.571 │
+└──────┴────────┘
+
+
+
+
+ +
+
+twists(q=None)[source]
+

Joint axes as twists

+
+
Parameters:
+

q – The joint configuration of the robot, defaults to zero

+
+
Returns:
+

a vector of joint axis twists

+
+
Return type:
+

Twist3 instance

+
+
Returns:
+

Pose of the tool

+
+
Return type:
+

SE3 instance

+
+
+
    +
  • tw, T0 = twists(q) calculates a vector of Twist objects (n) that +represent the axes of the joints for the robot with joint coordinates +q (n). Also returns T0 which is an SE3 object representing the +pose of the tool.

  • +
  • tw, T0 = twists() as above but the joint coordinates are taken +to be zero.

  • +
+

Example:

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> tw, T0 = robot.twists()
+>>> tw
+Twist3([
+  [-0, -0, -0, 0, 0, 1],
+  [0.67183, -0, -0, 0, -1, 6.1232e-17],
+  [0.67183, -2.644e-17, -0.4318, 0, -1, 6.1232e-17],
+  [-0.15005, -0.4521, 0, 0, 0, 1],
+  [1.1036, -2.7683e-17, -0.4521, 0, -1, 6.1232e-17],
+  [-0.15005, -0.4521, 0, 0, 0, 1]
+])
+>>> T0
+SE3(array([[ 1.    ,  0.    ,  0.    ,  0.4521],
+           [ 0.    ,  1.    ,  0.    , -0.15  ],
+           [ 0.    ,  0.    ,  1.    ,  1.1036],
+           [ 0.    ,  0.    ,  0.    ,  1.    ]]))
+
+
+
+ +
+
+ets(*args, **kwargs)[source]
+

Robot kinematics as an elemenary transform sequence

+
+
Returns:
+

elementary transform sequence

+
+
Return type:
+

ETS

+
+
+

Example:

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> puma.ets()
+[ET.Rz(jindex=0, qlim=array([-2.7925,  2.7925])), ET.tz(eta=0.6718299999999999), ET.Rx(eta=1.5707963267948966), ET.Rz(jindex=1, qlim=array([-1.9199,  1.9199])), ET.tx(eta=0.4318), ET.Rz(jindex=2, qlim=array([-2.3562,  2.3562])), ET.tz(eta=0.15005), ET.tx(eta=0.0203), ET.Rx(eta=-1.5707963267948966), ET.Rz(jindex=3, qlim=array([-4.6426,  4.6426])), ET.tz(eta=0.4318), ET.Rx(eta=1.5707963267948966), ET.Rz(jindex=4, qlim=array([-1.7453,  1.7453])), ET.Rx(eta=-1.5707963267948966), ET.Rz(jindex=5, qlim=array([-4.6426,  4.6426]))]
+
+
+
+ +
+
+fkine(q, **kwargs)[source]
+

Forward kinematics

+
+
Parameters:
+

q (ndarray(n) or ndarray(m,n)) – The joint configuration

+
+
Returns:
+

Forward kinematics as an SE(3) matrix

+
+
Return type:
+

SE3 instance

+
+
+
    +
  • robot.fkine(q) computes the forward kinematics for the robot at +joint configuration q.

  • +
+

If q is a 2D array, the rows are interpreted as the generalized joint +coordinates for a sequence of points along a trajectory. q[k,j] is +the j’th joint coordinate for the k’th trajectory configuration, and +the returned SE3 instance contains n values.

+

Example:

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> puma.fkine([0, 0, 0, 0, 0, 0])
+SE3(array([[ 1.    ,  0.    ,  0.    ,  0.4521],
+           [ 0.    ,  1.    ,  0.    , -0.15  ],
+           [ 0.    ,  0.    ,  1.    ,  1.1036],
+           [ 0.    ,  0.    ,  0.    ,  1.    ]]))
+
+
+
+

Note

+
    +
  • The robot’s base or tool transform, if present, are incorporated +into the result.

  • +
  • Joint offsets, if defined, are added to q before the forward +kinematics are computed.

  • +
+
+
+ +
+
+fkine_path(q, old=None)[source]
+

Compute the pose of every link frame

+
+
Parameters:
+

q (darray(n)) – The joint configuration

+
+
Returns:
+

Pose of all links

+
+
Return type:
+

SE3 instance

+
+
+

T = robot.fkine_path(q) is an SE3 instance with robot.nlinks + +1 values:

+
    +
  • T[0] is the base transform

  • +
  • T[i+1] is the pose of link whose number is i

  • +
+
+
References:
+
    +
  • Kinematic Derivatives using the Elementary Transform +Sequence, J. Haviland and P. Corke

  • +
+
+
+
+ +
+
+segments()[source]
+

Segments of branched robot

+

For a single-chain robot with structure:

+
L1 - L2 - L3
+
+
+

the return is [[None, L1, L2, L3]]

+

For a robot with structure:

+
L1 - L2 +-  L3 - L4
+        +- L5 - L6
+
+
+

the return is [[None, L1, L2], [L2, L3, L4], [L2, L5, L6]]

+
+
Returns:
+

Segment list

+
+
Return type:
+

segments

+
+
+

Notes

+
    +
  • the length of the list is the number of segments in the robot

  • +
  • +
    the first segment always starts with None which represents

    the base transform (since there is no base link)

    +
    +
    +
  • +
  • +
    the last link of one segment is also the first link of subsequent

    segments

    +
    +
    +
  • +
+
+ +
+
+fkine_all(q=None, old=True)[source]
+

Forward kinematics for all link frames

+
+
Parameters:
+
    +
  • q (ndarray(n) or ndarray(m,n)) – The joint configuration of the robot (Optional, +if not supplied will use the stored q values).

  • +
  • old (bool, optional) – “old” behaviour, defaults to True

  • +
+
+
Returns:
+

Forward kinematics as an SE(3) matrix

+
+
Return type:
+

SE3 instance with n values

+
+
+
    +
  • fkine_all(q) evaluates fkine for each joint within a robot and +returns a sequence of link frame poses.

  • +
  • fkine_all() as above except uses the stored q value of the +robot object.

  • +
+

Example:

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> T = puma.fkine_all([0, 0, 0, 0, 0, 0])
+>>> len(T)
+7
+
+
+
+

Note

+
    +
  • Old behaviour is to return a list of n frames {1} to {n}, but +if old=False it returns ``n``+1 frames {0} to {n}, ie. it +includes the base frame.

  • +
  • The robot’s base or tool transform, if present, are incorporated +into the result.

  • +
  • Joint offsets, if defined, are added to q before the forward +kinematics are computed.

  • +
+
+
+ +
+
+jacobe(q, half=None, **kwargs)[source]
+

Manipulator Jacobian in end-effector frame

+
+
Parameters:
+
    +
  • q (ndarray(n)) – Joint coordinate vector

  • +
  • half (str) – return half Jacobian: ‘trans’ or ‘rot’

  • +
+
+
Return J:
+

The manipulator Jacobian in the end-effector frame

+
+
Return type:
+

ndarray(6,n)

+
+
+
    +
  • robot.jacobe(q) is the manipulator Jacobian matrix which maps +joint velocity to end-effector spatial velocity.

  • +
+

End-effector spatial velocity \(\nu = (v_x, v_y, v_z, \omega_x, \omega_y, \omega_z)^T\) +is related to joint velocity by \({}^{E}\!\nu = \mathbf{J}_m(q) \dot{q}\).

+

Example:

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> puma.jacobe([0, 0, 0, 0, 0, 0])
+array([[ 0.15  , -0.4318, -0.4318,  0.    ,  0.    ,  0.    ],
+       [ 0.4521,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+       [ 0.    ,  0.4521,  0.0203,  0.    ,  0.    ,  0.    ],
+       [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+       [ 0.    , -1.    , -1.    ,  0.    , -1.    ,  0.    ],
+       [ 1.    ,  0.    ,  0.    ,  1.    ,  0.    ,  1.    ]])
+
+
+
+

Warning

+

This is the geometric Jacobian as described in texts by +Corke, Spong etal., Siciliano etal. The end-effector velocity is +described in terms of translational and angular velocity, not a +velocity twist as per the text by Lynch & Park.

+
+
+ +
+
+jacob0(q=None, T=None, half=None, start=None, end=None)[source]
+

Manipulator Jacobian in world frame

+
+
Parameters:
+
    +
  • q (ndarray(n)) – Joint coordinate vector

  • +
  • T (SE3 instance) – Forward kinematics if known, SE(3 matrix)

  • +
  • half (str) – return half Jacobian: ‘trans’ or ‘rot’

  • +
+
+
Return J:
+

The manipulator Jacobian in the world frame

+
+
Return type:
+

ndarray(6,n)

+
+
+
    +
  • robot.jacob0(q) is the manipulator geometric Jacobian matrix which maps +joint velocity to end-effector spatial velocity.

  • +
+

End-effector spatial velocity \(\nu = (v_x, v_y, v_z, \omega_x, \omega_y, \omega_z)^T\) +is related to joint velocity by \({}^{0}\!\nu = \mathbf{J}_0(q) \dot{q}\).

+

Example:

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> puma.jacob0([0, 0, 0, 0, 0, 0])
+array([[ 0.15  , -0.4318, -0.4318,  0.    ,  0.    ,  0.    ],
+       [ 0.4521,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+       [ 0.    ,  0.4521,  0.0203,  0.    ,  0.    ,  0.    ],
+       [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+       [ 0.    , -1.    , -1.    ,  0.    , -1.    ,  0.    ],
+       [ 1.    ,  0.    ,  0.    ,  1.    ,  0.    ,  1.    ]])
+
+
+
+

Warning

+

This is the geometric Jacobian is as described in texts by +Corke, Spong etal., Siciliano etal. The end-effector velocity is +described in terms of translational and angular velocity, not a +velocity twist as per the text by Lynch & Park.

+
+
+

Note

+

T can be passed in to save the cost of computing forward +kinematics which is needed to transform velocity from end-effector +frame to world frame.

+
+
+ +
+
+jacob0_analytical(q, representation=None, T=None)[source]
+

Manipulator Jacobian in world frame

+
+
Parameters:
+
    +
  • q (ndarray(n)) – Joint coordinate vector

  • +
  • representation (str) – return analytical Jacobian instead of geometric Jacobian

  • +
  • T (SE3 instance) – Forward kinematics if known, SE(3 matrix)

  • +
+
+
Return J:
+

The manipulator analytical Jacobian in the world frame

+
+
Return type:
+

ndarray(6,n)

+
+
+

Return the manipulator’s analytical Jacobian matrix which maps +joint velocity to end-effector spatial velocity.

+

End-effector spatial velocity \(\nu_a = (v_x, v_y, v_z, \dot{\Gamma}_1, \dot{\Gamma}_2, \dot{\Gamma}_3)^T\) +is related to joint velocity by \({}^{0}\!\nu_a = \mathbf{J}_{a,0}(q) \dot{q}\). +Where \(\dvec{\Gamma} = (\dot{\Gamma}_1, \dot{\Gamma}_2, \dot{\Gamma}_3)\) is +orientation rate expressed as one of:

+ + + + + + + + + + + + + + + + + + + + +

representation

Rotational representation

'rpy/xyz'

RPY angular rates in XYZ order

'rpy/zyx'

RPY angular rates in XYZ order

'eul'

Euler angular rates in ZYZ order

'exp'

exponential coordinate rates

+

Example:

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> puma.jacob0_analytical([0, 0, 0, 0, 0, 0], "rpy/xyz")
+array([[ 0.15  , -0.4318, -0.4318,  0.    ,  0.    ,  0.    ],
+       [ 0.4521,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+       [ 0.    ,  0.4521,  0.0203,  0.    ,  0.    ,  0.    ],
+       [ 1.    ,  0.    ,  0.    ,  1.    ,  0.    ,  1.    ],
+       [ 0.    , -1.    , -1.    ,  0.    , -1.    ,  0.    ],
+       [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ]])
+
+
+
+

Warning

+

The geometric Jacobian is as described in texts by +Corke, Spong etal., Siciliano etal. The end-effector velocity is +described in terms of translational and angular velocity, not a +velocity twist as per the text by Lynch & Park.

+
+
+

Note

+

T can be passed in to save the cost of computing forward +kinematics which is needed to transform velocity from end-effector +frame to world frame.

+
+
+ +
+
+hessian0(q=None, J0=None, start=None, end=None)[source]
+

Manipulator Hessian in base frame

+
+
Parameters:
+
    +
  • q (array_like) – joint coordinates

  • +
  • J0 (ndarray(6,n)) – Jacobian in {0} frame

  • +
+
+
Returns:
+

Hessian matrix

+
+
Return type:
+

ndarray(6,n,n)

+
+
+

This method calculcates the Hessisan in the base frame. One of J0 or +q is required. If J0 is already calculated for the joint +coordinates q it can be passed in to to save computation time.

+

If we take the time derivative of the differential kinematic +relationship

+
+\[\begin{split}\nu &= \mat{J}(\vec{q}) \dvec{q} \\ +\alpha &= \dmat{J} \dvec{q} + \mat{J} \ddvec{q}\end{split}\]
+

where

+
+\[\dmat{J} = \mat{H} \dvec{q}\]
+

and \(\mat{H} \in \mathbb{R}^{6\times n \times n}\) is the +Hessian tensor.

+

The elements of the Hessian are

+
+\[\mat{H}_{i,j,k} = \frac{d^2 u_i}{d q_j d q_k}\]
+

where \(u = \{t_x, t_y, t_z, r_x, r_y, r_z\}\) are the elements +of the spatial velocity vector.

+

Similarly, we can write

+
+\[\mat{J}_{i,j} = \frac{d u_i}{d q_j}\]
+
+
References:
+
    +
  • Kinematic Derivatives using the Elementary Transform +Sequence, J. Haviland and P. Corke

  • +
+
+
Seealso:
+

jacob0(), jacob_dot()

+
+
+
+ +
+
+delete_rne()[source]
+

Frees the memory holding the robot object in c if the robot object +has been initialised in c.

+
+ +
+
+rne(q, qd=None, qdd=None, gravity=None, fext=None, base_wrench=False)[source]
+

Inverse dynamics

+
+
Parameters:
+
    +
  • q (ndarray(n)) – Joint coordinates

  • +
  • qd (ndarray(n)) – Joint velocity

  • +
  • qdd (ndarray(n)) – The joint accelerations of the robot

  • +
  • gravity (ndarray(6)) – Gravitational acceleration to override robot’s gravity +value

  • +
  • fext (ndarray(6)) – Specify wrench acting on the end-effector +\(W=[F_x F_y F_z M_x M_y M_z]\)

  • +
+
+
+

tau = rne(q, qd, qdd, grav, fext) is the joint torque required for +the robot to achieve the specified joint position q (1xn), velocity +qd (1xn) and acceleration qdd (1xn), where n is the number of +robot joints. fext describes the wrench acting on the end-effector

+

Trajectory operation: +If q, qd and qdd (mxn) are matrices with m cols representing a +trajectory then tau (mxn) is a matrix with cols corresponding to each +trajectory step.

+
+

Note

+
    +
  • The torque computed contains a contribution due to armature +inertia and joint friction.

  • +
  • If a model has no dynamic parameters set the result is zero.

  • +
+
+
+
Seealso:
+

rne_python()

+
+
+
+ +
+
+rne_python(Q, QD=None, QDD=None, gravity=None, fext=None, debug=False, base_wrench=False)[source]
+

Compute inverse dynamics via recursive Newton-Euler formulation

+
+
Parameters:
+
    +
  • Q – Joint coordinates

  • +
  • QD – Joint velocity

  • +
  • QDD – Joint acceleration

  • +
  • gravity (array_like(3), optional) – gravitational acceleration, defaults to attribute +of self

  • +
  • fext (array-like(6), optional) – end-effector wrench, defaults to None

  • +
  • debug (bool, optional) – print debug information to console, defaults to False

  • +
  • base_wrench (bool, optional) – compute the base wrench, defaults to False

  • +
+
+
Raises:
+

ValueError – for misshaped inputs

+
+
Returns:
+

Joint force/torques

+
+
Return type:
+

NumPy array

+
+
+

Recursive Newton-Euler for standard Denavit-Hartenberg notation.

+
    +
  • rne_dh(q, qd, qdd) where the arguments have shape (n,) where n is +the number of robot joints. The result has shape (n,).

  • +
  • rne_dh(q, qd, qdd) where the arguments have shape (m,n) where n +is the number of robot joints and where m is the number of steps in +the joint trajectory. The result has shape (m,n).

  • +
  • rne_dh(p) where the input is a 1D array p = [q, qd, qdd] with +shape (3n,), and the result has shape (n,).

  • +
  • rne_dh(p) where the input is a 2D array p = [q, qd, qdd] with +shape (m,3n) and the result has shape (m,n).

  • +
+
+

Note

+
    +
  • This is a pure Python implementation and slower than the .rne()

  • +
+

which is written in C. +- This version supports symbolic model parameters +- Verified against MATLAB code

+
+
+
Seealso:
+

rne()

+
+
+
+ +
+
+ikine_6s(T, config, ikfunc)[source]
+
+ +
+
+config_validate(config, allowables)[source]
+

Validate a configuration string

+
+
Parameters:
+
    +
  • config (str) – a configuration string

  • +
  • allowable (tuple of str) – [description]

  • +
+
+
Raises:
+

ValueError – bad character in configuration string

+
+
Returns:
+

configuration string

+
+
Return type:
+

str

+
+
+

For analytic inverse kinematics the Toolbox uses a string whose +letters indicate particular solutions, eg. for the Puma 560

+
+
+ + + + + + + + + + + + + + + + + + + + + + + + + +

Character

Meaning

‘l’

lefty

‘r’

righty

‘u’

elbow up

‘d’

elbow down

‘n’

wrist not flipped

‘f’

wrist flipped

+
+

This method checks that the configuration string is valid and adds +default values for missing characters. For example:

+
+

config = self.config_validate(config, (‘lr’, ‘ud’, ‘nf’))

+
+

indicates the valid characters, and the first character in each +string is the default, ie. if neither ‘l’ or ‘r’ is given then +‘l’ will be added to the string.

+
+ +
+
+ik_lm_chan(Tep, q0=None, ilimit=30, slimit=100, tol=1e-06, reject_jl=True, we=None, λ=1.0)[source]
+

Numerical inverse kinematics by Levenberg-Marquadt optimization (Chan’s Method)

+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose or pose trajectory

  • +
  • q0 (Optional[ndarray]) – initial joint configuration (default to random valid joint +configuration contrained by the joint limits of the robot)

  • +
  • ilimit (int) – maximum number of iterations per search

  • +
  • slimit (int) – maximum number of search attempts

  • +
  • tol (float) – final error tolerance

  • +
  • reject_jl (bool) – constrain the solution to being within the joint limits of +the robot (reject solution with invalid joint configurations and perfrom +another search up to the slimit)

  • +
  • we (Optional[ndarray]) – a mask vector which weights the end-effector error priority. +Corresponds to translation in X, Y and Z and rotation about X, Y and Z +respectively

  • +
  • λ (float) – value of lambda for the damping matrix Wn

  • +
+
+
Returns:
+

inverse kinematic solution

+
+
Return type:
+

tuple (q, success, iterations, searches, residual)

+
+
+

sol = ets.ik_lm_chan(Tep) are the joint coordinates (n) corresponding +to the robot end-effector pose Tep which is an SE3 or ndarray object. +This method can be used for robots with any number of degrees of freedom. +The return value sol is a tuple with elements:

+

If success == 0 the q values will be valid numbers, but the +solution will be in error. The amount of error is indicated by +the residual.

+

Joint Limits:

+

sol = robot.ikine_LM(T, slimit=100) which is the deafualt for this method. +The solver will initialise a solution attempt with a random valid q0 and +perform a maximum of ilimit steps within this attempt. If a solution is not +found, this process is repeated up to slimit times.

+

Global search:

+

sol = robot.ikine_LM(T, reject_jl=True) is the deafualt for this method. +By setting reject_jl to True, the solver will discard any solution which +violates the defined joint limits of the robot. The solver will then +re-initialise with a new random q0 and repeat the process up to slimit times. +Note that finding a solution with valid joint coordinates takes longer than +without.

+

Underactuated robots:

+

For the case where the manipulator has fewer than 6 DOF the +solution space has more dimensions than can be spanned by the +manipulator joint coordinates.

+

In this case we specify the we option where the we vector +(6) specifies the Cartesian DOF (in the wrist coordinate frame) that +will be ignored in reaching a solution. The we vector has six +elements that correspond to translation in X, Y and Z, and rotation +about X, Y and Z respectively. The value can be 0 (for ignore) +or above to assign a priority relative to other Cartesian DoF. The number +of non-zero elements must equal the number of manipulator DOF.

+

For example when using a 3 DOF manipulator tool orientation might +be unimportant, in which case use the option we=[1, 1, 1, 0, 0, 0].

+
+

Note

+
    +
  • +
    See `Toolbox kinematics wiki page

    <https://github.com/petercorke/robotics-toolbox-python/wiki/Kinematics>`_

    +
    +
    +
  • +
  • Implements a Levenberg-Marquadt variable-damping solver.

  • +
  • +
    The tolerance is computed on the norm of the error between

    current and desired tool pose. This norm is computed from +distances and angles without any kind of weighting.

    +
    +
    +
  • +
  • +
    The inverse kinematic solution is generally not unique, and

    depends on the initial guess q0.

    +
    +
    +
  • +
+
+
+
References:
+

TODO

+
+
Seealso:
+

TODO

+
+
+
+ +
+
+ik_lm_wampler(Tep, q0=None, ilimit=30, slimit=100, tol=1e-06, reject_jl=True, we=None, λ=1.0)[source]
+

Numerical inverse kinematics by Levenberg-Marquadt optimization (Wamplers’s Method)

+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose or pose trajectory

  • +
  • q0 (Optional[ndarray]) – initial joint configuration (default to random valid joint +configuration contrained by the joint limits of the robot)

  • +
  • ilimit (int) – maximum number of iterations per search

  • +
  • slimit (int) – maximum number of search attempts

  • +
  • tol (float) – final error tolerance

  • +
  • reject_jl (bool) – constrain the solution to being within the joint limits of +the robot (reject solution with invalid joint configurations and perfrom +another search up to the slimit)

  • +
  • we (Optional[ndarray]) – a mask vector which weights the end-effector error priority. +Corresponds to translation in X, Y and Z and rotation about X, Y and Z +respectively

  • +
  • λ (float) – value of lambda for the damping matrix Wn

  • +
+
+
Returns:
+

inverse kinematic solution

+
+
Return type:
+

tuple (q, success, iterations, searches, residual)

+
+
+

sol = ets.ik_lm_chan(Tep) are the joint coordinates (n) corresponding +to the robot end-effector pose Tep which is an SE3 or ndarray object. +This method can be used for robots with any number of degrees of freedom. +The return value sol is a tuple with elements:

+

If success == 0 the q values will be valid numbers, but the +solution will be in error. The amount of error is indicated by +the residual.

+

Joint Limits:

+

sol = robot.ikine_LM(T, slimit=100) which is the deafualt for this method. +The solver will initialise a solution attempt with a random valid q0 and +perform a maximum of ilimit steps within this attempt. If a solution is not +found, this process is repeated up to slimit times.

+

Global search:

+

sol = robot.ikine_LM(T, reject_jl=True) is the deafualt for this method. +By setting reject_jl to True, the solver will discard any solution which +violates the defined joint limits of the robot. The solver will then +re-initialise with a new random q0 and repeat the process up to slimit times. +Note that finding a solution with valid joint coordinates takes longer than +without.

+

Underactuated robots:

+

For the case where the manipulator has fewer than 6 DOF the +solution space has more dimensions than can be spanned by the +manipulator joint coordinates.

+

In this case we specify the we option where the we vector +(6) specifies the Cartesian DOF (in the wrist coordinate frame) that +will be ignored in reaching a solution. The we vector has six +elements that correspond to translation in X, Y and Z, and rotation +about X, Y and Z respectively. The value can be 0 (for ignore) +or above to assign a priority relative to other Cartesian DoF. The number +of non-zero elements must equal the number of manipulator DOF.

+

For example when using a 3 DOF manipulator tool orientation might +be unimportant, in which case use the option we=[1, 1, 1, 0, 0, 0].

+
+

Note

+
    +
  • +
    See `Toolbox kinematics wiki page

    <https://github.com/petercorke/robotics-toolbox-python/wiki/Kinematics>`_

    +
    +
    +
  • +
  • Implements a Levenberg-Marquadt variable-damping solver.

  • +
  • +
    The tolerance is computed on the norm of the error between

    current and desired tool pose. This norm is computed from +distances and angles without any kind of weighting.

    +
    +
    +
  • +
  • +
    The inverse kinematic solution is generally not unique, and

    depends on the initial guess q0.

    +
    +
    +
  • +
+
+
+
References:
+

TODO

+
+
Seealso:
+

TODO

+
+
+
+ +
+
+ik_lm_sugihara(Tep, q0=None, ilimit=30, slimit=100, tol=1e-06, reject_jl=True, we=None, λ=1.0)[source]
+

Numerical inverse kinematics by Levenberg-Marquadt optimization (Sugihara’s Method)

+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose or pose trajectory

  • +
  • q0 (Optional[ndarray]) – initial joint configuration (default to random valid joint +configuration contrained by the joint limits of the robot)

  • +
  • ilimit (int) – maximum number of iterations per search

  • +
  • slimit (int) – maximum number of search attempts

  • +
  • tol (float) – final error tolerance

  • +
  • reject_jl (bool) – constrain the solution to being within the joint limits of +the robot (reject solution with invalid joint configurations and perfrom +another search up to the slimit)

  • +
  • we (Optional[ndarray]) – a mask vector which weights the end-effector error priority. +Corresponds to translation in X, Y and Z and rotation about X, Y and Z +respectively

  • +
  • λ (float) – value of lambda for the damping matrix Wn

  • +
+
+
Returns:
+

inverse kinematic solution

+
+
Return type:
+

tuple (q, success, iterations, searches, residual)

+
+
+

sol = ets.ik_lm_chan(Tep) are the joint coordinates (n) corresponding +to the robot end-effector pose Tep which is an SE3 or ndarray object. +This method can be used for robots with any number of degrees of freedom. +The return value sol is a tuple with elements:

+

If success == 0 the q values will be valid numbers, but the +solution will be in error. The amount of error is indicated by +the residual.

+

Joint Limits:

+

sol = robot.ikine_LM(T, slimit=100) which is the deafualt for this method. +The solver will initialise a solution attempt with a random valid q0 and +perform a maximum of ilimit steps within this attempt. If a solution is not +found, this process is repeated up to slimit times.

+

Global search:

+

sol = robot.ikine_LM(T, reject_jl=True) is the deafualt for this method. +By setting reject_jl to True, the solver will discard any solution which +violates the defined joint limits of the robot. The solver will then +re-initialise with a new random q0 and repeat the process up to slimit times. +Note that finding a solution with valid joint coordinates takes longer than +without.

+

Underactuated robots:

+

For the case where the manipulator has fewer than 6 DOF the +solution space has more dimensions than can be spanned by the +manipulator joint coordinates.

+

In this case we specify the we option where the we vector +(6) specifies the Cartesian DOF (in the wrist coordinate frame) that +will be ignored in reaching a solution. The we vector has six +elements that correspond to translation in X, Y and Z, and rotation +about X, Y and Z respectively. The value can be 0 (for ignore) +or above to assign a priority relative to other Cartesian DoF. The number +of non-zero elements must equal the number of manipulator DOF.

+

For example when using a 3 DOF manipulator tool orientation might +be unimportant, in which case use the option we=[1, 1, 1, 0, 0, 0].

+
+

Note

+
    +
  • +
    See `Toolbox kinematics wiki page

    <https://github.com/petercorke/robotics-toolbox-python/wiki/Kinematics>`_

    +
    +
    +
  • +
  • Implements a Levenberg-Marquadt variable-damping solver.

  • +
  • +
    The tolerance is computed on the norm of the error between

    current and desired tool pose. This norm is computed from +distances and angles without any kind of weighting.

    +
    +
    +
  • +
  • +
    The inverse kinematic solution is generally not unique, and

    depends on the initial guess q0.

    +
    +
    +
  • +
+
+
+
References:
+

TODO

+
+
Seealso:
+

TODO

+
+
+
+ +
+
+ik_nr(Tep, q0=None, ilimit=30, slimit=100, tol=1e-06, reject_jl=True, we=None, use_pinv=True, pinv_damping=0.0)[source]
+

Numerical inverse kinematics by Levenberg-Marquadt optimization (Newton-Raphson Method)

+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose or pose trajectory

  • +
  • q0 (Optional[ndarray]) – initial joint configuration (default to random valid joint +configuration contrained by the joint limits of the robot)

  • +
  • ilimit (int) – maximum number of iterations per search

  • +
  • slimit (int) – maximum number of search attempts

  • +
  • tol (float) – final error tolerance

  • +
  • reject_jl (bool) – constrain the solution to being within the joint limits of +the robot (reject solution with invalid joint configurations and perfrom +another search up to the slimit)

  • +
  • we (Optional[ndarray]) – a mask vector which weights the end-effector error priority. +Corresponds to translation in X, Y and Z and rotation about X, Y and Z +respectively

  • +
  • λ – value of lambda for the damping matrix Wn

  • +
+
+
Returns:
+

inverse kinematic solution

+
+
Return type:
+

tuple (q, success, iterations, searches, residual)

+
+
+

sol = ets.ik_lm_chan(Tep) are the joint coordinates (n) corresponding +to the robot end-effector pose Tep which is an SE3 or ndarray object. +This method can be used for robots with any number of degrees of freedom. +The return value sol is a tuple with elements:

+

If success == 0 the q values will be valid numbers, but the +solution will be in error. The amount of error is indicated by +the residual.

+

Joint Limits:

+

sol = robot.ikine_LM(T, slimit=100) which is the deafualt for this method. +The solver will initialise a solution attempt with a random valid q0 and +perform a maximum of ilimit steps within this attempt. If a solution is not +found, this process is repeated up to slimit times.

+

Global search:

+

sol = robot.ikine_LM(T, reject_jl=True) is the deafualt for this method. +By setting reject_jl to True, the solver will discard any solution which +violates the defined joint limits of the robot. The solver will then +re-initialise with a new random q0 and repeat the process up to slimit times. +Note that finding a solution with valid joint coordinates takes longer than +without.

+

Underactuated robots:

+

For the case where the manipulator has fewer than 6 DOF the +solution space has more dimensions than can be spanned by the +manipulator joint coordinates.

+

In this case we specify the we option where the we vector +(6) specifies the Cartesian DOF (in the wrist coordinate frame) that +will be ignored in reaching a solution. The we vector has six +elements that correspond to translation in X, Y and Z, and rotation +about X, Y and Z respectively. The value can be 0 (for ignore) +or above to assign a priority relative to other Cartesian DoF. The number +of non-zero elements must equal the number of manipulator DOF.

+

For example when using a 3 DOF manipulator tool orientation might +be unimportant, in which case use the option we=[1, 1, 1, 0, 0, 0].

+
+

Note

+
    +
  • +
    See `Toolbox kinematics wiki page

    <https://github.com/petercorke/robotics-toolbox-python/wiki/Kinematics>`_

    +
    +
    +
  • +
  • Implements a Levenberg-Marquadt variable-damping solver.

  • +
  • +
    The tolerance is computed on the norm of the error between

    current and desired tool pose. This norm is computed from +distances and angles without any kind of weighting.

    +
    +
    +
  • +
  • +
    The inverse kinematic solution is generally not unique, and

    depends on the initial guess q0.

    +
    +
    +
  • +
+
+
+
References:
+

TODO

+
+
Seealso:
+

TODO

+
+
+
+ +
+
+ik_gn(Tep, q0=None, ilimit=30, slimit=100, tol=1e-06, reject_jl=True, we=None, use_pinv=True, pinv_damping=0.0)[source]
+

Numerical inverse kinematics by Levenberg-Marquadt optimization (Gauss-Newton Method)

+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose or pose trajectory

  • +
  • q0 (Optional[ndarray]) – initial joint configuration (default to random valid joint +configuration contrained by the joint limits of the robot)

  • +
  • ilimit (int) – maximum number of iterations per search

  • +
  • slimit (int) – maximum number of search attempts

  • +
  • tol (float) – final error tolerance

  • +
  • reject_jl (bool) – constrain the solution to being within the joint limits of +the robot (reject solution with invalid joint configurations and perfrom +another search up to the slimit)

  • +
  • we (Optional[ndarray]) – a mask vector which weights the end-effector error priority. +Corresponds to translation in X, Y and Z and rotation about X, Y and Z +respectively

  • +
  • λ – value of lambda for the damping matrix Wn

  • +
+
+
Returns:
+

inverse kinematic solution

+
+
Return type:
+

tuple (q, success, iterations, searches, residual)

+
+
+

sol = ets.ik_lm_chan(Tep) are the joint coordinates (n) corresponding +to the robot end-effector pose Tep which is an SE3 or ndarray object. +This method can be used for robots with any number of degrees of freedom. +The return value sol is a tuple with elements:

+

If success == 0 the q values will be valid numbers, but the +solution will be in error. The amount of error is indicated by +the residual.

+

Joint Limits:

+

sol = robot.ikine_LM(T, slimit=100) which is the deafualt for this method. +The solver will initialise a solution attempt with a random valid q0 and +perform a maximum of ilimit steps within this attempt. If a solution is not +found, this process is repeated up to slimit times.

+

Global search:

+

sol = robot.ikine_LM(T, reject_jl=True) is the deafualt for this method. +By setting reject_jl to True, the solver will discard any solution which +violates the defined joint limits of the robot. The solver will then +re-initialise with a new random q0 and repeat the process up to slimit times. +Note that finding a solution with valid joint coordinates takes longer than +without.

+

Underactuated robots:

+

For the case where the manipulator has fewer than 6 DOF the +solution space has more dimensions than can be spanned by the +manipulator joint coordinates.

+

In this case we specify the we option where the we vector +(6) specifies the Cartesian DOF (in the wrist coordinate frame) that +will be ignored in reaching a solution. The we vector has six +elements that correspond to translation in X, Y and Z, and rotation +about X, Y and Z respectively. The value can be 0 (for ignore) +or above to assign a priority relative to other Cartesian DoF. The number +of non-zero elements must equal the number of manipulator DOF.

+

For example when using a 3 DOF manipulator tool orientation might +be unimportant, in which case use the option we=[1, 1, 1, 0, 0, 0].

+
+

Note

+
    +
  • +
    See `Toolbox kinematics wiki page

    <https://github.com/petercorke/robotics-toolbox-python/wiki/Kinematics>`_

    +
    +
    +
  • +
  • Implements a Levenberg-Marquadt variable-damping solver.

  • +
  • +
    The tolerance is computed on the norm of the error between

    current and desired tool pose. This norm is computed from +distances and angles without any kind of weighting.

    +
    +
    +
  • +
  • +
    The inverse kinematic solution is generally not unique, and

    depends on the initial guess q0.

    +
    +
    +
  • +
+
+
+
References:
+

TODO

+
+
Seealso:
+

TODO

+
+
+
+ +
+
+ikine_LM(Tep, q0=None, ilimit=30, slimit=100, tol=1e-06, joint_limits=False, mask=None, seed=None)[source]
+

Levenberg-Marquadt Numerical Inverse Kinematics Solver

+

A method which provides functionality to perform numerical inverse kinematics (IK) +using the Levemberg-Marquadt method.

+

See the Inverse Kinematics Docs Page for more details and for a +tutorial on numerical IK, see here.

+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose

  • +
  • end – the link considered as the end-effector

  • +
  • start – the link considered as the base frame, defaults to the robots’s base frame

  • +
  • q0 (Union[list, ndarray, tuple, set, None]) – The initial joint coordinate vector

  • +
  • ilimit (int) – How many iterations are allowed within a search before a new search +is started

  • +
  • slimit (int) – How many searches are allowed before being deemed unsuccessful

  • +
  • tol (float) – Maximum allowed residual error E

  • +
  • mask (Union[list, ndarray, tuple, set, None]) – A 6 vector which assigns weights to Cartesian degrees-of-freedom +error priority

  • +
  • joint_limits (bool) – Reject solutions with joint limit violations

  • +
  • seed (Optional[int]) – A seed for the private RNG used to generate random joint coordinate +vectors

  • +
  • k – Sets the gain value for the damping matrix Wn in the next iteration. See +synopsis

  • +
  • method – One of “chan”, “sugihara” or “wampler”. Defines which method is used +to calculate the damping matrix Wn in the step method

  • +
  • kq – The gain for joint limit avoidance. Setting to 0.0 will remove this +completely from the solution

  • +
  • km – The gain for maximisation. Setting to 0.0 will remove this completely +from the solution

  • +
  • ps – The minimum angle/distance (in radians or metres) in which the joint is +allowed to approach to its limit

  • +
  • pi – The influence angle/distance (in radians or metres) in null space motion +becomes active

  • +
+
+
+

Synopsis

+

The operation is defined by the choice of the method kwarg.

+

The step is deined as

+
+\[\begin{split}\vec{q}_{k+1} +&= +\vec{q}_k + +\left( + \mat{A}_k +\right)^{-1} +\bf{g}_k \\ +% +\mat{A}_k +&= +{\mat{J}(\vec{q}_k)}^\top +\mat{W}_e \ +{\mat{J}(\vec{q}_k)} ++ +\mat{W}_n\end{split}\]
+

where \(\mat{W}_n = \text{diag}(\vec{w_n})(\vec{w_n} \in \mathbb{R}^n_{>0})\) is a +diagonal damping matrix. The damping matrix ensures that \(\mat{A}_k\) is +non-singular and positive definite. The performance of the LM method largely depends +on the choice of \(\mat{W}_n\).

+

Chan’s Method

+

Chan proposed

+
+\[\mat{W}_n += +λ E_k \mat{1}_n\]
+

where λ is a constant which reportedly does not have much influence on performance. +Use the kwarg k to adjust the weighting term λ.

+

Sugihara’s Method

+

Sugihara proposed

+
+\[\mat{W}_n += +E_k \mat{1}_n + \text{diag}(\hat{\vec{w}}_n)\]
+

where \(\hat{\vec{w}}_n \in \mathbb{R}^n\), \(\hat{w}_{n_i} = l^2 \sim 0.01 l^2\), +and \(l\) is the length of a typical link within the manipulator. We provide the +variable k as a kwarg to adjust the value of \(w_n\).

+

Wampler’s Method

+

Wampler proposed \(\vec{w_n}\) to be a constant. This is set through the k kwarg.

+

Examples

+

The following example makes a panda robot object, makes a goal +pose Tep, and then solves for the joint coordinates which result in the pose +Tep using the ikine_LM method.

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda()
+>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> panda.ikine_LM(Tep)
+IKSolution(q=array([-2.036 ,  0.549 ,  2.2166, -2.1763, -0.4799,  1.8933,  1.1613]), success=True, iterations=11, searches=1, residual=2.2844759740298008e-07, reason='Success')
+
+
+

Notes

+

The value for the k kwarg will depend on the method chosen and the arm you are +using. Use the following as a rough guide chan, k = 1.0 - 0.01, +wampler, k = 0.01 - 0.0001, and sugihara, k = 0.1 - 0.0001

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian.

+

This class supports null-space motion to assist with maximising manipulability and +avoiding joint limits. These are enabled by setting kq and km to non-zero values.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
IK_LM

An IK Solver class which implements the Levemberg Marquadt optimisation technique

+
+
ikine_NR

Implements the IK_NR class as a method within the Robot class

+
+
ikine_GN

Implements the IK_GN class as a method within the Robot class

+
+
ikine_QP

Implements the IK_QP class as a method within the Robot class

+
+
+
+
+

Changed in version 1.0.4: Added the Levemberg-Marquadt IK solver method on the Robot class

+
+
+ +
+
+classmethod URDF(file_path, gripper=None)
+

Construct a Robot object from URDF file

+
+
Parameters:
+
    +
  • file_path (str) – the path to the URDF

  • +
  • gripper (Union[int, str, None]) – index or name of the gripper link(s)

  • +
+
+
Returns:
+

    +
  • If gripper is specified, links from that link outward are removed

  • +
  • from the rigid-body tree and folded into a Gripper object.

  • +
+

+
+
+
+ +
+
+static URDF_read(file_path, tld=None, xacro_tld=None)
+

Read a URDF file as Links

+

File should be specified relative to RTBDATA/URDF/xacro

+
+
Parameters:
+
    +
  • file_path – File path relative to the xacro folder

  • +
  • tld – A custom top-level directory which holds the xacro data, +defaults to None

  • +
  • xacro_tld – A custom top-level within the xacro data, +defaults to None

  • +
+
+
Return type:
+

Tuple[List[Link], str, str, Union[Path, PurePosixPath]]

+
+
Returns:
+

    +
  • links – a list of links

  • +
  • name – the name of the robot

  • +
  • urdf – a string representing the URDF

  • +
  • file_path – a path to the original file

  • +
+

+
+
+

Notes

+

If tld is not supplied, filepath pointing to xacro data should +be directly under RTBDATA/URDF/xacro OR under ./xacro relative +to the model file calling this method. If tld is supplied, then +`file_path` needs to be relative to tld

+
+ +
+
+__getitem__(i)
+

Get link

+

This also supports iterating over each link in the robot object, +from the base to the tool.

+
+
Parameters:
+

i (Union[int, str]) – link number or name

+
+
Returns:
+

i’th link or named link

+
+
Return type:
+

link

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> print(robot[1]) # print the 2nd link
+RevoluteDH:   θ=q,  d=0,  a=0.4318,  ⍺=0.0
+>>> print([link.a for link in robot])  # print all the a_j values
+[0, 0.4318, 0.0203, 0, 0, 0]
+
+
+

Notes

+
+
Robot supports link lookup by name,

eg. robot['link1']

+
+
+
+ +
+
+accel(q, qd, torque, gravity=None)
+

Compute acceleration due to applied torque

+

qdd = accel(q, qd, torque) calculates a vector (n) of joint +accelerations that result from applying the actuator force/torque (n) +to the manipulator in state q (n) and qd (n), and n is +the number of robot joints.

+
+\[\ddot{q} = \mathbf{M}^{-1} \left(\tau - \mathbf{C}(q)\dot{q} - \mathbf{g}(q)\right)\]
+

Trajectory operation

+

If q, qd, torque are matrices (m,n) then qdd is a matrix (m,n) +where each row is the acceleration corresponding to the equivalent rows +of q, qd, torque.

+
+
Parameters:
+
    +
  • q – Joint coordinates

  • +
  • qd – Joint velocity

  • +
  • torque – Joint torques of the robot

  • +
  • gravity – Gravitational acceleration (Optional, if not supplied will +use the gravity attribute of self).

  • +
+
+
Return type:
+

ndarray(n)

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> puma.accel(puma.qz, 0.5 * np.ones(6), np.zeros(6))
+array([ -7.5544, -12.22  ,  -6.4022,  -5.4303,  -4.9518,  -2.1178])
+
+
+

Notes

+
    +
  • +
    Useful for simulation of manipulator dynamics, in

    conjunction with a numerical integration function.

    +
    +
    +
  • +
  • +
    Uses the method 1 of Walker and Orin to compute the forward

    dynamics.

    +
    +
    +
  • +
  • +
    Featherstone’s method is more efficient for robots with large

    numbers of joints.

    +
    +
    +
  • +
  • Joint friction is considered.

  • +
+

References

+
    +
  • +
    Efficient dynamic computer simulation of robotic mechanisms,

    M. W. Walker and D. E. Orin, +ASME Journa of Dynamic Systems, Measurement and Control, vol. +104, no. 3, pp. 205-211, 1982.

    +
    +
    +
  • +
+
+ +
+
+accel_x(q, xd, wrench, gravity=None, pinv=False, representation='rpy/xyz')
+

Operational space acceleration due to applied wrench

+

xdd = accel_x(q, qd, wrench) is the operational space acceleration +due to wrench applied to the end-effector of a robot in joint +configuration q and joint velocity qd.

+
+\[\ddot{x} = \mathbf{J}(q) \mathbf{M}(q)^{-1} \left( + \mathbf{J}(q)^T w - \mathbf{C}(q)\dot{q} - \mathbf{g}(q) + \right)\]
+

Trajectory operation

+

If q, qd, torque are matrices (m,n) then qdd is a matrix (m,n) +where each row is the acceleration corresponding to the equivalent rows +of q, qd, wrench.

+
+
Parameters:
+
    +
  • q – Joint coordinates

  • +
  • qd – Joint velocity

  • +
  • wrench – Wrench applied to the end-effector

  • +
  • gravity – Gravitational acceleration (Optional, if not supplied will +use the gravity attribute of self).

  • +
  • pinv – use pseudo inverse rather than inverse

  • +
  • analytical – the type of analytical Jacobian to use, default is +‘rpy/xyz’

  • +
  • xd

  • +
  • representation – (Default value = “rpy/xyz”)

  • +
+
+
Returns:
+

Operational space accelerations of the end-effector

+
+
Return type:
+

accel

+
+
+

Examples

+
    ainv = _umath_linalg.inv(a, signature=signature, extobj=extobj)
+  File "/opt/hostedtoolcache/Python/3.7.17/x64/lib/python3.7/site-packages/numpy/linalg/linalg.py", line 88, in _raise_linalgerror_singular
+    raise LinAlgError("Singular matrix")
+numpy.linalg.LinAlgError: Singular matrix
+
+
+

Notes

+
    +
  • +
    Useful for simulation of manipulator dynamics, in

    conjunction with a numerical integration function.

    +
    +
    +
  • +
  • +
    Uses the method 1 of Walker and Orin to compute the forward

    dynamics.

    +
    +
    +
  • +
  • +
    Featherstone’s method is more efficient for robots with large

    numbers of joints.

    +
    +
    +
  • +
  • Joint friction is considered.

  • +
+
+

See also

+

accel()

+
+
+ +
+
+addconfiguration(name, q)
+

Add a named joint configuration

+

Add a named configuration to the robot instance’s dictionary of named +configurations.

+
+
Parameters:
+
    +
  • name (str) – Name of the joint configuration

  • +
  • q (ndarray) – Joint configuration

  • +
+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.addconfiguration_attr("mypos", [0.1, 0.2, 0.3, 0.4, 0.5, 0.6])
+>>> robot.configs["mypos"]
+array([0.1, 0.2, 0.3, 0.4, 0.5, 0.6])
+
+
+
+

See also

+

addconfiguration()

+
+
+ +
+
+addconfiguration_attr(name, q, unit='rad')
+

Add a named joint configuration as an attribute

+
+
Parameters:
+
+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.addconfiguration_attr("mypos", [0.1, 0.2, 0.3, 0.4, 0.5, 0.6])
+>>> robot.mypos
+array([0.1, 0.2, 0.3, 0.4, 0.5, 0.6])
+>>> robot.configs["mypos"]
+array([0.1, 0.2, 0.3, 0.4, 0.5, 0.6])
+
+
+

Notes

+
    +
  • Used in robot model init method to store the qr configuration

  • +
  • +
    Dynamically adding attributes to objects can cause issues with

    Python type checking.

    +
    +
    +
  • +
  • +
    Configuration is also added to the robot instance’s dictionary of

    named configurations.

    +
    +
    +
  • +
+
+

See also

+

addconfiguration()

+
+
+ +
+
+attach(object)
+
+ +
+
+attach_to(object)
+
+ +
+
+property base: SE3
+

Get/set robot base transform

+
    +
  • robot.base is the robot base transform

  • +
  • robot.base = ... checks and sets the robot base transform

  • +
+
+
Parameters:
+

base – the new robot base transform

+
+
Returns:
+

the current robot base transform

+
+
Return type:
+

base

+
+
+
+ +
+ +

Get the robot base link

+
    +
  • robot.base_link is the robot base link

  • +
+
+
Returns:
+

the first link in the robot tree

+
+
Return type:
+

base_link

+
+
+
+ +
+
+cinertia(q)
+

Deprecated, use inertia_x

+
+ +
+
+closest_point(q, shape, inf_dist=1.0, skip=False)
+

Find the closest point between robot and shape

+

closest_point(shape, inf_dist) returns the minimum euclidean +distance between this robot and shape, provided it is less than +inf_dist. It will also return the points on self and shape in the +world frame which connect the line of length distance between the +shapes. If the distance is negative then the shapes are collided.

+
+
+shape
+

The shape to compare distance to

+
+ +
+
+inf_dist
+

The minimum distance within which to consider +the shape

+
+ +
+
+skip
+

Skip setting all shape transforms based on q, use this +option if using this method in conjuction with Swift to save time

+
+ +
+
Return type:
+

Tuple[Optional[int], Optional[ndarray], Optional[ndarray]]

+
+
Returns:
+

    +
  • d – distance between the robot and shape

  • +
  • p1 – [x, y, z] point on the robot (in the world frame)

  • +
  • p2 – [x, y, z] point on the shape (in the world frame)

  • +
+

+
+
+
+ +
+
+collided(q, shape, skip=False)
+

Check if the robot is in collision with a shape

+

collided(shape) checks if this robot and shape have collided

+
+
+shape
+

The shape to compare distance to

+
+ +
+
+skip
+

Skip setting all shape transforms based on q, use this +option if using this method in conjuction with Swift to save time

+
+ +
+
Returns:
+

True if shapes have collided

+
+
Return type:
+

collided

+
+
+
+ +
+
+property comment: str
+

Get/set robot comment

+
    +
  • robot.comment is the robot comment

  • +
  • robot.comment = ... checks and sets the robot comment

  • +
+
+
Parameters:
+

name – the new robot comment

+
+
Returns:
+

robot comment

+
+
Return type:
+

comment

+
+
+
+ +
+
+property configs: Dict[str, ndarray]
+
+ +
+
+configurations_str(border='thin')
+
+ +
+
+property control_mode: str
+

Get/set robot control mode

+
    +
  • robot.control_type is the robot control mode

  • +
  • robot.control_type = ... checks and sets the robot control mode

  • +
+
+
Parameters:
+

control_mode – the new robot control mode

+
+
Returns:
+

the current robot control mode

+
+
Return type:
+

control_mode

+
+
+
+ +
+
+copy()
+
+ +
+
+coriolis(q, qd)
+

Coriolis and centripetal term

+

coriolis(q, qd) calculates the Coriolis/centripetal matrix (n,n) +for the robot in configuration q and velocity qd, where n +is the number of joints.

+

The product \(\mathbf{C} \dot{q}\) is the vector of joint +force/torque due to velocity coupling. The diagonal elements are due to +centripetal effects and the off-diagonal elements are due to Coriolis +effects. This matrix is also known as the velocity coupling matrix, +since it describes the disturbance forces on any joint due to +velocity of all other joints.

+

Trajectory operation

+

If q and qd are matrices (m,n), each row is interpretted as a +joint configuration, and the result (n,n,m) is a 3d-matrix where +each plane corresponds to a row of q and qd.

+
+
Parameters:
+
    +
  • q – Joint coordinates

  • +
  • qd – Joint velocity

  • +
+
+
Returns:
+

Velocity matrix

+
+
Return type:
+

coriolis

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> puma.coriolis(puma.qz, 0.5 * np.ones((6,)))
+array([[-0.4017, -0.5513, -0.2025, -0.0007, -0.0013,  0.    ],
+       [ 0.2023, -0.1937, -0.3868, -0.    , -0.002 ,  0.    ],
+       [ 0.1987,  0.193 , -0.    ,  0.    , -0.0001,  0.    ],
+       [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+       [ 0.0007,  0.0007,  0.0001,  0.    ,  0.    ,  0.    ],
+       [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ]])
+
+
+

Notes

+
    +
  • +
    Joint viscous friction is also a joint force proportional to

    velocity but it is eliminated in the computation of this value.

    +
    +
    +
  • +
  • Computationally slow, involves \(n^2/2\) invocations of RNE.

  • +
+
+ +
+
+coriolis_x(q, qd, pinv=False, representation='rpy/xyz', J=None, Ji=None, Jd=None, C=None, Mx=None)
+

Operational space Coriolis and centripetal term

+

coriolis_x(q, qd) is the Coriolis/centripetal matrix (m,m) +in operational space for the robot in configuration q and velocity +qd, where n is the number of joints.

+
+\[\mathbf{C}_x = \mathbf{J}(q)^{-T} \left( + \mathbf{C}(q) - \mathbf{M}_x(q) \mathbf{J})(q) + \right) \mathbf{J}(q)^{-1}\]
+

The product \(\mathbf{C} \dot{x}\) is the operational space wrench +due to joint velocity coupling. This matrix is also known as the +velocity coupling matrix, since it describes the disturbance forces on +any joint due to velocity of all other joints.

+

The transformation to operational space requires an analytical, rather +than geometric, Jacobian. analytical can be one of:

+
+
+ + + + + + + + + + + + + + + + + + + +

Value

Rotational representation

'rpy/xyz'

RPY angular rates in XYZ order (default)

'rpy/zyx'

RPY angular rates in XYZ order

'eul'

Euler angular rates in ZYZ order

'exp'

exponential coordinate rates

+
+

Trajectory operation

+

If q and qd are matrices (m,n), each row is interpretted as a +joint configuration, and the result (n,n,m) is a 3d-matrix where +each plane corresponds to a row of q and qd.

+
+
Parameters:
+
    +
  • q – Joint coordinates

  • +
  • qd – Joint velocity

  • +
  • pinv – use pseudo inverse rather than inverse (Default value = False)

  • +
  • analytical – the type of analytical Jacobian to use, default is +‘rpy/xyz’

  • +
  • representation – (Default value = “rpy/xyz”)

  • +
  • J

  • +
  • Ji

  • +
  • Jd

  • +
  • C

  • +
  • Mx

  • +
+
+
Returns:
+

Operational space velocity matrix

+
+
Return type:
+

ndarray(6,6) or ndarray(m,6,6)

+
+
+

Examples

+
    ainv = _umath_linalg.inv(a, signature=signature, extobj=extobj)
+  File "/opt/hostedtoolcache/Python/3.7.17/x64/lib/python3.7/site-packages/numpy/linalg/linalg.py", line 88, in _raise_linalgerror_singular
+    raise LinAlgError("Singular matrix")
+numpy.linalg.LinAlgError: Singular matrix
+
+
+

Notes

+
    +
  • +
    Joint viscous friction is also a joint force proportional to

    velocity but it is eliminated in the computation of this value.

    +
    +
    +
  • +
  • Computationally slow, involves \(n^2/2\) invocations of RNE.

  • +
  • If the robot is not 6 DOF the pinv option is set True.

  • +
  • pinv() is around 5x slower than inv()

  • +
+
+

Warning

+

Assumes that the operational space has 6 DOF.

+
+ +
+ +
+
+property default_backend
+

Get default graphical backend

+
    +
  • +
    robot.default_backend Get the default graphical backend, used when

    no explicit backend is passed to Robot.plot.

    +
    +
    +
  • +
  • +
    robot.default_backend = ... Set the default graphical backend, used when

    no explicit backend is passed to Robot.plot. The default set here will +be overridden if the particular Robot subclass cannot support it.

    +
    +
    +
  • +
+
+
Returns:
+

backend name

+
+
Return type:
+

default_backend

+
+
+
+ +
+ +

A link search method

+

Visit all links from start in depth-first order and will apply +func to each visited link

+
+
Parameters:
+
+
+
Returns:
+

A list of links

+
+
Return type:
+

links

+
+
+
+ +
+
+dotfile(filename, etsbox=False, ets='full', jtype=False, static=True)
+

Write a link transform graph as a GraphViz dot file

+
+
The file can be processed using dot:

% dot -Tpng -o out.png dotfile.dot

+
+
The nodes are:
    +
  • Base is shown as a grey square. This is the world frame origin, +but can be changed using the base attribute of the robot.

  • +
  • Link frames are indicated by circles

  • +
  • ETS transforms are indicated by rounded boxes

  • +
+
+
The edges are:
    +
  • an arrow if jtype is False or the joint is fixed

  • +
  • an arrow with a round head if jtype is True and the joint is +revolute

  • +
  • an arrow with a box head if jtype is True and the joint is +prismatic

  • +
+
+
+

Edge labels or nodes in blue have a fixed transformation to the +preceding link.

+
+

Note

+
+
If filename is a file object then the file will not

be closed after the GraphViz model is written.

+
+
+
+
+
Parameters:
+
    +
  • file – Name of file to write to

  • +
  • etsbox (bool) – Put the link ETS in a box, otherwise an edge label

  • +
  • ets (Literal['full', 'brief']) – Display the full ets with “full” or a brief version with “brief”

  • +
  • jtype (bool) – Arrowhead to node indicates revolute or prismatic type

  • +
  • static (bool) – Show static joints in blue and bold

  • +
+
+
+
+

See also

+

showgraph()

+
+
+ +
+
+dynamics()
+

Pretty print the dynamic parameters (Robot superclass)

+

The dynamic parameters (inertial and friction) are printed in a table, +with one row per link.

+

Examples

+

+
+
+
+ +
+
+dynamics_list()
+

Print dynamic parameters (Robot superclass)

+

Display the kinematic and dynamic parameters to the console in +reable format

+
+ +
+
+dynchanged(what=None)
+

Dynamic parameters have changed

+

Called from a property setter to inform the robot that the cache of +dynamic parameters is invalid.

+
+

See also

+

roboticstoolbox.Link._listen_dyn()

+
+
+ +
+ +
+ +
+
+fdyn(T, q0, Q=None, Q_args={}, qd0=None, solver='RK45', solver_args={}, dt=None, progress=False)
+

Integrate forward dynamics

+

tg = R.fdyn(T, q) integrates the dynamics of the robot with zero +input torques over the time interval 0 to T and returns the +trajectory as a namedtuple with elements:

+
    +
  • t the time vector (M,)

  • +
  • q the joint coordinates (M,n)

  • +
  • qd the joint velocities (M,n)

  • +
+

tg = R.fdyn(T, q, torqfun) as above but the torque applied to the +joints is given by the provided function:

+
tau = function(robot, t, q, qd, **args)
+
+
+

where the inputs are:

+
+
    +
  • the robot object

  • +
  • current time

  • +
  • current joint coordinates (n,)

  • +
  • current joint velocity (n,)

  • +
  • args, optional keyword arguments can be specified, these are

  • +
+

passed in from the targs kewyword argument.

+
+

The function must return a Numpy array (n,) of joint forces/torques.

+
+
Parameters:
+
+
+
Returns:
+

robot trajectory

+
+
Return type:
+

trajectory

+
+
+

Examples

+

To apply zero joint torque to the robot without Coulomb +friction:

+
>>> def myfunc(robot, t, q, qd):
+>>>     return np.zeros((robot.n,))
+
+
+
>>> tg = robot.nofriction().fdyn(5, q0, myfunc)
+
+
+
>>> plt.figure()
+>>> plt.plot(tg.t, tg.q)
+>>> plt.show()
+
+
+

We could also use a lambda function:

+
>>> tg = robot.nofriction().fdyn(
+>>>     5, q0, lambda r, t, q, qd: np.zeros((r.n,)))
+
+
+

The robot is controlled by a PD controller. We first define a +function to compute the control which has additional parameters for +the setpoint and control gains (qstar, P, D):

+
>>> def myfunc(robot, t, q, qd, qstar, P, D):
+>>>     return (qstar - q) * P + qd * D  # P, D are (6,)
+
+
+
>>> tg = robot.fdyn(10, q0, myfunc, torque_args=(qstar, P, D)) )
+
+
+

Many integrators have variable step length which is problematic if we +want to animate the result. If dt is specified then the solver +results are interpolated in time steps of dt.

+

Notes

+ + +
+ +
+
+fellipse(q, opt='trans', unit='rad', centre=[0, 0, 0])
+

Create a force ellipsoid object for plotting with PyPlot

+

robot.fellipse(q) creates a force ellipsoid for the robot at +pose q. The ellipsoid is centered at the origin.

+
+
+q
+

The joint configuration of the robot.

+
+ +
+
+opt
+

‘trans’ or ‘rot’ will plot either the translational or +rotational force ellipsoid

+
+ +
+
+unit
+

‘rad’ or ‘deg’ will plot the ellipsoid in radians or +degrees

+
+ +
+
+centre
+

The centre of the ellipsoid, either ‘ee’ for the end-effector +or a 3-vector [x, y, z] in the world frame

+
+ +
+
Returns:
+

An EllipsePlot object

+
+
Return type:
+

env

+
+
+

Notes

+
    +
  • +
    By default the ellipsoid related to translational motion is

    drawn. Use opt='rot' to draw the rotational velocity +ellipsoid.

    +
    +
    +
  • +
  • +
    By default the ellipsoid is drawn at the origin. The option

    centre allows its origin to set to set to the specified +3-vector, or the string “ee” ensures it is drawn at the +end-effector position.

    +
    +
    +
  • +
+
+ +
+
+friction(qd)
+

Manipulator joint friction (Robot superclass)

+

robot.friction(qd) is a vector of joint friction +forces/torques for the robot moving with joint velocities qd.

+

The friction model includes:

+
    +
  • Viscous friction which is a linear function of velocity.

  • +
  • Coulomb friction which is proportional to sign(qd).

  • +
+
+\[\begin{split}\tau_j = G^2 B \dot{q}_j + |G_j| \left\{ \begin{array}{ll} + \tau_{C,j}^+ & \mbox{if $\dot{q}_j > 0$} \\ + \tau_{C,j}^- & \mbox{if $\dot{q}_j < 0$} \end{array} \right.\end{split}\]
+
+
Parameters:
+

qd (ndarray) – The joint velocities of the robot

+
+
Returns:
+

The joint friction forces/torques for the robot

+
+
Return type:
+

friction

+
+
+

Notes

+
    +
  • +
    The friction value should be added to the motor output torque to

    determine the nett torque. It has a negative value when qd > 0.

    +
    +
    +
  • +
  • +
    The returned friction value is referred to the output of the

    gearbox.

    +
    +
    +
  • +
  • +
    The friction parameters in the Link object are referred to the

    motor.

    +
    +
    +
  • +
  • Motor viscous friction is scaled up by \(G^2\).

  • +
  • Motor Coulomb friction is scaled up by math:G.

  • +
  • +
    The appropriate Coulomb friction value to use in the

    non-symmetric case depends on the sign of the joint velocity, +not the motor velocity.

    +
    +
    +
  • +
  • +
    Coulomb friction is zero for zero joint velocity, stiction is

    not modeled.

    +
    +
    +
  • +
  • +
    The absolute value of the gear ratio is used. Negative gear

    ratios are tricky: the Puma560 robot has negative gear ratio for +joints 1 and 3.

    +
    +
    +
  • +
  • +
    The absolute value of the gear ratio is used. Negative gear

    ratios are tricky: the Puma560 has negative gear ratio for +joints 1 and 3.

    +
    +
    +
  • +
+
+

See also

+

Robot.nofriction(), Link.friction()

+
+
+ +
+
+get_path(end=None, start=None)
+

Find a path from start to end

+
+
Parameters:
+
+
+
Raises:
+

ValueError – link not known or ambiguous

+
+
Return type:
+

Tuple[List[TypeVar(LinkType, bound= BaseLink)], int, SE3]

+
+
Returns:
+

    +
  • path – the path from start to end

  • +
  • n – the number of joints in the path

  • +
  • T – the tool transform present after end

  • +
+

+
+
+
+ +
+
+property gravity: ndarray
+

Get/set default gravitational acceleration (Robot superclass)

+
    +
  • robot.name is the default gravitational acceleration

  • +
  • +
    robot.name = ... checks and sets default gravitational

    acceleration

    +
    +
    +
  • +
+
+
Parameters:
+

gravity – the new gravitational acceleration for this robot

+
+
Returns:
+

gravitational acceleration

+
+
Return type:
+

gravity

+
+
+

Notes

+

If the z-axis is upward, out of the Earth, this should be +a positive number.

+
+ +
+
+gravload(q=None, gravity=None)
+

Compute gravity load

+

robot.gravload(q) calculates the joint gravity loading (n) for +the robot in the joint configuration q and using the default +gravitational acceleration specified in the DHRobot object.

+

robot.gravload(q, gravity=g) as above except the gravitational +acceleration is explicitly specified as g.

+

Trajectory operation

+

If q is a matrix (nxm) each column is interpreted as a joint +configuration vector, and the result is a matrix (nxm) each column +being the corresponding joint torques.

+
+
Parameters:
+
+
+
Returns:
+

The generalised joint force/torques due to gravity

+
+
Return type:
+

gravload

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> puma.gravload(puma.qz)
+array([ 0.    , 37.4837,  0.2489,  0.    ,  0.    ,  0.    ])
+
+
+
+ +
+
+gravload_x(q=None, gravity=None, pinv=False, representation='rpy/xyz', Ji=None)
+

Operational space gravity load

+

robot.gravload_x(q) calculates the gravity wrench for +the robot in the joint configuration q and using the default +gravitational acceleration specified in the robot object.

+

robot.gravload_x(q, gravity=g) as above except the gravitational +acceleration is explicitly specified as g.

+
+\[\mathbf{G}_x = \mathbf{J}(q)^{-T} \mathbf{G}(q)\]
+

The transformation to operational space requires an analytical, rather +than geometric, Jacobian. analytical can be one of:

+
+
+ + + + + + + + + + + + + + + + + + + +

Value

Rotational representation

'rpy/xyz'

RPY angular rates in XYZ order (default)

'rpy/zyx'

RPY angular rates in XYZ order

'eul'

Euler angular rates in ZYZ order

'exp'

exponential coordinate rates

+
+

Trajectory operation

+

If q is a matrix (nxm) each column is interpreted as a joint +configuration vector, and the result is a matrix (nxm) each column +being the corresponding joint torques.

+
+
Parameters:
+
    +
  • q – Joint coordinates

  • +
  • gravity – Gravitational acceleration (Optional, if not supplied will +use the gravity attribute of self).

  • +
  • pinv – use pseudo inverse rather than inverse (Default value = False)

  • +
  • analytical – the type of analytical Jacobian to use, default is +‘rpy/xyz’

  • +
  • representation – (Default value = “rpy/xyz”)

  • +
  • Ji

  • +
+
+
Returns:
+

The operational space gravity wrench

+
+
Return type:
+

gravload

+
+
+

Examples

+
    ainv = _umath_linalg.inv(a, signature=signature, extobj=extobj)
+  File "/opt/hostedtoolcache/Python/3.7.17/x64/lib/python3.7/site-packages/numpy/linalg/linalg.py", line 88, in _raise_linalgerror_singular
+    raise LinAlgError("Singular matrix")
+numpy.linalg.LinAlgError: Singular matrix
+
+
+

Notes

+
    +
  • If the robot is not 6 DOF the pinv option is set True.

  • +
  • pinv() is around 5x slower than inv()

  • +
+
+

Warning

+

Assumes that the operational space has 6 DOF.

+
+
+

See also

+

gravload()

+
+
+ +
+
+property grippers: List[Gripper]
+

Grippers attached to the robot

+
+
Returns:
+

A list of grippers

+
+
Return type:
+

grippers

+
+
+
+ +
+
+property hascollision
+

Robot has collision model

+
+
Returns:
+

    +
  • hascollision – Robot has collision model

  • +
  • At least one link has associated collision model.

  • +
+

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.hascollision
+False
+
+
+
+

See also

+

hasgeometry()

+
+
+ +
+
+property hasdynamics
+

Robot has dynamic parameters

+
+
Returns:
+

    +
  • hasdynamics – Robot has dynamic parameters

  • +
  • At least one link has associated dynamic parameters.

  • +
+

+
+
+

Examples

+

+
+
+
+ +
+
+property hasgeometry
+

Robot has geometry model

+

At least one link has associated mesh to describe its shape.

+
+
Returns:
+

Robot has geometry model

+
+
Return type:
+

hasgeometry

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.hasgeometry
+True
+
+
+
+

See also

+

hascollision()

+
+
+ +
+
+hessiane(q=None, end=None, start=None, Je=None, tool=None)
+

Manipulator Hessian

+

The manipulator Hessian tensor maps joint acceleration to end-effector +spatial acceleration, expressed in the end coordinate frame. This +function calulcates this based on the ETS of the robot. One of J0 or q +is required. Supply J0 if already calculated to save computation time

+
+
Parameters:
+
    +
  • q – The joint angles/configuration of the robot (Optional, +if not supplied will use the stored q values).

  • +
  • end (Union[str, Link, Gripper, None]) – the final link/Gripper which the Hessian represents

  • +
  • start (Union[str, Link, Gripper, None]) – the first link which the Hessian represents

  • +
  • J0 – The manipulator Jacobian in the end frame

  • +
  • tool (Union[ndarray, SE3, None]) – a static tool transformation matrix to apply to the +end of end, defaults to None

  • +
+
+
Returns:
+

The manipulator Hessian in end frame

+
+
Return type:
+

he

+
+
+

Synopsis

+

This method computes the manipulator Hessian in the end frame. If +we take the time derivative of the differential kinematic relationship +.. math:

+
\nu    &= \mat{J}(\vec{q}) \dvec{q} \\
+\alpha &= \dmat{J} \dvec{q} + \mat{J} \ddvec{q}
+
+
+

where +.. math:

+
\dmat{J} = \mat{H} \dvec{q}
+
+
+

and \(\mat{H} \in \mathbb{R}^{6\times n \times n}\) is the +Hessian tensor.

+

The elements of the Hessian are +.. math:

+
\mat{H}_{i,j,k} =  \frac{d^2 u_i}{d q_j d q_k}
+
+
+

where \(u = \{t_x, t_y, t_z, r_x, r_y, r_z\}\) are the elements +of the spatial velocity vector.

+

Similarly, we can write +.. math:

+
\mat{J}_{i,j} = \frac{d u_i}{d q_j}
+
+
+

Examples

+

The following example makes a Panda robot object, and solves for the +end-effector frame Hessian at the given joint angle configuration

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda()
+>>> panda.hessiane([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+array([[[-0.4816, -0.    , -0.4835, -0.    , -0.1539, -0.    ,  0.    ],
+        [ 0.    , -0.0796,  0.    , -0.2466,  0.    , -0.2006,  0.    ],
+        [-0.0483, -0.    , -0.0485, -0.    , -0.0154, -0.    ,  0.    ],
+        [ 0.    , -0.995 ,  0.    ,  0.995 , -0.    ,  0.995 , -0.    ],
+        [ 0.    ,  0.    ,  0.2955, -0.    , -0.9463, -0.    , -0.0998],
+        [ 0.    , -0.0998,  0.    ,  0.0998, -0.    ,  0.0998,  0.    ]],
+
+       [[-0.    , -0.4896, -0.    ,  0.4715, -0.    ,  0.088 ,  0.    ],
+        [-0.0796,  0.    ,  0.    , -0.    ,  0.    , -0.    ,  0.    ],
+        [-0.    ,  0.0309,  0.    ,  0.2952,  0.    ,  0.2104, -0.    ],
+        [ 0.    ,  0.    ,  0.9801,  0.    , -0.4161,  0.    , -1.    ],
+        [ 0.    ,  0.    , -0.    , -0.    ,  0.    , -0.    ,  0.    ],
+        [ 0.    ,  0.    , -0.1987, -0.    ,  0.9093, -0.    ,  0.    ]],
+
+       [[-0.4835, -0.    , -0.4763, -0.    , -0.1516, -0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    , -0.383 ,  0.    , -0.2237,  0.    ],
+        [-0.0485,  0.    ,  0.0965, -0.    ,  0.0307, -0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.9801, -0.    ,  0.9801,  0.    ],
+        [ 0.    ,  0.    ,  0.    , -0.    , -0.8085, -0.    ,  0.1987],
+        [ 0.    ,  0.    ,  0.    , -0.1987, -0.    , -0.1987, -0.    ]],
+
+       [[-0.    ,  0.4715, -0.    , -0.4715,  0.    , -0.088 ,  0.    ],
+        [-0.2466, -0.    , -0.383 ,  0.    , -0.    ,  0.    , -0.    ],
+        [-0.    ,  0.2952, -0.    , -0.2952, -0.    , -0.2104,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.4161,  0.    ,  1.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    , -0.    ,  0.    , -0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    , -0.9093,  0.    ,  0.    ]],
+
+       [[-0.1539, -0.    , -0.1516,  0.    ,  0.0644,  0.    , -0.    ],
+        [ 0.    ,  0.    ,  0.    , -0.    , -0.    ,  0.1676, -0.    ],
+        [-0.0154,  0.    ,  0.0307, -0.    , -0.1407, -0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    , -0.4161, -0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    , -0.9093],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.9093,  0.    ]],
+
+       [[-0.    ,  0.088 , -0.    , -0.088 ,  0.    , -0.088 ,  0.    ],
+        [-0.2006, -0.    , -0.2237,  0.    ,  0.1676,  0.    , -0.    ],
+        [-0.    ,  0.2104, -0.    , -0.2104, -0.    , -0.2104,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  1.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    , -0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ]],
+
+       [[ 0.    ,  0.    ,  0.    ,  0.    , -0.    ,  0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    , -0.    , -0.    , -0.    , -0.    ],
+        [ 0.    , -0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ]]])
+
+
+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+ +
+
+hierarchy()
+

Pretty print the robot link hierachy

+
+
Return type:
+

Pretty print of the robot model

+
+
+

Examples

+

Makes a robot and prints the heirachy

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.URDF.Panda()
+>>> robot.hierarchy()
+ panda_link0
+   panda_link1
+     panda_link2
+       panda_link3
+         panda_link4
+           panda_link5
+             panda_link6
+               panda_link7
+                 panda_link8
+                   panda_hand
+                     panda_leftfinger
+                     panda_rightfinger
+
+
+
+ +
+
+ik_GN(Tep, end=None, start=None, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, pinv=True, pinv_damping=0.0)
+

Fast numerical inverse kinematics by Gauss-Newton optimization

+

sol = ets.ik_GN(Tep) are the joint coordinates (n) corresponding +to the robot end-effector pose Tep which is an SE3 or ndarray object. +This method can be used for robots with any number of degrees of freedom. This +is a fast solver implemented in C++.

+

See the Inverse Kinematics Docs Page for more details and for a +tutorial on numerical IK, see here.

+
+

Note

+

When using this method with redundant robots (>6 DoF), pinv must be set to True

+
+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose or pose trajectory

  • +
  • end (Union[str, Link, Gripper, None]) – the link considered as the end-effector

  • +
  • start (Union[str, Link, Gripper, None]) – the link considered as the base frame, defaults to the robots’s base frame

  • +
  • q0 (Optional[ndarray]) – initial joint configuration (default to random valid joint +configuration contrained by the joint limits of the robot)

  • +
  • ilimit (int) – maximum number of iterations per search

  • +
  • slimit (int) – maximum number of search attempts

  • +
  • tol (float) – final error tolerance

  • +
  • mask (Optional[ndarray]) – a mask vector which weights the end-effector error priority. +Corresponds to translation in X, Y and Z and rotation about X, Y and Z +respectively

  • +
  • joint_limits (bool) – constrain the solution to being within the joint limits of +the robot (reject solution with invalid joint configurations and perfrom +another search up to the slimit)

  • +
  • pinv (int) – Use the psuedo-inverse instad of the normal matrix inverse

  • +
  • pinv_damping (float) – Damping factor for the psuedo-inverse

  • +
+
+
Return type:
+

Tuple[ndarray, int, int, int, float]

+
+
Returns:
+

    +
  • sol – tuple (q, success, iterations, searches, residual)

  • +
  • The return value sol is a tuple with elements

  • +
  • ============ ========== ===============================================

  • +
  • Element Type Description

  • +
  • ============ ========== ===============================================

  • +
  • q ndarray(n) joint coordinates in units of radians or metres

  • +
  • success int whether a solution was found

  • +
  • iterations int total number of iterations

  • +
  • searches int total number of searches

  • +
  • residual float final value of cost function

  • +
  • ============ ========== ===============================================

  • +
  • If success == 0 the q values will be valid numbers, but the

  • +
  • solution will be in error. The amount of error is indicated by

  • +
  • the residual.

  • +
+

+
+
+

Synopsis

+

Each iteration uses the Gauss-Newton optimisation method

+
+\[\begin{split}\vec{q}_{k+1} &= \vec{q}_k + +\left( +{\mat{J}(\vec{q}_k)}^\top +\mat{W}_e \ +{\mat{J}(\vec{q}_k)} +\right)^{-1} +\bf{g}_k \\ +\bf{g}_k &= +{\mat{J}(\vec{q}_k)}^\top +\mat{W}_e +\vec{e}_k\end{split}\]
+

where \(\mat{J} = {^0\mat{J}}\) is the base-frame manipulator Jacobian. If +\(\mat{J}(\vec{q}_k)\) is non-singular, and \(\mat{W}_e = \mat{1}_n\), then +the above provides the pseudoinverse solution. However, if \(\mat{J}(\vec{q}_k)\) +is singular, the above can not be computed and the GN solution is infeasible.

+

Examples

+

The following example gets a panda robot object, makes a goal +pose Tep, and then solves for the joint coordinates which result in the pose +Tep using the ikine_GN method.

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda()
+>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> panda.ik_GN(Tep)
+(array([-1.0805, -0.5328,  0.9086, -2.1781,  0.4612,  1.9018,  0.4239]), 1, 510, 32, 2.803306327008683e-09)
+
+
+

Notes

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
ik_NR

A fast numerical inverse kinematics solver using Newton-Raphson optimisation

+
+
ik_GN

A fast numerical inverse kinematics solver using Gauss-Newton optimisation

+
+
+
+
+ +
+
+ik_LM(Tep, end=None, start=None, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, k=1.0, method='chan')
+

Fast levenberg-Marquadt Numerical Inverse Kinematics Solver

+

A method which provides functionality to perform numerical inverse kinematics (IK) +using the Levemberg-Marquadt method. This +is a fast solver implemented in C++.

+

See the Inverse Kinematics Docs Page for more details and for a +tutorial on numerical IK, see here.

+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose

  • +
  • end (Union[str, Link, Gripper, None]) – the link considered as the end-effector

  • +
  • start (Union[str, Link, Gripper, None]) – the link considered as the base frame, defaults to the robots’s base frame

  • +
  • q0 (Optional[ndarray]) – The initial joint coordinate vector

  • +
  • ilimit (int) – How many iterations are allowed within a search before a new search +is started

  • +
  • slimit (int) – How many searches are allowed before being deemed unsuccessful

  • +
  • tol (float) – Maximum allowed residual error E

  • +
  • mask (Optional[ndarray]) – A 6 vector which assigns weights to Cartesian degrees-of-freedom +error priority

  • +
  • joint_limits (bool) – Reject solutions with joint limit violations

  • +
  • seed – A seed for the private RNG used to generate random joint coordinate +vectors

  • +
  • k (float) – Sets the gain value for the damping matrix Wn in the next iteration. See +synopsis

  • +
  • method (Literal['chan', 'wampler', 'sugihara']) – One of “chan”, “sugihara” or “wampler”. Defines which method is used +to calculate the damping matrix Wn in the step method

  • +
+
+
Return type:
+

Tuple[ndarray, int, int, int, float]

+
+
+

Synopsis

+

The operation is defined by the choice of the method kwarg.

+

The step is deined as

+
+\[\begin{split}\vec{q}_{k+1} +&= +\vec{q}_k + +\left( + \mat{A}_k +\right)^{-1} +\bf{g}_k \\ +% +\mat{A}_k +&= +{\mat{J}(\vec{q}_k)}^\top +\mat{W}_e \ +{\mat{J}(\vec{q}_k)} ++ +\mat{W}_n\end{split}\]
+

where \(\mat{W}_n = \text{diag}(\vec{w_n})(\vec{w_n} \in \mathbb{R}^n_{>0})\) is a +diagonal damping matrix. The damping matrix ensures that \(\mat{A}_k\) is +non-singular and positive definite. The performance of the LM method largely depends +on the choice of \(\mat{W}_n\).

+

Chan’s Method

+

Chan proposed

+
+\[\mat{W}_n += +λ E_k \mat{1}_n\]
+

where λ is a constant which reportedly does not have much influence on performance. +Use the kwarg k to adjust the weighting term λ.

+

Sugihara’s Method

+

Sugihara proposed

+
+\[\mat{W}_n += +E_k \mat{1}_n + \text{diag}(\hat{\vec{w}}_n)\]
+

where \(\hat{\vec{w}}_n \in \mathbb{R}^n\), \(\hat{w}_{n_i} = l^2 \sim 0.01 l^2\), +and \(l\) is the length of a typical link within the manipulator. We provide the +variable k as a kwarg to adjust the value of \(w_n\).

+

Wampler’s Method

+

Wampler proposed \(\vec{w_n}\) to be a constant. This is set through the k kwarg.

+

Examples

+

The following example makes a panda robot object, makes a goal +pose Tep, and then solves for the joint coordinates which result in the pose +Tep using the ikine_LM method.

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda()
+>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> panda.ikine_LM(Tep)
+IKSolution(q=array([-0.8539, -0.4228,  0.742 , -2.1893,  0.3191,  1.954 ,  0.5343]), success=True, iterations=61, searches=4, residual=4.257612407309886e-07, reason='Success')
+
+
+

Notes

+

The value for the k kwarg will depend on the method chosen and the arm you are +using. Use the following as a rough guide chan, k = 1.0 - 0.01, +wampler, k = 0.01 - 0.0001, and sugihara, k = 0.1 - 0.0001

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian.

+

This class supports null-space motion to assist with maximising manipulability and +avoiding joint limits. These are enabled by setting kq and km to non-zero values.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
ik_NR

A fast numerical inverse kinematics solver using Newton-Raphson optimisation

+
+
ik_GN

A fast numerical inverse kinematics solver using Gauss-Newton optimisation

+
+
+
+
+

Changed in version 1.0.4: Merged the Levemberg-Marquadt IK solvers into the ik_LM method

+
+
+ +
+
+ik_NR(Tep, end=None, start=None, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, pinv=True, pinv_damping=0.0)
+

Fast numerical inverse kinematics using Newton-Raphson optimization

+

sol = ets.ik_NR(Tep) are the joint coordinates (n) corresponding +to the robot end-effector pose Tep which is an SE3 or ndarray object. +This method can be used for robots with any number of degrees of freedom. This +is a fast solver implemented in C++.

+

See the Inverse Kinematics Docs Page for more details and for a +tutorial on numerical IK, see here.

+
+

Note

+

When using this method with redundant robots (>6 DoF), pinv must be set to True

+
+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose or pose trajectory

  • +
  • end (Union[str, Link, Gripper, None]) – the link considered as the end-effector

  • +
  • start (Union[str, Link, Gripper, None]) – the link considered as the base frame, defaults to the robots’s base frame

  • +
  • q0 (Optional[ndarray]) – initial joint configuration (default to random valid joint +configuration contrained by the joint limits of the robot)

  • +
  • ilimit (int) – maximum number of iterations per search

  • +
  • slimit (int) – maximum number of search attempts

  • +
  • tol (float) – final error tolerance

  • +
  • mask (Optional[ndarray]) – a mask vector which weights the end-effector error priority. +Corresponds to translation in X, Y and Z and rotation about X, Y and Z +respectively

  • +
  • joint_limits (bool) – constrain the solution to being within the joint limits of +the robot (reject solution with invalid joint configurations and perfrom +another search up to the slimit)

  • +
  • pinv (int) – Use the psuedo-inverse instad of the normal matrix inverse

  • +
  • pinv_damping (float) – Damping factor for the psuedo-inverse

  • +
+
+
Return type:
+

Tuple[ndarray, int, int, int, float]

+
+
Returns:
+

    +
  • sol – tuple (q, success, iterations, searches, residual)

  • +
  • The return value sol is a tuple with elements

  • +
  • ============ ========== ===============================================

  • +
  • Element Type Description

  • +
  • ============ ========== ===============================================

  • +
  • q ndarray(n) joint coordinates in units of radians or metres

  • +
  • success int whether a solution was found

  • +
  • iterations int total number of iterations

  • +
  • searches int total number of searches

  • +
  • residual float final value of cost function

  • +
  • ============ ========== ===============================================

  • +
  • If success == 0 the q values will be valid numbers, but the

  • +
  • solution will be in error. The amount of error is indicated by

  • +
  • the residual.

  • +
+

+
+
+

Synopsis

+

Each iteration uses the Newton-Raphson optimisation method

+
+\[\vec{q}_{k+1} = \vec{q}_k + {^0\mat{J}(\vec{q}_k)}^{-1} \vec{e}_k\]
+

Examples

+

The following example gets a panda robot object, makes a goal +pose Tep, and then solves for the joint coordinates which result in the pose +Tep using the ikine_GN method.

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda()
+>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> panda.ik_NR(Tep)
+(array([-1.0805, -0.5328,  0.9086, -2.1781,  0.4612,  1.9018,  0.4239]), 1, 489, 32, 2.8033063270234757e-09)
+
+
+

Notes

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
ik_LM

A fast numerical inverse kinematics solver using Levenberg-Marquadt optimisation

+
+
ik_GN

A fast numerical inverse kinematics solver using Gauss-Newton optimisation

+
+
+
+
+ +
+
+ikine_GN(Tep, end=None, start=None, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, seed=None, pinv=False, kq=0.0, km=0.0, ps=0.0, pi=0.3, **kwargs)
+

Gauss-Newton Numerical Inverse Kinematics Solver

+

A method which provides functionality to perform numerical inverse kinematics (IK) +using the Gauss-Newton method.

+

See the Inverse Kinematics Docs Page for more details and for a +tutorial on numerical IK, see here.

+
+

Note

+

When using this method with redundant robots (>6 DoF), pinv must be set to True

+
+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose

  • +
  • end (Union[str, Link, Gripper, None]) – the link considered as the end-effector

  • +
  • start (Union[str, Link, Gripper, None]) – the link considered as the base frame, defaults to the robots’s base frame

  • +
  • q0 (Union[ndarray, List[float], Tuple[float], Set[float], None]) – The initial joint coordinate vector

  • +
  • ilimit (int) – How many iterations are allowed within a search before a new search +is started

  • +
  • slimit (int) – How many searches are allowed before being deemed unsuccessful

  • +
  • tol (float) – Maximum allowed residual error E

  • +
  • mask (Union[ndarray, List[float], Tuple[float], Set[float], None]) – A 6 vector which assigns weights to Cartesian degrees-of-freedom +error priority

  • +
  • joint_limits (bool) – Reject solutions with joint limit violations

  • +
  • seed (Optional[int]) – A seed for the private RNG used to generate random joint coordinate +vectors

  • +
  • pinv (bool) – If True, will use the psuedoinverse in the step method instead of +the normal inverse

  • +
  • kq (float) – The gain for joint limit avoidance. Setting to 0.0 will remove this +completely from the solution

  • +
  • km (float) – The gain for maximisation. Setting to 0.0 will remove this completely +from the solution

  • +
  • ps (float) – The minimum angle/distance (in radians or metres) in which the joint is +allowed to approach to its limit

  • +
  • pi (Union[ndarray, float]) – The influence angle/distance (in radians or metres) in null space motion +becomes active

  • +
+
+
+

Synopsis

+

Each iteration uses the Gauss-Newton optimisation method

+
+\[\begin{split}\vec{q}_{k+1} &= \vec{q}_k + +\left( +{\mat{J}(\vec{q}_k)}^\top +\mat{W}_e \ +{\mat{J}(\vec{q}_k)} +\right)^{-1} +\bf{g}_k \\ +\bf{g}_k &= +{\mat{J}(\vec{q}_k)}^\top +\mat{W}_e +\vec{e}_k\end{split}\]
+

where \(\mat{J} = {^0\mat{J}}\) is the base-frame manipulator Jacobian. If +\(\mat{J}(\vec{q}_k)\) is non-singular, and \(\mat{W}_e = \mat{1}_n\), then +the above provides the pseudoinverse solution. However, if \(\mat{J}(\vec{q}_k)\) +is singular, the above can not be computed and the GN solution is infeasible.

+

Examples

+

The following example gets a panda robot object, makes a goal +pose Tep, and then solves for the joint coordinates which result in the pose +Tep using the ikine_GN method.

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda()
+>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> panda.ikine_GN(Tep)
+IKSolution(q=array([-0.603 , -1.0082, -2.0555, -0.5958, -2.191 ,  0.6342, -1.7251]), success=False, iterations=100, searches=100, residual=0.0, reason='iteration and search limit reached, 100 numpy.LinAlgError encountered')
+
+
+

Notes

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian.

+

This class supports null-space motion to assist with maximising manipulability and +avoiding joint limits. These are enabled by setting kq and km to non-zero values.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
IK_NR

An IK Solver class which implements the Newton-Raphson optimisation technique

+
+
ikine_LM

Implements the IK_LM class as a method within the ETS class

+
+
ikine_NR

Implements the IK_NR class as a method within the ETS class

+
+
ikine_QP

Implements the IK_QP class as a method within the ETS class

+
+
+
+
+

Changed in version 1.0.4: Added the Gauss-Newton IK solver method on the Robot class

+
+
+ +
+
+ikine_NR(Tep, end=None, start=None, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, seed=None, pinv=False, kq=0.0, km=0.0, ps=0.0, pi=0.3, **kwargs)
+

Newton-Raphson Numerical Inverse Kinematics Solver

+

A method which provides functionality to perform numerical inverse kinematics (IK) +using the Newton-Raphson method.

+

See the Inverse Kinematics Docs Page for more details and for a +tutorial on numerical IK, see here.

+
+

Note

+

When using this method with redundant robots (>6 DoF), pinv must be set to True

+
+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose

  • +
  • end (Union[str, Link, Gripper, None]) – the link considered as the end-effector

  • +
  • start (Union[str, Link, Gripper, None]) – the link considered as the base frame, defaults to the robots’s base frame

  • +
  • q0 (Union[ndarray, List[float], Tuple[float], Set[float], None]) – The initial joint coordinate vector

  • +
  • ilimit (int) – How many iterations are allowed within a search before a new search +is started

  • +
  • slimit (int) – How many searches are allowed before being deemed unsuccessful

  • +
  • tol (float) – Maximum allowed residual error E

  • +
  • mask (Union[ndarray, List[float], Tuple[float], Set[float], None]) – A 6 vector which assigns weights to Cartesian degrees-of-freedom +error priority

  • +
  • joint_limits (bool) – Reject solutions with joint limit violations

  • +
  • seed (Optional[int]) – A seed for the private RNG used to generate random joint coordinate +vectors

  • +
  • pinv (bool) – If True, will use the psuedoinverse in the step method instead of +the normal inverse

  • +
  • kq (float) – The gain for joint limit avoidance. Setting to 0.0 will remove this +completely from the solution

  • +
  • km (float) – The gain for maximisation. Setting to 0.0 will remove this completely +from the solution

  • +
  • ps (float) – The minimum angle/distance (in radians or metres) in which the joint is +allowed to approach to its limit

  • +
  • pi (Union[ndarray, float]) – The influence angle/distance (in radians or metres) in null space motion +becomes active

  • +
+
+
+

Synopsis

+

Each iteration uses the Newton-Raphson optimisation method

+
+\[\vec{q}_{k+1} = \vec{q}_k + {^0\mat{J}(\vec{q}_k)}^{-1} \vec{e}_k\]
+

Examples

+

The following example gets a panda robot object, makes a goal +pose Tep, and then solves for the joint coordinates which result in the pose +Tep using the ikine_NR method.

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda()
+>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> panda.ikine_NR(Tep)
+IKSolution(q=array([-2.2737, -1.4516, -0.5652, -2.2696, -0.8241,  0.3567, -2.4835]), success=False, iterations=100, searches=100, residual=0.0, reason='iteration and search limit reached, 100 numpy.LinAlgError encountered')
+
+
+

Notes

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian.

+

This class supports null-space motion to assist with maximising manipulability and +avoiding joint limits. These are enabled by setting kq and km to non-zero values.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
IK_NR

An IK Solver class which implements the Newton-Raphson optimisation technique

+
+
ikine_LM

Implements the IK_LM class as a method within the ETS class

+
+
ikine_GN

Implements the IK_GN class as a method within the ETS class

+
+
ikine_QP

Implements the IK_QP class as a method within the ETS class

+
+
+
+
+

Changed in version 1.0.4: Added the Newton-Raphson IK solver method on the Robot class

+
+
+ +
+
+ikine_QP(Tep, end=None, start=None, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, seed=None, kj=1.0, ks=1.0, kq=0.0, km=0.0, ps=0.0, pi=0.3, **kwargs)
+

Quadratic Programming Numerical Inverse Kinematics Solver

+

A method that provides functionality to perform numerical inverse kinematics +(IK) using a quadratic progamming approach.

+

See the Inverse Kinematics Docs Page for more details and for a +tutorial on numerical IK, see here.

+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose

  • +
  • end (Union[str, Link, Gripper, None]) – the link considered as the end-effector

  • +
  • start (Union[str, Link, Gripper, None]) – the link considered as the base frame, defaults to the robots’s base frame

  • +
  • q0 (Union[ndarray, List[float], Tuple[float], Set[float], None]) – The initial joint coordinate vector

  • +
  • ilimit (int) – How many iterations are allowed within a search before a new search +is started

  • +
  • slimit (int) – How many searches are allowed before being deemed unsuccessful

  • +
  • tol (float) – Maximum allowed residual error E

  • +
  • mask (Union[ndarray, List[float], Tuple[float], Set[float], None]) – A 6 vector which assigns weights to Cartesian degrees-of-freedom +error priority

  • +
  • joint_limits (bool) – Reject solutions with joint limit violations

  • +
  • seed (Optional[int]) – A seed for the private RNG used to generate random joint coordinate +vectors

  • +
  • kj – A gain for joint velocity norm minimisation

  • +
  • ks – A gain which adjusts the cost of slack (intentional error)

  • +
  • kq (float) – The gain for joint limit avoidance. Setting to 0.0 will remove this +completely from the solution

  • +
  • km (float) – The gain for maximisation. Setting to 0.0 will remove this completely +from the solution

  • +
  • ps (float) – The minimum angle/distance (in radians or metres) in which the joint is +allowed to approach to its limit

  • +
  • pi (Union[ndarray, float]) – The influence angle/distance (in radians or metres) in null space motion +becomes active

  • +
+
+
Raises:
+

ImportError – If the package qpsolvers is not installed

+
+
+

Synopsis

+

Each iteration uses the following approach

+
+\[\vec{q}_{k+1} = \vec{q}_{k} + \dot{\vec{q}}.\]
+

where the QP is defined as

+
+\[\begin{split}\min_x \quad f_o(\vec{x}) &= \frac{1}{2} \vec{x}^\top \mathcal{Q} \vec{x}+ \mathcal{C}^\top \vec{x}, \\ +\text{subject to} \quad \mathcal{J} \vec{x} &= \vec{\nu}, \\ +\mathcal{A} \vec{x} &\leq \mathcal{B}, \\ +\vec{x}^- &\leq \vec{x} \leq \vec{x}^+\end{split}\]
+

with

+
+\[\begin{split}\vec{x} &= +\begin{pmatrix} + \dvec{q} \\ \vec{\delta} +\end{pmatrix} \in \mathbb{R}^{(n+6)} \\ +\mathcal{Q} &= +\begin{pmatrix} + \lambda_q \mat{1}_{n} & \mathbf{0}_{6 \times 6} \\ \mathbf{0}_{n \times n} & \lambda_\delta \mat{1}_{6} +\end{pmatrix} \in \mathbb{R}^{(n+6) \times (n+6)} \\ +\mathcal{J} &= +\begin{pmatrix} + \mat{J}(\vec{q}) & \mat{1}_{6} +\end{pmatrix} \in \mathbb{R}^{6 \times (n+6)} \\ +\mathcal{C} &= +\begin{pmatrix} + \mat{J}_m \\ \bf{0}_{6 \times 1} +\end{pmatrix} \in \mathbb{R}^{(n + 6)} \\ +\mathcal{A} &= +\begin{pmatrix} + \mat{1}_{n \times n + 6} \\ +\end{pmatrix} \in \mathbb{R}^{(l + n) \times (n + 6)} \\ +\mathcal{B} &= +\eta +\begin{pmatrix} + \frac{\rho_0 - \rho_s} + {\rho_i - \rho_s} \\ + \vdots \\ + \frac{\rho_n - \rho_s} + {\rho_i - \rho_s} +\end{pmatrix} \in \mathbb{R}^{n} \\ +\vec{x}^{-, +} &= +\begin{pmatrix} + \dvec{q}^{-, +} \\ + \vec{\delta}^{-, +} +\end{pmatrix} \in \mathbb{R}^{(n+6)},\end{split}\]
+

where \(\vec{\delta} \in \mathbb{R}^6\) is the slack vector, +\(\lambda_\delta \in \mathbb{R}^+\) is a gain term which adjusts the +cost of the norm of the slack vector in the optimiser, +\(\dvec{q}^{-,+}\) are the minimum and maximum joint velocities, and +\(\dvec{\delta}^{-,+}\) are the minimum and maximum slack velocities.

+

Examples

+

The following example gets a panda robot object, makes a goal +pose Tep, and then solves for the joint coordinates which result in the pose +Tep using the ikine_QP method.

+
  File "/opt/hostedtoolcache/Python/3.7.17/x64/lib/python3.7/site-packages/roboticstoolbox/robot/IK.py", line 1331, in __init__
+    "the package qpsolvers is required for this class. \nInstall using 'pip"
+ImportError: the package qpsolvers is required for this class. 
+Install using 'pip install qpsolvers'
+
+
+

Notes

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian.

+

This class supports null-space motion to assist with maximising manipulability and +avoiding joint limits. These are enabled by setting kq and km to non-zero values.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
IK_NR

An IK Solver class which implements the Newton-Raphson optimisation technique

+
+
ikine_LM

Implements the IK_LM class as a method within the ETS class

+
+
ikine_GN

Implements the IK_GN class as a method within the ETS class

+
+
ikine_NR

Implements the IK_NR class as a method within the ETS class

+
+
+
+
+

Changed in version 1.0.4: Added the Quadratic Programming IK solver method on the Robot class

+
+
+ +
+
+inertia(q)
+

Manipulator inertia matrix +inertia(q) is the symmetric joint inertia matrix (n,n) which +relates joint torque to joint acceleration for the robot at joint +configuration q.

+

Trajectory operation

+

If q is a matrix (m,n), each row is interpretted as a joint state +vector, and the result is a 3d-matrix (nxnxk) where each plane +corresponds to the inertia for the corresponding row of q.

+
+
Parameters:
+

q (ndarray) – Joint coordinates

+
+
Returns:
+

The inertia matrix

+
+
Return type:
+

inertia

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> puma.inertia(puma.qz)
+array([[ 3.9611, -0.1627, -0.1389,  0.0016, -0.0004,  0.    ],
+       [-0.1627,  4.4566,  0.3727,  0.    ,  0.0019,  0.    ],
+       [-0.1389,  0.3727,  0.9387,  0.    ,  0.0019,  0.    ],
+       [ 0.0016,  0.    ,  0.    ,  0.1924,  0.    ,  0.    ],
+       [-0.0004,  0.0019,  0.0019,  0.    ,  0.1713,  0.    ],
+       [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.1941]])
+
+
+

Notes

+
    +
  • +
    The diagonal elements M[j,j] are the inertia seen by joint

    actuator j.

    +
    +
    +
  • +
  • +
    The off-diagonal elements M[j,k] are coupling inertias that

    relate acceleration on joint j to force/torque on +joint k.

    +
    +
    +
  • +
  • +
    The diagonal terms include the motor inertia reflected through

    the gear ratio.

    +
    +
    +
  • +
+
+

See also

+

cinertia()

+
+
+ +
+
+inertia_x(q=None, pinv=False, representation='rpy/xyz', Ji=None)
+

Operational space inertia matrix

+

robot.inertia_x(q) is the operational space (Cartesian) inertia +matrix which relates Cartesian force/torque to Cartesian +acceleration at the joint configuration q.

+
+\[\mathbf{M}_x = \mathbf{J}(q)^{-T} \mathbf{M}(q) \mathbf{J}(q)^{-1}\]
+

The transformation to operational space requires an analytical, rather +than geometric, Jacobian. analytical can be one of:

+
+
+ + + + + + + + + + + + + + + + + + + +

Value

Rotational representation

'rpy/xyz'

RPY angular rates in XYZ order (default)

'rpy/zyx'

RPY angular rates in XYZ order

'eul'

Euler angular rates in ZYZ order

'exp'

exponential coordinate rates

+
+

Trajectory operation

+

If q is a matrix (m,n), each row is interpretted as a joint state +vector, and the result is a 3d-matrix (m,n,n) where each plane +corresponds to the Cartesian inertia for the corresponding +row of q.

+
+
Parameters:
+
    +
  • q – Joint coordinates

  • +
  • pinv – use pseudo inverse rather than inverse (Default value = False)

  • +
  • analytical – the type of analytical Jacobian to use, default is +‘rpy/xyz’

  • +
  • representation – (Default value = “rpy/xyz”)

  • +
  • Ji – The inverse analytical Jacobian (base-frame)

  • +
+
+
Returns:
+

The operational space inertia matrix

+
+
Return type:
+

inertia_x

+
+
+

Examples

+
    ainv = _umath_linalg.inv(a, signature=signature, extobj=extobj)
+  File "/opt/hostedtoolcache/Python/3.7.17/x64/lib/python3.7/site-packages/numpy/linalg/linalg.py", line 88, in _raise_linalgerror_singular
+    raise LinAlgError("Singular matrix")
+numpy.linalg.LinAlgError: Singular matrix
+
+
+

Notes

+
    +
  • If the robot is not 6 DOF the pinv option is set True.

  • +
  • pinv() is around 5x slower than inv()

  • +
+
+

Warning

+

Assumes that the operational space has 6 DOF.

+
+
+

See also

+

inertia()

+
+
+ +
+
+iscollided(q, shape, skip=False)
+

Check if the robot is in collision with a shape

+

iscollided(shape) checks if this robot and shape have collided

+
+
+shape
+

The shape to compare distance to

+
+ +
+
+skip
+

Skip setting all shape transforms based on q, use this +option if using this method in conjuction with Swift to save time

+
+ +
+
Returns:
+

True if shapes have collided

+
+
Return type:
+

iscollided

+
+
+
+ +
+
+isprismatic(j)
+

Check if joint is prismatic

+
+
Returns:
+

True if prismatic

+
+
Return type:
+

j

+
+
+

Examples

+

+
+
+
+

See also

+

Link.isprismatic(), prismaticjoints()

+
+
+ +
+
+isrevolute(j)
+

Check if joint is revolute

+
+
Returns:
+

True if revolute

+
+
Return type:
+

j

+
+
+

Examples

+

+
+
+
+

See also

+

Link.isrevolute(), revolutejoints()

+
+
+ +
+
+itorque(q, qdd)
+

Inertia torque

+

itorque(q, qdd) is the inertia force/torque vector (n) at +the specified joint configuration q (n) and acceleration qdd (n), and +n is the number of robot joints. +It is \(\mathbf{I}(q) \ddot{q}\).

+

Trajectory operation

+

If q and qdd are matrices (m,n), each row is interpretted as a +joint configuration, and the result is a matrix (m,n) where each row is +the corresponding joint torques.

+
+
Parameters:
+
    +
  • q – Joint coordinates

  • +
  • qdd – Joint acceleration

  • +
+
+
Returns:
+

The inertia torque vector

+
+
Return type:
+

itorque

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> puma.itorque(puma.qz, 0.5 * np.ones((6,)))
+array([1.8304, 2.3343, 0.5872, 0.0971, 0.0873, 0.0971])
+
+
+

Notes

+
    +
  • +
    If the robot model contains non-zero motor inertia then this

    will be included in the result.

    +
    +
    +
  • +
+
+

See also

+

inertia()

+
+
+ +
+
+jacob0_dot(q, qd, J0=None, representation=None)
+

Derivative of Jacobian

+

robot.jacob_dot(q, qd) computes the rate of change of the +Jacobian elements

+
+\[\dmat{J} = \frac{d \mat{J}}{d \vec{q}} \frac{d \vec{q}}{dt}\]
+

where the first term is the rank-3 Hessian.

+
+
Parameters:
+
    +
  • q

  • +
  • robot (The joint configuration of the) –

  • +
  • qd (Union[ndarray, List[float], Tuple[float], Set[float]]) – The joint velocity of the robot

  • +
  • J0 – Jacobian in {0} frame

  • +
  • representation (Optional[Literal['rpy/xyz', 'rpy/zyx', 'eul', 'exp']]) – angular representation

  • +
+
+
Returns:
+

The derivative of the manipulator Jacobian

+
+
Return type:
+

jdot

+
+
+

Synopsis

+

If J0 is already calculated for the joint +coordinates q it can be passed in to to save computation time.

+

It is computed as the mode-3 product of the Hessian tensor and the +velocity vector.

+

The derivative of an analytical Jacobian can be obtained by setting +representation as

+

|``representation`` | Rotational representation | +|---------------------|————————————-| +|'rpy/xyz' | RPY angular rates in XYZ order | +|'rpy/zyx' | RPY angular rates in XYZ order | +|'eul' | Euler angular rates in ZYZ order | +|'exp' | exponential coordinate rates |

+

References

+
    +
  • +
    Kinematic Derivatives using the Elementary Transform

    Sequence, J. Haviland and P. Corke

    +
    +
    +
  • +
+
+

See also

+

jacob0(), hessian0()

+
+
+ +
+
+jacobm(q=None, J=None, H=None, end=None, start=None, axes='all')
+

The manipulability Jacobian

+

This measure relates the rate of change of the manipulability to the +joint velocities of the robot. One of J or q is required. Supply J +and H if already calculated to save computation time

+
+
Parameters:
+
    +
  • q – The joint angles/configuration of the robot (Optional, +if not supplied will use the stored q values).

  • +
  • J – The manipulator Jacobian in any frame

  • +
  • H – The manipulator Hessian in any frame

  • +
  • end (Union[str, Link, Gripper, None]) – the final link or Gripper which the Hessian represents

  • +
  • start (Union[str, Link, Gripper, None]) – the first link which the Hessian represents

  • +
+
+
Returns:
+

The manipulability Jacobian

+
+
Return type:
+

jacobm

+
+
+

Synopsis

+

Yoshikawa’s manipulability measure

+
+\[m(\vec{q}) = \sqrt{\mat{J}(\vec{q}) \mat{J}(\vec{q})^T}\]
+

This method returns its Jacobian with respect to configuration

+
+\[\frac{\partial m(\vec{q})}{\partial \vec{q}}\]
+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+ +
+
+joint_velocity_damper(ps=0.05, pi=0.1, n=None, gain=1.0)
+

Compute the joint velocity damper for QP motion control

+

Formulates an inequality contraint which, when optimised for will +make it impossible for the robot to run into joint limits. Requires +the joint limits of the robot to be specified. See examples/mmc.py +for use case

+
+
+ps
+

The minimum angle (in radians) in which the joint is +allowed to approach to its limit

+
+ +
+
+pi
+

The influence angle (in radians) in which the velocity +damper becomes active

+
+ +
+
+n
+

The number of joints to consider. Defaults to all joints

+
+ +
+
+gain
+

The gain for the velocity damper

+
+ +
+
Return type:
+

Tuple[ndarray, ndarray]

+
+
Returns:
+

    +
  • Ain – A (6,) vector inequality contraint for an optisator

  • +
  • Bin – b (6,) vector inequality contraint for an optisator

  • +
+

+
+
+
+ +
+
+jointdynamics(q, qd=None)
+

Transfer function of joint actuator

+

tf = jointdynamics(qd, q) calculates a vector of n +continuous-time transfer functions that represent the transfer +function 1/(Js+B) for each joint based on the dynamic parameters +of the robot and the configuration q (n). n is the number of robot +joints. The result is a list of tuples (J, B) for each joint.

+

tf = jointdynamics(q, qd) as above but include the linearized +effects of Coulomb friction when operating at joint velocity QD +(1xN).

+
+
Parameters:
+
    +
  • q – Joint coordinates

  • +
  • qd – Joint velocity

  • +
+
+
Returns:
+

transfer function denominators

+
+
Return type:
+

list of 2-tuples

+
+
+
+ +
+
+jtraj(T1, T2, t, **kwargs)
+

Joint-space trajectory between SE(3) poses

+

The initial and final poses are mapped to joint space using inverse +kinematics:

+
    +
  • if the object has an analytic solution ikine_a that will be used,

  • +
  • otherwise the general numerical algorithm ikine_lm will be used.

  • +
+

traj = obot.jtraj(T1, T2, t) is a trajectory object whose +attribute traj.q is a row-wise joint-space trajectory.

+
+
Parameters:
+
    +
  • T1 (Union[ndarray, SE3]) – initial end-effector pose

  • +
  • T2 (Union[ndarray, SE3]) – final end-effector pose

  • +
  • t (Union[ndarray, int]) – time vector or number of steps

  • +
  • kwargs – arguments passed to the IK solver

  • +
+
+
Return type:
+

trajectory

+
+
+
+ +
+
+property keywords: List[str]
+
+ +
+ +

Compute a collision constrain for QP motion control

+

Formulates an inequality contraint which, when optimised for will +make it impossible for the robot to run into a collision. Requires +See examples/neo.py for use case

+
+
+ds
+

The minimum distance in which a joint is allowed to +approach the collision object shape

+
+ +
+
+di
+

The influence distance in which the velocity +damper becomes active

+
+ +
+
+xi
+

The gain for the velocity damper

+
+ +
+
+end
+

The end link of the robot to consider

+
+ +
+
+start
+

The start link of the robot to consider

+
+ +
+
+collision_list
+

A list of shapes to consider for collision

+
+ +
+
Returns:
+

    +
  • Ain – A (6,) vector inequality contraint for an optisator

  • +
  • Bin – b (6,) vector inequality contraint for an optisator

  • +
+

+
+
+
+ +
+ +
+ +
+
+linkcolormap(linkcolors='viridis')
+

Create a colormap for robot joints

+
    +
  • cm = robot.linkcolormap() is an n-element colormap that gives a +unique color for every link. The RGBA colors for link j are +cm(j).

  • +
  • cm = robot.linkcolormap(cmap) as above but cmap is the name +of a valid matplotlib colormap. The default, example above, is the +viridis colormap.

  • +
  • cm = robot.linkcolormap(list of colors) as above but a +colormap is created from a list of n color names given as strings, +tuples or hexstrings.

  • +
+
+
Parameters:
+

linkcolors (Union[List[Any], str]) – list of colors or colormap, defaults to “viridis”

+
+
Returns:
+

the color map

+
+
Return type:
+

color map

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> cm = robot.linkcolormap("inferno")
+>>> print(cm(range(6))) # cm(i) is 3rd color in colormap
+[[0.0015 0.0005 0.0139 1.    ]
+ [0.2582 0.0386 0.4065 1.    ]
+ [0.5783 0.148  0.4044 1.    ]
+ [0.865  0.3168 0.2261 1.    ]
+ [0.9876 0.6453 0.0399 1.    ]
+ [0.9884 0.9984 0.6449 1.    ]]
+>>> cm = robot.linkcolormap(
+...     ['red', 'g', (0,0.5,0), '#0f8040', 'yellow', 'cyan'])
+>>> print(cm(range(6)))
+[[1.     0.     0.     1.    ]
+ [0.     0.5    0.     1.    ]
+ [0.     0.5    0.     1.    ]
+ [0.0588 0.502  0.251  1.    ]
+ [1.     1.     0.     1.    ]
+ [0.     1.     1.     1.    ]]
+
+
+

Notes

+ +
+ +
+ +

Robot links

+
+
Returns:
+

A list of link objects

+
+
Return type:
+

links

+
+
+

Notes

+

It is probably more concise to index the robot object rather +than the list of links, ie. the following are equivalent: +- robot.links[i] +- robot[i]

+
+ +
+
+manipulability(q=None, J=None, end=None, start=None, method='yoshikawa', axes='all', **kwargs)
+

Manipulability measure

+

manipulability(q) is the scalar manipulability index +for the robot at the joint configuration q. It indicates +dexterity, that is, how well conditioned the robot is for motion +with respect to the 6 degrees of Cartesian motion. The values is +zero if the robot is at a singularity.

+
+
Parameters:
+
    +
  • q – Joint coordinates, one of J or q required

  • +
  • J – Jacobian in base frame if already computed, one of J or +q required

  • +
  • method (Literal['yoshikawa', 'asada', 'minsingular', 'invcondition']) – method to use, “yoshikawa” (default), “invcondition”, +“minsingular” or “asada”

  • +
  • axes (Union[Literal['all', 'trans', 'rot'], List[bool]]) – Task space axes to consider: “all” [default], +“trans”, or “rot”

  • +
+
+
Return type:
+

manipulability

+
+
+

Synopsis

+

Various measures are supported:

+
+
Measure | Description |
+
+

|-------------------|————————————————-| +| "yoshikawa" | Volume of the velocity ellipsoid, distance | +| | from singularity [Yoshikawa85] | +| "invcondition"``| Inverse condition number of Jacobian, isotropy  | +|                   | of the velocity ellipsoid [Klein87]_            | +| ``"minsingular" | Minimum singular value of the Jacobian, | +| | distance from singularity [Klein87] | +| "asada" | Isotropy of the task-space acceleration | +| | ellipsoid which is a function of the Cartesian | +| | inertia matrix which depends on the inertial | +| | parameters [Asada83] |

+

Trajectory operation:

+

If q is a matrix (m,n) then the result (m,) is a vector of +manipulability indices for each joint configuration specified by a row +of q.

+

Notes

+
    +
  • Invokes the jacob0 method of the robot if J is not passed

  • +
  • +
    The “all” option includes rotational and translational

    dexterity, but this involves adding different units. It can be +more useful to look at the translational and rotational +manipulability separately.

    +
    +
    +
  • +
  • +
    Examples in the RVC book (1st edition) can be replicated by

    using the “all” option

    +
    +
    +
  • +
  • +
    Asada’s measure requires inertial a robot model with inertial

    parameters.

    +
    +
    +
  • +
+

References

+
+
+[Yoshikawa85] +

Manipulability of Robotic Mechanisms. Yoshikawa T., +The International Journal of Robotics Research. +1985;4(2):3-9. doi:10.1177/027836498500400201

+
+
+[Asada83] +

A geometrical representation of manipulator dynamics and +its application to arm design, H. Asada, +Journal of Dynamic Systems, Measurement, and Control, +vol. 105, p. 131, 1983.

+
+
+[Klein87] +

Dexterity Measures for the Design and Control of +Kinematically Redundant Manipulators. Klein CA, Blaho BE. +The International Journal of Robotics Research. +1987;6(2):72-83. doi:10.1177/027836498700600206

+
+
+
    +
  • Robotics, Vision & Control, Chap 8, P. Corke, Springer 2011.

  • +
+
+

Changed in version 1.0.3: Removed ‘both’ option for axes, added a custom list option.

+
+
+ +
+
+property manufacturer
+

Get/set robot manufacturer’s name

+
    +
  • robot.manufacturer is the robot manufacturer’s name

  • +
  • robot.manufacturer = ... checks and sets the manufacturer’s name

  • +
+
+
Returns:
+

robot manufacturer’s name

+
+
Return type:
+

manufacturer

+
+
+
+ +
+
+property n: int
+

Number of joints

+
+
Returns:
+

Number of joints

+
+
Return type:
+

n

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.n
+6
+
+
+
+

See also

+

nlinks(), nbranches()

+
+
+ +
+
+property name: str
+

Get/set robot name

+
    +
  • robot.name is the robot name

  • +
  • robot.name = ... checks and sets the robot name

  • +
+
+
Parameters:
+

name – the new robot name

+
+
Returns:
+

the current robot name

+
+
Return type:
+

name

+
+
+
+ +
+ +

Number of links

+

The returned number is the total of both variable joints and +static links

+
+
Returns:
+

Number of links

+
+
Return type:
+

nlinks

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.nlinks
+6
+
+
+
+

See also

+

n(), nbranches()

+
+
+ +
+
+nofriction(coulomb=True, viscous=False)
+

Remove manipulator joint friction

+

nofriction() copies the robot and returns +a robot with the same link parameters except the Coulomb and/or viscous +friction parameter are set to zero.

+
+
Parameters:
+
    +
  • coulomb (bool) – set the Coulomb friction to 0

  • +
  • viscous (bool) – set the viscous friction to 0

  • +
+
+
Returns:
+

A copy of the robot with dynamic parameters perturbed

+
+
Return type:
+

robot

+
+
+
+

See also

+

Robot.friction(), Link.nofriction()

+
+
+ +
+
+partial_fkine0(q, n=3, end=None, start=None)
+

Manipulator Forward Kinematics nth Partial Derivative

+

This method computes the nth derivative of the forward kinematics where n is +greater than or equal to 3. This is an extension of the differential kinematics +where the Jacobian is the first partial derivative and the Hessian is the +second.

+
+
Parameters:
+
    +
  • q (Union[ndarray, List[float], Tuple[float], Set[float]]) – The joint angles/configuration of the robot (Optional, +if not supplied will use the stored q values).

  • +
  • end (Union[str, Link, Gripper, None]) – the final link/Gripper which the Hessian represents

  • +
  • start (Union[str, Link, Gripper, None]) – the first link which the Hessian represents

  • +
  • tool – a static tool transformation matrix to apply to the +end of end, defaults to None

  • +
+
+
Returns:
+

The nth Partial Derivative of the forward kinematics

+
+
Return type:
+

A

+
+
+

Examples

+

The following example makes a Panda robot object, and solves for the +base-effector frame 4th defivative of the forward kinematics at the given +joint angle configuration

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda()
+>>> panda.partial_fkine0([0, -0.3, 0, -2.2, 0, 2, 0.7854], n=4)
+array([[[[[ 0.484 ,  0.    ,  0.486 , ...,  0.1547, -0.    ,  0.    ],
+          [-0.    , -0.0796, -0.    , ..., -0.    , -0.2006,  0.    ],
+          [-0.    , -0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
+          [ 0.    ,  1.    ,  0.    , ..., -0.    , -1.    , -0.    ],
+          [ 0.    ,  0.    ,  0.2955, ..., -0.9463,  0.    , -0.0998],
+          [ 0.    , -0.    ,  0.    , ..., -0.    ,  0.    , -0.    ]],
+
+         [[ 0.    ,  0.484 ,  0.    , ...,  0.    , -0.1086,  0.    ],
+          [-0.    , -0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
+          [ 0.    , -0.    , -0.    , ..., -0.    ,  0.    ,  0.    ],
+          [-1.    ,  0.    , -0.9553, ...,  0.3233, -0.    ,  0.995 ],
+          [-0.    ,  0.    ,  0.    , ..., -0.    ,  0.    , -0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.    ]],
+
+         [[ 0.4624,  0.    ,  0.4643, ...,  0.1478, -0.    ,  0.    ],
+          [-0.    ,  0.067 , -0.    , ..., -0.    , -0.2237,  0.    ],
+          [-0.    ,  0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
+          [-0.    ,  0.9553,  0.    , ..., -0.    , -0.9553, -0.    ],
+          [-0.2955, -0.    ,  0.    , ..., -0.8085,  0.    ,  0.1987],
+          [-0.    , -0.    ,  0.    , ..., -0.    ,  0.    ,  0.    ]],
+
+         ...,
+
+         [[-0.1565, -0.    , -0.1571, ..., -0.05  ,  0.    ,  0.    ],
+          [ 0.    , -0.4323,  0.    , ..., -0.    ,  0.1676,  0.    ],
+          [ 0.    , -0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    , -0.3233,  0.    , ...,  0.    ,  0.3233, -0.    ],
+          [ 0.9463,  0.    ,  0.8085, ...,  0.    ,  0.    , -0.9093],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.    ]],
+
+         [[ 0.    , -0.484 ,  0.    , ..., -0.    ,  0.1086,  0.    ],
+          [-0.    ,  0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
+          [ 0.    ,  0.    , -0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 1.    ,  0.    ,  0.9553, ..., -0.3233,  0.    , -0.995 ],
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+
+        [[[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         ...,
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]]],
+
+
+        [[[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         ...,
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]]],
+
+
+        ...,
+
+
+        [[[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         ...,
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]]],
+
+
+        [[[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         ...,
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]]],
+
+
+        [[[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         ...,
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]]]]])
+
+
+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+ +
+
+pay(W, q=None, J=None, frame=1)
+

Generalised joint force/torque due to a payload wrench

+

tau = pay(W, J) Returns the generalised joint force/torques due to a +payload wrench W applied to the end-effector. Where the manipulator +Jacobian is J (6xn), and n is the number of robot joints.

+

tau = pay(W, q, frame) as above but the Jacobian is calculated at pose +q in the frame given by frame which is 0 for base frame, 1 for +end-effector frame.

+

Uses the formula tau = J’W, where W is a wrench vector applied at the +end effector, W = [Fx Fy Fz Mx My Mz]’.

+
+
Trajectory operation:

In the case q is nxm or J is 6xnxm then tau is nxm where each row +is the generalised force/torque at the pose given by corresponding +row of q.

+
+
+
+
Parameters:
+
    +
  • W (Union[ndarray, List[float], Tuple[float], Set[float]]) – A wrench vector applied at the end effector, +W = [Fx Fy Fz Mx My Mz]

  • +
  • q (Optional[ndarray]) – Joint coordinates

  • +
  • J (Optional[ndarray]) – The manipulator Jacobian (Optional, if not supplied will +use the q value).

  • +
  • frame (int) – The frame in which to torques are expressed in when J +is not supplied. 0 means base frame of the robot, 1 means end- +effector frame

  • +
+
+
Returns:
+

Joint forces/torques due to w

+
+
Return type:
+

t

+
+
+

Notes

+
    +
  • +
    Wrench vector and Jacobian must be from the same reference

    frame.

    +
    +
    +
  • +
  • Tool transforms are taken into consideration when frame=1.

  • +
  • +
    Must have a constant wrench - no trajectory support for this

    yet.

    +
    +
    +
  • +
+
+ +
+
+paycap(w, tauR, frame=1, q=None)
+

Static payload capacity of a robot

+

wmax, joint = paycap(q, w, f, tauR) returns the maximum permissible +payload wrench wmax (6) applied at the end-effector, and the index +of the joint (zero indexed) which hits its force/torque limit at that +wrench. q (n) is the manipulator pose, w the payload wrench +(6), f the wrench reference frame and tauR (nx2) is a matrix of +joint forces/torques (first col is maximum, second col minimum).

+

Trajectory operation:

+

In the case q is nxm then wmax is Mx6 and J is Mx1 where the rows are +the results at the pose given by corresponding row of q.

+
+
Parameters:
+
    +
  • w (ndarray) – The payload wrench

  • +
  • tauR (ndarray) – Joint torque matrix minimum and maximums

  • +
  • frame (int) – The frame in which to torques are expressed in when J +is not supplied. ‘base’ means base frame of the robot, ‘ee’ means +end-effector frame

  • +
  • q (Union[ndarray, List[float], Tuple[float], Set[float], None]) – Joint coordinates

  • +
+
+
Returns:
+

The maximum permissible payload wrench

+
+
Return type:
+

ndarray(6)

+
+
+

Notes

+
    +
  • Wrench vector and Jacobian must be from the same reference frame

  • +
  • Tool transforms are taken into consideration for frame=1.

  • +
+
+ +
+
+payload(m, p=array([0.0, 0.0, 0.0]))
+

Add a payload to the end-effector

+

payload(m, p) adds payload mass adds a payload with point mass m at +position p in the end-effector coordinate frame.

+

payload(m) adds payload mass adds a payload with point mass m at +in the end-effector coordinate frame.

+

payload(0) removes added payload.

+
+
Parameters:
+
    +
  • m (float) – mass (kg)

  • +
  • p – position in end-effector frame

  • +
+
+
+
+ +
+
+perturb(p=0.1)
+

Perturb robot parameters

+

rp = perturb(p) is a new robot object in which the dynamic parameters +(link mass and inertia) have been perturbed. The perturbation is +multiplicative so that values are multiplied by random numbers in the +interval (1-p) to (1+p). The name string of the perturbed robot is +prefixed by ‘P/’.

+

Useful for investigating the robustness of various model-based control +schemes. For example to vary parameters in the range +/- 10 percent +is: r2 = puma.perturb(0.1)

+
+
Parameters:
+

p – The percent (+/-) to be perturbed. Default 10%

+
+
Returns:
+

A copy of the robot with dynamic parameters perturbed

+
+
Return type:
+

DHRobot

+
+
+
+ +
+
+plot(q, backend=None, block=False, dt=0.05, limits=None, vellipse=False, fellipse=False, fig=None, movie=None, loop=False, **kwargs)
+

Graphical display and animation

+

robot.plot(q, 'pyplot') displays a graphical view of a robot +based on the kinematic model and the joint configuration q. +This is a stick figure polyline which joins the origins of the +link coordinate frames. The plot will autoscale with an aspect +ratio of 1.

+

If q (m,n) representing a joint-space trajectory it will create an +animation with a pause of dt seconds between each frame.

+
+
+q
+

The joint configuration of the robot.

+
+ +
+
+backend
+

The graphical backend to use, currently ‘swift’ +and ‘pyplot’ are implemented. Defaults to ‘swift’ of a Robot +and ‘pyplot` for a DHRobot

+
+ +
+
+block
+

Block operation of the code and keep the figure open

+
+ +
+
+dt
+

if q is a trajectory, this describes the delay in +seconds between frames

+
+ +
+
+limits
+

Custom view limits for the plot. If not supplied will +autoscale, [x1, x2, y1, y2, z1, z2] +(this option is for ‘pyplot’ only)

+
+ +
+
+vellipse
+

(Plot Option) Plot the velocity ellipse at the +end-effector (this option is for ‘pyplot’ only)

+
+ +
+
+fellipse
+

(Plot Option) Plot the force ellipse at the +end-effector (this option is for ‘pyplot’ only)

+
+ +
+
+fig
+

(Plot Option) The figure label to plot in (this option is for +‘pyplot’ only)

+
+ +
+
+movie
+

(Plot Option) The filename to save the movie to (this option is for +‘pyplot’ only)

+
+ +
+
+loop
+

(Plot Option) Loop the movie (this option is for +‘pyplot’ only)

+
+ +
+
+jointaxes
+

(Plot Option) Plot an arrow indicating the axes in +which the joint revolves around(revolute joint) or translates +along (prosmatic joint) (this option is for ‘pyplot’ only)

+
+ +
+
+eeframe
+

(Plot Option) Plot the end-effector coordinate frame +at the location of the end-effector. Uses three arrows, red, +green and blue to indicate the x, y, and z-axes. +(this option is for ‘pyplot’ only)

+
+ +
+
+shadow
+

(Plot Option) Plot a shadow of the robot in the x-y +plane. (this option is for ‘pyplot’ only)

+
+ +
+
+name
+

(Plot Option) Plot the name of the robot near its base +(this option is for ‘pyplot’ only)

+
+ +
+
Returns:
+

A reference to the environment object which controls the +figure

+
+
Return type:
+

env

+
+
+

Notes

+
    +
  • +
    By default this method will block until the figure is dismissed.

    To avoid this set block=False.

    +
    +
    +
  • +
  • +
    For PyPlot, the polyline joins the origins of the link frames,

    but for some Denavit-Hartenberg models those frames may not +actually be on the robot, ie. the lines to not neccessarily +represent the links of the robot.

    +
    +
    +
  • +
+
+

See also

+

teach()

+
+
+ +
+
+plot_ellipse(ellipse, block=True, limits=None, jointaxes=True, eeframe=True, shadow=True, name=True)
+

Plot the an ellipsoid

+

robot.plot_ellipse(ellipsoid) displays the ellipsoid.

+
+
+ellipse
+

the ellipsoid to plot

+
+ +
+
+block
+

Block operation of the code and keep the figure open

+
+ +
+
+limits
+

Custom view limits for the plot. If not supplied will +autoscale, [x1, x2, y1, y2, z1, z2]

+
+ +
+
+jointaxes
+

(Plot Option) Plot an arrow indicating the axes in +which the joint revolves around(revolute joint) or translates +along (prosmatic joint)

+
+ +
+
+eeframe
+

(Plot Option) Plot the end-effector coordinate frame +at the location of the end-effector. Uses three arrows, red, +green and blue to indicate the x, y, and z-axes.

+
+ +
+
+shadow
+

(Plot Option) Plot a shadow of the robot in the x-y +plane

+
+ +
+
+name
+

(Plot Option) Plot the name of the robot near its base

+
+ +
+
Returns:
+

A reference to the PyPlot object which controls the +matplotlib figure

+
+
Return type:
+

env

+
+
+
+ +
+
+plot_fellipse(q, block=True, fellipse=None, limits=None, opt='trans', centre=[0, 0, 0], jointaxes=True, eeframe=True, shadow=True, name=True)
+

Plot the force ellipsoid for manipulator

+

robot.plot_fellipse(q) displays the velocity ellipsoid for the +robot at pose q. The plot will autoscale with an aspect ratio +of 1.

+

robot.plot_fellipse(vellipse) specifies a custon ellipse to plot.

+
+
+q
+

The joint configuration of the robot

+
+ +
+
+block
+

Block operation of the code and keep the figure open

+
+ +
+
+fellipse
+

the vellocity ellipsoid to plot

+
+ +
+
+limits
+

Custom view limits for the plot. If not supplied will +autoscale, [x1, x2, y1, y2, z1, z2]

+
+ +
+
+opt
+

‘trans’ or ‘rot’ will plot either the translational or +rotational force ellipsoid

+
+ +
+
+centre
+

The coordinates to plot the fellipse [x, y, z] or “ee” +to plot at the end-effector location

+
+ +
+
+jointaxes
+

(Plot Option) Plot an arrow indicating the axes in +which the joint revolves around(revolute joint) or translates +along (prosmatic joint)

+
+ +
+
+eeframe
+

(Plot Option) Plot the end-effector coordinate frame +at the location of the end-effector. Uses three arrows, red, +green and blue to indicate the x, y, and z-axes.

+
+ +
+
+shadow
+

(Plot Option) Plot a shadow of the robot in the x-y +plane

+
+ +
+
+name
+

(Plot Option) Plot the name of the robot near its base

+
+ +
+
Raises:
+

ValueError – q or fellipse must be supplied

+
+
Returns:
+

A reference to the PyPlot object which controls the +matplotlib figure

+
+
Return type:
+

env

+
+
+

Notes

+
    +
  • +
    By default the ellipsoid related to translational motion is

    drawn. Use opt='rot' to draw the rotational velocity +ellipsoid.

    +
    +
    +
  • +
  • +
    By default the ellipsoid is drawn at the origin. The option

    centre allows its origin to set to set to the specified +3-vector, or the string “ee” ensures it is drawn at the +end-effector position.

    +
    +
    +
  • +
+
+ +
+
+plot_vellipse(q, block=True, vellipse=None, limits=None, opt='trans', centre=[0, 0, 0], jointaxes=True, eeframe=True, shadow=True, name=True)
+

Plot the velocity ellipsoid for manipulator

+

robot.plot_vellipse(q) displays the velocity ellipsoid for the +robot at pose q. The plot will autoscale with an aspect ratio +of 1.

+

robot.plot_vellipse(vellipse) specifies a custon ellipse to plot.

+
+
block

Block operation of the code and keep the figure open

+
+
q

The joint configuration of the robot

+
+
vellipse

the vellocity ellipsoid to plot

+
+
limits

Custom view limits for the plot. If not supplied will +autoscale, [x1, x2, y1, y2, z1, z2]

+
+
opt

‘trans’ or ‘rot’ will plot either the translational or +rotational velocity ellipsoid

+
+
centre

The coordinates to plot the vellipse [x, y, z] or “ee” +to plot at the end-effector location

+
+
jointaxes

(Plot Option) Plot an arrow indicating the axes in +which the joint revolves around(revolute joint) or translates +along (prosmatic joint)

+
+
eeframe

(Plot Option) Plot the end-effector coordinate frame +at the location of the end-effector. Uses three arrows, red, +green and blue to indicate the x, y, and z-axes.

+
+
shadow

(Plot Option) Plot a shadow of the robot in the x-y +plane

+
+
name

(Plot Option) Plot the name of the robot near its base

+
+
+
+
Raises:
+

ValueError – q or fellipse must be supplied

+
+
Returns:
+

A reference to the PyPlot object which controls the +matplotlib figure

+
+
Return type:
+

env

+
+
+

Notes

+
    +
  • +
    By default the ellipsoid related to translational motion is

    drawn. Use opt='rot' to draw the rotational velocity +ellipsoid.

    +
    +
    +
  • +
  • +
    By default the ellipsoid is drawn at the origin. The option

    centre allows its origin to set to set to the specified +3-vector, or the string “ee” ensures it is drawn at the +end-effector position.

    +
    +
    +
  • +
+
+ +
+
+property prismaticjoints: List[bool]
+

Revolute joints as bool array

+
+
Returns:
+

array of joint type, True if prismatic

+
+
Return type:
+

prismaticjoints

+
+
+

Examples

+

+
+
+

Notes

+

Fixed joints, that maintain a constant link relative pose, +are not included.

+
+

See also

+

Link.isprismatic(), revolutejoints()

+
+
+ +
+
+property q: ndarray
+

Get/set robot joint configuration

+
    +
  • robot.q is the robot joint configuration

  • +
  • robot.q = ... checks and sets the joint configuration

  • +
+
+
Parameters:
+

q – the new robot joint configuration

+
+
Returns:
+

robot joint configuration

+
+
Return type:
+

q

+
+
+
+ +
+
+property qd: ndarray
+

Get/set robot joint velocity

+
    +
  • robot.qd is the robot joint velocity

  • +
  • robot.qd = ... checks and sets the joint velocity

  • +
+
+
Returns:
+

robot joint velocity

+
+
Return type:
+

qd

+
+
+
+ +
+
+property qdd: ndarray
+

Get/set robot joint acceleration

+
    +
  • robot.qdd is the robot joint acceleration

  • +
  • robot.qdd = ... checks and sets the robot joint acceleration

  • +
+
+
Returns:
+

robot joint acceleration

+
+
Return type:
+

qdd

+
+
+
+ +
+
+property qlim: ndarray
+

Joint limits

+

Limits are extracted from the link objects. If joints limits are +not set for:

+
    +
  • a revolute joint [-𝜋. 𝜋] is returned

  • +
  • a prismatic joint an exception is raised

  • +
+
+
+qlim
+

An array of joints limits (2, n)

+
+ +
+
Raises:
+

ValueError – unset limits for a prismatic joint

+
+
Returns:
+

Array of joint limit values

+
+
Return type:
+

qlim

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.qlim
+array([[-2.7925, -1.9199, -2.3562, -4.6426, -1.7453, -4.6426],
+       [ 2.7925,  1.9199,  2.3562,  4.6426,  1.7453,  4.6426]])
+
+
+
+ +
+
+random_q()
+

Return a random joint configuration

+

The value for each joint is uniform randomly distributed between the +limits set for the robot.

+
+

Note

+

The joint limit for all joints must be set.

+
+
+
Returns:
+

Random joint configuration :rtype: ndarray(n)

+
+
Return type:
+

q

+
+
+
+

See also

+

Robot.qlim(), Link.qlim()

+
+
+ +
+
+property revolutejoints: List[bool]
+

Revolute joints as bool array

+
+
Returns:
+

array of joint type, True if revolute

+
+
Return type:
+

revolutejoints

+
+
+

Examples

+

+
+
+

Notes

+

Fixed joints, that maintain a constant link relative pose, +are not included.

+
+

See also

+

Link.isrevolute(), prismaticjoints()

+
+
+ +
+
+property scene_children: List[SceneNode]
+

Returns the child nodes of this object

+
+ +
+
+property scene_parent: Type[SceneNode]
+

Returns the parent node of this object

+
+ +
+
+showgraph(display_graph=True, **kwargs)
+

Display a link transform graph in browser

+

robot.showgraph() displays a graph of the robot’s link frames +and the ETS between them. It uses GraphViz dot.

+
+
The nodes are:
    +
  • Base is shown as a grey square. This is the world frame origin, +but can be changed using the base attribute of the robot.

  • +
  • Link frames are indicated by circles

  • +
  • ETS transforms are indicated by rounded boxes

  • +
+
+
The edges are:
    +
  • an arrow if jtype is False or the joint is fixed

  • +
  • an arrow with a round head if jtype is True and the joint is +revolute

  • +
  • an arrow with a box head if jtype is True and the joint is +prismatic

  • +
+
+
+

Edge labels or nodes in blue have a fixed transformation to the +preceding link.

+
+
Parameters:
+
    +
  • display_graph (bool) – Open the graph in a browser if True. Otherwise will return the +file path

  • +
  • etsbox – Put the link ETS in a box, otherwise an edge label

  • +
  • jtype – Arrowhead to node indicates revolute or prismatic type

  • +
  • static – Show static joints in blue and bold

  • +
+
+
Return type:
+

Optional[str]

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.URDF.Panda()
+>>> panda.showgraph()
+
+
+_images/panda-graph.svg +
+

See also

+

dotfile()

+
+
+ +
+
+property structure: str
+

Return the joint structure string

+

A string with one letter per joint: R for a revolute +joint, and P for a prismatic joint.

+
+
Returns:
+

joint configuration string

+
+
Return type:
+

structure

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> puma.structure
+'RRRRRR'
+>>> stanford = rtb.models.DH.Stanford()
+>>> stanford.structure
+'RRPRRR'
+
+
+

Notes

+

Fixed joints, that maintain a constant link relative pose, +are not included. +len(self.structure) == self.n.

+
+ +
+
+property symbolic: bool
+
+ +
+
+teach(q, block=True, limits=None, vellipse=False, fellipse=False, backend=None)
+

Graphical teach pendant

+

robot.teach(q) creates a matplotlib plot which allows the user to +“drive” a graphical robot using a graphical slider panel. The robot’s +inital joint configuration is q. The plot will autoscale with an +aspect ratio of 1.

+

robot.teach() as above except the robot’s stored value of q +is used.

+
+
q

The joint configuration of the robot (Optional, +if not supplied will use the stored q values).

+
+
block

Block operation of the code and keep the figure open

+
+
limits

Custom view limits for the plot. If not supplied will +autoscale, [x1, x2, y1, y2, z1, z2]

+
+
vellipse

(Plot Option) Plot the velocity ellipse at the +end-effector (this option is for ‘pyplot’ only)

+
+
fellipse

(Plot Option) Plot the force ellipse at the +end-effector (this option is for ‘pyplot’ only)

+
+
+
+
Returns:
+

A reference to the PyPlot object which controls the +matplotlib figure

+
+
Return type:
+

env

+
+
+

Notes

+
    +
  • +
    Program execution is blocked until the teach window is

    dismissed. If block=False the method is non-blocking but +you need to poll the window manager to ensure that the window +remains responsive.

    +
    +
    +
  • +
  • +
    The slider limits are derived from the joint limit properties.

    If not set then: +- For revolute joints they are assumed to be [-pi, +pi] +- For prismatic joint they are assumed unknown and an error

    +
    +

    occurs.

    +
    +
    +
    +
  • +
+
+ +
+
+todegrees(q)
+

Convert joint angles to degrees

+
+
Parameters:
+

q – The joint configuration of the robot

+
+
Return type:
+

ndarray

+
+
Returns:
+

    +
  • q – a vector of joint coordinates in degrees and metres

  • +
  • robot.todegrees(q) converts joint coordinates q to degrees

  • +
  • taking into account whether elements of q correspond to revolute

  • +
  • or prismatic joints, ie. prismatic joint values are not converted.

  • +
  • If q is a matrix, with one column per joint, the conversion is

  • +
  • performed columnwise.

  • +
+

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> from math import pi
+>>> stanford = rtb.models.DH.Stanford()
+>>> stanford.todegrees([pi/4, pi/8, 2, -pi/4, pi/6, pi/3])
+array([ 45. ,  22.5,   2. , -45. ,  30. ,  60. ])
+
+
+
+ +
+
+property tool: SE3
+

Get/set robot tool transform

+
    +
  • robot.tool is the robot tool transform as an SE3 object

  • +
  • robot._tool is the robot tool transform as a numpy array

  • +
  • robot.tool = ... checks and sets the robot tool transform

  • +
+
+
Parameters:
+

tool – the new robot tool transform (as an SE(3))

+
+
Returns:
+

robot tool transform

+
+
Return type:
+

tool

+
+
+
+ +
+
+toradians(q)
+

Convert joint angles to radians

+

robot.toradians(q) converts joint coordinates q to radians +taking into account whether elements of q correspond to revolute +or prismatic joints, ie. prismatic joint values are not converted.

+

If q is a matrix, with one column per joint, the conversion is +performed columnwise.

+
+
Parameters:
+

q – The joint configuration of the robot

+
+
Returns:
+

a vector of joint coordinates in radians and metres

+
+
Return type:
+

q

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> stanford = rtb.models.DH.Stanford()
+>>> stanford.toradians([10, 20, 2, 30, 40, 50])
+array([0.1745, 0.3491, 2.    , 0.5236, 0.6981, 0.8727])
+
+
+
+ +
+
+property urdf_filepath: str
+
+ +
+
+property urdf_string: str
+
+ +
+
+vellipse(q, opt='trans', unit='rad', centre=[0, 0, 0], scale=0.1)
+

Create a velocity ellipsoid object for plotting with PyPlot

+

robot.vellipse(q) creates a force ellipsoid for the robot at +pose q. The ellipsoid is centered at the origin.

+
+
+q
+

The joint configuration of the robot.

+
+ +
+
+opt
+

‘trans’ or ‘rot’ will plot either the translational or +rotational force ellipsoid

+
+ +
+
+unit
+

‘rad’ or ‘deg’ will plot the ellipsoid in radians or +degrees

+
+ +
+
+centre
+

The centre of the ellipsoid, either ‘ee’ for the end-effector +or a 3-vector [x, y, z] in the world frame

+
+ +
+
+scale
+

The scale factor for the ellipsoid

+
+ +
+
Returns:
+

An EllipsePlot object

+
+
Return type:
+

env

+
+
+

Notes

+
    +
  • +
    By default the ellipsoid related to translational motion is

    drawn. Use opt='rot' to draw the rotational velocity +ellipsoid.

    +
    +
    +
  • +
  • +
    By default the ellipsoid is drawn at the origin. The option

    centre allows its origin to set to set to the specified +3-vector, or the string “ee” ensures it is drawn at the +end-effector position.

    +
    +
    +
  • +
+
+ +
+
+vision_collision_damper(shape, camera=None, camera_n=0, q=None, di=0.3, ds=0.05, xi=1.0, end=None, start=None, collision_list=None)
+

Compute a vision collision constrain for QP motion control

+

Formulates an inequality contraint which, when optimised for will +make it impossible for the robot to run into a line of sight. +See examples/fetch_vision.py for use case

+
+
+camera
+

The camera link, either as a robotic link or SE3 +pose

+
+ +
+
+camera_n
+

Degrees of freedom of the camera link

+
+ +
+
+ds
+

The minimum distance in which a joint is allowed to +approach the collision object shape

+
+ +
+
+di
+

The influence distance in which the velocity +damper becomes active

+
+ +
+
+xi
+

The gain for the velocity damper

+
+ +
+
+end
+

The end link of the robot to consider

+
+ +
+
+start
+

The start link of the robot to consider

+
+ +
+
+collision_list
+

A list of shapes to consider for collision

+
+ +
+
Returns:
+

    +
  • Ain – A (6,) vector inequality contraint for an optisator

  • +
  • Bin – b (6,) vector inequality contraint for an optisator

  • +
+

+
+
+
+ +
+ +
+ +
+

Models

+

A number of models are defined in terms of Denavit-Hartenberg parameters, either +standard or modified. They can be listed by:

+

+
+
+
+
+class roboticstoolbox.models.DH.Panda[source]
+

Bases: DHRobot

+

A class representing the Panda robot arm.

+

Panda() is a class which models a Franka-Emika Panda robot and +describes its kinematic characteristics using modified DH +conventions.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Panda()
+>>> print(robot)
+DHRobot: Panda (by Franka Emika), 7 joints (RRRRRRR), dynamics, geometry, modified DH parameters
+┌────────┬────────┬─────┬───────┬─────────┬────────┐
+│ aⱼ₋₁   │  ⍺ⱼ₋₁  │ θⱼ  │  dⱼ   │   q⁻    │   q⁺   │
+├────────┼────────┼─────┼───────┼─────────┼────────┤
+│    0.0 │   0.0° │  q1 │ 0.333 │ -166.0° │ 166.0° │
+│    0.0 │ -90.0° │  q2 │   0.0 │ -101.0° │ 101.0° │
+│    0.0 │  90.0° │  q3 │ 0.316 │ -166.0° │ 166.0° │
+│ 0.0825 │  90.0° │  q4 │   0.0 │ -176.0° │  -4.0° │
+│-0.0825 │ -90.0° │  q5 │ 0.384 │ -166.0° │ 166.0° │
+│    0.0 │  90.0° │  q6 │   0.0 │   -1.0° │ 215.0° │
+│  0.088 │  90.0° │  q7 │ 0.107 │ -166.0° │ 166.0° │
+└────────┴────────┴─────┴───────┴─────────┴────────┘
+
+┌─────┬───────────────────────────────────────┐
+│tool │ t = 0, 0, 0.1; rpy/xyz = -45°, 0°, 0° │
+└─────┴───────────────────────────────────────┘
+
+┌─────┬─────┬────────┬─────┬───────┬─────┬───────┬──────┐
+│name │ q0  │ q1     │ q2  │ q3    │ q4  │ q5    │ q6   │
+├─────┼─────┼────────┼─────┼───────┼─────┼───────┼──────┤
+│  qr │  0° │ -17.2° │  0° │ -126° │  0° │  115° │  45° │
+│  qz │  0° │  0°    │  0° │  0°   │  0° │  0°   │  0°  │
+└─────┴─────┴────────┴─────┴───────┴─────┴───────┴──────┘
+
+
+
+
+

Note

+
    +
  • SI units of metres are used.

  • +
  • The model includes a tool offset.

  • +
+
+
+
References:
+
+
+
+

Code author: Samuel Drew

+

Code author: Peter Corke

+
+ +
+
+class roboticstoolbox.models.DH.Puma560(symbolic=False)[source]
+

Bases: DHRobot

+

Class that models a Puma 560 manipulator

+
+
Parameters:
+

symbolic (bool) – use symbolic constants

+
+
+

Puma560() is an object which models a Unimation Puma560 robot and +describes its kinematic and dynamic characteristics using standard DH +conventions.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> print(robot)
+DHRobot: Puma 560 (by Unimation), 6 joints (RRRRRR), dynamics, geometry, standard DH parameters
+┌────┬────────┬────────┬────────┬─────────┬────────┐
+│θⱼ  │   dⱼ   │   aⱼ   │   ⍺ⱼ   │   q⁻    │   q⁺   │
+├────┼────────┼────────┼────────┼─────────┼────────┤
+│ q1 │ 0.6718 │      0 │  90.0° │ -160.0° │ 160.0° │
+│ q2 │      0 │ 0.4318 │   0.0° │ -110.0° │ 110.0° │
+│ q3 │   0.15 │ 0.0203 │ -90.0° │ -135.0° │ 135.0° │
+│ q4 │ 0.4318 │      0 │  90.0° │ -266.0° │ 266.0° │
+│ q5 │      0 │      0 │ -90.0° │ -100.0° │ 100.0° │
+│ q6 │      0 │      0 │   0.0° │ -266.0° │ 266.0° │
+└────┴────────┴────────┴────────┴─────────┴────────┘
+
+┌─┬──┐
+└─┴──┘
+
+┌─────┬─────┬──────┬───────┬─────┬──────┬─────┐
+│name │ q0  │ q1   │ q2    │ q3  │ q4   │ q5  │
+├─────┼─────┼──────┼───────┼─────┼──────┼─────┤
+│  qr │  0° │  90° │ -90°  │  0° │  0°  │  0° │
+│  qz │  0° │  0°  │  0°   │  0° │  0°  │  0° │
+│  qn │  0° │  45° │  180° │  0° │  45° │  0° │
+│  qs │  0° │  0°  │ -90°  │  0° │  0°  │  0° │
+└─────┴─────┴──────┴───────┴─────┴──────┴─────┘
+
+
+
+

Defined joint configurations are:

+
    +
  • qz, zero joint angle configuration, ‘L’ shaped configuration

  • +
  • qr, vertical ‘READY’ configuration

  • +
  • qs, arm is stretched out in the x-direction

  • +
  • qn, arm is at a nominal non-singular configuration

  • +
+
+

Note

+
    +
  • SI units are used.

  • +
  • The model includes armature inertia and gear ratios.

  • +
  • The value of m1 is given as 0 here. Armstrong found no value for it +and it does not appear in the equation for tau1 after the +substituion is made to inertia about link frame rather than COG +frame.

  • +
  • Gravity load torque is the motor torque necessary to keep the joint +static, and is thus -ve of the gravity caused torque.

  • +
+
+
+

Warning

+

Compared to the MATLAB version of the Toolbox this model +includes the pedestal, making the z-coordinates 26 inches larger.

+
+
+
References:
+
    +
  • “A search for consensus among model parameters reported for the PUMA +560 robot”, P. Corke and B. Armstrong-Helouvry, +Proc. IEEE Int. Conf. Robotics and Automation, (San Diego), +pp. 1608-1613, May 1994. (for kinematic and dynamic parameters)

  • +
  • “A combined optimization method for solving the inverse kinematics +problem”, Wang & Chen, IEEE Trans. RA 7(4) 1991 pp 489-. +(for joint angle limits)

  • +
  • https://github.com/4rtur1t0/ARTE/blob/master/robots/UNIMATE/puma560/parameters.m

  • +
+
+
+

Code author: Peter Corke

+
+
+ikine_a(T, config='lun')[source]
+

Analytic inverse kinematic solution

+
+
Parameters:
+
    +
  • T (SE3) – end-effector pose

  • +
  • config (str, optional) – arm configuration, defaults to “lun”

  • +
+
+
Returns:
+

joint angle vector in radians

+
+
Return type:
+

ndarray(6)

+
+
+

robot.ikine_a(T, config) is the joint angle vector which achieves the +end-effector pose T`. The configuration string selects the specific +solution and is a sting comprising the following letters:

+ + + + + + + + + + + + + + + + + + + + + + + + + + +

Letter

Meaning

l

Choose the left-handed configuration

r

Choose the right-handed configuration

u

Choose the elbow up configuration

d

Choose the elbow down configuration

n

Choose the wrist not-flipped configuration

f

Choose the wrist flipped configuration

+
+
Reference:
+
    +
  • Inverse kinematics for a PUMA 560, +Paul and Zhang, +The International Journal of Robotics Research, +Vol. 5, No. 2, Summer 1986, p. 32-44

  • +
+
+
Author:
+

based on MATLAB code by Robert Biro with Gary Von McMurray, +GTRI/ATRP/IIMB, Georgia Institute of Technology, 2/13/95

+
+
+
+ +
+ +
+
+class roboticstoolbox.models.DH.Stanford[source]
+

Bases: DHRobot

+

Class that models a Stanford arm manipulator

+

Stanford() is a class which models a Unimation Puma560 robot and +describes its kinematic and dynamic characteristics using standard DH +conventions.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Stanford()
+>>> print(robot)
+DHRobot: Stanford arm (by Victor Scheinman), 6 joints (RRPRRR), dynamics, standard DH parameters
+┌───────┬───────┬────────┬────────┬─────────────────────┬────────┐
+│  θⱼ   │  dⱼ   │   aⱼ   │   ⍺ⱼ   │         q⁻          │   q⁺   │
+├───────┼───────┼────────┼────────┼─────────────────────┼────────┤
+│ q1    │ 0.412 │      0 │ -90.0° │             -170.0° │ 170.0° │
+│ q2    │ 0.154 │      0 │  90.0° │             -170.0° │ 170.0° │
+│-90.0° │    q3 │ 0.0203 │   0.0° │ 0.30479999999999996 │   1.27 │
+│ q4    │     0 │      0 │ -90.0° │             -170.0° │ 170.0° │
+│ q5    │     0 │      0 │  90.0° │              -90.0° │  90.0° │
+│ q6    │     0 │      0 │   0.0° │             -170.0° │ 170.0° │
+└───────┴───────┴────────┴────────┴─────────────────────┴────────┘
+
+┌─┬──┐
+└─┴──┘
+
+┌─────┬─────┬─────┬────┬─────┬─────┬─────┐
+│name │ q0  │ q1  │ q2 │ q3  │ q4  │ q5  │
+├─────┼─────┼─────┼────┼─────┼─────┼─────┤
+│  qr │  0° │  0° │  0 │  0° │  0° │  0° │
+│  qz │  0° │  0° │  0 │  0° │  0° │  0° │
+└─────┴─────┴─────┴────┴─────┴─────┴─────┘
+
+
+
+

Defined joint configurations are:

+
+
    +
  • qz, zero joint angle configuration

  • +
+
+
+

Note

+
    +
  • SI units are used.

  • +
  • Gear ratios not currently known, though reflected armature inertia +is known, so gear ratios are set to 1.

  • +
+
+
+
References:
+
    +
  • Kinematic data from “Modelling, Trajectory calculation and Servoing +of a computer controlled arm”. Stanford AIM-177. Figure 2.3

  • +
  • Dynamic data from “Robot manipulators: mathematics, programming and +control” +Paul 1981, Tables 6.5, 6.6

  • +
  • Dobrotin & Scheinman, “Design of a computer controlled manipulator +for robot research”, IJCAI, 1973.

  • +
+
+
+

Code author: Peter Corke

+
+ +
+
+class roboticstoolbox.models.DH.Ball(N=10)[source]
+

Bases: DHRobot

+

Class that models a ball manipulator

+
+
Parameters:
+
    +
  • N (int, optional) – number of links, defaults to 10

  • +
  • symbolic (bool, optional) – [description], defaults to False

  • +
+
+
+

The ball robot is an abstract robot with an arbitrary number of joints. +At zero joint angles it is straight along the x-axis, and as the joint +angles are increased (equally) it wraps up into a 3D ball shape.

+
    +
  • Ball() is an object which describes the kinematic characteristics of +a ball robot using standard DH conventions.

  • +
  • Ball(N) as above, but models a robot with N joints.

  • +
+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Ball()
+>>> print(robot)
+DHRobot: ball, 10 joints (RRRRRRRRRR), dynamics, standard DH parameters
+┌─────┬────┬─────┬───────┐
+│ θⱼ  │ dⱼ │ aⱼ  │  ⍺ⱼ   │
+├─────┼────┼─────┼───────┤
+│ q1  │  0 │ 0.1 │ 90.0° │
+│ q2  │  0 │ 0.1 │ 90.0° │
+│ q3  │  0 │ 0.1 │ 90.0° │
+│ q4  │  0 │ 0.1 │ 90.0° │
+│ q5  │  0 │ 0.1 │ 90.0° │
+│ q6  │  0 │ 0.1 │ 90.0° │
+│ q7  │  0 │ 0.1 │ 90.0° │
+│ q8  │  0 │ 0.1 │ 90.0° │
+│ q9  │  0 │ 0.1 │ 90.0° │
+│ q10 │  0 │ 0.1 │ 90.0° │
+└─────┴────┴─────┴───────┘
+
+┌─┬──┐
+└─┴──┘
+
+┌─────┬────────┬────────┬────────┬────────┬────────┬────────┬────────┬────────┬────────┬────────┐
+│name │ q0     │ q1     │ q2     │ q3     │ q4     │ q5     │ q6     │ q7     │ q8     │ q9     │
+├─────┼────────┼────────┼────────┼────────┼────────┼────────┼────────┼────────┼────────┼────────┤
+│  qr │  57.3° │  57.3° │ -38.2° │  57.3° │  57.3° │ -38.2° │ -38.2° │  57.3° │ -38.2° │  57.3° │
+│  qz │  0°    │  0°    │  0°    │  0°    │  0°    │  0°    │  0°    │  0°    │  0°    │  0°    │
+└─────┴────────┴────────┴────────┴────────┴────────┴────────┴────────┴────────┴────────┴────────┘
+
+
+
+

Defined joint configurations are:

+
+
    +
  • qz, zero joint angles

  • +
  • q1, ball shaped configuration

  • +
+
+
+
References:
+
    +
  • “A divide and conquer articulated-body algorithm for parallel +O(log(n)) calculation of rigid body dynamics, Part 2”, +Int. J. Robotics Research, 18(9), pp 876-892.

  • +
+
+
Seealso:
+

Hyper(), Ball()

+
+
+

Code author: Peter Corke

+
+ +
+
+class roboticstoolbox.models.DH.Coil(N=10, symbolic=False)[source]
+

Bases: DHRobot

+

Create model of a coil manipulator

+
+
Parameters:
+
    +
  • N (int, optional) – number of links, defaults to 10

  • +
  • symbolic (bool, optional) – [description], defaults to False

  • +
+
+
+

The coil robot is an abstract robot with an arbitrary number of joints +that folds into a helix shape. At zero joint angles it is straight along +the x-axis, and as the joint angles are increased (equally) it wraps up into +a 3D helix shape.

+
    +
  • Coil() is an object which describes the kinematic characteristics of +a coil robot using standard DH conventions.

  • +
  • Coil(N) as above, but models a robot with N joints.

  • +
+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Coil()
+>>> print(robot)
+DHRobot: Coil10, 10 joints (RRRRRRRRRR), dynamics, standard DH parameters
+┌─────┬────┬─────┬───────┐
+│ θⱼ  │ dⱼ │ aⱼ  │  ⍺ⱼ   │
+├─────┼────┼─────┼───────┤
+│ q1  │  0 │ 0.1 │ 90.0° │
+│ q2  │  0 │ 0.1 │ 90.0° │
+│ q3  │  0 │ 0.1 │ 90.0° │
+│ q4  │  0 │ 0.1 │ 90.0° │
+│ q5  │  0 │ 0.1 │ 90.0° │
+│ q6  │  0 │ 0.1 │ 90.0° │
+│ q7  │  0 │ 0.1 │ 90.0° │
+│ q8  │  0 │ 0.1 │ 90.0° │
+│ q9  │  0 │ 0.1 │ 90.0° │
+│ q10 │  0 │ 0.1 │ 90.0° │
+└─────┴────┴─────┴───────┘
+
+┌─┬──┐
+└─┴──┘
+
+┌─────┬───────┬───────┬───────┬───────┬───────┬───────┬───────┬───────┬───────┬───────┐
+│name │ q0    │ q1    │ q2    │ q3    │ q4    │ q5    │ q6    │ q7    │ q8    │ q9    │
+├─────┼───────┼───────┼───────┼───────┼───────┼───────┼───────┼───────┼───────┼───────┤
+│  qr │  180° │  180° │  180° │  180° │  180° │  180° │  180° │  180° │  180° │  180° │
+│  qz │  0°   │  0°   │  0°   │  0°   │  0°   │  0°   │  0°   │  0°   │  0°   │  0°   │
+└─────┴───────┴───────┴───────┴───────┴───────┴───────┴───────┴───────┴───────┴───────┘
+
+
+
+

Defined joint configurations are:

+
    +
  • qz, zero joint angle configuration

  • +
+
+
References:
+
    +
  • “A divide and conquer articulated-body algorithm for parallel O(log(n)) +calculation of rigid body dynamics, Part 2”, +Int. J. Robotics Research, 18(9), pp 876-892.

  • +
+
+
Seealso:
+

Hyper(), Ball()

+
+
+

Code author: Peter Corke

+
+ +
+
+class roboticstoolbox.models.DH.Hyper(N=10, a=None, symbolic=False)[source]
+

Bases: DHRobot

+

Create model of a hyper redundant planar manipulator

+
+
Parameters:
+
    +
  • N (int, optional) – number of links, defaults to 10

  • +
  • a (int or symbolic, optional) – link length, defaults total 1/N giving a reach of 1

  • +
  • symbolic (bool, optional) – [description], defaults to False

  • +
+
+
+
    +
  • Hyper() is an object which describes the kinematics of a serial link +manipulator with 10 joints which moves in the xy-plane, using standard DH +conventions. At zero angles it forms a straight line along the x-axis.

  • +
  • Hyper(N) as above, but models a robot with N joints.

  • +
+
IndexError: too many indices for array: array is 0-dimensional, but 1 were indexed
+
+
+

Defined joint configurations are:

+
    +
  • qz, zero joint angle configuration

  • +
+
+
References:
+
    +
  • “A divide and conquer articulated-body algorithm for parallel O(log(n))

  • +
+

calculation of rigid body dynamics, Part 2”, +Int. J. Robotics Research, 18(9), pp 876-892.

+
+
Seealso:
+

Coil(), Ball()

+
+
+

Code author: Peter Corke

+
+ +
+
+class roboticstoolbox.models.DH.Hyper3d(N=10, a=None, symbolic=False)[source]
+

Bases: DHRobot

+

Create model of a hyper redundant manipulator

+
+
Parameters:
+
    +
  • N (int, optional) – number of links, defaults to 10

  • +
  • a (int or symbolic, optional) – link length, defaults total 1/N giving a reach of 1

  • +
  • symbolic (bool, optional) – [description], defaults to False

  • +
+
+
+
    +
  • Hyper3d() is an object which describes the kinematics of a serial link +manipulator with 10 joints which moves in the xy-plane, using standard DH +conventions. At zero angles it forms a straight line along the x-axis.

  • +
  • Hyper3d(N) as above, but models a robot with N joints.

  • +
+
IndexError: too many indices for array: array is 0-dimensional, but 1 were indexed
+
+
+

Defined joint configurations are:

+
    +
  • qz, zero joint angle configuration

  • +
+
+
References:
+
    +
  • “A divide and conquer articulated-body algorithm for parallel O(log(n))

  • +
+

calculation of rigid body dynamics, Part 2”, +Int. J. Robotics Research, 18(9), pp 876-892.

+
+
Seealso:
+

Coil(), Ball()

+
+
+

Code author: Peter Corke

+
+ +
+
+class roboticstoolbox.models.DH.Cobra600[source]
+

Bases: DHRobot

+

Class that models a Adept Cobra 600 SCARA manipulator

+

Cobra600() is a class which models an Adept Cobra 600 SCARA robot and +describes its kinematic characteristics using standard DH +conventions.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Cobra600()
+>>> print(robot)
+DHRobot: Cobra600 (by Omron), 4 joints (RRPR), dynamics, standard DH parameters
+┌─────┬───────┬───────┬────────┬─────────┬────────┐
+│ θⱼ  │  dⱼ   │  aⱼ   │   ⍺ⱼ   │   q⁻    │   q⁺   │
+├─────┼───────┼───────┼────────┼─────────┼────────┤
+│ q1  │ 0.387 │ 0.325 │   0.0° │  -50.0° │  50.0° │
+│ q2  │     0 │ 0.275 │ 180.0° │  -88.0° │  88.0° │
+│0.0° │    q3 │     0 │   0.0° │     0.0 │   0.21 │
+│ q4  │     0 │     0 │   0.0° │ -180.0° │ 180.0° │
+└─────┴───────┴───────┴────────┴─────────┴────────┘
+
+┌─┬──┐
+└─┴──┘
+
+┌─────┬─────┬─────┬────┬─────┐
+│name │ q0  │ q1  │ q2 │ q3  │
+├─────┼─────┼─────┼────┼─────┤
+│  qz │  0° │  0° │  0 │  0° │
+│  qr │  0° │  0° │  0 │  0° │
+└─────┴─────┴─────┴────┴─────┘
+
+
+
+

Defined joint configurations are:

+
    +
  • qz, zero joint angle configuration, ‘L’ shaped configuration

  • +
+
+

Note

+
    +
  • SI units are used.

  • +
  • Robot has only 4 DoF.

  • +
+
+

Code author: Peter Corke

+
+ +
+
+class roboticstoolbox.models.DH.IRB140[source]
+

Bases: DHRobot

+

Class that models an ABB IRB140 manipulator

+

IRB140() is a class which models a Unimation Puma560 robot and +describes its kinematic and dynamic characteristics using standard DH +conventions.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.IRB140()
+>>> print(robot)
+DHRobot: IRB 140 (by ABB), 6 joints (RRRRRR), dynamics, geometry, standard DH parameters
+┌────┬───────┬──────┬────────┬─────────┬────────┐
+│θⱼ  │  dⱼ   │  aⱼ  │   ⍺ⱼ   │   q⁻    │   q⁺   │
+├────┼───────┼──────┼────────┼─────────┼────────┤
+│ q1 │ 0.352 │ 0.07 │ -90.0° │ -180.0° │ 180.0° │
+│ q2 │     0 │ 0.36 │   0.0° │ -100.0° │ 100.0° │
+│ q3 │     0 │    0 │ -90.0° │ -220.0° │  60.0° │
+│ q4 │  0.38 │    0 │  90.0° │ -200.0° │ 200.0° │
+│ q5 │     0 │    0 │ -90.0° │ -120.0° │ 120.0° │
+│ q6 │ 0.065 │    0 │   0.0° │ -400.0° │ 400.0° │
+└────┴───────┴──────┴────────┴─────────┴────────┘
+
+┌─┬──┐
+└─┴──┘
+
+┌─────┬─────┬──────┬───────┬─────┬──────┬──────┐
+│name │ q0  │ q1   │ q2    │ q3  │ q4   │ q5   │
+├─────┼─────┼──────┼───────┼─────┼──────┼──────┤
+│  qr │  0° │ -90° │  90°  │  0° │  90° │ -90° │
+│  qz │  0° │  0°  │  0°   │  0° │  0°  │  0°  │
+│  qd │  0° │ -90° │  180° │  0° │  0°  │ -90° │
+└─────┴─────┴──────┴───────┴─────┴──────┴──────┘
+
+
+
+

Defined joint configurations are:

+
+
    +
  • qz, zero joint angle configuration

  • +
  • qr, vertical ‘READY’ configuration

  • +
  • qd, lower arm horizontal as per data sheet

  • +
+
+
+

Note

+

SI units of metres are used.

+
+
+
References:
+
+
+
+

Code author: Peter Corke

+
+ +
+
+class roboticstoolbox.models.DH.KR5[source]
+

Bases: DHRobot

+

Class that models a Kuka KR5 manipulator

+

KR5() is a class which models a Kuka KR5 robot and +describes its kinematic characteristics using standard DH +conventions.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.KR5()
+>>> print(robot)
+DHRobot: KR5 (by KUKA), 6 joints (RRRRRR), dynamics, geometry, standard DH parameters
+┌────┬────────┬──────┬────────┬─────────┬────────┐
+│θⱼ  │   dⱼ   │  aⱼ  │   ⍺ⱼ   │   q⁻    │   q⁺   │
+├────┼────────┼──────┼────────┼─────────┼────────┤
+│ q1 │    0.4 │ 0.18 │ -90.0° │ -155.0° │ 155.0° │
+│ q2 │      0 │  0.6 │   0.0° │ -180.0° │  65.0° │
+│ q3 │      0 │ 0.12 │  90.0° │  -15.0° │ 158.0° │
+│ q4 │  -0.62 │    0 │ -90.0° │ -350.0° │ 350.0° │
+│ q5 │      0 │    0 │  90.0° │ -130.0° │ 130.0° │
+│ q6 │ -0.115 │    0 │ 180.0° │ -350.0° │ 350.0° │
+└────┴────────┴──────┴────────┴─────────┴────────┘
+
+┌─┬──┐
+└─┴──┘
+
+┌─────┬──────┬──────┬──────┬──────┬──────┬──────┐
+│name │ q0   │ q1   │ q2   │ q3   │ q4   │ q5   │
+├─────┼──────┼──────┼──────┼──────┼──────┼──────┤
+│  qr │  45° │  60° │  45° │  30° │  45° │  30° │
+│  qz │  0°  │  0°  │  0°  │  0°  │  0°  │  0°  │
+│ qk1 │  45° │  60° │  45° │  30° │  45° │  30° │
+│ qk2 │  45° │  60° │  30° │  60° │  45° │  30° │
+│ qk3 │  30° │  60° │  30° │  60° │  30° │  60° │
+└─────┴──────┴──────┴──────┴──────┴──────┴──────┘
+
+
+
+

Defined joint configurations are:

+
+
    +
  • qk1, nominal working position 1

  • +
  • qk2, nominal working position 2

  • +
  • qk3, nominal working position 3

  • +
+
+
+

Note

+
    +
  • SI units of metres are used.

  • +
  • Includes an 11.5cm tool in the z-direction

  • +
+
+
+
References:
+
+
+
+

Code author: Gautam Sinha, Indian Institute of Technology, Kanpur (original MATLAB version)

+

Code author: Samuel Drew

+

Code author: Peter Corke

+
+ +
+
+class roboticstoolbox.models.DH.Orion5(base=None)[source]
+

Bases: DHRobot

+

Class that models a RAWR robotics Orion5 manipulator

+

Orion5() is a class which models a RAWR Robotics Orion5 robot and +describes its kinematic characteristics using standard DH +conventions.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Orion5()
+>>> print(robot)
+DHRobot: Orion 5 (by RAWR Robotics), 4 joints (RRRR), dynamics, standard DH parameters
+┌────┬───────┬─────────┬───────┬─────────┬────────┐
+│θⱼ  │  dⱼ   │   aⱼ    │  ⍺ⱼ   │   q⁻    │   q⁺   │
+├────┼───────┼─────────┼───────┼─────────┼────────┤
+│ q1 │ 0.053 │       0 │ 90.0° │ -180.0° │ 180.0° │
+│ q2 │     0 │  0.1704 │  0.0° │   10.0° │ 122.5° │
+│ q3 │     0 │ -0.1363 │  0.0° │   20.0° │ 340.0° │
+│ q4 │     0 │   0.126 │  0.0° │   45.0° │ 315.0° │
+└────┴───────┴─────────┴───────┴─────────┴────────┘
+
+┌─────┬────────────────────────────────────┐
+│base │ None                               │
+│tool │ t = 0, 0, 0; rpy/xyz = 0°, 90°, 0° │
+└─────┴────────────────────────────────────┘
+
+┌─────┬─────┬──────┬───────┬───────┐
+│name │ q0  │ q1   │ q2    │ q3    │
+├─────┼─────┼──────┼───────┼───────┤
+│  qr │  0° │  0°  │  180° │  90°  │
+│  qz │  0° │  0°  │  0°   │  0°   │
+│  qv │  0° │  90° │  180° │  180° │
+│  qh │  0° │  0°  │  180° │  90°  │
+└─────┴─────┴──────┴───────┴───────┘
+
+
+
+

Defined joint configurations are:

+
+
    +
  • qz, zero angles, all folded up

  • +
  • qv, stretched out vertically

  • +
  • qh, arm horizontal, hand down

  • +
+
+
+

Note

+
    +
  • SI units of metres are used.

  • +
  • Robot has only 4 DoF.

  • +
+
+
+
References:
+
+
+
+

Code author: Aditya Dua

+

Code author: Peter Corke

+
+ +
+
+class roboticstoolbox.models.DH.Planar3[source]
+

Bases: DHRobot

+

Class that models a planar 3-link robot

+

Planar2() is a class which models a 3-link planar robot and +describes its kinematic characteristics using standard DH +conventions.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Planar3()
+>>> print(robot)
+DHRobot: Planar 3 link, 3 joints (RRR), dynamics, standard DH parameters
+┌────┬────┬────┬──────┐
+│θⱼ  │ dⱼ │ aⱼ │  ⍺ⱼ  │
+├────┼────┼────┼──────┤
+│ q1 │  0 │  1 │ 0.0° │
+│ q2 │  0 │  1 │ 0.0° │
+│ q3 │  0 │  1 │ 0.0° │
+└────┴────┴────┴──────┘
+
+┌─┬──┐
+└─┴──┘
+
+┌─────┬─────┬─────┬─────┐
+│name │ q0  │ q1  │ q2  │
+├─────┼─────┼─────┼─────┤
+│  qr │  0° │  0° │  0° │
+│  qz │  0° │  0° │  0° │
+└─────┴─────┴─────┴─────┘
+
+
+
+

Defined joint configurations are:

+
+
    +
  • qz, zero angles, all folded up

  • +
+
+
+

Note

+
    +
  • Robot has only 3 DoF.

  • +
+
+

Code author: Peter Corke

+
+ +
+
+class roboticstoolbox.models.DH.Planar2(symbolic=False)[source]
+

Bases: DHRobot

+

Class that models a planar 2-link robot

+

Planar2() is a class which models a 2-link planar robot and +describes its kinematic characteristics using standard DH +conventions.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Planar2()
+>>> print(robot)
+DHRobot: Planar 2 link, 2 joints (RR), dynamics, standard DH parameters
+┌────┬────┬────┬──────┐
+│θⱼ  │ dⱼ │ aⱼ │  ⍺ⱼ  │
+├────┼────┼────┼──────┤
+│ q1 │  0 │  1 │ 0.0° │
+│ q2 │  0 │  1 │ 0.0° │
+└────┴────┴────┴──────┘
+
+┌─┬──┐
+└─┴──┘
+
+┌─────┬──────┬──────┐
+│name │ q0   │ q1   │
+├─────┼──────┼──────┤
+│  qr │  0°  │  90° │
+│  qz │  0°  │  0°  │
+│  q1 │  0°  │  90° │
+│  q2 │  90° │ -90° │
+└─────┴──────┴──────┘
+
+
+
+

Defined joint configurations are:

+
+
    +
  • qz, zero angles, all folded up

  • +
  • q1, links are horizontal and vertical respectively

  • +
  • q2, links are vertical and horizontal respectively

  • +
+
+
+

Note

+
    +
  • Robot has only 2 DoF.

  • +
+
+

Code author: Peter Corke

+
+ +
+
+class roboticstoolbox.models.DH.LWR4[source]
+

Bases: DHRobot

+

Class that models a LWR-IV manipulator

+

LWR4() is a class which models a Kuka LWR-IV robot and +describes its kinematic characteristics using standard DH +conventions.

+
IndexError: too many indices for array: array is 0-dimensional, but 1 were indexed
+
+
+

Defined joint configurations are:

+
    +
  • qz, zero joint angle configuration, ‘L’ shaped configuration

  • +
  • qr, vertical ‘READY’ configuration

  • +
  • qs, arm is stretched out in the X direction

  • +
  • qn, arm is at a nominal non-singular configuration

  • +
+
+

Note

+

SI units are used.

+
+
+
References:
+
+
+
+

Code author: Peter Corke

+
+ +
+
+class roboticstoolbox.models.DH.UR3(symbolic=False)[source]
+

Bases: DHRobot

+

Class that models a Universal Robotics UR3 manipulator

+
+
Parameters:
+

symbolic (bool) – use symbolic constants

+
+
+

UR3() is an object which models a Unimation Puma560 robot and +describes its kinematic and dynamic characteristics using standard DH +conventions.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.UR3()
+>>> print(robot)
+DHRobot: UR3 (by Universal Robotics), 6 joints (RRRRRR), dynamics, standard DH parameters
+┌────┬─────────┬─────────┬────────┐
+│θⱼ  │   dⱼ    │   aⱼ    │   ⍺ⱼ   │
+├────┼─────────┼─────────┼────────┤
+│ q1 │  0.1519 │       0 │  90.0° │
+│ q2 │       0 │ -0.2437 │   0.0° │
+│ q3 │       0 │ -0.2132 │   0.0° │
+│ q4 │  0.1124 │       0 │  90.0° │
+│ q5 │ 0.08535 │       0 │ -90.0° │
+│ q6 │  0.0819 │       0 │   0.0° │
+└────┴─────────┴─────────┴────────┘
+
+┌─┬──┐
+└─┴──┘
+
+┌─────┬───────┬─────┬─────┬─────┬──────┬─────┐
+│name │ q0    │ q1  │ q2  │ q3  │ q4   │ q5  │
+├─────┼───────┼─────┼─────┼─────┼──────┼─────┤
+│  qr │  180° │  0° │  0° │  0° │  90° │  0° │
+│  qz │  0°   │  0° │  0° │  0° │  0°  │  0° │
+└─────┴───────┴─────┴─────┴─────┴──────┴─────┘
+
+
+
+

Defined joint configurations are:

+
    +
  • qz, zero joint angle configuration

  • +
  • qr, arm horizontal along x-axis

  • +
+
+

Note

+
    +
  • SI units are used.

  • +
+
+
+
References:
+
+
+
Sealso:
+

UR5(), UR10()

+
+
+

Code author: Peter Corke

+
+ +
+
+class roboticstoolbox.models.DH.UR5(symbolic=False)[source]
+

Bases: DHRobot

+

Class that models a Universal Robotics UR5 manipulator

+
+
Parameters:
+

symbolic (bool) – use symbolic constants

+
+
+

UR5() is an object which models a Unimation Puma560 robot and +describes its kinematic and dynamic characteristics using standard DH +conventions.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.UR5()
+>>> print(robot)
+DHRobot: UR5 (by Universal Robotics), 6 joints (RRRRRR), dynamics, standard DH parameters
+┌────┬─────────┬─────────┬────────┐
+│θⱼ  │   dⱼ    │   aⱼ    │   ⍺ⱼ   │
+├────┼─────────┼─────────┼────────┤
+│ q1 │ 0.08946 │       0 │  90.0° │
+│ q2 │       0 │  -0.425 │   0.0° │
+│ q3 │       0 │ -0.3922 │   0.0° │
+│ q4 │  0.1091 │       0 │  90.0° │
+│ q5 │ 0.09465 │       0 │ -90.0° │
+│ q6 │  0.0823 │       0 │   0.0° │
+└────┴─────────┴─────────┴────────┘
+
+┌─┬──┐
+└─┴──┘
+
+┌─────┬───────┬─────┬─────┬─────┬──────┬─────┐
+│name │ q0    │ q1  │ q2  │ q3  │ q4   │ q5  │
+├─────┼───────┼─────┼─────┼─────┼──────┼─────┤
+│  qr │  180° │  0° │  0° │  0° │  90° │  0° │
+│  qz │  0°   │  0° │  0° │  0° │  0°  │  0° │
+└─────┴───────┴─────┴─────┴─────┴──────┴─────┘
+
+
+
+

Defined joint configurations are:

+
    +
  • qz, zero joint angle configuration

  • +
  • qr, arm horizontal along x-axis

  • +
+
+

Note

+
    +
  • SI units are used.

  • +
+
+
+
References:
+
+
+
Sealso:
+

UR4(), UR10()

+
+
+

Code author: Peter Corke

+
+ +
+
+class roboticstoolbox.models.DH.UR10(symbolic=False)[source]
+

Bases: DHRobot

+

Class that models a Universal Robotics UR10 manipulator

+
+
Parameters:
+

symbolic (bool) – use symbolic constants

+
+
+

UR10() is an object which models a Unimation Puma560 robot and +describes its kinematic and dynamic characteristics using standard DH +conventions.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.UR10()
+>>> print(robot)
+DHRobot: UR10 (by Universal Robotics), 6 joints (RRRRRR), dynamics, standard DH parameters
+┌────┬────────┬─────────┬────────┐
+│θⱼ  │   dⱼ   │   aⱼ    │   ⍺ⱼ   │
+├────┼────────┼─────────┼────────┤
+│ q1 │ 0.1273 │       0 │  90.0° │
+│ q2 │      0 │  -0.612 │   0.0° │
+│ q3 │      0 │ -0.5723 │   0.0° │
+│ q4 │ 0.1639 │       0 │  90.0° │
+│ q5 │ 0.1157 │       0 │ -90.0° │
+│ q6 │ 0.0922 │       0 │   0.0° │
+└────┴────────┴─────────┴────────┘
+
+┌─┬──┐
+└─┴──┘
+
+┌─────┬───────┬─────┬─────┬─────┬──────┬─────┐
+│name │ q0    │ q1  │ q2  │ q3  │ q4   │ q5  │
+├─────┼───────┼─────┼─────┼─────┼──────┼─────┤
+│  qr │  180° │  0° │  0° │  0° │  90° │  0° │
+│  qz │  0°   │  0° │  0° │  0° │  0°  │  0° │
+└─────┴───────┴─────┴─────┴─────┴──────┴─────┘
+
+
+
+

Defined joint configurations are:

+
    +
  • qz, zero joint angle configuration

  • +
  • qr, arm horizontal along x-axis

  • +
+
+

Note

+
    +
  • SI units are used.

  • +
+
+
+
References:
+
+
+
Sealso:
+

UR3(), UR5()

+
+
+

Code author: Peter Corke

+
+ +
+
+class roboticstoolbox.models.DH.Sawyer(symbolic=False)[source]
+

Bases: DHRobot

+

Class that models a Sawyer manipulator

+
+
Parameters:
+

symbolic (bool) – use symbolic constants

+
+
+

Sawyer() is an object which models a Rethink Sawyer robot and +describes its kinematic characteristics using standard DH +conventions.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Sawyer()
+>>> print(robot)
+DHRobot: Sawyer (by Rethink Robotics), 7 joints (RRRRRRR), dynamics, standard DH parameters
+┌────┬────────┬───────┬────────┐
+│θⱼ  │   dⱼ   │  aⱼ   │   ⍺ⱼ   │
+├────┼────────┼───────┼────────┤
+│ q1 │  0.317 │ 0.081 │ -90.0° │
+│ q2 │ 0.1925 │     0 │ -90.0° │
+│ q3 │    0.4 │     0 │ -90.0° │
+│ q4 │ 0.1685 │     0 │ -90.0° │
+│ q5 │    0.4 │     0 │ -90.0° │
+│ q6 │ 0.1363 │     0 │ -90.0° │
+│ q7 │ 0.1338 │     0 │   0.0° │
+└────┴────────┴───────┴────────┘
+
+┌─┬──┐
+└─┴──┘
+
+┌─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┐
+│name │ q0  │ q1  │ q2  │ q3  │ q4  │ q5  │ q6  │
+├─────┼─────┼─────┼─────┼─────┼─────┼─────┼─────┤
+│  qr │  0° │  0° │  0° │  0° │  0° │  0° │  0° │
+│  qz │  0° │  0° │  0° │  0° │  0° │  0° │  0° │
+└─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┘
+
+
+
+

Defined joint configurations are:

+
    +
  • qz, zero joint angle configuration, ‘L’ shaped configuration

  • +
+
+
References:
+

-Layeghi, Daniel. “Dynamic and Kinematic Modelling of the Sawyer Arm ” Google Sites, 20 Nov. 2017

+
+
+
+

Note

+

SI units of metres are used.

+
+
+ +
+
+class roboticstoolbox.models.DH.Mico(symbolic=False)[source]
+

Bases: DHRobot

+

Class that models a Kinova Mico manipulator

+
+
Parameters:
+

symbolic (bool) – use symbolic constants

+
+
+

Mico() is an object which models a Kinova Mico robot and +describes its kinematic characteristics using standard DH +conventions.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> print(robot)
+DHRobot: Puma 560 (by Unimation), 6 joints (RRRRRR), dynamics, geometry, standard DH parameters
+┌────┬────────┬────────┬────────┬─────────┬────────┐
+│θⱼ  │   dⱼ   │   aⱼ   │   ⍺ⱼ   │   q⁻    │   q⁺   │
+├────┼────────┼────────┼────────┼─────────┼────────┤
+│ q1 │ 0.6718 │      0 │  90.0° │ -160.0° │ 160.0° │
+│ q2 │      0 │ 0.4318 │   0.0° │ -110.0° │ 110.0° │
+│ q3 │   0.15 │ 0.0203 │ -90.0° │ -135.0° │ 135.0° │
+│ q4 │ 0.4318 │      0 │  90.0° │ -266.0° │ 266.0° │
+│ q5 │      0 │      0 │ -90.0° │ -100.0° │ 100.0° │
+│ q6 │      0 │      0 │   0.0° │ -266.0° │ 266.0° │
+└────┴────────┴────────┴────────┴─────────┴────────┘
+
+┌─┬──┐
+└─┴──┘
+
+┌─────┬─────┬──────┬───────┬─────┬──────┬─────┐
+│name │ q0  │ q1   │ q2    │ q3  │ q4   │ q5  │
+├─────┼─────┼──────┼───────┼─────┼──────┼─────┤
+│  qr │  0° │  90° │ -90°  │  0° │  0°  │  0° │
+│  qz │  0° │  0°  │  0°   │  0° │  0°  │  0° │
+│  qn │  0° │  45° │  180° │  0° │  45° │  0° │
+│  qs │  0° │  0°  │ -90°  │  0° │  0°  │  0° │
+└─────┴─────┴──────┴───────┴─────┴──────┴─────┘
+
+
+
+

Defined joint configurations are:

+
    +
  • qz, zero joint angle configuration, ‘L’ shaped configuration

  • +
  • qr, vertical ‘READY’ configuration

  • +
  • qs, arm is stretched out in the x-direction

  • +
  • qn, arm is at a nominal non-singular configuration

  • +
+
+

Note

+
    +
  • SI units are used.

  • +
+
+
+
References:
+
    +
  • “DH Parameters of Mico” Version 1.0.8, July 25, 2013.

  • +
+
+
Seealso:
+

Mico()

+
+
+
+ +
+
+class roboticstoolbox.models.DH.Jaco(symbolic=False)[source]
+

Bases: DHRobot

+

Class that models a Kinova Jaco manipulator

+
+
Parameters:
+

symbolic (bool) – use symbolic constants

+
+
+

Jaco() is an object which models a Kinova Jaco robot and +describes its kinematic characteristics using standard DH +conventions.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> print(robot)
+DHRobot: Puma 560 (by Unimation), 6 joints (RRRRRR), dynamics, geometry, standard DH parameters
+┌────┬────────┬────────┬────────┬─────────┬────────┐
+│θⱼ  │   dⱼ   │   aⱼ   │   ⍺ⱼ   │   q⁻    │   q⁺   │
+├────┼────────┼────────┼────────┼─────────┼────────┤
+│ q1 │ 0.6718 │      0 │  90.0° │ -160.0° │ 160.0° │
+│ q2 │      0 │ 0.4318 │   0.0° │ -110.0° │ 110.0° │
+│ q3 │   0.15 │ 0.0203 │ -90.0° │ -135.0° │ 135.0° │
+│ q4 │ 0.4318 │      0 │  90.0° │ -266.0° │ 266.0° │
+│ q5 │      0 │      0 │ -90.0° │ -100.0° │ 100.0° │
+│ q6 │      0 │      0 │   0.0° │ -266.0° │ 266.0° │
+└────┴────────┴────────┴────────┴─────────┴────────┘
+
+┌─┬──┐
+└─┴──┘
+
+┌─────┬─────┬──────┬───────┬─────┬──────┬─────┐
+│name │ q0  │ q1   │ q2    │ q3  │ q4   │ q5  │
+├─────┼─────┼──────┼───────┼─────┼──────┼─────┤
+│  qr │  0° │  90° │ -90°  │  0° │  0°  │  0° │
+│  qz │  0° │  0°  │  0°   │  0° │  0°  │  0° │
+│  qn │  0° │  45° │  180° │  0° │  45° │  0° │
+│  qs │  0° │  0°  │ -90°  │  0° │  0°  │  0° │
+└─────┴─────┴──────┴───────┴─────┴──────┴─────┘
+
+
+
+

Defined joint configurations are:

+
    +
  • qz, zero joint angle configuration, ‘L’ shaped configuration

  • +
  • qr, vertical ‘READY’ configuration

  • +
  • qs, arm is stretched out in the x-direction

  • +
  • qn, arm is at a nominal non-singular configuration

  • +
+
+

Note

+
    +
  • SI units are used.

  • +
+
+
+
References:
+
    +
  • “DH Parameters of Jaco” Version 1.0.8, July 25, 2013.

  • +
+
+
Seealso:
+

Mico()

+
+
+
+ +
+
+class roboticstoolbox.models.DH.Baxter(arm='left')[source]
+

Bases: DHRobot

+

Class that models a Baxter manipulator

+
+
Parameters:
+

symbolic (bool) – use symbolic constants

+
+
+

Baxter() is an object which models the left arm of the two 7-joint +arms of a Rethink Robotics Baxter robot using standard DH conventions.

+

Baxter(which) as above but models the specified arm and which is +either ‘left’ or ‘right’.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Baxter()
+>>> print(robot)
+DHRobot: Baxter-left (by Rethink Robotics), 7 joints (RRRRRRR), dynamics, standard DH parameters
+┌──────────┬───────┬───────┬────────┐
+│   θⱼ     │  dⱼ   │  aⱼ   │   ⍺ⱼ   │
+├──────────┼───────┼───────┼────────┤
+│ q1       │  0.27 │ 0.069 │ -90.0° │
+│ q2 + 90° │     0 │     0 │  90.0° │
+│ q3       │ 0.364 │ 0.069 │ -90.0° │
+│ q4       │     0 │     0 │  90.0° │
+│ q5       │ 0.374 │  0.01 │ -90.0° │
+│ q6       │     0 │     0 │  90.0° │
+│ q7       │  0.28 │     0 │   0.0° │
+└──────────┴───────┴───────┴────────┘
+
+┌─────┬──────┐
+│base │ None │
+└─────┴──────┘
+
+┌─────┬─────┬──────┬──────┬─────┬──────┬─────┬─────┐
+│name │ q0  │ q1   │ q2   │ q3  │ q4   │ q5  │ q6  │
+├─────┼─────┼──────┼──────┼─────┼──────┼─────┼─────┤
+│  qr │  0° │ -90° │ -90° │  0° │  0°  │  0° │  0° │
+│  qz │  0° │  0°  │  0°  │  0° │  0°  │  0° │  0° │
+│  qs │  0° │  0°  │ -90° │  0° │  0°  │  0° │  0° │
+│  qn │  0° │  45° │  90° │  0° │  45° │  0° │  0° │
+└─────┴─────┴──────┴──────┴─────┴──────┴─────┴─────┘
+
+
+
+

Defined joint configurations are:

+
    +
  • qz, zero joint angle configuration

  • +
  • qr, vertical ‘READY’ configuration

  • +
  • qs, arm is stretched out in the X direction

  • +
  • qd, lower arm horizontal as per data sheet

  • +
+
+

Note

+

SI units are used.

+
+
+

Warning

+

The base transform is set to reflect the pose of the arm’s +shoulder with respect to the base. Changing the base attribute of the +arm will overwrite this, IT DOES NOT CHANGE THE POSE OF BAXTER’s base. +Instead do baxter.base = newbase * baxter.base.

+
+
+
References:
+
    +
  • “Kinematics Modeling and Experimental Verification of Baxter Robot” +Z. Ju, C. Yang, H. Ma, Chinese Control Conf, 2015.

  • +
+
+
+

Code author: Peter Corke

+
+ +
+ +

Bases: DHRobot

+

Class that models a 2-link robot moving in the vertical plane

+
+
Parameters:
+

symbolic (bool) – use symbolic constants

+
+
+

TwoLink() is a class which models a 2-link planar robot and +describes its kinematic and dynamic characteristics using standard DH +conventions. of a simple planar 2-link mechanism moving in the xz-plane, it experiences gravity loading. +All mass is concentrated at the joints.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.TwoLink()
+>>> print(robot)
+DHRobot: 2 link, 2 joints (RR), dynamics, standard DH parameters
+┌────┬────┬────┬──────┐
+│θⱼ  │ dⱼ │ aⱼ │  ⍺ⱼ  │
+├────┼────┼────┼──────┤
+│ q1 │  0 │  1 │ 0.0° │
+│ q2 │  0 │  1 │ 0.0° │
+└────┴────┴────┴──────┘
+
+┌─────┬──────┐
+│base │ None │
+└─────┴──────┘
+
+┌─────┬──────┬──────┐
+│name │ q0   │ q1   │
+├─────┼──────┼──────┤
+│  qr │  30° │ -30° │
+│  qz │  0°  │  0°  │
+│  q1 │  0°  │  90° │
+│  q2 │  90° │ -90° │
+│  qn │  30° │ -30° │
+└─────┴──────┴──────┘
+
+
+
+

The parameters values depend on the symbolic parameter

+ + + + + + + + + + + + + + + + + + + + + + + + + +

Parameters

Numeric values

Symbolic values

link lengths

1, 1

a1, a2

link masses

1, 1

m1, m2

link CoMs in the link frame x-direction

-0.5, -0.5

c1, c2

gravitational acceleration

9.8

g

+

Defined joint configurations are:

+
+
    +
  • qz, zero angles, all folded up

  • +
  • q1, links are horizontal and vertical respectively

  • +
  • q2, links are vertical and horizontal respectively

  • +
  • qn, nominal working configuration

  • +
+
+
+

Note

+
    +
  • Robot has only 2 DoF.

  • +
  • Motor inertia is 0.

  • +
  • Link inertias are 0.

  • +
  • Viscous and Coulomb friction is 0.

  • +
+
+
+
Reference:
+

Based on Fig 3-6 (p73) of Spong and Vidyasagar (1st edition).

+
+
+

Code author: Peter Corke

+
+ +
+
+class roboticstoolbox.models.DH.P8[source]
+

Bases: DHRobot

+

Class that models a Puma robot on an XY base

+

P8() is an object which models an 8-axis robot comprising a Puma 560 +robot on an XY base. Joints 0 and 1 are the base, joints 2-7 are the robot +arm.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> print(robot)
+DHRobot: Puma 560 (by Unimation), 6 joints (RRRRRR), dynamics, geometry, standard DH parameters
+┌────┬────────┬────────┬────────┬─────────┬────────┐
+│θⱼ  │   dⱼ   │   aⱼ   │   ⍺ⱼ   │   q⁻    │   q⁺   │
+├────┼────────┼────────┼────────┼─────────┼────────┤
+│ q1 │ 0.6718 │      0 │  90.0° │ -160.0° │ 160.0° │
+│ q2 │      0 │ 0.4318 │   0.0° │ -110.0° │ 110.0° │
+│ q3 │   0.15 │ 0.0203 │ -90.0° │ -135.0° │ 135.0° │
+│ q4 │ 0.4318 │      0 │  90.0° │ -266.0° │ 266.0° │
+│ q5 │      0 │      0 │ -90.0° │ -100.0° │ 100.0° │
+│ q6 │      0 │      0 │   0.0° │ -266.0° │ 266.0° │
+└────┴────────┴────────┴────────┴─────────┴────────┘
+
+┌─┬──┐
+└─┴──┘
+
+┌─────┬─────┬──────┬───────┬─────┬──────┬─────┐
+│name │ q0  │ q1   │ q2    │ q3  │ q4   │ q5  │
+├─────┼─────┼──────┼───────┼─────┼──────┼─────┤
+│  qr │  0° │  90° │ -90°  │  0° │  0°  │  0° │
+│  qz │  0° │  0°  │  0°   │  0° │  0°  │  0° │
+│  qn │  0° │  45° │  180° │  0° │  45° │  0° │
+│  qs │  0° │  0°  │ -90°  │  0° │  0°  │  0° │
+└─────┴─────┴──────┴───────┴─────┴──────┴─────┘
+
+
+
+

Defined joint configurations are:

+
    +
  • qz, zero joint angle configuration, ‘L’ shaped configuration

  • +
+
+

Note

+
    +
  • SI units are used.

  • +
+
+

Code author: Peter Corke

+
+
Seealso:
+

models.DH.Puma560()

+
+
+
+ +
+
+class roboticstoolbox.models.DH.AL5D(symbolic=False)[source]
+

Bases: DHRobot

+

Class that models a Lynxmotion AL5D manipulator

+
+
Parameters:
+

symbolic (bool) – use symbolic constants

+
+
+

AL5D() is an object which models a Lynxmotion AL5D robot and +describes its kinematic and dynamic characteristics using modified DH +conventions.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.AL5D()
+>>> print(robot)
+DHRobot: AL5D (by Lynxmotion), 3 joints (RRR), dynamics, modified DH parameters
+┌────────┬────────┬─────────────┬──────────┬────────┬───────┐
+│ aⱼ₋₁   │  ⍺ⱼ₋₁  │     θⱼ      │    dⱼ    │   q⁻   │  q⁺   │
+├────────┼────────┼─────────────┼──────────┼────────┼───────┤
+│      0 │ 180.0° │    q1 + 90° │ -0.06858 │ -90.0° │ 90.0° │
+│  0.002 │  90.0° │   q2 + 180° │        0 │ -90.0° │ 90.0° │
+│0.14679 │ 180.0° │  q3 - 2.45° │        0 │ -90.0° │ 90.0° │
+└────────┴────────┴─────────────┴──────────┴────────┴───────┘
+
+┌─────┬───────────────────────────────────────┐
+│tool │ t = 0.077, 0, 0; rpy/xyz = 0°, 0°, 0° │
+└─────┴───────────────────────────────────────┘
+
+┌─────┬──────┬──────┬──────┐
+│name │ q0   │ q1   │ q2   │
+├─────┼──────┼──────┼──────┤
+│home │  90° │  90° │  90° │
+└─────┴──────┴──────┴──────┘
+
+
+
+

Defined joint configurations are:

+
    +
  • qz, zero joint angle configuration

  • +
+
+

Note

+
    +
  • SI units are used.

  • +
+
+
+
References:
+
+
+
+

Code author: Tassos Natsakis

+
+ +
+
+ + +
+
+
+ +
+ +
+

© Copyright 2023, Jesse Haviland and Peter Corke. + Last updated on 11-May-2024. +

+
+ + + +
+
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/arm_erobot.html b/arm_erobot.html new file mode 100644 index 00000000..284dc792 --- /dev/null +++ b/arm_erobot.html @@ -0,0 +1,12536 @@ + + + + + + + + + + + ERobot models — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

ERobot models

+

Code author: Jesse Haviland

+
+

ERobot

+
Inheritance diagram of roboticstoolbox.ERobot
+ + + +

The various models ERobot models all subclass this class.

+

@author: Jesse Haviland

+
+
+class roboticstoolbox.robot.ERobot.ERobot(*args, **kwargs)[source]
+

Bases: Robot

+
+
+classmethod URDF(file_path, gripper=None)
+

Construct a Robot object from URDF file

+
+
Parameters:
+
    +
  • file_path (str) – the path to the URDF

  • +
  • gripper (Union[int, str, None]) – index or name of the gripper link(s)

  • +
+
+
Returns:
+

    +
  • If gripper is specified, links from that link outward are removed

  • +
  • from the rigid-body tree and folded into a Gripper object.

  • +
+

+
+
+
+ +
+
+static URDF_read(file_path, tld=None, xacro_tld=None)
+

Read a URDF file as Links

+

File should be specified relative to RTBDATA/URDF/xacro

+
+
Parameters:
+
    +
  • file_path – File path relative to the xacro folder

  • +
  • tld – A custom top-level directory which holds the xacro data, +defaults to None

  • +
  • xacro_tld – A custom top-level within the xacro data, +defaults to None

  • +
+
+
Return type:
+

Tuple[List[Link], str, str, Union[Path, PurePosixPath]]

+
+
Returns:
+

    +
  • links – a list of links

  • +
  • name – the name of the robot

  • +
  • urdf – a string representing the URDF

  • +
  • file_path – a path to the original file

  • +
+

+
+
+

Notes

+

If tld is not supplied, filepath pointing to xacro data should +be directly under RTBDATA/URDF/xacro OR under ./xacro relative +to the model file calling this method. If tld is supplied, then +`file_path` needs to be relative to tld

+
+ +
+
+__getitem__(i)
+

Get link

+

This also supports iterating over each link in the robot object, +from the base to the tool.

+
+
Parameters:
+

i (Union[int, str]) – link number or name

+
+
Returns:
+

i’th link or named link

+
+
Return type:
+

link

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> print(robot[1]) # print the 2nd link
+RevoluteDH:   θ=q,  d=0,  a=0.4318,  ⍺=0.0
+>>> print([link.a for link in robot])  # print all the a_j values
+[0, 0.4318, 0.0203, 0, 0, 0]
+
+
+

Notes

+
+
Robot supports link lookup by name,

eg. robot['link1']

+
+
+
+ +
+
+__str__()
+

Pretty prints the ETS Model of the robot.

+
+
Returns:
+

Pretty print of the robot model

+
+
Return type:
+

str

+
+
+

Notes

+
    +
  • Constant links are shown in blue.

  • +
  • End-effector links are prefixed with an @

  • +
  • Angles in degrees

  • +
  • +
    The robot base frame is denoted as BASE and is equal to the

    robot’s base attribute.

    +
    +
    +
  • +
+
+ +
+
+accel(q, qd, torque, gravity=None)
+

Compute acceleration due to applied torque

+

qdd = accel(q, qd, torque) calculates a vector (n) of joint +accelerations that result from applying the actuator force/torque (n) +to the manipulator in state q (n) and qd (n), and n is +the number of robot joints.

+
+\[\ddot{q} = \mathbf{M}^{-1} \left(\tau - \mathbf{C}(q)\dot{q} - \mathbf{g}(q)\right)\]
+

Trajectory operation

+

If q, qd, torque are matrices (m,n) then qdd is a matrix (m,n) +where each row is the acceleration corresponding to the equivalent rows +of q, qd, torque.

+
+
Parameters:
+
    +
  • q – Joint coordinates

  • +
  • qd – Joint velocity

  • +
  • torque – Joint torques of the robot

  • +
  • gravity – Gravitational acceleration (Optional, if not supplied will +use the gravity attribute of self).

  • +
+
+
Return type:
+

ndarray(n)

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> puma.accel(puma.qz, 0.5 * np.ones(6), np.zeros(6))
+array([ -7.5544, -12.22  ,  -6.4022,  -5.4303,  -4.9518,  -2.1178])
+
+
+

Notes

+
    +
  • +
    Useful for simulation of manipulator dynamics, in

    conjunction with a numerical integration function.

    +
    +
    +
  • +
  • +
    Uses the method 1 of Walker and Orin to compute the forward

    dynamics.

    +
    +
    +
  • +
  • +
    Featherstone’s method is more efficient for robots with large

    numbers of joints.

    +
    +
    +
  • +
  • Joint friction is considered.

  • +
+

References

+
    +
  • +
    Efficient dynamic computer simulation of robotic mechanisms,

    M. W. Walker and D. E. Orin, +ASME Journa of Dynamic Systems, Measurement and Control, vol. +104, no. 3, pp. 205-211, 1982.

    +
    +
    +
  • +
+
+ +
+
+accel_x(q, xd, wrench, gravity=None, pinv=False, representation='rpy/xyz')
+

Operational space acceleration due to applied wrench

+

xdd = accel_x(q, qd, wrench) is the operational space acceleration +due to wrench applied to the end-effector of a robot in joint +configuration q and joint velocity qd.

+
+\[\ddot{x} = \mathbf{J}(q) \mathbf{M}(q)^{-1} \left( + \mathbf{J}(q)^T w - \mathbf{C}(q)\dot{q} - \mathbf{g}(q) + \right)\]
+

Trajectory operation

+

If q, qd, torque are matrices (m,n) then qdd is a matrix (m,n) +where each row is the acceleration corresponding to the equivalent rows +of q, qd, wrench.

+
+
Parameters:
+
    +
  • q – Joint coordinates

  • +
  • qd – Joint velocity

  • +
  • wrench – Wrench applied to the end-effector

  • +
  • gravity – Gravitational acceleration (Optional, if not supplied will +use the gravity attribute of self).

  • +
  • pinv – use pseudo inverse rather than inverse

  • +
  • analytical – the type of analytical Jacobian to use, default is +‘rpy/xyz’

  • +
  • xd

  • +
  • representation – (Default value = “rpy/xyz”)

  • +
+
+
Returns:
+

Operational space accelerations of the end-effector

+
+
Return type:
+

accel

+
+
+

Examples

+
    ainv = _umath_linalg.inv(a, signature=signature, extobj=extobj)
+  File "/opt/hostedtoolcache/Python/3.7.17/x64/lib/python3.7/site-packages/numpy/linalg/linalg.py", line 88, in _raise_linalgerror_singular
+    raise LinAlgError("Singular matrix")
+numpy.linalg.LinAlgError: Singular matrix
+
+
+

Notes

+
    +
  • +
    Useful for simulation of manipulator dynamics, in

    conjunction with a numerical integration function.

    +
    +
    +
  • +
  • +
    Uses the method 1 of Walker and Orin to compute the forward

    dynamics.

    +
    +
    +
  • +
  • +
    Featherstone’s method is more efficient for robots with large

    numbers of joints.

    +
    +
    +
  • +
  • Joint friction is considered.

  • +
+
+

See also

+

accel()

+
+
+ +
+
+addconfiguration(name, q)
+

Add a named joint configuration

+

Add a named configuration to the robot instance’s dictionary of named +configurations.

+
+
Parameters:
+
    +
  • name (str) – Name of the joint configuration

  • +
  • q (ndarray) – Joint configuration

  • +
+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.addconfiguration_attr("mypos", [0.1, 0.2, 0.3, 0.4, 0.5, 0.6])
+>>> robot.configs["mypos"]
+array([0.1, 0.2, 0.3, 0.4, 0.5, 0.6])
+
+
+
+

See also

+

addconfiguration()

+
+
+ +
+
+addconfiguration_attr(name, q, unit='rad')
+

Add a named joint configuration as an attribute

+
+
Parameters:
+
+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.addconfiguration_attr("mypos", [0.1, 0.2, 0.3, 0.4, 0.5, 0.6])
+>>> robot.mypos
+array([0.1, 0.2, 0.3, 0.4, 0.5, 0.6])
+>>> robot.configs["mypos"]
+array([0.1, 0.2, 0.3, 0.4, 0.5, 0.6])
+
+
+

Notes

+
    +
  • Used in robot model init method to store the qr configuration

  • +
  • +
    Dynamically adding attributes to objects can cause issues with

    Python type checking.

    +
    +
    +
  • +
  • +
    Configuration is also added to the robot instance’s dictionary of

    named configurations.

    +
    +
    +
  • +
+
+

See also

+

addconfiguration()

+
+
+ +
+
+attach(object)
+
+ +
+
+attach_to(object)
+
+ +
+
+property base: SE3
+

Get/set robot base transform

+
    +
  • robot.base is the robot base transform

  • +
  • robot.base = ... checks and sets the robot base transform

  • +
+
+
Parameters:
+

base – the new robot base transform

+
+
Returns:
+

the current robot base transform

+
+
Return type:
+

base

+
+
+
+ +
+ +

Get the robot base link

+
    +
  • robot.base_link is the robot base link

  • +
+
+
Returns:
+

the first link in the robot tree

+
+
Return type:
+

base_link

+
+
+
+ +
+
+cinertia(q)
+

Deprecated, use inertia_x

+
+ +
+
+closest_point(q, shape, inf_dist=1.0, skip=False)
+

Find the closest point between robot and shape

+

closest_point(shape, inf_dist) returns the minimum euclidean +distance between this robot and shape, provided it is less than +inf_dist. It will also return the points on self and shape in the +world frame which connect the line of length distance between the +shapes. If the distance is negative then the shapes are collided.

+
+
+shape
+

The shape to compare distance to

+
+ +
+
+inf_dist
+

The minimum distance within which to consider +the shape

+
+ +
+
+skip
+

Skip setting all shape transforms based on q, use this +option if using this method in conjuction with Swift to save time

+
+ +
+
Return type:
+

Tuple[Optional[int], Optional[ndarray], Optional[ndarray]]

+
+
Returns:
+

    +
  • d – distance between the robot and shape

  • +
  • p1 – [x, y, z] point on the robot (in the world frame)

  • +
  • p2 – [x, y, z] point on the shape (in the world frame)

  • +
+

+
+
+
+ +
+
+collided(q, shape, skip=False)
+

Check if the robot is in collision with a shape

+

collided(shape) checks if this robot and shape have collided

+
+
+shape
+

The shape to compare distance to

+
+ +
+
+skip
+

Skip setting all shape transforms based on q, use this +option if using this method in conjuction with Swift to save time

+
+ +
+
Returns:
+

True if shapes have collided

+
+
Return type:
+

collided

+
+
+
+ +
+
+property comment: str
+

Get/set robot comment

+
    +
  • robot.comment is the robot comment

  • +
  • robot.comment = ... checks and sets the robot comment

  • +
+
+
Parameters:
+

name – the new robot comment

+
+
Returns:
+

robot comment

+
+
Return type:
+

comment

+
+
+
+ +
+
+property configs: Dict[str, ndarray]
+
+ +
+
+configurations_str(border='thin')
+
+ +
+
+property control_mode: str
+

Get/set robot control mode

+
    +
  • robot.control_type is the robot control mode

  • +
  • robot.control_type = ... checks and sets the robot control mode

  • +
+
+
Parameters:
+

control_mode – the new robot control mode

+
+
Returns:
+

the current robot control mode

+
+
Return type:
+

control_mode

+
+
+
+ +
+
+copy()
+
+ +
+
+coriolis(q, qd)
+

Coriolis and centripetal term

+

coriolis(q, qd) calculates the Coriolis/centripetal matrix (n,n) +for the robot in configuration q and velocity qd, where n +is the number of joints.

+

The product \(\mathbf{C} \dot{q}\) is the vector of joint +force/torque due to velocity coupling. The diagonal elements are due to +centripetal effects and the off-diagonal elements are due to Coriolis +effects. This matrix is also known as the velocity coupling matrix, +since it describes the disturbance forces on any joint due to +velocity of all other joints.

+

Trajectory operation

+

If q and qd are matrices (m,n), each row is interpretted as a +joint configuration, and the result (n,n,m) is a 3d-matrix where +each plane corresponds to a row of q and qd.

+
+
Parameters:
+
    +
  • q – Joint coordinates

  • +
  • qd – Joint velocity

  • +
+
+
Returns:
+

Velocity matrix

+
+
Return type:
+

coriolis

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> puma.coriolis(puma.qz, 0.5 * np.ones((6,)))
+array([[-0.4017, -0.5513, -0.2025, -0.0007, -0.0013,  0.    ],
+       [ 0.2023, -0.1937, -0.3868, -0.    , -0.002 ,  0.    ],
+       [ 0.1987,  0.193 , -0.    ,  0.    , -0.0001,  0.    ],
+       [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+       [ 0.0007,  0.0007,  0.0001,  0.    ,  0.    ,  0.    ],
+       [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ]])
+
+
+

Notes

+
    +
  • +
    Joint viscous friction is also a joint force proportional to

    velocity but it is eliminated in the computation of this value.

    +
    +
    +
  • +
  • Computationally slow, involves \(n^2/2\) invocations of RNE.

  • +
+
+ +
+
+coriolis_x(q, qd, pinv=False, representation='rpy/xyz', J=None, Ji=None, Jd=None, C=None, Mx=None)
+

Operational space Coriolis and centripetal term

+

coriolis_x(q, qd) is the Coriolis/centripetal matrix (m,m) +in operational space for the robot in configuration q and velocity +qd, where n is the number of joints.

+
+\[\mathbf{C}_x = \mathbf{J}(q)^{-T} \left( + \mathbf{C}(q) - \mathbf{M}_x(q) \mathbf{J})(q) + \right) \mathbf{J}(q)^{-1}\]
+

The product \(\mathbf{C} \dot{x}\) is the operational space wrench +due to joint velocity coupling. This matrix is also known as the +velocity coupling matrix, since it describes the disturbance forces on +any joint due to velocity of all other joints.

+

The transformation to operational space requires an analytical, rather +than geometric, Jacobian. analytical can be one of:

+
+
+ + + + + + + + + + + + + + + + + + + +

Value

Rotational representation

'rpy/xyz'

RPY angular rates in XYZ order (default)

'rpy/zyx'

RPY angular rates in XYZ order

'eul'

Euler angular rates in ZYZ order

'exp'

exponential coordinate rates

+
+

Trajectory operation

+

If q and qd are matrices (m,n), each row is interpretted as a +joint configuration, and the result (n,n,m) is a 3d-matrix where +each plane corresponds to a row of q and qd.

+
+
Parameters:
+
    +
  • q – Joint coordinates

  • +
  • qd – Joint velocity

  • +
  • pinv – use pseudo inverse rather than inverse (Default value = False)

  • +
  • analytical – the type of analytical Jacobian to use, default is +‘rpy/xyz’

  • +
  • representation – (Default value = “rpy/xyz”)

  • +
  • J

  • +
  • Ji

  • +
  • Jd

  • +
  • C

  • +
  • Mx

  • +
+
+
Returns:
+

Operational space velocity matrix

+
+
Return type:
+

ndarray(6,6) or ndarray(m,6,6)

+
+
+

Examples

+
    ainv = _umath_linalg.inv(a, signature=signature, extobj=extobj)
+  File "/opt/hostedtoolcache/Python/3.7.17/x64/lib/python3.7/site-packages/numpy/linalg/linalg.py", line 88, in _raise_linalgerror_singular
+    raise LinAlgError("Singular matrix")
+numpy.linalg.LinAlgError: Singular matrix
+
+
+

Notes

+
    +
  • +
    Joint viscous friction is also a joint force proportional to

    velocity but it is eliminated in the computation of this value.

    +
    +
    +
  • +
  • Computationally slow, involves \(n^2/2\) invocations of RNE.

  • +
  • If the robot is not 6 DOF the pinv option is set True.

  • +
  • pinv() is around 5x slower than inv()

  • +
+
+

Warning

+

Assumes that the operational space has 6 DOF.

+
+ +
+ +
+
+property default_backend
+

Get default graphical backend

+
    +
  • +
    robot.default_backend Get the default graphical backend, used when

    no explicit backend is passed to Robot.plot.

    +
    +
    +
  • +
  • +
    robot.default_backend = ... Set the default graphical backend, used when

    no explicit backend is passed to Robot.plot. The default set here will +be overridden if the particular Robot subclass cannot support it.

    +
    +
    +
  • +
+
+
Returns:
+

backend name

+
+
Return type:
+

default_backend

+
+
+
+ +
+ +

A link search method

+

Visit all links from start in depth-first order and will apply +func to each visited link

+
+
Parameters:
+
+
+
Returns:
+

A list of links

+
+
Return type:
+

links

+
+
+
+ +
+
+dotfile(filename, etsbox=False, ets='full', jtype=False, static=True)
+

Write a link transform graph as a GraphViz dot file

+
+
The file can be processed using dot:

% dot -Tpng -o out.png dotfile.dot

+
+
The nodes are:
    +
  • Base is shown as a grey square. This is the world frame origin, +but can be changed using the base attribute of the robot.

  • +
  • Link frames are indicated by circles

  • +
  • ETS transforms are indicated by rounded boxes

  • +
+
+
The edges are:
    +
  • an arrow if jtype is False or the joint is fixed

  • +
  • an arrow with a round head if jtype is True and the joint is +revolute

  • +
  • an arrow with a box head if jtype is True and the joint is +prismatic

  • +
+
+
+

Edge labels or nodes in blue have a fixed transformation to the +preceding link.

+
+

Note

+
+
If filename is a file object then the file will not

be closed after the GraphViz model is written.

+
+
+
+
+
Parameters:
+
    +
  • file – Name of file to write to

  • +
  • etsbox (bool) – Put the link ETS in a box, otherwise an edge label

  • +
  • ets (Literal['full', 'brief']) – Display the full ets with “full” or a brief version with “brief”

  • +
  • jtype (bool) – Arrowhead to node indicates revolute or prismatic type

  • +
  • static (bool) – Show static joints in blue and bold

  • +
+
+
+
+

See also

+

showgraph()

+
+
+ +
+
+dynamics()
+

Pretty print the dynamic parameters (Robot superclass)

+

The dynamic parameters (inertial and friction) are printed in a table, +with one row per link.

+

Examples

+

+
+
+
+ +
+
+dynamics_list()
+

Print dynamic parameters (Robot superclass)

+

Display the kinematic and dynamic parameters to the console in +reable format

+
+ +
+
+dynchanged(what=None)
+

Dynamic parameters have changed

+

Called from a property setter to inform the robot that the cache of +dynamic parameters is invalid.

+
+

See also

+

roboticstoolbox.Link._listen_dyn()

+
+
+ +
+ +
+ +
+
+ets(start=None, end=None)
+

Robot to ETS

+

robot.ets() is an ETS representing the kinematics from base to +end-effector.

+

robot.ets(end=link) is an ETS representing the kinematics from +base to the link link specified as a Link reference or a name.

+

robot.ets(start=l1, end=l2) is an ETS representing the kinematics +from link l1 to link l2.

+

:param : +:type : param start: start of path, defaults to base_link +:param : +:type : param end: end of path, defaults to end-effector

+
+
Raises:
+
+
+
Returns:
+

elementary transform sequence

+
+
Return type:
+

ets

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.ETS.Panda()
+>>> panda.ets()
+[ET.tz(eta=0.333), ET.Rz(jindex=0), ET.Rx(eta=-1.5707963267948966), ET.Rz(jindex=1), ET.Rx(eta=1.5707963267948966), ET.tz(eta=0.316), ET.Rz(jindex=2), ET.tx(eta=0.0825), ET.Rx(eta=1.5707963267948966), ET.Rz(jindex=3), ET.tx(eta=-0.0825), ET.Rx(eta=-1.5707963267948966), ET.tz(eta=0.384), ET.Rz(jindex=4), ET.Rx(eta=1.5707963267948966), ET.Rz(jindex=5), ET.tx(eta=0.088), ET.Rx(eta=1.5707963267948966), ET.tz(eta=0.107), ET.Rz(jindex=6), ET.tz(eta=0.10300000000000001), ET.Rz(eta=-0.7853981633974483)]
+
+
+
+ +
+
+fdyn(T, q0, Q=None, Q_args={}, qd0=None, solver='RK45', solver_args={}, dt=None, progress=False)
+

Integrate forward dynamics

+

tg = R.fdyn(T, q) integrates the dynamics of the robot with zero +input torques over the time interval 0 to T and returns the +trajectory as a namedtuple with elements:

+
    +
  • t the time vector (M,)

  • +
  • q the joint coordinates (M,n)

  • +
  • qd the joint velocities (M,n)

  • +
+

tg = R.fdyn(T, q, torqfun) as above but the torque applied to the +joints is given by the provided function:

+
tau = function(robot, t, q, qd, **args)
+
+
+

where the inputs are:

+
+
    +
  • the robot object

  • +
  • current time

  • +
  • current joint coordinates (n,)

  • +
  • current joint velocity (n,)

  • +
  • args, optional keyword arguments can be specified, these are

  • +
+

passed in from the targs kewyword argument.

+
+

The function must return a Numpy array (n,) of joint forces/torques.

+
+
Parameters:
+
+
+
Returns:
+

robot trajectory

+
+
Return type:
+

trajectory

+
+
+

Examples

+

To apply zero joint torque to the robot without Coulomb +friction:

+
>>> def myfunc(robot, t, q, qd):
+>>>     return np.zeros((robot.n,))
+
+
+
>>> tg = robot.nofriction().fdyn(5, q0, myfunc)
+
+
+
>>> plt.figure()
+>>> plt.plot(tg.t, tg.q)
+>>> plt.show()
+
+
+

We could also use a lambda function:

+
>>> tg = robot.nofriction().fdyn(
+>>>     5, q0, lambda r, t, q, qd: np.zeros((r.n,)))
+
+
+

The robot is controlled by a PD controller. We first define a +function to compute the control which has additional parameters for +the setpoint and control gains (qstar, P, D):

+
>>> def myfunc(robot, t, q, qd, qstar, P, D):
+>>>     return (qstar - q) * P + qd * D  # P, D are (6,)
+
+
+
>>> tg = robot.fdyn(10, q0, myfunc, torque_args=(qstar, P, D)) )
+
+
+

Many integrators have variable step length which is problematic if we +want to animate the result. If dt is specified then the solver +results are interpolated in time steps of dt.

+

Notes

+ +
+

See also

+

DHRobot.accel(), DHRobot.nofriction(), DHRobot.rne()

+
+
+ +
+
+fellipse(q, opt='trans', unit='rad', centre=[0, 0, 0])
+

Create a force ellipsoid object for plotting with PyPlot

+

robot.fellipse(q) creates a force ellipsoid for the robot at +pose q. The ellipsoid is centered at the origin.

+
+
+q
+

The joint configuration of the robot.

+
+ +
+
+opt
+

‘trans’ or ‘rot’ will plot either the translational or +rotational force ellipsoid

+
+ +
+
+unit
+

‘rad’ or ‘deg’ will plot the ellipsoid in radians or +degrees

+
+ +
+
+centre
+

The centre of the ellipsoid, either ‘ee’ for the end-effector +or a 3-vector [x, y, z] in the world frame

+
+ +
+
Returns:
+

An EllipsePlot object

+
+
Return type:
+

env

+
+
+

Notes

+
    +
  • +
    By default the ellipsoid related to translational motion is

    drawn. Use opt='rot' to draw the rotational velocity +ellipsoid.

    +
    +
    +
  • +
  • +
    By default the ellipsoid is drawn at the origin. The option

    centre allows its origin to set to set to the specified +3-vector, or the string “ee” ensures it is drawn at the +end-effector position.

    +
    +
    +
  • +
+
+ +
+
+fkine(q, end=None, start=None, tool=None, include_base=True)
+

Forward kinematics

+

T = robot.fkine(q) evaluates forward kinematics for the robot at +joint configuration q.

+

Trajectory operation: +If q has multiple rows (mxn), it is considered a trajectory and the +result is an SE3 instance with m values.

+
+
+q
+

Joint coordinates

+
+ +
+
+end
+

end-effector or gripper to compute forward kinematics to

+
+ +
+
+start
+

the link to compute forward kinematics from

+
+ +
+
+tool
+

tool transform, optional

+
+ +
+
Return type:
+

SE3

+
+
Returns:
+

The transformation matrix representing the pose of the +end-effector

+
+
+

Examples

+

The following example makes a panda robot object, and solves for the +forward kinematics at the listed configuration.

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda()
+>>> panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+SE3(array([[ 0.995 , -0.    ,  0.0998,  0.484 ],
+           [-0.    , -1.    ,  0.    ,  0.    ],
+           [ 0.0998, -0.    , -0.995 ,  0.4126],
+           [ 0.    ,  0.    ,  0.    ,  1.    ]]))
+
+
+

Notes

+
    +
  • +
    For a robot with a single end-effector there is no need to

    specify end

    +
    +
    +
  • +
  • +
    For a robot with multiple end-effectors, the end must

    be specified.

    +
    +
    +
  • +
  • +
    The robot’s base tool transform, if set, is incorporated

    into the result.

    +
    +
    +
  • +
  • A tool transform, if provided, is incorporated into the result.

  • +
  • Works from the end-effector link to the base

  • +
+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+ +
+
+fkine_all(q)
+

Compute the pose of every link frame

+

T = robot.fkine_all(q) is an SE3 instance with robot.nlinks + +1 values:

+
    +
  • T[0] is the base transform

  • +
  • T[i] is the pose of link whose number is i

  • +
+
+
Parameters:
+

q (Union[ndarray, List[float], Tuple[float], Set[float]]) – The joint configuration

+
+
Returns:
+

Pose of all links

+
+
Return type:
+

fkine_all

+
+
+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
+
+ +
+
+friction(qd)
+

Manipulator joint friction (Robot superclass)

+

robot.friction(qd) is a vector of joint friction +forces/torques for the robot moving with joint velocities qd.

+

The friction model includes:

+
    +
  • Viscous friction which is a linear function of velocity.

  • +
  • Coulomb friction which is proportional to sign(qd).

  • +
+
+\[\begin{split}\tau_j = G^2 B \dot{q}_j + |G_j| \left\{ \begin{array}{ll} + \tau_{C,j}^+ & \mbox{if $\dot{q}_j > 0$} \\ + \tau_{C,j}^- & \mbox{if $\dot{q}_j < 0$} \end{array} \right.\end{split}\]
+
+
Parameters:
+

qd (ndarray) – The joint velocities of the robot

+
+
Returns:
+

The joint friction forces/torques for the robot

+
+
Return type:
+

friction

+
+
+

Notes

+
    +
  • +
    The friction value should be added to the motor output torque to

    determine the nett torque. It has a negative value when qd > 0.

    +
    +
    +
  • +
  • +
    The returned friction value is referred to the output of the

    gearbox.

    +
    +
    +
  • +
  • +
    The friction parameters in the Link object are referred to the

    motor.

    +
    +
    +
  • +
  • Motor viscous friction is scaled up by \(G^2\).

  • +
  • Motor Coulomb friction is scaled up by math:G.

  • +
  • +
    The appropriate Coulomb friction value to use in the

    non-symmetric case depends on the sign of the joint velocity, +not the motor velocity.

    +
    +
    +
  • +
  • +
    Coulomb friction is zero for zero joint velocity, stiction is

    not modeled.

    +
    +
    +
  • +
  • +
    The absolute value of the gear ratio is used. Negative gear

    ratios are tricky: the Puma560 robot has negative gear ratio for +joints 1 and 3.

    +
    +
    +
  • +
  • +
    The absolute value of the gear ratio is used. Negative gear

    ratios are tricky: the Puma560 has negative gear ratio for +joints 1 and 3.

    +
    +
    +
  • +
+
+

See also

+

Robot.nofriction(), Link.friction()

+
+
+ +
+
+get_path(end=None, start=None)
+

Find a path from start to end

+
+
Parameters:
+
+
+
Raises:
+

ValueError – link not known or ambiguous

+
+
Return type:
+

Tuple[List[TypeVar(LinkType, bound= BaseLink)], int, SE3]

+
+
Returns:
+

    +
  • path – the path from start to end

  • +
  • n – the number of joints in the path

  • +
  • T – the tool transform present after end

  • +
+

+
+
+
+ +
+
+property gravity: ndarray
+

Get/set default gravitational acceleration (Robot superclass)

+
    +
  • robot.name is the default gravitational acceleration

  • +
  • +
    robot.name = ... checks and sets default gravitational

    acceleration

    +
    +
    +
  • +
+
+
Parameters:
+

gravity – the new gravitational acceleration for this robot

+
+
Returns:
+

gravitational acceleration

+
+
Return type:
+

gravity

+
+
+

Notes

+

If the z-axis is upward, out of the Earth, this should be +a positive number.

+
+ +
+
+gravload(q=None, gravity=None)
+

Compute gravity load

+

robot.gravload(q) calculates the joint gravity loading (n) for +the robot in the joint configuration q and using the default +gravitational acceleration specified in the DHRobot object.

+

robot.gravload(q, gravity=g) as above except the gravitational +acceleration is explicitly specified as g.

+

Trajectory operation

+

If q is a matrix (nxm) each column is interpreted as a joint +configuration vector, and the result is a matrix (nxm) each column +being the corresponding joint torques.

+
+
Parameters:
+
+
+
Returns:
+

The generalised joint force/torques due to gravity

+
+
Return type:
+

gravload

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> puma.gravload(puma.qz)
+array([ 0.    , 37.4837,  0.2489,  0.    ,  0.    ,  0.    ])
+
+
+
+ +
+
+gravload_x(q=None, gravity=None, pinv=False, representation='rpy/xyz', Ji=None)
+

Operational space gravity load

+

robot.gravload_x(q) calculates the gravity wrench for +the robot in the joint configuration q and using the default +gravitational acceleration specified in the robot object.

+

robot.gravload_x(q, gravity=g) as above except the gravitational +acceleration is explicitly specified as g.

+
+\[\mathbf{G}_x = \mathbf{J}(q)^{-T} \mathbf{G}(q)\]
+

The transformation to operational space requires an analytical, rather +than geometric, Jacobian. analytical can be one of:

+
+
+ + + + + + + + + + + + + + + + + + + +

Value

Rotational representation

'rpy/xyz'

RPY angular rates in XYZ order (default)

'rpy/zyx'

RPY angular rates in XYZ order

'eul'

Euler angular rates in ZYZ order

'exp'

exponential coordinate rates

+
+

Trajectory operation

+

If q is a matrix (nxm) each column is interpreted as a joint +configuration vector, and the result is a matrix (nxm) each column +being the corresponding joint torques.

+
+
Parameters:
+
    +
  • q – Joint coordinates

  • +
  • gravity – Gravitational acceleration (Optional, if not supplied will +use the gravity attribute of self).

  • +
  • pinv – use pseudo inverse rather than inverse (Default value = False)

  • +
  • analytical – the type of analytical Jacobian to use, default is +‘rpy/xyz’

  • +
  • representation – (Default value = “rpy/xyz”)

  • +
  • Ji

  • +
+
+
Returns:
+

The operational space gravity wrench

+
+
Return type:
+

gravload

+
+
+

Examples

+
    ainv = _umath_linalg.inv(a, signature=signature, extobj=extobj)
+  File "/opt/hostedtoolcache/Python/3.7.17/x64/lib/python3.7/site-packages/numpy/linalg/linalg.py", line 88, in _raise_linalgerror_singular
+    raise LinAlgError("Singular matrix")
+numpy.linalg.LinAlgError: Singular matrix
+
+
+

Notes

+
    +
  • If the robot is not 6 DOF the pinv option is set True.

  • +
  • pinv() is around 5x slower than inv()

  • +
+
+

Warning

+

Assumes that the operational space has 6 DOF.

+
+
+

See also

+

gravload()

+
+
+ +
+
+property grippers: List[Gripper]
+

Grippers attached to the robot

+
+
Returns:
+

A list of grippers

+
+
Return type:
+

grippers

+
+
+
+ +
+
+property hascollision
+

Robot has collision model

+
+
Returns:
+

    +
  • hascollision – Robot has collision model

  • +
  • At least one link has associated collision model.

  • +
+

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.hascollision
+False
+
+
+
+

See also

+

hasgeometry()

+
+
+ +
+
+property hasdynamics
+

Robot has dynamic parameters

+
+
Returns:
+

    +
  • hasdynamics – Robot has dynamic parameters

  • +
  • At least one link has associated dynamic parameters.

  • +
+

+
+
+

Examples

+

+
+
+
+ +
+
+property hasgeometry
+

Robot has geometry model

+

At least one link has associated mesh to describe its shape.

+
+
Returns:
+

Robot has geometry model

+
+
Return type:
+

hasgeometry

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.hasgeometry
+True
+
+
+
+

See also

+

hascollision()

+
+
+ +
+
+hessian0(q=None, end=None, start=None, J0=None, tool=None)
+

Manipulator Hessian

+

The manipulator Hessian tensor maps joint acceleration to end-effector +spatial acceleration, expressed in the start frame. This +function calulcates this based on the ETS of the robot. One of J0 or q +is required. Supply J0 if already calculated to save computation time

+
+
Parameters:
+
    +
  • q – The joint angles/configuration of the robot (Optional, +if not supplied will use the stored q values).

  • +
  • end (Union[str, Link, Gripper, None]) – the final link/Gripper which the Hessian represents

  • +
  • start (Union[str, Link, Gripper, None]) – the first link which the Hessian represents

  • +
  • J0 – The manipulator Jacobian in the start frame

  • +
  • tool (Union[ndarray, SE3, None]) – a static tool transformation matrix to apply to the +end of end, defaults to None

  • +
+
+
Returns:
+

The manipulator Hessian in the start frame

+
+
Return type:
+

h0

+
+
+

Synopsis

+

This method computes the manipulator Hessian in the start frame. If +we take the time derivative of the differential kinematic relationship +.. math:

+
\nu    &= \mat{J}(\vec{q}) \dvec{q} \\
+\alpha &= \dmat{J} \dvec{q} + \mat{J} \ddvec{q}
+
+
+

where +.. math:

+
\dmat{J} = \mat{H} \dvec{q}
+
+
+

and \(\mat{H} \in \mathbb{R}^{6\times n \times n}\) is the +Hessian tensor.

+

The elements of the Hessian are +.. math:

+
\mat{H}_{i,j,k} =  \frac{d^2 u_i}{d q_j d q_k}
+
+
+

where \(u = \{t_x, t_y, t_z, r_x, r_y, r_z\}\) are the elements +of the spatial velocity vector.

+

Similarly, we can write +.. math:

+
\mat{J}_{i,j} = \frac{d u_i}{d q_j}
+
+
+

Examples

+

The following example makes a Panda robot object, and solves for the +base frame Hessian at the given joint angle configuration

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda()
+>>> panda.hessian0([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+array([[[-0.484 , -0.    , -0.486 , -0.    , -0.1547,  0.    , -0.    ],
+        [ 0.    ,  0.0796,  0.    ,  0.2466,  0.    ,  0.2006,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+        [ 0.    , -1.    , -0.    ,  1.    ,  0.    ,  1.    ,  0.    ],
+        [ 0.    , -0.    , -0.2955, -0.    ,  0.9463, -0.    ,  0.0998],
+        [ 0.    ,  0.    , -0.    , -0.    ,  0.    , -0.    ,  0.    ]],
+
+       [[-0.    , -0.484 , -0.    ,  0.4986, -0.    ,  0.1086,  0.    ],
+        [ 0.0796,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+        [ 0.    , -0.0796, -0.    , -0.2466, -0.    , -0.2006, -0.    ],
+        [ 0.    ,  0.    ,  0.9553,  0.    , -0.3233,  0.    , -0.995 ],
+        [ 0.    ,  0.    , -0.    , -0.    ,  0.    , -0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.2955,  0.    , -0.9463,  0.    , -0.0998]],
+
+       [[-0.486 , -0.    , -0.4643, -0.    , -0.1478,  0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.383 ,  0.    ,  0.2237,  0.    ],
+        [ 0.    , -0.    , -0.1436, -0.    , -0.0457, -0.    , -0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.9553,  0.    ,  0.9553,  0.    ],
+        [ 0.    ,  0.    ,  0.    , -0.    ,  0.8085, -0.    , -0.1987],
+        [ 0.    ,  0.    ,  0.    ,  0.2955,  0.    ,  0.2955,  0.    ]],
+
+       [[-0.    ,  0.4986, -0.    , -0.4986,  0.    , -0.1086, -0.    ],
+        [ 0.2466,  0.    ,  0.383 ,  0.    ,  0.    ,  0.    ,  0.    ],
+        [ 0.    , -0.2466, -0.    ,  0.2466,  0.    ,  0.2006,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.3233,  0.    ,  0.995 ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    , -0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.9463,  0.    ,  0.0998]],
+
+       [[-0.1547, -0.    , -0.1478,  0.    ,  0.05  , -0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    , -0.1676, -0.    ],
+        [ 0.    , -0.    , -0.0457,  0.    ,  0.1464,  0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    , -0.3233,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    , -0.    ,  0.9093],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    , -0.9463, -0.    ]],
+
+       [[ 0.    ,  0.1086,  0.    , -0.1086, -0.    , -0.1086, -0.    ],
+        [ 0.2006,  0.    ,  0.2237,  0.    , -0.1676,  0.    ,  0.    ],
+        [ 0.    , -0.2006, -0.    ,  0.2006,  0.    ,  0.2006,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.995 ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    , -0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.0998]],
+
+       [[-0.    ,  0.    ,  0.    , -0.    ,  0.    , -0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    , -0.    ,  0.    , -0.    ],
+        [ 0.    , -0.    , -0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ]]])
+
+
+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+ +
+
+hessiane(q=None, end=None, start=None, Je=None, tool=None)
+

Manipulator Hessian

+

The manipulator Hessian tensor maps joint acceleration to end-effector +spatial acceleration, expressed in the end coordinate frame. This +function calulcates this based on the ETS of the robot. One of J0 or q +is required. Supply J0 if already calculated to save computation time

+
+
Parameters:
+
    +
  • q – The joint angles/configuration of the robot (Optional, +if not supplied will use the stored q values).

  • +
  • end (Union[str, Link, Gripper, None]) – the final link/Gripper which the Hessian represents

  • +
  • start (Union[str, Link, Gripper, None]) – the first link which the Hessian represents

  • +
  • J0 – The manipulator Jacobian in the end frame

  • +
  • tool (Union[ndarray, SE3, None]) – a static tool transformation matrix to apply to the +end of end, defaults to None

  • +
+
+
Returns:
+

The manipulator Hessian in end frame

+
+
Return type:
+

he

+
+
+

Synopsis

+

This method computes the manipulator Hessian in the end frame. If +we take the time derivative of the differential kinematic relationship +.. math:

+
\nu    &= \mat{J}(\vec{q}) \dvec{q} \\
+\alpha &= \dmat{J} \dvec{q} + \mat{J} \ddvec{q}
+
+
+

where +.. math:

+
\dmat{J} = \mat{H} \dvec{q}
+
+
+

and \(\mat{H} \in \mathbb{R}^{6\times n \times n}\) is the +Hessian tensor.

+

The elements of the Hessian are +.. math:

+
\mat{H}_{i,j,k} =  \frac{d^2 u_i}{d q_j d q_k}
+
+
+

where \(u = \{t_x, t_y, t_z, r_x, r_y, r_z\}\) are the elements +of the spatial velocity vector.

+

Similarly, we can write +.. math:

+
\mat{J}_{i,j} = \frac{d u_i}{d q_j}
+
+
+

Examples

+

The following example makes a Panda robot object, and solves for the +end-effector frame Hessian at the given joint angle configuration

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda()
+>>> panda.hessiane([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+array([[[-0.4816, -0.    , -0.4835, -0.    , -0.1539, -0.    ,  0.    ],
+        [ 0.    , -0.0796,  0.    , -0.2466,  0.    , -0.2006,  0.    ],
+        [-0.0483, -0.    , -0.0485, -0.    , -0.0154, -0.    ,  0.    ],
+        [ 0.    , -0.995 ,  0.    ,  0.995 , -0.    ,  0.995 , -0.    ],
+        [ 0.    ,  0.    ,  0.2955, -0.    , -0.9463, -0.    , -0.0998],
+        [ 0.    , -0.0998,  0.    ,  0.0998, -0.    ,  0.0998,  0.    ]],
+
+       [[-0.    , -0.4896, -0.    ,  0.4715, -0.    ,  0.088 ,  0.    ],
+        [-0.0796,  0.    ,  0.    , -0.    ,  0.    , -0.    ,  0.    ],
+        [-0.    ,  0.0309,  0.    ,  0.2952,  0.    ,  0.2104, -0.    ],
+        [ 0.    ,  0.    ,  0.9801,  0.    , -0.4161,  0.    , -1.    ],
+        [ 0.    ,  0.    , -0.    , -0.    ,  0.    , -0.    ,  0.    ],
+        [ 0.    ,  0.    , -0.1987, -0.    ,  0.9093, -0.    ,  0.    ]],
+
+       [[-0.4835, -0.    , -0.4763, -0.    , -0.1516, -0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    , -0.383 ,  0.    , -0.2237,  0.    ],
+        [-0.0485,  0.    ,  0.0965, -0.    ,  0.0307, -0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.9801, -0.    ,  0.9801,  0.    ],
+        [ 0.    ,  0.    ,  0.    , -0.    , -0.8085, -0.    ,  0.1987],
+        [ 0.    ,  0.    ,  0.    , -0.1987, -0.    , -0.1987, -0.    ]],
+
+       [[-0.    ,  0.4715, -0.    , -0.4715,  0.    , -0.088 ,  0.    ],
+        [-0.2466, -0.    , -0.383 ,  0.    , -0.    ,  0.    , -0.    ],
+        [-0.    ,  0.2952, -0.    , -0.2952, -0.    , -0.2104,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.4161,  0.    ,  1.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    , -0.    ,  0.    , -0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    , -0.9093,  0.    ,  0.    ]],
+
+       [[-0.1539, -0.    , -0.1516,  0.    ,  0.0644,  0.    , -0.    ],
+        [ 0.    ,  0.    ,  0.    , -0.    , -0.    ,  0.1676, -0.    ],
+        [-0.0154,  0.    ,  0.0307, -0.    , -0.1407, -0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    , -0.4161, -0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    , -0.9093],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.9093,  0.    ]],
+
+       [[-0.    ,  0.088 , -0.    , -0.088 ,  0.    , -0.088 ,  0.    ],
+        [-0.2006, -0.    , -0.2237,  0.    ,  0.1676,  0.    , -0.    ],
+        [-0.    ,  0.2104, -0.    , -0.2104, -0.    , -0.2104,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  1.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    , -0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ]],
+
+       [[ 0.    ,  0.    ,  0.    ,  0.    , -0.    ,  0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    , -0.    , -0.    , -0.    , -0.    ],
+        [ 0.    , -0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ]]])
+
+
+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+ +
+
+hierarchy()
+

Pretty print the robot link hierachy

+
+
Return type:
+

Pretty print of the robot model

+
+
+

Examples

+

Makes a robot and prints the heirachy

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.URDF.Panda()
+>>> robot.hierarchy()
+ panda_link0
+   panda_link1
+     panda_link2
+       panda_link3
+         panda_link4
+           panda_link5
+             panda_link6
+               panda_link7
+                 panda_link8
+                   panda_hand
+                     panda_leftfinger
+                     panda_rightfinger
+
+
+
+ +
+
+ik_GN(Tep, end=None, start=None, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, pinv=True, pinv_damping=0.0)
+

Fast numerical inverse kinematics by Gauss-Newton optimization

+

sol = ets.ik_GN(Tep) are the joint coordinates (n) corresponding +to the robot end-effector pose Tep which is an SE3 or ndarray object. +This method can be used for robots with any number of degrees of freedom. This +is a fast solver implemented in C++.

+

See the Inverse Kinematics Docs Page for more details and for a +tutorial on numerical IK, see here.

+
+

Note

+

When using this method with redundant robots (>6 DoF), pinv must be set to True

+
+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose or pose trajectory

  • +
  • end (Union[str, Link, Gripper, None]) – the link considered as the end-effector

  • +
  • start (Union[str, Link, Gripper, None]) – the link considered as the base frame, defaults to the robots’s base frame

  • +
  • q0 (Optional[ndarray]) – initial joint configuration (default to random valid joint +configuration contrained by the joint limits of the robot)

  • +
  • ilimit (int) – maximum number of iterations per search

  • +
  • slimit (int) – maximum number of search attempts

  • +
  • tol (float) – final error tolerance

  • +
  • mask (Optional[ndarray]) – a mask vector which weights the end-effector error priority. +Corresponds to translation in X, Y and Z and rotation about X, Y and Z +respectively

  • +
  • joint_limits (bool) – constrain the solution to being within the joint limits of +the robot (reject solution with invalid joint configurations and perfrom +another search up to the slimit)

  • +
  • pinv (int) – Use the psuedo-inverse instad of the normal matrix inverse

  • +
  • pinv_damping (float) – Damping factor for the psuedo-inverse

  • +
+
+
Return type:
+

Tuple[ndarray, int, int, int, float]

+
+
Returns:
+

    +
  • sol – tuple (q, success, iterations, searches, residual)

  • +
  • The return value sol is a tuple with elements

  • +
  • ============ ========== ===============================================

  • +
  • Element Type Description

  • +
  • ============ ========== ===============================================

  • +
  • q ndarray(n) joint coordinates in units of radians or metres

  • +
  • success int whether a solution was found

  • +
  • iterations int total number of iterations

  • +
  • searches int total number of searches

  • +
  • residual float final value of cost function

  • +
  • ============ ========== ===============================================

  • +
  • If success == 0 the q values will be valid numbers, but the

  • +
  • solution will be in error. The amount of error is indicated by

  • +
  • the residual.

  • +
+

+
+
+

Synopsis

+

Each iteration uses the Gauss-Newton optimisation method

+
+\[\begin{split}\vec{q}_{k+1} &= \vec{q}_k + +\left( +{\mat{J}(\vec{q}_k)}^\top +\mat{W}_e \ +{\mat{J}(\vec{q}_k)} +\right)^{-1} +\bf{g}_k \\ +\bf{g}_k &= +{\mat{J}(\vec{q}_k)}^\top +\mat{W}_e +\vec{e}_k\end{split}\]
+

where \(\mat{J} = {^0\mat{J}}\) is the base-frame manipulator Jacobian. If +\(\mat{J}(\vec{q}_k)\) is non-singular, and \(\mat{W}_e = \mat{1}_n\), then +the above provides the pseudoinverse solution. However, if \(\mat{J}(\vec{q}_k)\) +is singular, the above can not be computed and the GN solution is infeasible.

+

Examples

+

The following example gets a panda robot object, makes a goal +pose Tep, and then solves for the joint coordinates which result in the pose +Tep using the ikine_GN method.

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda()
+>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> panda.ik_GN(Tep)
+(array([-1.0805, -0.5328,  0.9086, -2.1781,  0.4612,  1.9018,  0.4239]), 1, 510, 32, 2.803306327008683e-09)
+
+
+

Notes

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
ik_NR

A fast numerical inverse kinematics solver using Newton-Raphson optimisation

+
+
ik_GN

A fast numerical inverse kinematics solver using Gauss-Newton optimisation

+
+
+
+
+ +
+
+ik_LM(Tep, end=None, start=None, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, k=1.0, method='chan')
+

Fast levenberg-Marquadt Numerical Inverse Kinematics Solver

+

A method which provides functionality to perform numerical inverse kinematics (IK) +using the Levemberg-Marquadt method. This +is a fast solver implemented in C++.

+

See the Inverse Kinematics Docs Page for more details and for a +tutorial on numerical IK, see here.

+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose

  • +
  • end (Union[str, Link, Gripper, None]) – the link considered as the end-effector

  • +
  • start (Union[str, Link, Gripper, None]) – the link considered as the base frame, defaults to the robots’s base frame

  • +
  • q0 (Optional[ndarray]) – The initial joint coordinate vector

  • +
  • ilimit (int) – How many iterations are allowed within a search before a new search +is started

  • +
  • slimit (int) – How many searches are allowed before being deemed unsuccessful

  • +
  • tol (float) – Maximum allowed residual error E

  • +
  • mask (Optional[ndarray]) – A 6 vector which assigns weights to Cartesian degrees-of-freedom +error priority

  • +
  • joint_limits (bool) – Reject solutions with joint limit violations

  • +
  • seed – A seed for the private RNG used to generate random joint coordinate +vectors

  • +
  • k (float) – Sets the gain value for the damping matrix Wn in the next iteration. See +synopsis

  • +
  • method (Literal['chan', 'wampler', 'sugihara']) – One of “chan”, “sugihara” or “wampler”. Defines which method is used +to calculate the damping matrix Wn in the step method

  • +
+
+
Return type:
+

Tuple[ndarray, int, int, int, float]

+
+
+

Synopsis

+

The operation is defined by the choice of the method kwarg.

+

The step is deined as

+
+\[\begin{split}\vec{q}_{k+1} +&= +\vec{q}_k + +\left( + \mat{A}_k +\right)^{-1} +\bf{g}_k \\ +% +\mat{A}_k +&= +{\mat{J}(\vec{q}_k)}^\top +\mat{W}_e \ +{\mat{J}(\vec{q}_k)} ++ +\mat{W}_n\end{split}\]
+

where \(\mat{W}_n = \text{diag}(\vec{w_n})(\vec{w_n} \in \mathbb{R}^n_{>0})\) is a +diagonal damping matrix. The damping matrix ensures that \(\mat{A}_k\) is +non-singular and positive definite. The performance of the LM method largely depends +on the choice of \(\mat{W}_n\).

+

Chan’s Method

+

Chan proposed

+
+\[\mat{W}_n += +λ E_k \mat{1}_n\]
+

where λ is a constant which reportedly does not have much influence on performance. +Use the kwarg k to adjust the weighting term λ.

+

Sugihara’s Method

+

Sugihara proposed

+
+\[\mat{W}_n += +E_k \mat{1}_n + \text{diag}(\hat{\vec{w}}_n)\]
+

where \(\hat{\vec{w}}_n \in \mathbb{R}^n\), \(\hat{w}_{n_i} = l^2 \sim 0.01 l^2\), +and \(l\) is the length of a typical link within the manipulator. We provide the +variable k as a kwarg to adjust the value of \(w_n\).

+

Wampler’s Method

+

Wampler proposed \(\vec{w_n}\) to be a constant. This is set through the k kwarg.

+

Examples

+

The following example makes a panda robot object, makes a goal +pose Tep, and then solves for the joint coordinates which result in the pose +Tep using the ikine_LM method.

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda()
+>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> panda.ikine_LM(Tep)
+IKSolution(q=array([ 1.1107, -0.5525, -0.9283, -2.1759, -0.4838,  1.8915,  1.1643]), success=True, iterations=11, searches=1, residual=4.681925120567035e-12, reason='Success')
+
+
+

Notes

+

The value for the k kwarg will depend on the method chosen and the arm you are +using. Use the following as a rough guide chan, k = 1.0 - 0.01, +wampler, k = 0.01 - 0.0001, and sugihara, k = 0.1 - 0.0001

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian.

+

This class supports null-space motion to assist with maximising manipulability and +avoiding joint limits. These are enabled by setting kq and km to non-zero values.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
ik_NR

A fast numerical inverse kinematics solver using Newton-Raphson optimisation

+
+
ik_GN

A fast numerical inverse kinematics solver using Gauss-Newton optimisation

+
+
+
+
+

Changed in version 1.0.4: Merged the Levemberg-Marquadt IK solvers into the ik_LM method

+
+
+ +
+
+ik_NR(Tep, end=None, start=None, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, pinv=True, pinv_damping=0.0)
+

Fast numerical inverse kinematics using Newton-Raphson optimization

+

sol = ets.ik_NR(Tep) are the joint coordinates (n) corresponding +to the robot end-effector pose Tep which is an SE3 or ndarray object. +This method can be used for robots with any number of degrees of freedom. This +is a fast solver implemented in C++.

+

See the Inverse Kinematics Docs Page for more details and for a +tutorial on numerical IK, see here.

+
+

Note

+

When using this method with redundant robots (>6 DoF), pinv must be set to True

+
+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose or pose trajectory

  • +
  • end (Union[str, Link, Gripper, None]) – the link considered as the end-effector

  • +
  • start (Union[str, Link, Gripper, None]) – the link considered as the base frame, defaults to the robots’s base frame

  • +
  • q0 (Optional[ndarray]) – initial joint configuration (default to random valid joint +configuration contrained by the joint limits of the robot)

  • +
  • ilimit (int) – maximum number of iterations per search

  • +
  • slimit (int) – maximum number of search attempts

  • +
  • tol (float) – final error tolerance

  • +
  • mask (Optional[ndarray]) – a mask vector which weights the end-effector error priority. +Corresponds to translation in X, Y and Z and rotation about X, Y and Z +respectively

  • +
  • joint_limits (bool) – constrain the solution to being within the joint limits of +the robot (reject solution with invalid joint configurations and perfrom +another search up to the slimit)

  • +
  • pinv (int) – Use the psuedo-inverse instad of the normal matrix inverse

  • +
  • pinv_damping (float) – Damping factor for the psuedo-inverse

  • +
+
+
Return type:
+

Tuple[ndarray, int, int, int, float]

+
+
Returns:
+

    +
  • sol – tuple (q, success, iterations, searches, residual)

  • +
  • The return value sol is a tuple with elements

  • +
  • ============ ========== ===============================================

  • +
  • Element Type Description

  • +
  • ============ ========== ===============================================

  • +
  • q ndarray(n) joint coordinates in units of radians or metres

  • +
  • success int whether a solution was found

  • +
  • iterations int total number of iterations

  • +
  • searches int total number of searches

  • +
  • residual float final value of cost function

  • +
  • ============ ========== ===============================================

  • +
  • If success == 0 the q values will be valid numbers, but the

  • +
  • solution will be in error. The amount of error is indicated by

  • +
  • the residual.

  • +
+

+
+
+

Synopsis

+

Each iteration uses the Newton-Raphson optimisation method

+
+\[\vec{q}_{k+1} = \vec{q}_k + {^0\mat{J}(\vec{q}_k)}^{-1} \vec{e}_k\]
+

Examples

+

The following example gets a panda robot object, makes a goal +pose Tep, and then solves for the joint coordinates which result in the pose +Tep using the ikine_GN method.

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda()
+>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> panda.ik_NR(Tep)
+(array([-1.0805, -0.5328,  0.9086, -2.1781,  0.4612,  1.9018,  0.4239]), 1, 489, 32, 2.8033063270234757e-09)
+
+
+

Notes

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
ik_LM

A fast numerical inverse kinematics solver using Levenberg-Marquadt optimisation

+
+
ik_GN

A fast numerical inverse kinematics solver using Gauss-Newton optimisation

+
+
+
+
+ +
+
+ikine_GN(Tep, end=None, start=None, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, seed=None, pinv=False, kq=0.0, km=0.0, ps=0.0, pi=0.3, **kwargs)
+

Gauss-Newton Numerical Inverse Kinematics Solver

+

A method which provides functionality to perform numerical inverse kinematics (IK) +using the Gauss-Newton method.

+

See the Inverse Kinematics Docs Page for more details and for a +tutorial on numerical IK, see here.

+
+

Note

+

When using this method with redundant robots (>6 DoF), pinv must be set to True

+
+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose

  • +
  • end (Union[str, Link, Gripper, None]) – the link considered as the end-effector

  • +
  • start (Union[str, Link, Gripper, None]) – the link considered as the base frame, defaults to the robots’s base frame

  • +
  • q0 (Union[ndarray, List[float], Tuple[float], Set[float], None]) – The initial joint coordinate vector

  • +
  • ilimit (int) – How many iterations are allowed within a search before a new search +is started

  • +
  • slimit (int) – How many searches are allowed before being deemed unsuccessful

  • +
  • tol (float) – Maximum allowed residual error E

  • +
  • mask (Union[ndarray, List[float], Tuple[float], Set[float], None]) – A 6 vector which assigns weights to Cartesian degrees-of-freedom +error priority

  • +
  • joint_limits (bool) – Reject solutions with joint limit violations

  • +
  • seed (Optional[int]) – A seed for the private RNG used to generate random joint coordinate +vectors

  • +
  • pinv (bool) – If True, will use the psuedoinverse in the step method instead of +the normal inverse

  • +
  • kq (float) – The gain for joint limit avoidance. Setting to 0.0 will remove this +completely from the solution

  • +
  • km (float) – The gain for maximisation. Setting to 0.0 will remove this completely +from the solution

  • +
  • ps (float) – The minimum angle/distance (in radians or metres) in which the joint is +allowed to approach to its limit

  • +
  • pi (Union[ndarray, float]) – The influence angle/distance (in radians or metres) in null space motion +becomes active

  • +
+
+
+

Synopsis

+

Each iteration uses the Gauss-Newton optimisation method

+
+\[\begin{split}\vec{q}_{k+1} &= \vec{q}_k + +\left( +{\mat{J}(\vec{q}_k)}^\top +\mat{W}_e \ +{\mat{J}(\vec{q}_k)} +\right)^{-1} +\bf{g}_k \\ +\bf{g}_k &= +{\mat{J}(\vec{q}_k)}^\top +\mat{W}_e +\vec{e}_k\end{split}\]
+

where \(\mat{J} = {^0\mat{J}}\) is the base-frame manipulator Jacobian. If +\(\mat{J}(\vec{q}_k)\) is non-singular, and \(\mat{W}_e = \mat{1}_n\), then +the above provides the pseudoinverse solution. However, if \(\mat{J}(\vec{q}_k)\) +is singular, the above can not be computed and the GN solution is infeasible.

+

Examples

+

The following example gets a panda robot object, makes a goal +pose Tep, and then solves for the joint coordinates which result in the pose +Tep using the ikine_GN method.

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda()
+>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> panda.ikine_GN(Tep)
+IKSolution(q=array([ 1.9971,  0.6991,  2.5847, -1.9158, -1.8717,  3.2556,  1.3953]), success=False, iterations=100, searches=100, residual=0.0, reason='iteration and search limit reached, 100 numpy.LinAlgError encountered')
+
+
+

Notes

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian.

+

This class supports null-space motion to assist with maximising manipulability and +avoiding joint limits. These are enabled by setting kq and km to non-zero values.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
IK_NR

An IK Solver class which implements the Newton-Raphson optimisation technique

+
+
ikine_LM

Implements the IK_LM class as a method within the ETS class

+
+
ikine_NR

Implements the IK_NR class as a method within the ETS class

+
+
ikine_QP

Implements the IK_QP class as a method within the ETS class

+
+
+
+
+

Changed in version 1.0.4: Added the Gauss-Newton IK solver method on the Robot class

+
+
+ +
+
+ikine_LM(Tep, end=None, start=None, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, seed=None, k=1.0, method='chan', kq=0.0, km=0.0, ps=0.0, pi=0.3, **kwargs)
+

Levenberg-Marquadt Numerical Inverse Kinematics Solver

+

A method which provides functionality to perform numerical inverse kinematics (IK) +using the Levemberg-Marquadt method.

+

See the Inverse Kinematics Docs Page for more details and for a +tutorial on numerical IK, see here.

+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose

  • +
  • end (Union[str, Link, Gripper, None]) – the link considered as the end-effector

  • +
  • start (Union[str, Link, Gripper, None]) – the link considered as the base frame, defaults to the robots’s base frame

  • +
  • q0 (Union[ndarray, List[float], Tuple[float], Set[float], None]) – The initial joint coordinate vector

  • +
  • ilimit (int) – How many iterations are allowed within a search before a new search +is started

  • +
  • slimit (int) – How many searches are allowed before being deemed unsuccessful

  • +
  • tol (float) – Maximum allowed residual error E

  • +
  • mask (Union[ndarray, List[float], Tuple[float], Set[float], None]) – A 6 vector which assigns weights to Cartesian degrees-of-freedom +error priority

  • +
  • joint_limits (bool) – Reject solutions with joint limit violations

  • +
  • seed (Optional[int]) – A seed for the private RNG used to generate random joint coordinate +vectors

  • +
  • k (float) – Sets the gain value for the damping matrix Wn in the next iteration. See +synopsis

  • +
  • method (Literal['chan', 'wampler', 'sugihara']) – One of “chan”, “sugihara” or “wampler”. Defines which method is used +to calculate the damping matrix Wn in the step method

  • +
  • kq (float) – The gain for joint limit avoidance. Setting to 0.0 will remove this +completely from the solution

  • +
  • km (float) – The gain for maximisation. Setting to 0.0 will remove this completely +from the solution

  • +
  • ps (float) – The minimum angle/distance (in radians or metres) in which the joint is +allowed to approach to its limit

  • +
  • pi (Union[ndarray, float]) – The influence angle/distance (in radians or metres) in null space motion +becomes active

  • +
+
+
+

Synopsis

+

The operation is defined by the choice of the method kwarg.

+

The step is deined as

+
+\[\begin{split}\vec{q}_{k+1} +&= +\vec{q}_k + +\left( + \mat{A}_k +\right)^{-1} +\bf{g}_k \\ +% +\mat{A}_k +&= +{\mat{J}(\vec{q}_k)}^\top +\mat{W}_e \ +{\mat{J}(\vec{q}_k)} ++ +\mat{W}_n\end{split}\]
+

where \(\mat{W}_n = \text{diag}(\vec{w_n})(\vec{w_n} \in \mathbb{R}^n_{>0})\) is a +diagonal damping matrix. The damping matrix ensures that \(\mat{A}_k\) is +non-singular and positive definite. The performance of the LM method largely depends +on the choice of \(\mat{W}_n\).

+

Chan’s Method

+

Chan proposed

+
+\[\mat{W}_n += +λ E_k \mat{1}_n\]
+

where λ is a constant which reportedly does not have much influence on performance. +Use the kwarg k to adjust the weighting term λ.

+

Sugihara’s Method

+

Sugihara proposed

+
+\[\mat{W}_n += +E_k \mat{1}_n + \text{diag}(\hat{\vec{w}}_n)\]
+

where \(\hat{\vec{w}}_n \in \mathbb{R}^n\), \(\hat{w}_{n_i} = l^2 \sim 0.01 l^2\), +and \(l\) is the length of a typical link within the manipulator. We provide the +variable k as a kwarg to adjust the value of \(w_n\).

+

Wampler’s Method

+

Wampler proposed \(\vec{w_n}\) to be a constant. This is set through the k kwarg.

+

Examples

+

The following example makes a panda robot object, makes a goal +pose Tep, and then solves for the joint coordinates which result in the pose +Tep using the ikine_LM method.

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda()
+>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> panda.ikine_LM(Tep)
+IKSolution(q=array([ 1.3588,  1.388 , -2.1523, -2.0854,  1.1755,  1.2453, -0.0636]), success=True, iterations=12, searches=1, residual=1.6826908252399263e-10, reason='Success')
+
+
+

Notes

+

The value for the k kwarg will depend on the method chosen and the arm you are +using. Use the following as a rough guide chan, k = 1.0 - 0.01, +wampler, k = 0.01 - 0.0001, and sugihara, k = 0.1 - 0.0001

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian.

+

This class supports null-space motion to assist with maximising manipulability and +avoiding joint limits. These are enabled by setting kq and km to non-zero values.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
IK_LM

An IK Solver class which implements the Levemberg Marquadt optimisation technique

+
+
ikine_NR

Implements the IK_NR class as a method within the Robot class

+
+
ikine_GN

Implements the IK_GN class as a method within the Robot class

+
+
ikine_QP

Implements the IK_QP class as a method within the Robot class

+
+
+
+
+

Changed in version 1.0.4: Added the Levemberg-Marquadt IK solver method on the Robot class

+
+
+ +
+
+ikine_NR(Tep, end=None, start=None, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, seed=None, pinv=False, kq=0.0, km=0.0, ps=0.0, pi=0.3, **kwargs)
+

Newton-Raphson Numerical Inverse Kinematics Solver

+

A method which provides functionality to perform numerical inverse kinematics (IK) +using the Newton-Raphson method.

+

See the Inverse Kinematics Docs Page for more details and for a +tutorial on numerical IK, see here.

+
+

Note

+

When using this method with redundant robots (>6 DoF), pinv must be set to True

+
+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose

  • +
  • end (Union[str, Link, Gripper, None]) – the link considered as the end-effector

  • +
  • start (Union[str, Link, Gripper, None]) – the link considered as the base frame, defaults to the robots’s base frame

  • +
  • q0 (Union[ndarray, List[float], Tuple[float], Set[float], None]) – The initial joint coordinate vector

  • +
  • ilimit (int) – How many iterations are allowed within a search before a new search +is started

  • +
  • slimit (int) – How many searches are allowed before being deemed unsuccessful

  • +
  • tol (float) – Maximum allowed residual error E

  • +
  • mask (Union[ndarray, List[float], Tuple[float], Set[float], None]) – A 6 vector which assigns weights to Cartesian degrees-of-freedom +error priority

  • +
  • joint_limits (bool) – Reject solutions with joint limit violations

  • +
  • seed (Optional[int]) – A seed for the private RNG used to generate random joint coordinate +vectors

  • +
  • pinv (bool) – If True, will use the psuedoinverse in the step method instead of +the normal inverse

  • +
  • kq (float) – The gain for joint limit avoidance. Setting to 0.0 will remove this +completely from the solution

  • +
  • km (float) – The gain for maximisation. Setting to 0.0 will remove this completely +from the solution

  • +
  • ps (float) – The minimum angle/distance (in radians or metres) in which the joint is +allowed to approach to its limit

  • +
  • pi (Union[ndarray, float]) – The influence angle/distance (in radians or metres) in null space motion +becomes active

  • +
+
+
+

Synopsis

+

Each iteration uses the Newton-Raphson optimisation method

+
+\[\vec{q}_{k+1} = \vec{q}_k + {^0\mat{J}(\vec{q}_k)}^{-1} \vec{e}_k\]
+

Examples

+

The following example gets a panda robot object, makes a goal +pose Tep, and then solves for the joint coordinates which result in the pose +Tep using the ikine_NR method.

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda()
+>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> panda.ikine_NR(Tep)
+IKSolution(q=array([ 1.5006, -1.2116, -1.5987, -1.9942, -0.0664,  3.1527, -0.6222]), success=False, iterations=100, searches=100, residual=0.0, reason='iteration and search limit reached, 100 numpy.LinAlgError encountered')
+
+
+

Notes

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian.

+

This class supports null-space motion to assist with maximising manipulability and +avoiding joint limits. These are enabled by setting kq and km to non-zero values.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
IK_NR

An IK Solver class which implements the Newton-Raphson optimisation technique

+
+
ikine_LM

Implements the IK_LM class as a method within the ETS class

+
+
ikine_GN

Implements the IK_GN class as a method within the ETS class

+
+
ikine_QP

Implements the IK_QP class as a method within the ETS class

+
+
+
+
+

Changed in version 1.0.4: Added the Newton-Raphson IK solver method on the Robot class

+
+
+ +
+
+ikine_QP(Tep, end=None, start=None, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, seed=None, kj=1.0, ks=1.0, kq=0.0, km=0.0, ps=0.0, pi=0.3, **kwargs)
+

Quadratic Programming Numerical Inverse Kinematics Solver

+

A method that provides functionality to perform numerical inverse kinematics +(IK) using a quadratic progamming approach.

+

See the Inverse Kinematics Docs Page for more details and for a +tutorial on numerical IK, see here.

+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose

  • +
  • end (Union[str, Link, Gripper, None]) – the link considered as the end-effector

  • +
  • start (Union[str, Link, Gripper, None]) – the link considered as the base frame, defaults to the robots’s base frame

  • +
  • q0 (Union[ndarray, List[float], Tuple[float], Set[float], None]) – The initial joint coordinate vector

  • +
  • ilimit (int) – How many iterations are allowed within a search before a new search +is started

  • +
  • slimit (int) – How many searches are allowed before being deemed unsuccessful

  • +
  • tol (float) – Maximum allowed residual error E

  • +
  • mask (Union[ndarray, List[float], Tuple[float], Set[float], None]) – A 6 vector which assigns weights to Cartesian degrees-of-freedom +error priority

  • +
  • joint_limits (bool) – Reject solutions with joint limit violations

  • +
  • seed (Optional[int]) – A seed for the private RNG used to generate random joint coordinate +vectors

  • +
  • kj – A gain for joint velocity norm minimisation

  • +
  • ks – A gain which adjusts the cost of slack (intentional error)

  • +
  • kq (float) – The gain for joint limit avoidance. Setting to 0.0 will remove this +completely from the solution

  • +
  • km (float) – The gain for maximisation. Setting to 0.0 will remove this completely +from the solution

  • +
  • ps (float) – The minimum angle/distance (in radians or metres) in which the joint is +allowed to approach to its limit

  • +
  • pi (Union[ndarray, float]) – The influence angle/distance (in radians or metres) in null space motion +becomes active

  • +
+
+
Raises:
+

ImportError – If the package qpsolvers is not installed

+
+
+

Synopsis

+

Each iteration uses the following approach

+
+\[\vec{q}_{k+1} = \vec{q}_{k} + \dot{\vec{q}}.\]
+

where the QP is defined as

+
+\[\begin{split}\min_x \quad f_o(\vec{x}) &= \frac{1}{2} \vec{x}^\top \mathcal{Q} \vec{x}+ \mathcal{C}^\top \vec{x}, \\ +\text{subject to} \quad \mathcal{J} \vec{x} &= \vec{\nu}, \\ +\mathcal{A} \vec{x} &\leq \mathcal{B}, \\ +\vec{x}^- &\leq \vec{x} \leq \vec{x}^+\end{split}\]
+

with

+
+\[\begin{split}\vec{x} &= +\begin{pmatrix} + \dvec{q} \\ \vec{\delta} +\end{pmatrix} \in \mathbb{R}^{(n+6)} \\ +\mathcal{Q} &= +\begin{pmatrix} + \lambda_q \mat{1}_{n} & \mathbf{0}_{6 \times 6} \\ \mathbf{0}_{n \times n} & \lambda_\delta \mat{1}_{6} +\end{pmatrix} \in \mathbb{R}^{(n+6) \times (n+6)} \\ +\mathcal{J} &= +\begin{pmatrix} + \mat{J}(\vec{q}) & \mat{1}_{6} +\end{pmatrix} \in \mathbb{R}^{6 \times (n+6)} \\ +\mathcal{C} &= +\begin{pmatrix} + \mat{J}_m \\ \bf{0}_{6 \times 1} +\end{pmatrix} \in \mathbb{R}^{(n + 6)} \\ +\mathcal{A} &= +\begin{pmatrix} + \mat{1}_{n \times n + 6} \\ +\end{pmatrix} \in \mathbb{R}^{(l + n) \times (n + 6)} \\ +\mathcal{B} &= +\eta +\begin{pmatrix} + \frac{\rho_0 - \rho_s} + {\rho_i - \rho_s} \\ + \vdots \\ + \frac{\rho_n - \rho_s} + {\rho_i - \rho_s} +\end{pmatrix} \in \mathbb{R}^{n} \\ +\vec{x}^{-, +} &= +\begin{pmatrix} + \dvec{q}^{-, +} \\ + \vec{\delta}^{-, +} +\end{pmatrix} \in \mathbb{R}^{(n+6)},\end{split}\]
+

where \(\vec{\delta} \in \mathbb{R}^6\) is the slack vector, +\(\lambda_\delta \in \mathbb{R}^+\) is a gain term which adjusts the +cost of the norm of the slack vector in the optimiser, +\(\dvec{q}^{-,+}\) are the minimum and maximum joint velocities, and +\(\dvec{\delta}^{-,+}\) are the minimum and maximum slack velocities.

+

Examples

+

The following example gets a panda robot object, makes a goal +pose Tep, and then solves for the joint coordinates which result in the pose +Tep using the ikine_QP method.

+
  File "/opt/hostedtoolcache/Python/3.7.17/x64/lib/python3.7/site-packages/roboticstoolbox/robot/IK.py", line 1331, in __init__
+    "the package qpsolvers is required for this class. \nInstall using 'pip"
+ImportError: the package qpsolvers is required for this class. 
+Install using 'pip install qpsolvers'
+
+
+

Notes

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian.

+

This class supports null-space motion to assist with maximising manipulability and +avoiding joint limits. These are enabled by setting kq and km to non-zero values.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
IK_NR

An IK Solver class which implements the Newton-Raphson optimisation technique

+
+
ikine_LM

Implements the IK_LM class as a method within the ETS class

+
+
ikine_GN

Implements the IK_GN class as a method within the ETS class

+
+
ikine_NR

Implements the IK_NR class as a method within the ETS class

+
+
+
+
+

Changed in version 1.0.4: Added the Quadratic Programming IK solver method on the Robot class

+
+
+ +
+
+inertia(q)
+

Manipulator inertia matrix +inertia(q) is the symmetric joint inertia matrix (n,n) which +relates joint torque to joint acceleration for the robot at joint +configuration q.

+

Trajectory operation

+

If q is a matrix (m,n), each row is interpretted as a joint state +vector, and the result is a 3d-matrix (nxnxk) where each plane +corresponds to the inertia for the corresponding row of q.

+
+
Parameters:
+

q (ndarray) – Joint coordinates

+
+
Returns:
+

The inertia matrix

+
+
Return type:
+

inertia

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> puma.inertia(puma.qz)
+array([[ 3.9611, -0.1627, -0.1389,  0.0016, -0.0004,  0.    ],
+       [-0.1627,  4.4566,  0.3727,  0.    ,  0.0019,  0.    ],
+       [-0.1389,  0.3727,  0.9387,  0.    ,  0.0019,  0.    ],
+       [ 0.0016,  0.    ,  0.    ,  0.1924,  0.    ,  0.    ],
+       [-0.0004,  0.0019,  0.0019,  0.    ,  0.1713,  0.    ],
+       [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.1941]])
+
+
+

Notes

+
    +
  • +
    The diagonal elements M[j,j] are the inertia seen by joint

    actuator j.

    +
    +
    +
  • +
  • +
    The off-diagonal elements M[j,k] are coupling inertias that

    relate acceleration on joint j to force/torque on +joint k.

    +
    +
    +
  • +
  • +
    The diagonal terms include the motor inertia reflected through

    the gear ratio.

    +
    +
    +
  • +
+
+

See also

+

cinertia()

+
+
+ +
+
+inertia_x(q=None, pinv=False, representation='rpy/xyz', Ji=None)
+

Operational space inertia matrix

+

robot.inertia_x(q) is the operational space (Cartesian) inertia +matrix which relates Cartesian force/torque to Cartesian +acceleration at the joint configuration q.

+
+\[\mathbf{M}_x = \mathbf{J}(q)^{-T} \mathbf{M}(q) \mathbf{J}(q)^{-1}\]
+

The transformation to operational space requires an analytical, rather +than geometric, Jacobian. analytical can be one of:

+
+
+ + + + + + + + + + + + + + + + + + + +

Value

Rotational representation

'rpy/xyz'

RPY angular rates in XYZ order (default)

'rpy/zyx'

RPY angular rates in XYZ order

'eul'

Euler angular rates in ZYZ order

'exp'

exponential coordinate rates

+
+

Trajectory operation

+

If q is a matrix (m,n), each row is interpretted as a joint state +vector, and the result is a 3d-matrix (m,n,n) where each plane +corresponds to the Cartesian inertia for the corresponding +row of q.

+
+
Parameters:
+
    +
  • q – Joint coordinates

  • +
  • pinv – use pseudo inverse rather than inverse (Default value = False)

  • +
  • analytical – the type of analytical Jacobian to use, default is +‘rpy/xyz’

  • +
  • representation – (Default value = “rpy/xyz”)

  • +
  • Ji – The inverse analytical Jacobian (base-frame)

  • +
+
+
Returns:
+

The operational space inertia matrix

+
+
Return type:
+

inertia_x

+
+
+

Examples

+
    ainv = _umath_linalg.inv(a, signature=signature, extobj=extobj)
+  File "/opt/hostedtoolcache/Python/3.7.17/x64/lib/python3.7/site-packages/numpy/linalg/linalg.py", line 88, in _raise_linalgerror_singular
+    raise LinAlgError("Singular matrix")
+numpy.linalg.LinAlgError: Singular matrix
+
+
+

Notes

+
    +
  • If the robot is not 6 DOF the pinv option is set True.

  • +
  • pinv() is around 5x slower than inv()

  • +
+
+

Warning

+

Assumes that the operational space has 6 DOF.

+
+
+

See also

+

inertia()

+
+
+ +
+
+iscollided(q, shape, skip=False)
+

Check if the robot is in collision with a shape

+

iscollided(shape) checks if this robot and shape have collided

+
+
+shape
+

The shape to compare distance to

+
+ +
+
+skip
+

Skip setting all shape transforms based on q, use this +option if using this method in conjuction with Swift to save time

+
+ +
+
Returns:
+

True if shapes have collided

+
+
Return type:
+

iscollided

+
+
+
+ +
+
+isprismatic(j)
+

Check if joint is prismatic

+
+
Returns:
+

True if prismatic

+
+
Return type:
+

j

+
+
+

Examples

+

+
+
+
+

See also

+

Link.isprismatic(), prismaticjoints()

+
+
+ +
+
+isrevolute(j)
+

Check if joint is revolute

+
+
Returns:
+

True if revolute

+
+
Return type:
+

j

+
+
+

Examples

+

+
+
+
+

See also

+

Link.isrevolute(), revolutejoints()

+
+
+ +
+
+itorque(q, qdd)
+

Inertia torque

+

itorque(q, qdd) is the inertia force/torque vector (n) at +the specified joint configuration q (n) and acceleration qdd (n), and +n is the number of robot joints. +It is \(\mathbf{I}(q) \ddot{q}\).

+

Trajectory operation

+

If q and qdd are matrices (m,n), each row is interpretted as a +joint configuration, and the result is a matrix (m,n) where each row is +the corresponding joint torques.

+
+
Parameters:
+
    +
  • q – Joint coordinates

  • +
  • qdd – Joint acceleration

  • +
+
+
Returns:
+

The inertia torque vector

+
+
Return type:
+

itorque

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> puma.itorque(puma.qz, 0.5 * np.ones((6,)))
+array([1.8304, 2.3343, 0.5872, 0.0971, 0.0873, 0.0971])
+
+
+

Notes

+
    +
  • +
    If the robot model contains non-zero motor inertia then this

    will be included in the result.

    +
    +
    +
  • +
+
+

See also

+

inertia()

+
+
+ +
+
+jacob0(q, end=None, start=None, tool=None)
+

Manipulator geometric Jacobian in the start frame

+

robot.jacobo(q) is the manipulator Jacobian matrix which maps +joint velocity to end-effector spatial velocity expressed in the +base frame.

+

End-effector spatial velocity \(\nu = (v_x, v_y, v_z, \omega_x, \omega_y, \omega_z)^T\) +is related to joint velocity by \({}^{E}\!\nu = \mathbf{J}_m(q) \dot{q}\).

+
+
Parameters:
+
    +
  • q (Union[ndarray, List[float], Tuple[float], Set[float]]) – Joint coordinate vector

  • +
  • end (Union[str, Link, Gripper, None]) – the particular link or gripper whose velocity the Jacobian +describes, defaults to the end-effector if only one is present

  • +
  • start (Union[str, Link, Gripper, None]) – the link considered as the base frame, defaults to the robots’s base frame

  • +
  • tool (Union[ndarray, SE3, None]) – a static tool transformation matrix to apply to the +end of end, defaults to None

  • +
+
+
Returns:
+

Manipulator Jacobian in the start frame

+
+
Return type:
+

J0

+
+
+

Examples

+

The following example makes a Puma560 robot object, and solves for the +base-frame Jacobian at the zero joint angle configuration

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.Puma560()
+>>> puma.jacob0([0, 0, 0, 0, 0, 0])
+array([[ 0.1295, -0.4854, -0.4854, -0.    , -0.0533,  0.    ],
+       [ 0.4318,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+       [-0.    ,  0.4318,  0.    ,  0.    ,  0.    ,  0.    ],
+       [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+       [ 0.    , -1.    , -1.    ,  0.    , -1.    ,  0.    ],
+       [ 1.    ,  0.    ,  0.    ,  1.    ,  0.    ,  1.    ]])
+
+
+

Notes

+
    +
  • +
    This is the geometric Jacobian as described in texts by

    Corke, Spong etal., Siciliano etal. The end-effector velocity is +described in terms of translational and angular velocity, not a +velocity twist as per the text by Lynch & Park.

    +
    +
    +
  • +
+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
+
+ +
+
+jacob0_analytical(q, representation='rpy/xyz', end=None, start=None, tool=None)
+

Manipulator analytical Jacobian in the start frame

+

robot.jacob0_analytical(q) is the manipulator Jacobian matrix which maps +joint velocity to end-effector spatial velocity expressed in the +start frame.

+
+
Parameters:
+
    +
  • q (Union[ndarray, List[float], Tuple[float], Set[float]]) – Joint coordinate vector

  • +
  • representation (Literal['rpy/xyz', 'rpy/zyx', 'eul', 'exp']) – angular representation

  • +
  • end (Union[str, Link, Gripper, None]) – the particular link or gripper whose velocity the Jacobian +describes, defaults to the base link

  • +
  • start (Union[str, Link, Gripper, None]) – the link considered as the end-effector, defaults to the robots’s end-effector

  • +
  • tool (Union[ndarray, SE3, None]) – a static tool transformation matrix to apply to the +end of end, defaults to None

  • +
+
+
Returns:
+

Manipulator Jacobian in the start frame

+
+
Return type:
+

jacob0

+
+
+

Synopsis

+

End-effector spatial velocity \(\nu = (v_x, v_y, v_z, \omega_x, \omega_y, \omega_z)^T\) +is related to joint velocity by \({}^{E}\!\nu = \mathbf{J}_m(q) \dot{q}\).

+

|``representation`` | Rotational representation | +|---------------------|————————————-| +|'rpy/xyz' | RPY angular rates in XYZ order | +|'rpy/zyx' | RPY angular rates in XYZ order | +|'eul' | Euler angular rates in ZYZ order | +|'exp' | exponential coordinate rates |

+

Examples

+

Makes a robot object and computes the analytic Jacobian for the given +joint configuration

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.ETS.Puma560()
+>>> puma.jacob0_analytical([0, 0, 0, 0, 0, 0])
+array([[ 0.15  ,  0.8636,  0.4318,  0.    ,  0.    ,  0.    ],
+       [ 0.2203,  0.    ,  0.    ,  0.2   ,  0.    ,  0.2   ],
+       [ 0.    , -0.2203, -0.2   ,  0.    , -0.2   ,  0.    ],
+       [ 1.    ,  0.    ,  0.    ,  1.    ,  0.    ,  1.    ],
+       [ 0.    ,  1.    ,  1.    ,  0.    ,  1.    ,  0.    ],
+       [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ]])
+
+
+
+ +
+
+jacob0_dot(q, qd, J0=None, representation=None)
+

Derivative of Jacobian

+

robot.jacob_dot(q, qd) computes the rate of change of the +Jacobian elements

+
+\[\dmat{J} = \frac{d \mat{J}}{d \vec{q}} \frac{d \vec{q}}{dt}\]
+

where the first term is the rank-3 Hessian.

+
+
Parameters:
+
    +
  • q

  • +
  • robot (The joint configuration of the) –

  • +
  • qd (Union[ndarray, List[float], Tuple[float], Set[float]]) – The joint velocity of the robot

  • +
  • J0 – Jacobian in {0} frame

  • +
  • representation (Optional[Literal['rpy/xyz', 'rpy/zyx', 'eul', 'exp']]) – angular representation

  • +
+
+
Returns:
+

The derivative of the manipulator Jacobian

+
+
Return type:
+

jdot

+
+
+

Synopsis

+

If J0 is already calculated for the joint +coordinates q it can be passed in to to save computation time.

+

It is computed as the mode-3 product of the Hessian tensor and the +velocity vector.

+

The derivative of an analytical Jacobian can be obtained by setting +representation as

+

|``representation`` | Rotational representation | +|---------------------|————————————-| +|'rpy/xyz' | RPY angular rates in XYZ order | +|'rpy/zyx' | RPY angular rates in XYZ order | +|'eul' | Euler angular rates in ZYZ order | +|'exp' | exponential coordinate rates |

+

References

+
    +
  • +
    Kinematic Derivatives using the Elementary Transform

    Sequence, J. Haviland and P. Corke

    +
    +
    +
  • +
+
+

See also

+

jacob0(), hessian0()

+
+
+ +
+
+jacobe(q, end=None, start=None, tool=None)
+

Manipulator geometric Jacobian in the end-effector frame

+

robot.jacobe(q) is the manipulator Jacobian matrix which maps +joint velocity to end-effector spatial velocity expressed in the +end frame.

+

End-effector spatial velocity \(\nu = (v_x, v_y, v_z, \omega_x, \omega_y, \omega_z)^T\) +is related to joint velocity by \({}^{E}\!\nu = \mathbf{J}_m(q) \dot{q}\).

+
+
Parameters:
+
    +
  • q (Union[ndarray, List[float], Tuple[float], Set[float]]) – Joint coordinate vector

  • +
  • end (Union[str, Link, Gripper, None]) – the particular link or gripper whose velocity the Jacobian +describes, defaults to the end-effector if only one is present

  • +
  • start (Union[str, Link, Gripper, None]) – the link considered as the base frame, defaults to the robots’s base frame

  • +
  • tool (Union[ndarray, SE3, None]) – a static tool transformation matrix to apply to the +end of end, defaults to None

  • +
+
+
Returns:
+

Manipulator Jacobian in the end frame

+
+
Return type:
+

Je

+
+
+

Examples

+

The following example makes a Puma560 robot object, and solves for the +end-effector frame Jacobian at the zero joint angle configuration

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.Puma560()
+>>> puma.jacobe([0, 0, 0, 0, 0, 0])
+array([[ 0.1295, -0.4854, -0.4854, -0.    , -0.0533,  0.    ],
+       [ 0.4318,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+       [-0.    ,  0.4318,  0.    ,  0.    ,  0.    ,  0.    ],
+       [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+       [ 0.    , -1.    , -1.    ,  0.    , -1.    ,  0.    ],
+       [ 1.    ,  0.    ,  0.    ,  1.    ,  0.    ,  1.    ]])
+
+
+

Notes

+
    +
  • +
    This is the geometric Jacobian as described in texts by

    Corke, Spong etal., Siciliano etal. The end-effector velocity is +described in terms of translational and angular velocity, not a +velocity twist as per the text by Lynch & Park.

    +
    +
    +
  • +
+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
+
+ +
+
+jacobm(q=None, J=None, H=None, end=None, start=None, axes='all')
+

The manipulability Jacobian

+

This measure relates the rate of change of the manipulability to the +joint velocities of the robot. One of J or q is required. Supply J +and H if already calculated to save computation time

+
+
Parameters:
+
    +
  • q – The joint angles/configuration of the robot (Optional, +if not supplied will use the stored q values).

  • +
  • J – The manipulator Jacobian in any frame

  • +
  • H – The manipulator Hessian in any frame

  • +
  • end (Union[str, Link, Gripper, None]) – the final link or Gripper which the Hessian represents

  • +
  • start (Union[str, Link, Gripper, None]) – the first link which the Hessian represents

  • +
+
+
Returns:
+

The manipulability Jacobian

+
+
Return type:
+

jacobm

+
+
+

Synopsis

+

Yoshikawa’s manipulability measure

+
+\[m(\vec{q}) = \sqrt{\mat{J}(\vec{q}) \mat{J}(\vec{q})^T}\]
+

This method returns its Jacobian with respect to configuration

+
+\[\frac{\partial m(\vec{q})}{\partial \vec{q}}\]
+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+ +
+
+joint_velocity_damper(ps=0.05, pi=0.1, n=None, gain=1.0)
+

Compute the joint velocity damper for QP motion control

+

Formulates an inequality contraint which, when optimised for will +make it impossible for the robot to run into joint limits. Requires +the joint limits of the robot to be specified. See examples/mmc.py +for use case

+
+
+ps
+

The minimum angle (in radians) in which the joint is +allowed to approach to its limit

+
+ +
+
+pi
+

The influence angle (in radians) in which the velocity +damper becomes active

+
+ +
+
+n
+

The number of joints to consider. Defaults to all joints

+
+ +
+
+gain
+

The gain for the velocity damper

+
+ +
+
Return type:
+

Tuple[ndarray, ndarray]

+
+
Returns:
+

    +
  • Ain – A (6,) vector inequality contraint for an optisator

  • +
  • Bin – b (6,) vector inequality contraint for an optisator

  • +
+

+
+
+
+ +
+
+jointdynamics(q, qd=None)
+

Transfer function of joint actuator

+

tf = jointdynamics(qd, q) calculates a vector of n +continuous-time transfer functions that represent the transfer +function 1/(Js+B) for each joint based on the dynamic parameters +of the robot and the configuration q (n). n is the number of robot +joints. The result is a list of tuples (J, B) for each joint.

+

tf = jointdynamics(q, qd) as above but include the linearized +effects of Coulomb friction when operating at joint velocity QD +(1xN).

+
+
Parameters:
+
    +
  • q – Joint coordinates

  • +
  • qd – Joint velocity

  • +
+
+
Returns:
+

transfer function denominators

+
+
Return type:
+

list of 2-tuples

+
+
+
+ +
+
+jtraj(T1, T2, t, **kwargs)
+

Joint-space trajectory between SE(3) poses

+

The initial and final poses are mapped to joint space using inverse +kinematics:

+
    +
  • if the object has an analytic solution ikine_a that will be used,

  • +
  • otherwise the general numerical algorithm ikine_lm will be used.

  • +
+

traj = obot.jtraj(T1, T2, t) is a trajectory object whose +attribute traj.q is a row-wise joint-space trajectory.

+
+
Parameters:
+
    +
  • T1 (Union[ndarray, SE3]) – initial end-effector pose

  • +
  • T2 (Union[ndarray, SE3]) – final end-effector pose

  • +
  • t (Union[ndarray, int]) – time vector or number of steps

  • +
  • kwargs – arguments passed to the IK solver

  • +
+
+
Return type:
+

trajectory

+
+
+
+ +
+
+property keywords: List[str]
+
+ +
+ +

Compute a collision constrain for QP motion control

+

Formulates an inequality contraint which, when optimised for will +make it impossible for the robot to run into a collision. Requires +See examples/neo.py for use case

+
+
+ds
+

The minimum distance in which a joint is allowed to +approach the collision object shape

+
+ +
+
+di
+

The influence distance in which the velocity +damper becomes active

+
+ +
+
+xi
+

The gain for the velocity damper

+
+ +
+
+end
+

The end link of the robot to consider

+
+ +
+
+start
+

The start link of the robot to consider

+
+ +
+
+collision_list
+

A list of shapes to consider for collision

+
+ +
+
Returns:
+

    +
  • Ain – A (6,) vector inequality contraint for an optisator

  • +
  • Bin – b (6,) vector inequality contraint for an optisator

  • +
+

+
+
+
+ +
+ +
+ +
+
+linkcolormap(linkcolors='viridis')
+

Create a colormap for robot joints

+
    +
  • cm = robot.linkcolormap() is an n-element colormap that gives a +unique color for every link. The RGBA colors for link j are +cm(j).

  • +
  • cm = robot.linkcolormap(cmap) as above but cmap is the name +of a valid matplotlib colormap. The default, example above, is the +viridis colormap.

  • +
  • cm = robot.linkcolormap(list of colors) as above but a +colormap is created from a list of n color names given as strings, +tuples or hexstrings.

  • +
+
+
Parameters:
+

linkcolors (Union[List[Any], str]) – list of colors or colormap, defaults to “viridis”

+
+
Returns:
+

the color map

+
+
Return type:
+

color map

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> cm = robot.linkcolormap("inferno")
+>>> print(cm(range(6))) # cm(i) is 3rd color in colormap
+[[0.0015 0.0005 0.0139 1.    ]
+ [0.2582 0.0386 0.4065 1.    ]
+ [0.5783 0.148  0.4044 1.    ]
+ [0.865  0.3168 0.2261 1.    ]
+ [0.9876 0.6453 0.0399 1.    ]
+ [0.9884 0.9984 0.6449 1.    ]]
+>>> cm = robot.linkcolormap(
+...     ['red', 'g', (0,0.5,0), '#0f8040', 'yellow', 'cyan'])
+>>> print(cm(range(6)))
+[[1.     0.     0.     1.    ]
+ [0.     0.5    0.     1.    ]
+ [0.     0.5    0.     1.    ]
+ [0.0588 0.502  0.251  1.    ]
+ [1.     1.     0.     1.    ]
+ [0.     1.     1.     1.    ]]
+
+
+

Notes

+ +
+ +
+ +

Robot links

+
+
Returns:
+

A list of link objects

+
+
Return type:
+

links

+
+
+

Notes

+

It is probably more concise to index the robot object rather +than the list of links, ie. the following are equivalent: +- robot.links[i] +- robot[i]

+
+ +
+
+manipulability(q=None, J=None, end=None, start=None, method='yoshikawa', axes='all', **kwargs)
+

Manipulability measure

+

manipulability(q) is the scalar manipulability index +for the robot at the joint configuration q. It indicates +dexterity, that is, how well conditioned the robot is for motion +with respect to the 6 degrees of Cartesian motion. The values is +zero if the robot is at a singularity.

+
+
Parameters:
+
    +
  • q – Joint coordinates, one of J or q required

  • +
  • J – Jacobian in base frame if already computed, one of J or +q required

  • +
  • method (Literal['yoshikawa', 'asada', 'minsingular', 'invcondition']) – method to use, “yoshikawa” (default), “invcondition”, +“minsingular” or “asada”

  • +
  • axes (Union[Literal['all', 'trans', 'rot'], List[bool]]) – Task space axes to consider: “all” [default], +“trans”, or “rot”

  • +
+
+
Return type:
+

manipulability

+
+
+

Synopsis

+

Various measures are supported:

+
+
Measure | Description |
+
+

|-------------------|————————————————-| +| "yoshikawa" | Volume of the velocity ellipsoid, distance | +| | from singularity [Yoshikawa85] | +| "invcondition"``| Inverse condition number of Jacobian, isotropy  | +|                   | of the velocity ellipsoid [Klein87]_            | +| ``"minsingular" | Minimum singular value of the Jacobian, | +| | distance from singularity [Klein87] | +| "asada" | Isotropy of the task-space acceleration | +| | ellipsoid which is a function of the Cartesian | +| | inertia matrix which depends on the inertial | +| | parameters [Asada83] |

+

Trajectory operation:

+

If q is a matrix (m,n) then the result (m,) is a vector of +manipulability indices for each joint configuration specified by a row +of q.

+

Notes

+
    +
  • Invokes the jacob0 method of the robot if J is not passed

  • +
  • +
    The “all” option includes rotational and translational

    dexterity, but this involves adding different units. It can be +more useful to look at the translational and rotational +manipulability separately.

    +
    +
    +
  • +
  • +
    Examples in the RVC book (1st edition) can be replicated by

    using the “all” option

    +
    +
    +
  • +
  • +
    Asada’s measure requires inertial a robot model with inertial

    parameters.

    +
    +
    +
  • +
+

References

+
+
+[Yoshikawa85] +

Manipulability of Robotic Mechanisms. Yoshikawa T., +The International Journal of Robotics Research. +1985;4(2):3-9. doi:10.1177/027836498500400201

+
+
+[Asada83] +

A geometrical representation of manipulator dynamics and +its application to arm design, H. Asada, +Journal of Dynamic Systems, Measurement, and Control, +vol. 105, p. 131, 1983.

+
+
+[Klein87] +

Dexterity Measures for the Design and Control of +Kinematically Redundant Manipulators. Klein CA, Blaho BE. +The International Journal of Robotics Research. +1987;6(2):72-83. doi:10.1177/027836498700600206

+
+
+
    +
  • Robotics, Vision & Control, Chap 8, P. Corke, Springer 2011.

  • +
+
+

Changed in version 1.0.3: Removed ‘both’ option for axes, added a custom list option.

+
+
+ +
+
+property manufacturer
+

Get/set robot manufacturer’s name

+
    +
  • robot.manufacturer is the robot manufacturer’s name

  • +
  • robot.manufacturer = ... checks and sets the manufacturer’s name

  • +
+
+
Returns:
+

robot manufacturer’s name

+
+
Return type:
+

manufacturer

+
+
+
+ +
+
+property n: int
+

Number of joints

+
+
Returns:
+

Number of joints

+
+
Return type:
+

n

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.n
+6
+
+
+
+

See also

+

nlinks(), nbranches()

+
+
+ +
+
+property name: str
+

Get/set robot name

+
    +
  • robot.name is the robot name

  • +
  • robot.name = ... checks and sets the robot name

  • +
+
+
Parameters:
+

name – the new robot name

+
+
Returns:
+

the current robot name

+
+
Return type:
+

name

+
+
+
+ +
+
+property nbranches: int
+

Number of branches

+

Number of branches in this robot. Computed as the number of links with +zero children

+
+
Returns:
+

number of branches in the robot’s kinematic tree

+
+
Return type:
+

nbranches

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.ETS.Panda()
+>>> robot.nbranches
+1
+
+
+
+

See also

+

n(), nlinks()

+
+
+ +
+ +

Number of links

+

The returned number is the total of both variable joints and +static links

+
+
Returns:
+

Number of links

+
+
Return type:
+

nlinks

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.nlinks
+6
+
+
+
+

See also

+

n(), nbranches()

+
+
+ +
+
+nofriction(coulomb=True, viscous=False)
+

Remove manipulator joint friction

+

nofriction() copies the robot and returns +a robot with the same link parameters except the Coulomb and/or viscous +friction parameter are set to zero.

+
+
Parameters:
+
    +
  • coulomb (bool) – set the Coulomb friction to 0

  • +
  • viscous (bool) – set the viscous friction to 0

  • +
+
+
Returns:
+

A copy of the robot with dynamic parameters perturbed

+
+
Return type:
+

robot

+
+
+
+

See also

+

Robot.friction(), Link.nofriction()

+
+
+ +
+
+partial_fkine0(q, n=3, end=None, start=None)
+

Manipulator Forward Kinematics nth Partial Derivative

+

This method computes the nth derivative of the forward kinematics where n is +greater than or equal to 3. This is an extension of the differential kinematics +where the Jacobian is the first partial derivative and the Hessian is the +second.

+
+
Parameters:
+
    +
  • q (Union[ndarray, List[float], Tuple[float], Set[float]]) – The joint angles/configuration of the robot (Optional, +if not supplied will use the stored q values).

  • +
  • end (Union[str, Link, Gripper, None]) – the final link/Gripper which the Hessian represents

  • +
  • start (Union[str, Link, Gripper, None]) – the first link which the Hessian represents

  • +
  • tool – a static tool transformation matrix to apply to the +end of end, defaults to None

  • +
+
+
Returns:
+

The nth Partial Derivative of the forward kinematics

+
+
Return type:
+

A

+
+
+

Examples

+

The following example makes a Panda robot object, and solves for the +base-effector frame 4th defivative of the forward kinematics at the given +joint angle configuration

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda()
+>>> panda.partial_fkine0([0, -0.3, 0, -2.2, 0, 2, 0.7854], n=4)
+array([[[[[ 0.484 ,  0.    ,  0.486 , ...,  0.1547, -0.    ,  0.    ],
+          [-0.    , -0.0796, -0.    , ..., -0.    , -0.2006,  0.    ],
+          [-0.    , -0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
+          [ 0.    ,  1.    ,  0.    , ..., -0.    , -1.    , -0.    ],
+          [ 0.    ,  0.    ,  0.2955, ..., -0.9463,  0.    , -0.0998],
+          [ 0.    , -0.    ,  0.    , ..., -0.    ,  0.    , -0.    ]],
+
+         [[ 0.    ,  0.484 ,  0.    , ...,  0.    , -0.1086,  0.    ],
+          [-0.    , -0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
+          [ 0.    , -0.    , -0.    , ..., -0.    ,  0.    ,  0.    ],
+          [-1.    ,  0.    , -0.9553, ...,  0.3233, -0.    ,  0.995 ],
+          [-0.    ,  0.    ,  0.    , ..., -0.    ,  0.    , -0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.    ]],
+
+         [[ 0.4624,  0.    ,  0.4643, ...,  0.1478, -0.    ,  0.    ],
+          [-0.    ,  0.067 , -0.    , ..., -0.    , -0.2237,  0.    ],
+          [-0.    ,  0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
+          [-0.    ,  0.9553,  0.    , ..., -0.    , -0.9553, -0.    ],
+          [-0.2955, -0.    ,  0.    , ..., -0.8085,  0.    ,  0.1987],
+          [-0.    , -0.    ,  0.    , ..., -0.    ,  0.    ,  0.    ]],
+
+         ...,
+
+         [[-0.1565, -0.    , -0.1571, ..., -0.05  ,  0.    ,  0.    ],
+          [ 0.    , -0.4323,  0.    , ..., -0.    ,  0.1676,  0.    ],
+          [ 0.    , -0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    , -0.3233,  0.    , ...,  0.    ,  0.3233, -0.    ],
+          [ 0.9463,  0.    ,  0.8085, ...,  0.    ,  0.    , -0.9093],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.    ]],
+
+         [[ 0.    , -0.484 ,  0.    , ..., -0.    ,  0.1086,  0.    ],
+          [-0.    ,  0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
+          [ 0.    ,  0.    , -0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 1.    ,  0.    ,  0.9553, ..., -0.3233,  0.    , -0.995 ],
+          [-0.    , -0.    , -0.    , ..., -0.    ,  0.    ,  0.    ],
+          [-0.    , -0.    , -0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[-0.4816, -0.    , -0.4835, ..., -0.1539,  0.    ,  0.    ],
+          [ 0.    ,  0.0309,  0.    , ...,  0.    ,  0.2104,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    , -0.995 ,  0.    , ...,  0.    ,  0.995 ,  0.    ],
+          [ 0.0998,  0.    , -0.1987, ...,  0.9093, -0.    ,  0.    ],
+          [ 0.    ,  0.    , -0.    , ...,  0.    , -0.    ,  0.    ]]],
+
+
+        [[[ 0.    ,  0.    ,  0.    , ...,  0.    , -0.    ,  0.    ],
+          [-0.    , -0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
+          [-0.    , -0.0796, -0.    , ..., -0.    , -0.2006,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ..., -0.    , -0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.    , -0.    ],
+          [ 0.    , -0.    ,  0.2955, ..., -0.9463,  0.    , -0.0998]],
+
+         [[-0.    ,  0.    ,  0.    , ...,  0.    , -0.    ,  0.    ],
+          [ 0.    , -0.0796, -0.    , ..., -0.    , -0.2006,  0.    ],
+          [-0.0796, -0.    ,  0.    , ...,  0.    , -0.    ,  0.    ],
+          [-0.    ,  0.    , -0.    , ...,  0.    , -0.    ,  0.    ],
+          [-0.    ,  0.    ,  0.2955, ..., -0.9463,  0.    , -0.0998],
+          [ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.    , -0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [-0.143 , -0.    , -0.1436, ..., -0.0457, -0.    ,  0.    ],
+          [-0.    ,  0.    , -0.    , ..., -0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    , -0.    ,  0.    ],
+          [-0.    , -0.2955,  0.    , ..., -0.    ,  0.2955,  0.    ],
+          [-0.2955, -0.    ,  0.    , ..., -0.    ,  0.    ,  0.    ]],
+
+         ...,
+
+         [[ 0.    ,  0.    , -0.    , ..., -0.    ,  0.    ,  0.    ],
+          [ 0.4581,  0.    ,  0.4599, ...,  0.1464,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    , -0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.9463,  0.    , ...,  0.    , -0.9463, -0.    ],
+          [ 0.9463,  0.    ,  0.    , ...,  0.    , -0.    ,  0.    ]],
+
+         [[ 0.    , -0.    ,  0.    , ..., -0.    ,  0.    ,  0.    ],
+          [-0.    ,  0.0796,  0.    , ...,  0.    ,  0.2006,  0.    ],
+          [ 0.0796,  0.    , -0.    , ..., -0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.    , -0.    ],
+          [-0.    , -0.    , -0.2955, ...,  0.9463,  0.    ,  0.0998],
+          [-0.    , -0.    , -0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[-0.    ,  0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
+          [ 0.0483,  0.    ,  0.0485, ...,  0.0154,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    , -0.    ,  0.    ],
+          [ 0.    , -0.    ,  0.    , ..., -0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.0998, -0.    , ...,  0.    , -0.0998,  0.    ],
+          [ 0.0998,  0.    ,  0.    , ...,  0.    , -0.    ,  0.    ]]],
+
+
+        [[[ 0.4624,  0.    ,  0.4643, ...,  0.1478, -0.    ,  0.    ],
+          [-0.    , -0.0761, -0.    , ..., -0.    , -0.1916,  0.    ],
+          [-0.143 , -0.    , -0.1436, ..., -0.0457, -0.    ,  0.    ],
+          [ 0.    ,  0.9553,  0.    , ..., -0.    , -0.9553, -0.    ],
+          [ 0.    ,  0.    ,  0.2823, ..., -0.904 ,  0.    , -0.0954],
+          [ 0.    , -0.2955,  0.    , ...,  0.    ,  0.2955,  0.    ]],
+
+         [[ 0.    ,  0.4624,  0.    , ...,  0.    , -0.1037,  0.    ],
+          [-0.    , -0.    , -0.1436, ..., -0.0457, -0.    ,  0.    ],
+          [ 0.    ,  0.0091,  0.    , ...,  0.    ,  0.1916,  0.    ],
+          [-0.9553,  0.    , -0.9127, ...,  0.3089, -0.    ,  0.9506],
+          [-0.    ,  0.    ,  0.    , ..., -0.    ,  0.2955,  0.    ],
+          [ 0.2955,  0.    , -0.2823, ...,  0.904 , -0.    ,  0.0954]],
+
+         [[ 0.4418,  0.    ,  0.486 , ...,  0.1547, -0.    ,  0.    ],
+          [-0.    ,  0.064 , -0.    , ..., -0.    , -0.2137,  0.    ],
+          [-0.007 ,  0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
+          [-0.    ,  0.9127,  0.    , ..., -0.    , -1.    , -0.    ],
+          [-0.2823, -0.    ,  0.    , ..., -0.7724,  0.    ,  0.1898],
+          [-0.    ,  0.2823,  0.    , ..., -0.    ,  0.    ,  0.    ]],
+
+         ...,
+
+         [[-0.1495, -0.    , -0.286 , ..., -0.091 ,  0.    ,  0.    ],
+          [ 0.    , -0.413 ,  0.    , ..., -0.    ,  0.1601,  0.    ],
+          [ 0.0223, -0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    , -0.3089,  0.    , ..., -0.    ,  0.5885, -0.    ],
+          [ 0.904 ,  0.    ,  0.7724, ...,  0.    ,  0.    , -0.8687],
+          [-0.    , -0.904 ,  0.    , ...,  0.    , -0.    , -0.    ]],
+
+         [[ 0.    , -0.486 ,  0.    , ..., -0.    ,  0.0444,  0.    ],
+          [-0.143 ,  0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
+          [-0.    ,  0.067 ,  0.    , ...,  0.    , -0.    ,  0.    ],
+          [ 0.9553,  0.    ,  1.    , ..., -0.5885,  0.    , -0.9801],
+          [-0.    , -0.2955, -0.    , ..., -0.    ,  0.    ,  0.    ],
+          [-0.2955,  0.    , -0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[-0.4601, -0.    , -0.4763, ..., -0.1516,  0.    ,  0.    ],
+          [ 0.    ,  0.0295,  0.    , ...,  0.    ,  0.201 ,  0.    ],
+          [ 0.0023,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    , -0.9506,  0.    , ...,  0.    ,  0.9801,  0.    ],
+          [ 0.0954, -0.    , -0.1898, ...,  0.8687, -0.    ,  0.    ],
+          [-0.    , -0.0954, -0.    , ...,  0.    , -0.    ,  0.    ]]],
+
+
+        ...,
+
+
+        [[[-0.1565, -0.    , -0.1571, ..., -0.05  ,  0.    ,  0.    ],
+          [ 0.    ,  0.0257,  0.    , ...,  0.    ,  0.0648,  0.    ],
+          [ 0.4581,  0.    ,  0.4599, ...,  0.1464,  0.    ,  0.    ],
+          [ 0.    , -0.3233, -0.    , ...,  0.    ,  0.3233,  0.    ],
+          [ 0.    , -0.    , -0.0955, ...,  0.3059, -0.    ,  0.0323],
+          [ 0.    ,  0.9463, -0.    , ...,  0.    , -0.9463, -0.    ]],
+
+         [[-0.    , -0.1565, -0.    , ..., -0.    ,  0.0351,  0.    ],
+          [ 0.    , -0.    , -0.0457, ...,  0.1464,  0.    ,  0.    ],
+          [-0.    ,  0.4066, -0.    , ...,  0.    , -0.0648,  0.    ],
+          [ 0.3233,  0.    ,  0.3089, ..., -0.1045,  0.    , -0.3217],
+          [ 0.    ,  0.    , -0.    , ...,  0.    , -0.9463, -0.    ],
+          [-0.9463,  0.    , -0.713 , ..., -0.3059, -0.    , -0.0323]],
+
+         [[-0.1495, -0.    , -0.1366, ..., -0.091 , -0.    ,  0.    ],
+          [ 0.    , -0.0217,  0.    , ...,  0.    ,  0.0723,  0.    ],
+          [ 0.2994,  0.    ,  0.3028, ...,  0.1251,  0.    ,  0.    ],
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+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]]],
+
+
+        [[[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         ...,
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]]],
+
+
+        [[[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         ...,
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]]]]])
+
+
+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+ +
+
+pay(W, q=None, J=None, frame=1)
+

Generalised joint force/torque due to a payload wrench

+

tau = pay(W, J) Returns the generalised joint force/torques due to a +payload wrench W applied to the end-effector. Where the manipulator +Jacobian is J (6xn), and n is the number of robot joints.

+

tau = pay(W, q, frame) as above but the Jacobian is calculated at pose +q in the frame given by frame which is 0 for base frame, 1 for +end-effector frame.

+

Uses the formula tau = J’W, where W is a wrench vector applied at the +end effector, W = [Fx Fy Fz Mx My Mz]’.

+
+
Trajectory operation:

In the case q is nxm or J is 6xnxm then tau is nxm where each row +is the generalised force/torque at the pose given by corresponding +row of q.

+
+
+
+
Parameters:
+
    +
  • W (Union[ndarray, List[float], Tuple[float], Set[float]]) – A wrench vector applied at the end effector, +W = [Fx Fy Fz Mx My Mz]

  • +
  • q (Optional[ndarray]) – Joint coordinates

  • +
  • J (Optional[ndarray]) – The manipulator Jacobian (Optional, if not supplied will +use the q value).

  • +
  • frame (int) – The frame in which to torques are expressed in when J +is not supplied. 0 means base frame of the robot, 1 means end- +effector frame

  • +
+
+
Returns:
+

Joint forces/torques due to w

+
+
Return type:
+

t

+
+
+

Notes

+
    +
  • +
    Wrench vector and Jacobian must be from the same reference

    frame.

    +
    +
    +
  • +
  • Tool transforms are taken into consideration when frame=1.

  • +
  • +
    Must have a constant wrench - no trajectory support for this

    yet.

    +
    +
    +
  • +
+
+ +
+
+paycap(w, tauR, frame=1, q=None)
+

Static payload capacity of a robot

+

wmax, joint = paycap(q, w, f, tauR) returns the maximum permissible +payload wrench wmax (6) applied at the end-effector, and the index +of the joint (zero indexed) which hits its force/torque limit at that +wrench. q (n) is the manipulator pose, w the payload wrench +(6), f the wrench reference frame and tauR (nx2) is a matrix of +joint forces/torques (first col is maximum, second col minimum).

+

Trajectory operation:

+

In the case q is nxm then wmax is Mx6 and J is Mx1 where the rows are +the results at the pose given by corresponding row of q.

+
+
Parameters:
+
    +
  • w (ndarray) – The payload wrench

  • +
  • tauR (ndarray) – Joint torque matrix minimum and maximums

  • +
  • frame (int) – The frame in which to torques are expressed in when J +is not supplied. ‘base’ means base frame of the robot, ‘ee’ means +end-effector frame

  • +
  • q (Union[ndarray, List[float], Tuple[float], Set[float], None]) – Joint coordinates

  • +
+
+
Returns:
+

The maximum permissible payload wrench

+
+
Return type:
+

ndarray(6)

+
+
+

Notes

+
    +
  • Wrench vector and Jacobian must be from the same reference frame

  • +
  • Tool transforms are taken into consideration for frame=1.

  • +
+
+ +
+
+payload(m, p=array([0.0, 0.0, 0.0]))
+

Add a payload to the end-effector

+

payload(m, p) adds payload mass adds a payload with point mass m at +position p in the end-effector coordinate frame.

+

payload(m) adds payload mass adds a payload with point mass m at +in the end-effector coordinate frame.

+

payload(0) removes added payload.

+
+
Parameters:
+
    +
  • m (float) – mass (kg)

  • +
  • p – position in end-effector frame

  • +
+
+
+
+ +
+
+perturb(p=0.1)
+

Perturb robot parameters

+

rp = perturb(p) is a new robot object in which the dynamic parameters +(link mass and inertia) have been perturbed. The perturbation is +multiplicative so that values are multiplied by random numbers in the +interval (1-p) to (1+p). The name string of the perturbed robot is +prefixed by ‘P/’.

+

Useful for investigating the robustness of various model-based control +schemes. For example to vary parameters in the range +/- 10 percent +is: r2 = puma.perturb(0.1)

+
+
Parameters:
+

p – The percent (+/-) to be perturbed. Default 10%

+
+
Returns:
+

A copy of the robot with dynamic parameters perturbed

+
+
Return type:
+

DHRobot

+
+
+
+ +
+
+plot(q, backend=None, block=False, dt=0.05, limits=None, vellipse=False, fellipse=False, fig=None, movie=None, loop=False, **kwargs)
+

Graphical display and animation

+

robot.plot(q, 'pyplot') displays a graphical view of a robot +based on the kinematic model and the joint configuration q. +This is a stick figure polyline which joins the origins of the +link coordinate frames. The plot will autoscale with an aspect +ratio of 1.

+

If q (m,n) representing a joint-space trajectory it will create an +animation with a pause of dt seconds between each frame.

+
+
+q
+

The joint configuration of the robot.

+
+ +
+
+backend
+

The graphical backend to use, currently ‘swift’ +and ‘pyplot’ are implemented. Defaults to ‘swift’ of a Robot +and ‘pyplot` for a DHRobot

+
+ +
+
+block
+

Block operation of the code and keep the figure open

+
+ +
+
+dt
+

if q is a trajectory, this describes the delay in +seconds between frames

+
+ +
+
+limits
+

Custom view limits for the plot. If not supplied will +autoscale, [x1, x2, y1, y2, z1, z2] +(this option is for ‘pyplot’ only)

+
+ +
+
+vellipse
+

(Plot Option) Plot the velocity ellipse at the +end-effector (this option is for ‘pyplot’ only)

+
+ +
+
+fellipse
+

(Plot Option) Plot the force ellipse at the +end-effector (this option is for ‘pyplot’ only)

+
+ +
+
+fig
+

(Plot Option) The figure label to plot in (this option is for +‘pyplot’ only)

+
+ +
+
+movie
+

(Plot Option) The filename to save the movie to (this option is for +‘pyplot’ only)

+
+ +
+
+loop
+

(Plot Option) Loop the movie (this option is for +‘pyplot’ only)

+
+ +
+
+jointaxes
+

(Plot Option) Plot an arrow indicating the axes in +which the joint revolves around(revolute joint) or translates +along (prosmatic joint) (this option is for ‘pyplot’ only)

+
+ +
+
+eeframe
+

(Plot Option) Plot the end-effector coordinate frame +at the location of the end-effector. Uses three arrows, red, +green and blue to indicate the x, y, and z-axes. +(this option is for ‘pyplot’ only)

+
+ +
+
+shadow
+

(Plot Option) Plot a shadow of the robot in the x-y +plane. (this option is for ‘pyplot’ only)

+
+ +
+
+name
+

(Plot Option) Plot the name of the robot near its base +(this option is for ‘pyplot’ only)

+
+ +
+
Returns:
+

A reference to the environment object which controls the +figure

+
+
Return type:
+

env

+
+
+

Notes

+
    +
  • +
    By default this method will block until the figure is dismissed.

    To avoid this set block=False.

    +
    +
    +
  • +
  • +
    For PyPlot, the polyline joins the origins of the link frames,

    but for some Denavit-Hartenberg models those frames may not +actually be on the robot, ie. the lines to not neccessarily +represent the links of the robot.

    +
    +
    +
  • +
+
+

See also

+

teach()

+
+
+ +
+
+plot_ellipse(ellipse, block=True, limits=None, jointaxes=True, eeframe=True, shadow=True, name=True)
+

Plot the an ellipsoid

+

robot.plot_ellipse(ellipsoid) displays the ellipsoid.

+
+
+ellipse
+

the ellipsoid to plot

+
+ +
+
+block
+

Block operation of the code and keep the figure open

+
+ +
+
+limits
+

Custom view limits for the plot. If not supplied will +autoscale, [x1, x2, y1, y2, z1, z2]

+
+ +
+
+jointaxes
+

(Plot Option) Plot an arrow indicating the axes in +which the joint revolves around(revolute joint) or translates +along (prosmatic joint)

+
+ +
+
+eeframe
+

(Plot Option) Plot the end-effector coordinate frame +at the location of the end-effector. Uses three arrows, red, +green and blue to indicate the x, y, and z-axes.

+
+ +
+
+shadow
+

(Plot Option) Plot a shadow of the robot in the x-y +plane

+
+ +
+
+name
+

(Plot Option) Plot the name of the robot near its base

+
+ +
+
Returns:
+

A reference to the PyPlot object which controls the +matplotlib figure

+
+
Return type:
+

env

+
+
+
+ +
+
+plot_fellipse(q, block=True, fellipse=None, limits=None, opt='trans', centre=[0, 0, 0], jointaxes=True, eeframe=True, shadow=True, name=True)
+

Plot the force ellipsoid for manipulator

+

robot.plot_fellipse(q) displays the velocity ellipsoid for the +robot at pose q. The plot will autoscale with an aspect ratio +of 1.

+

robot.plot_fellipse(vellipse) specifies a custon ellipse to plot.

+
+
+q
+

The joint configuration of the robot

+
+ +
+
+block
+

Block operation of the code and keep the figure open

+
+ +
+
+fellipse
+

the vellocity ellipsoid to plot

+
+ +
+
+limits
+

Custom view limits for the plot. If not supplied will +autoscale, [x1, x2, y1, y2, z1, z2]

+
+ +
+
+opt
+

‘trans’ or ‘rot’ will plot either the translational or +rotational force ellipsoid

+
+ +
+
+centre
+

The coordinates to plot the fellipse [x, y, z] or “ee” +to plot at the end-effector location

+
+ +
+
+jointaxes
+

(Plot Option) Plot an arrow indicating the axes in +which the joint revolves around(revolute joint) or translates +along (prosmatic joint)

+
+ +
+
+eeframe
+

(Plot Option) Plot the end-effector coordinate frame +at the location of the end-effector. Uses three arrows, red, +green and blue to indicate the x, y, and z-axes.

+
+ +
+
+shadow
+

(Plot Option) Plot a shadow of the robot in the x-y +plane

+
+ +
+
+name
+

(Plot Option) Plot the name of the robot near its base

+
+ +
+
Raises:
+

ValueError – q or fellipse must be supplied

+
+
Returns:
+

A reference to the PyPlot object which controls the +matplotlib figure

+
+
Return type:
+

env

+
+
+

Notes

+
    +
  • +
    By default the ellipsoid related to translational motion is

    drawn. Use opt='rot' to draw the rotational velocity +ellipsoid.

    +
    +
    +
  • +
  • +
    By default the ellipsoid is drawn at the origin. The option

    centre allows its origin to set to set to the specified +3-vector, or the string “ee” ensures it is drawn at the +end-effector position.

    +
    +
    +
  • +
+
+ +
+
+plot_vellipse(q, block=True, vellipse=None, limits=None, opt='trans', centre=[0, 0, 0], jointaxes=True, eeframe=True, shadow=True, name=True)
+

Plot the velocity ellipsoid for manipulator

+

robot.plot_vellipse(q) displays the velocity ellipsoid for the +robot at pose q. The plot will autoscale with an aspect ratio +of 1.

+

robot.plot_vellipse(vellipse) specifies a custon ellipse to plot.

+
+
block

Block operation of the code and keep the figure open

+
+
q

The joint configuration of the robot

+
+
vellipse

the vellocity ellipsoid to plot

+
+
limits

Custom view limits for the plot. If not supplied will +autoscale, [x1, x2, y1, y2, z1, z2]

+
+
opt

‘trans’ or ‘rot’ will plot either the translational or +rotational velocity ellipsoid

+
+
centre

The coordinates to plot the vellipse [x, y, z] or “ee” +to plot at the end-effector location

+
+
jointaxes

(Plot Option) Plot an arrow indicating the axes in +which the joint revolves around(revolute joint) or translates +along (prosmatic joint)

+
+
eeframe

(Plot Option) Plot the end-effector coordinate frame +at the location of the end-effector. Uses three arrows, red, +green and blue to indicate the x, y, and z-axes.

+
+
shadow

(Plot Option) Plot a shadow of the robot in the x-y +plane

+
+
name

(Plot Option) Plot the name of the robot near its base

+
+
+
+
Raises:
+

ValueError – q or fellipse must be supplied

+
+
Returns:
+

A reference to the PyPlot object which controls the +matplotlib figure

+
+
Return type:
+

env

+
+
+

Notes

+
    +
  • +
    By default the ellipsoid related to translational motion is

    drawn. Use opt='rot' to draw the rotational velocity +ellipsoid.

    +
    +
    +
  • +
  • +
    By default the ellipsoid is drawn at the origin. The option

    centre allows its origin to set to set to the specified +3-vector, or the string “ee” ensures it is drawn at the +end-effector position.

    +
    +
    +
  • +
+
+ +
+
+property prismaticjoints: List[bool]
+

Revolute joints as bool array

+
+
Returns:
+

array of joint type, True if prismatic

+
+
Return type:
+

prismaticjoints

+
+
+

Examples

+

+
+
+

Notes

+

Fixed joints, that maintain a constant link relative pose, +are not included.

+
+

See also

+

Link.isprismatic(), revolutejoints()

+
+
+ +
+
+property q: ndarray
+

Get/set robot joint configuration

+
    +
  • robot.q is the robot joint configuration

  • +
  • robot.q = ... checks and sets the joint configuration

  • +
+
+
Parameters:
+

q – the new robot joint configuration

+
+
Returns:
+

robot joint configuration

+
+
Return type:
+

q

+
+
+
+ +
+
+property qd: ndarray
+

Get/set robot joint velocity

+
    +
  • robot.qd is the robot joint velocity

  • +
  • robot.qd = ... checks and sets the joint velocity

  • +
+
+
Returns:
+

robot joint velocity

+
+
Return type:
+

qd

+
+
+
+ +
+
+property qdd: ndarray
+

Get/set robot joint acceleration

+
    +
  • robot.qdd is the robot joint acceleration

  • +
  • robot.qdd = ... checks and sets the robot joint acceleration

  • +
+
+
Returns:
+

robot joint acceleration

+
+
Return type:
+

qdd

+
+
+
+ +
+
+property qlim: ndarray
+

Joint limits

+

Limits are extracted from the link objects. If joints limits are +not set for:

+
    +
  • a revolute joint [-𝜋. 𝜋] is returned

  • +
  • a prismatic joint an exception is raised

  • +
+
+
+qlim
+

An array of joints limits (2, n)

+
+ +
+
Raises:
+

ValueError – unset limits for a prismatic joint

+
+
Returns:
+

Array of joint limit values

+
+
Return type:
+

qlim

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.qlim
+array([[-2.7925, -1.9199, -2.3562, -4.6426, -1.7453, -4.6426],
+       [ 2.7925,  1.9199,  2.3562,  4.6426,  1.7453,  4.6426]])
+
+
+
+ +
+
+random_q()
+

Return a random joint configuration

+

The value for each joint is uniform randomly distributed between the +limits set for the robot.

+
+

Note

+

The joint limit for all joints must be set.

+
+
+
Returns:
+

Random joint configuration :rtype: ndarray(n)

+
+
Return type:
+

q

+
+
+
+

See also

+

Robot.qlim(), Link.qlim()

+
+
+ +
+
+property reach: float
+

Reach of the robot

+

A conservative estimate of the reach of the robot. It is computed as +the sum of the translational ETs that define the link transform.

+
+

Note

+

Computed on the first access. If kinematic parameters +subsequently change this will not be reflected.

+
+
+
Returns:
+

Maximum reach of the robot

+
+
Return type:
+

reach

+
+
+

Notes

+
    +
  • Probably an overestimate of reach

  • +
  • Used by numerical inverse kinematics to scale translational +error.

  • +
  • For a prismatic joint, uses qlim if it is set

  • +
+
+ +
+
+property revolutejoints: List[bool]
+

Revolute joints as bool array

+
+
Returns:
+

array of joint type, True if revolute

+
+
Return type:
+

revolutejoints

+
+
+

Examples

+

+
+
+

Notes

+

Fixed joints, that maintain a constant link relative pose, +are not included.

+
+

See also

+

Link.isrevolute(), prismaticjoints()

+
+
+ +
+
+rne(q, qd, qdd, symbolic=False, gravity=None)
+

Compute inverse dynamics via recursive Newton-Euler formulation

+

rne_dh(q, qd, qdd) where the arguments have shape (n,) where n is +the number of robot joints. The result has shape (n,).

+

rne_dh(q, qd, qdd) where the arguments have shape (m,n) where n +is the number of robot joints and where m is the number of steps in +the joint trajectory. The result has shape (m,n).

+

rne_dh(p) where the input is a 1D array p = [q, qd, qdd] with +shape (3n,), and the result has shape (n,).

+

rne_dh(p) where the input is a 2D array p = [q, qd, qdd] with +shape (m,3n) and the result has shape (m,n).

+
+
Parameters:
+
+
+
Returns:
+

Joint force/torques

+
+
Return type:
+

tau

+
+
+

Notes

+
    +
  • This version supports symbolic model parameters

  • +
  • Verified against MATLAB code

  • +
+
+ +
+
+property scene_children: List[SceneNode]
+

Returns the child nodes of this object

+
+ +
+
+property scene_parent: Type[SceneNode]
+

Returns the parent node of this object

+
+ +
+
+segments()
+

Segments of branched robot

+

For a single-chain robot with structure:

+
L1 - L2 - L3
+
+
+

the return is [[None, L1, L2, L3]]

+

For a robot with structure:

+
L1 - L2 +-  L3 - L4
+        +- L5 - L6
+
+
+

the return is [[None, L1, L2], [L2, L3, L4], [L2, L5, L6]]

+
+
Returns:
+

Segment list

+
+
Return type:
+

segments

+
+
+

Notes

+
    +
  • the length of the list is the number of segments in the robot

  • +
  • +
    the first segment always starts with None which represents

    the base transform (since there is no base link)

    +
    +
    +
  • +
  • +
    the last link of one segment is also the first link of subsequent

    segments

    +
    +
    +
  • +
+
+ +
+
+showgraph(display_graph=True, **kwargs)
+

Display a link transform graph in browser

+

robot.showgraph() displays a graph of the robot’s link frames +and the ETS between them. It uses GraphViz dot.

+
+
The nodes are:
    +
  • Base is shown as a grey square. This is the world frame origin, +but can be changed using the base attribute of the robot.

  • +
  • Link frames are indicated by circles

  • +
  • ETS transforms are indicated by rounded boxes

  • +
+
+
The edges are:
    +
  • an arrow if jtype is False or the joint is fixed

  • +
  • an arrow with a round head if jtype is True and the joint is +revolute

  • +
  • an arrow with a box head if jtype is True and the joint is +prismatic

  • +
+
+
+

Edge labels or nodes in blue have a fixed transformation to the +preceding link.

+
+
Parameters:
+
    +
  • display_graph (bool) – Open the graph in a browser if True. Otherwise will return the +file path

  • +
  • etsbox – Put the link ETS in a box, otherwise an edge label

  • +
  • jtype – Arrowhead to node indicates revolute or prismatic type

  • +
  • static – Show static joints in blue and bold

  • +
+
+
Return type:
+

Optional[str]

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.URDF.Panda()
+>>> panda.showgraph()
+
+
+_images/panda-graph.svg +
+

See also

+

dotfile()

+
+
+ +
+
+property structure: str
+

Return the joint structure string

+

A string with one letter per joint: R for a revolute +joint, and P for a prismatic joint.

+
+
Returns:
+

joint configuration string

+
+
Return type:
+

structure

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> puma.structure
+'RRRRRR'
+>>> stanford = rtb.models.DH.Stanford()
+>>> stanford.structure
+'RRPRRR'
+
+
+

Notes

+

Fixed joints, that maintain a constant link relative pose, +are not included. +len(self.structure) == self.n.

+
+ +
+
+property symbolic: bool
+
+ +
+
+teach(q, block=True, limits=None, vellipse=False, fellipse=False, backend=None)
+

Graphical teach pendant

+

robot.teach(q) creates a matplotlib plot which allows the user to +“drive” a graphical robot using a graphical slider panel. The robot’s +inital joint configuration is q. The plot will autoscale with an +aspect ratio of 1.

+

robot.teach() as above except the robot’s stored value of q +is used.

+
+
q

The joint configuration of the robot (Optional, +if not supplied will use the stored q values).

+
+
block

Block operation of the code and keep the figure open

+
+
limits

Custom view limits for the plot. If not supplied will +autoscale, [x1, x2, y1, y2, z1, z2]

+
+
vellipse

(Plot Option) Plot the velocity ellipse at the +end-effector (this option is for ‘pyplot’ only)

+
+
fellipse

(Plot Option) Plot the force ellipse at the +end-effector (this option is for ‘pyplot’ only)

+
+
+
+
Returns:
+

A reference to the PyPlot object which controls the +matplotlib figure

+
+
Return type:
+

env

+
+
+

Notes

+
    +
  • +
    Program execution is blocked until the teach window is

    dismissed. If block=False the method is non-blocking but +you need to poll the window manager to ensure that the window +remains responsive.

    +
    +
    +
  • +
  • +
    The slider limits are derived from the joint limit properties.

    If not set then: +- For revolute joints they are assumed to be [-pi, +pi] +- For prismatic joint they are assumed unknown and an error

    +
    +

    occurs.

    +
    +
    +
    +
  • +
+
+ +
+
+todegrees(q)
+

Convert joint angles to degrees

+
+
Parameters:
+

q – The joint configuration of the robot

+
+
Return type:
+

ndarray

+
+
Returns:
+

    +
  • q – a vector of joint coordinates in degrees and metres

  • +
  • robot.todegrees(q) converts joint coordinates q to degrees

  • +
  • taking into account whether elements of q correspond to revolute

  • +
  • or prismatic joints, ie. prismatic joint values are not converted.

  • +
  • If q is a matrix, with one column per joint, the conversion is

  • +
  • performed columnwise.

  • +
+

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> from math import pi
+>>> stanford = rtb.models.DH.Stanford()
+>>> stanford.todegrees([pi/4, pi/8, 2, -pi/4, pi/6, pi/3])
+array([ 45. ,  22.5,   2. , -45. ,  30. ,  60. ])
+
+
+
+ +
+
+property tool: SE3
+

Get/set robot tool transform

+
    +
  • robot.tool is the robot tool transform as an SE3 object

  • +
  • robot._tool is the robot tool transform as a numpy array

  • +
  • robot.tool = ... checks and sets the robot tool transform

  • +
+
+
Parameters:
+

tool – the new robot tool transform (as an SE(3))

+
+
Returns:
+

robot tool transform

+
+
Return type:
+

tool

+
+
+
+ +
+
+toradians(q)
+

Convert joint angles to radians

+

robot.toradians(q) converts joint coordinates q to radians +taking into account whether elements of q correspond to revolute +or prismatic joints, ie. prismatic joint values are not converted.

+

If q is a matrix, with one column per joint, the conversion is +performed columnwise.

+
+
Parameters:
+

q – The joint configuration of the robot

+
+
Returns:
+

a vector of joint coordinates in radians and metres

+
+
Return type:
+

q

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> stanford = rtb.models.DH.Stanford()
+>>> stanford.toradians([10, 20, 2, 30, 40, 50])
+array([0.1745, 0.3491, 2.    , 0.5236, 0.6981, 0.8727])
+
+
+
+ +
+
+property urdf_filepath: str
+
+ +
+
+property urdf_string: str
+
+ +
+
+vellipse(q, opt='trans', unit='rad', centre=[0, 0, 0], scale=0.1)
+

Create a velocity ellipsoid object for plotting with PyPlot

+

robot.vellipse(q) creates a force ellipsoid for the robot at +pose q. The ellipsoid is centered at the origin.

+
+
+q
+

The joint configuration of the robot.

+
+ +
+
+opt
+

‘trans’ or ‘rot’ will plot either the translational or +rotational force ellipsoid

+
+ +
+
+unit
+

‘rad’ or ‘deg’ will plot the ellipsoid in radians or +degrees

+
+ +
+
+centre
+

The centre of the ellipsoid, either ‘ee’ for the end-effector +or a 3-vector [x, y, z] in the world frame

+
+ +
+
+scale
+

The scale factor for the ellipsoid

+
+ +
+
Returns:
+

An EllipsePlot object

+
+
Return type:
+

env

+
+
+

Notes

+
    +
  • +
    By default the ellipsoid related to translational motion is

    drawn. Use opt='rot' to draw the rotational velocity +ellipsoid.

    +
    +
    +
  • +
  • +
    By default the ellipsoid is drawn at the origin. The option

    centre allows its origin to set to set to the specified +3-vector, or the string “ee” ensures it is drawn at the +end-effector position.

    +
    +
    +
  • +
+
+ +
+
+vision_collision_damper(shape, camera=None, camera_n=0, q=None, di=0.3, ds=0.05, xi=1.0, end=None, start=None, collision_list=None)
+

Compute a vision collision constrain for QP motion control

+

Formulates an inequality contraint which, when optimised for will +make it impossible for the robot to run into a line of sight. +See examples/fetch_vision.py for use case

+
+
+camera
+

The camera link, either as a robotic link or SE3 +pose

+
+ +
+
+camera_n
+

Degrees of freedom of the camera link

+
+ +
+
+ds
+

The minimum distance in which a joint is allowed to +approach the collision object shape

+
+ +
+
+di
+

The influence distance in which the velocity +damper becomes active

+
+ +
+
+xi
+

The gain for the velocity damper

+
+ +
+
+end
+

The end link of the robot to consider

+
+ +
+
+start
+

The start link of the robot to consider

+
+ +
+
+collision_list
+

A list of shapes to consider for collision

+
+ +
+
Returns:
+

    +
  • Ain – A (6,) vector inequality contraint for an optisator

  • +
  • Bin – b (6,) vector inequality contraint for an optisator

  • +
+

+
+
+
+ +
+ +
+
+class roboticstoolbox.robot.ERobot.ERobot2(*args, **kwargs)[source]
+

Bases: Robot2

+
+
+__getitem__(i)
+

Get link

+

This also supports iterating over each link in the robot object, +from the base to the tool.

+
+
Parameters:
+

i (Union[int, str]) – link number or name

+
+
Returns:
+

i’th link or named link

+
+
Return type:
+

link

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> print(robot[1]) # print the 2nd link
+RevoluteDH:   θ=q,  d=0,  a=0.4318,  ⍺=0.0
+>>> print([link.a for link in robot])  # print all the a_j values
+[0, 0.4318, 0.0203, 0, 0, 0]
+
+
+

Notes

+
+
Robot supports link lookup by name,

eg. robot['link1']

+
+
+
+ +
+
+__str__()
+

Pretty prints the ETS Model of the robot.

+
+
Returns:
+

Pretty print of the robot model

+
+
Return type:
+

str

+
+
+

Notes

+
    +
  • Constant links are shown in blue.

  • +
  • End-effector links are prefixed with an @

  • +
  • Angles in degrees

  • +
  • +
    The robot base frame is denoted as BASE and is equal to the

    robot’s base attribute.

    +
    +
    +
  • +
+
+ +
+
+accel(q, qd, torque, gravity=None)
+

Compute acceleration due to applied torque

+

qdd = accel(q, qd, torque) calculates a vector (n) of joint +accelerations that result from applying the actuator force/torque (n) +to the manipulator in state q (n) and qd (n), and n is +the number of robot joints.

+
+\[\ddot{q} = \mathbf{M}^{-1} \left(\tau - \mathbf{C}(q)\dot{q} - \mathbf{g}(q)\right)\]
+

Trajectory operation

+

If q, qd, torque are matrices (m,n) then qdd is a matrix (m,n) +where each row is the acceleration corresponding to the equivalent rows +of q, qd, torque.

+
+
Parameters:
+
    +
  • q – Joint coordinates

  • +
  • qd – Joint velocity

  • +
  • torque – Joint torques of the robot

  • +
  • gravity – Gravitational acceleration (Optional, if not supplied will +use the gravity attribute of self).

  • +
+
+
Return type:
+

ndarray(n)

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> puma.accel(puma.qz, 0.5 * np.ones(6), np.zeros(6))
+array([ -7.5544, -12.22  ,  -6.4022,  -5.4303,  -4.9518,  -2.1178])
+
+
+

Notes

+
    +
  • +
    Useful for simulation of manipulator dynamics, in

    conjunction with a numerical integration function.

    +
    +
    +
  • +
  • +
    Uses the method 1 of Walker and Orin to compute the forward

    dynamics.

    +
    +
    +
  • +
  • +
    Featherstone’s method is more efficient for robots with large

    numbers of joints.

    +
    +
    +
  • +
  • Joint friction is considered.

  • +
+

References

+
    +
  • +
    Efficient dynamic computer simulation of robotic mechanisms,

    M. W. Walker and D. E. Orin, +ASME Journa of Dynamic Systems, Measurement and Control, vol. +104, no. 3, pp. 205-211, 1982.

    +
    +
    +
  • +
+
+ +
+
+accel_x(q, xd, wrench, gravity=None, pinv=False, representation='rpy/xyz')
+

Operational space acceleration due to applied wrench

+

xdd = accel_x(q, qd, wrench) is the operational space acceleration +due to wrench applied to the end-effector of a robot in joint +configuration q and joint velocity qd.

+
+\[\ddot{x} = \mathbf{J}(q) \mathbf{M}(q)^{-1} \left( + \mathbf{J}(q)^T w - \mathbf{C}(q)\dot{q} - \mathbf{g}(q) + \right)\]
+

Trajectory operation

+

If q, qd, torque are matrices (m,n) then qdd is a matrix (m,n) +where each row is the acceleration corresponding to the equivalent rows +of q, qd, wrench.

+
+
Parameters:
+
    +
  • q – Joint coordinates

  • +
  • qd – Joint velocity

  • +
  • wrench – Wrench applied to the end-effector

  • +
  • gravity – Gravitational acceleration (Optional, if not supplied will +use the gravity attribute of self).

  • +
  • pinv – use pseudo inverse rather than inverse

  • +
  • analytical – the type of analytical Jacobian to use, default is +‘rpy/xyz’

  • +
  • xd

  • +
  • representation – (Default value = “rpy/xyz”)

  • +
+
+
Returns:
+

Operational space accelerations of the end-effector

+
+
Return type:
+

accel

+
+
+

Examples

+
    ainv = _umath_linalg.inv(a, signature=signature, extobj=extobj)
+  File "/opt/hostedtoolcache/Python/3.7.17/x64/lib/python3.7/site-packages/numpy/linalg/linalg.py", line 88, in _raise_linalgerror_singular
+    raise LinAlgError("Singular matrix")
+numpy.linalg.LinAlgError: Singular matrix
+
+
+

Notes

+
    +
  • +
    Useful for simulation of manipulator dynamics, in

    conjunction with a numerical integration function.

    +
    +
    +
  • +
  • +
    Uses the method 1 of Walker and Orin to compute the forward

    dynamics.

    +
    +
    +
  • +
  • +
    Featherstone’s method is more efficient for robots with large

    numbers of joints.

    +
    +
    +
  • +
  • Joint friction is considered.

  • +
+
+

See also

+

accel()

+
+
+ +
+
+addconfiguration(name, q)
+

Add a named joint configuration

+

Add a named configuration to the robot instance’s dictionary of named +configurations.

+
+
Parameters:
+
    +
  • name (str) – Name of the joint configuration

  • +
  • q (ndarray) – Joint configuration

  • +
+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.addconfiguration_attr("mypos", [0.1, 0.2, 0.3, 0.4, 0.5, 0.6])
+>>> robot.configs["mypos"]
+array([0.1, 0.2, 0.3, 0.4, 0.5, 0.6])
+
+
+
+

See also

+

addconfiguration()

+
+
+ +
+
+addconfiguration_attr(name, q, unit='rad')
+

Add a named joint configuration as an attribute

+
+
Parameters:
+
+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.addconfiguration_attr("mypos", [0.1, 0.2, 0.3, 0.4, 0.5, 0.6])
+>>> robot.mypos
+array([0.1, 0.2, 0.3, 0.4, 0.5, 0.6])
+>>> robot.configs["mypos"]
+array([0.1, 0.2, 0.3, 0.4, 0.5, 0.6])
+
+
+

Notes

+
    +
  • Used in robot model init method to store the qr configuration

  • +
  • +
    Dynamically adding attributes to objects can cause issues with

    Python type checking.

    +
    +
    +
  • +
  • +
    Configuration is also added to the robot instance’s dictionary of

    named configurations.

    +
    +
    +
  • +
+
+

See also

+

addconfiguration()

+
+
+ +
+
+attach(object)
+
+ +
+
+attach_to(object)
+
+ +
+
+property base: SE2
+

Get/set robot base transform (Robot superclass)

+

robot.base is the robot base transform

+
+
Returns:
+

    +
  • base – robot tool transform

  • +
    • +
    • robot.base = ... checks and sets the robot base transform

    • +
    +
  • +
+

+
+
+

Notes

+
    +
  • +
    The private attribute _base will be None in the case of

    no base transform, but this property will return SE3() which +is an identity matrix.

    +
    +
    +
  • +
+
+ +
+ +

Get the robot base link

+
    +
  • robot.base_link is the robot base link

  • +
+
+
Returns:
+

the first link in the robot tree

+
+
Return type:
+

base_link

+
+
+
+ +
+
+cinertia(q)
+

Deprecated, use inertia_x

+
+ +
+
+property comment: str
+

Get/set robot comment

+
    +
  • robot.comment is the robot comment

  • +
  • robot.comment = ... checks and sets the robot comment

  • +
+
+
Parameters:
+

name – the new robot comment

+
+
Returns:
+

robot comment

+
+
Return type:
+

comment

+
+
+
+ +
+
+property configs: Dict[str, ndarray]
+
+ +
+
+configurations_str(border='thin')
+
+ +
+
+property control_mode: str
+

Get/set robot control mode

+
    +
  • robot.control_type is the robot control mode

  • +
  • robot.control_type = ... checks and sets the robot control mode

  • +
+
+
Parameters:
+

control_mode – the new robot control mode

+
+
Returns:
+

the current robot control mode

+
+
Return type:
+

control_mode

+
+
+
+ +
+
+copy()
+
+ +
+
+coriolis(q, qd)
+

Coriolis and centripetal term

+

coriolis(q, qd) calculates the Coriolis/centripetal matrix (n,n) +for the robot in configuration q and velocity qd, where n +is the number of joints.

+

The product \(\mathbf{C} \dot{q}\) is the vector of joint +force/torque due to velocity coupling. The diagonal elements are due to +centripetal effects and the off-diagonal elements are due to Coriolis +effects. This matrix is also known as the velocity coupling matrix, +since it describes the disturbance forces on any joint due to +velocity of all other joints.

+

Trajectory operation

+

If q and qd are matrices (m,n), each row is interpretted as a +joint configuration, and the result (n,n,m) is a 3d-matrix where +each plane corresponds to a row of q and qd.

+
+
Parameters:
+
    +
  • q – Joint coordinates

  • +
  • qd – Joint velocity

  • +
+
+
Returns:
+

Velocity matrix

+
+
Return type:
+

coriolis

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> puma.coriolis(puma.qz, 0.5 * np.ones((6,)))
+array([[-0.4017, -0.5513, -0.2025, -0.0007, -0.0013,  0.    ],
+       [ 0.2023, -0.1937, -0.3868, -0.    , -0.002 ,  0.    ],
+       [ 0.1987,  0.193 , -0.    ,  0.    , -0.0001,  0.    ],
+       [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+       [ 0.0007,  0.0007,  0.0001,  0.    ,  0.    ,  0.    ],
+       [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ]])
+
+
+

Notes

+
    +
  • +
    Joint viscous friction is also a joint force proportional to

    velocity but it is eliminated in the computation of this value.

    +
    +
    +
  • +
  • Computationally slow, involves \(n^2/2\) invocations of RNE.

  • +
+
+ +
+
+coriolis_x(q, qd, pinv=False, representation='rpy/xyz', J=None, Ji=None, Jd=None, C=None, Mx=None)
+

Operational space Coriolis and centripetal term

+

coriolis_x(q, qd) is the Coriolis/centripetal matrix (m,m) +in operational space for the robot in configuration q and velocity +qd, where n is the number of joints.

+
+\[\mathbf{C}_x = \mathbf{J}(q)^{-T} \left( + \mathbf{C}(q) - \mathbf{M}_x(q) \mathbf{J})(q) + \right) \mathbf{J}(q)^{-1}\]
+

The product \(\mathbf{C} \dot{x}\) is the operational space wrench +due to joint velocity coupling. This matrix is also known as the +velocity coupling matrix, since it describes the disturbance forces on +any joint due to velocity of all other joints.

+

The transformation to operational space requires an analytical, rather +than geometric, Jacobian. analytical can be one of:

+
+
+ + + + + + + + + + + + + + + + + + + +

Value

Rotational representation

'rpy/xyz'

RPY angular rates in XYZ order (default)

'rpy/zyx'

RPY angular rates in XYZ order

'eul'

Euler angular rates in ZYZ order

'exp'

exponential coordinate rates

+
+

Trajectory operation

+

If q and qd are matrices (m,n), each row is interpretted as a +joint configuration, and the result (n,n,m) is a 3d-matrix where +each plane corresponds to a row of q and qd.

+
+
Parameters:
+
    +
  • q – Joint coordinates

  • +
  • qd – Joint velocity

  • +
  • pinv – use pseudo inverse rather than inverse (Default value = False)

  • +
  • analytical – the type of analytical Jacobian to use, default is +‘rpy/xyz’

  • +
  • representation – (Default value = “rpy/xyz”)

  • +
  • J

  • +
  • Ji

  • +
  • Jd

  • +
  • C

  • +
  • Mx

  • +
+
+
Returns:
+

Operational space velocity matrix

+
+
Return type:
+

ndarray(6,6) or ndarray(m,6,6)

+
+
+

Examples

+
    ainv = _umath_linalg.inv(a, signature=signature, extobj=extobj)
+  File "/opt/hostedtoolcache/Python/3.7.17/x64/lib/python3.7/site-packages/numpy/linalg/linalg.py", line 88, in _raise_linalgerror_singular
+    raise LinAlgError("Singular matrix")
+numpy.linalg.LinAlgError: Singular matrix
+
+
+

Notes

+
    +
  • +
    Joint viscous friction is also a joint force proportional to

    velocity but it is eliminated in the computation of this value.

    +
    +
    +
  • +
  • Computationally slow, involves \(n^2/2\) invocations of RNE.

  • +
  • If the robot is not 6 DOF the pinv option is set True.

  • +
  • pinv() is around 5x slower than inv()

  • +
+
+

Warning

+

Assumes that the operational space has 6 DOF.

+
+
+

See also

+

coriolis(), inertia_x(), hessian0()

+
+
+ +
+
+property default_backend
+

Get default graphical backend

+
    +
  • +
    robot.default_backend Get the default graphical backend, used when

    no explicit backend is passed to Robot.plot.

    +
    +
    +
  • +
  • +
    robot.default_backend = ... Set the default graphical backend, used when

    no explicit backend is passed to Robot.plot. The default set here will +be overridden if the particular Robot subclass cannot support it.

    +
    +
    +
  • +
+
+
Returns:
+

backend name

+
+
Return type:
+

default_backend

+
+
+
+ +
+ +

A link search method

+

Visit all links from start in depth-first order and will apply +func to each visited link

+
+
Parameters:
+
+
+
Returns:
+

A list of links

+
+
Return type:
+

links

+
+
+
+ +
+
+dotfile(filename, etsbox=False, ets='full', jtype=False, static=True)
+

Write a link transform graph as a GraphViz dot file

+
+
The file can be processed using dot:

% dot -Tpng -o out.png dotfile.dot

+
+
The nodes are:
    +
  • Base is shown as a grey square. This is the world frame origin, +but can be changed using the base attribute of the robot.

  • +
  • Link frames are indicated by circles

  • +
  • ETS transforms are indicated by rounded boxes

  • +
+
+
The edges are:
    +
  • an arrow if jtype is False or the joint is fixed

  • +
  • an arrow with a round head if jtype is True and the joint is +revolute

  • +
  • an arrow with a box head if jtype is True and the joint is +prismatic

  • +
+
+
+

Edge labels or nodes in blue have a fixed transformation to the +preceding link.

+
+

Note

+
+
If filename is a file object then the file will not

be closed after the GraphViz model is written.

+
+
+
+
+
Parameters:
+
    +
  • file – Name of file to write to

  • +
  • etsbox (bool) – Put the link ETS in a box, otherwise an edge label

  • +
  • ets (Literal['full', 'brief']) – Display the full ets with “full” or a brief version with “brief”

  • +
  • jtype (bool) – Arrowhead to node indicates revolute or prismatic type

  • +
  • static (bool) – Show static joints in blue and bold

  • +
+
+
+
+

See also

+

showgraph()

+
+
+ +
+
+dynamics()
+

Pretty print the dynamic parameters (Robot superclass)

+

The dynamic parameters (inertial and friction) are printed in a table, +with one row per link.

+

Examples

+

+
+
+
+ +
+
+dynamics_list()
+

Print dynamic parameters (Robot superclass)

+

Display the kinematic and dynamic parameters to the console in +reable format

+
+ +
+
+dynchanged(what=None)
+

Dynamic parameters have changed

+

Called from a property setter to inform the robot that the cache of +dynamic parameters is invalid.

+
+

See also

+

roboticstoolbox.Link._listen_dyn()

+
+
+ +
+ +
+ +
+
+ets(start=None, end=None)
+

Robot to ETS

+

robot.ets() is an ETS representing the kinematics from base to +end-effector.

+

robot.ets(end=link) is an ETS representing the kinematics from +base to the link link specified as a Link reference or a name.

+

robot.ets(start=l1, end=l2) is an ETS representing the kinematics +from link l1 to link l2.

+

:param : +:type : param start: start of path, defaults to base_link +:param : +:type : param end: end of path, defaults to end-effector

+
+
Raises:
+
+
+
Returns:
+

elementary transform sequence

+
+
Return type:
+

ets

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.ETS.Panda()
+>>> panda.ets()
+[ET.tz(eta=0.333), ET.Rz(jindex=0), ET.Rx(eta=-1.5707963267948966), ET.Rz(jindex=1), ET.Rx(eta=1.5707963267948966), ET.tz(eta=0.316), ET.Rz(jindex=2), ET.tx(eta=0.0825), ET.Rx(eta=1.5707963267948966), ET.Rz(jindex=3), ET.tx(eta=-0.0825), ET.Rx(eta=-1.5707963267948966), ET.tz(eta=0.384), ET.Rz(jindex=4), ET.Rx(eta=1.5707963267948966), ET.Rz(jindex=5), ET.tx(eta=0.088), ET.Rx(eta=1.5707963267948966), ET.tz(eta=0.107), ET.Rz(jindex=6), ET.tz(eta=0.10300000000000001), ET.Rz(eta=-0.7853981633974483)]
+
+
+
+ +
+
+fdyn(T, q0, Q=None, Q_args={}, qd0=None, solver='RK45', solver_args={}, dt=None, progress=False)
+

Integrate forward dynamics

+

tg = R.fdyn(T, q) integrates the dynamics of the robot with zero +input torques over the time interval 0 to T and returns the +trajectory as a namedtuple with elements:

+
    +
  • t the time vector (M,)

  • +
  • q the joint coordinates (M,n)

  • +
  • qd the joint velocities (M,n)

  • +
+

tg = R.fdyn(T, q, torqfun) as above but the torque applied to the +joints is given by the provided function:

+
tau = function(robot, t, q, qd, **args)
+
+
+

where the inputs are:

+
+
    +
  • the robot object

  • +
  • current time

  • +
  • current joint coordinates (n,)

  • +
  • current joint velocity (n,)

  • +
  • args, optional keyword arguments can be specified, these are

  • +
+

passed in from the targs kewyword argument.

+
+

The function must return a Numpy array (n,) of joint forces/torques.

+
+
Parameters:
+
+
+
Returns:
+

robot trajectory

+
+
Return type:
+

trajectory

+
+
+

Examples

+

To apply zero joint torque to the robot without Coulomb +friction:

+
>>> def myfunc(robot, t, q, qd):
+>>>     return np.zeros((robot.n,))
+
+
+
>>> tg = robot.nofriction().fdyn(5, q0, myfunc)
+
+
+
>>> plt.figure()
+>>> plt.plot(tg.t, tg.q)
+>>> plt.show()
+
+
+

We could also use a lambda function:

+
>>> tg = robot.nofriction().fdyn(
+>>>     5, q0, lambda r, t, q, qd: np.zeros((r.n,)))
+
+
+

The robot is controlled by a PD controller. We first define a +function to compute the control which has additional parameters for +the setpoint and control gains (qstar, P, D):

+
>>> def myfunc(robot, t, q, qd, qstar, P, D):
+>>>     return (qstar - q) * P + qd * D  # P, D are (6,)
+
+
+
>>> tg = robot.fdyn(10, q0, myfunc, torque_args=(qstar, P, D)) )
+
+
+

Many integrators have variable step length which is problematic if we +want to animate the result. If dt is specified then the solver +results are interpolated in time steps of dt.

+

Notes

+ +
+

See also

+

DHRobot.accel(), DHRobot.nofriction(), DHRobot.rne()

+
+
+ +
+
+fellipse(q, opt='trans', unit='rad', centre=[0, 0, 0])
+

Create a force ellipsoid object for plotting with PyPlot

+

robot.fellipse(q) creates a force ellipsoid for the robot at +pose q. The ellipsoid is centered at the origin.

+
+
+q
+

The joint configuration of the robot.

+
+ +
+
+opt
+

‘trans’ or ‘rot’ will plot either the translational or +rotational force ellipsoid

+
+ +
+
+unit
+

‘rad’ or ‘deg’ will plot the ellipsoid in radians or +degrees

+
+ +
+
+centre
+

The centre of the ellipsoid, either ‘ee’ for the end-effector +or a 3-vector [x, y, z] in the world frame

+
+ +
+
Returns:
+

An EllipsePlot object

+
+
Return type:
+

env

+
+
+

Notes

+
    +
  • +
    By default the ellipsoid related to translational motion is

    drawn. Use opt='rot' to draw the rotational velocity +ellipsoid.

    +
    +
    +
  • +
  • +
    By default the ellipsoid is drawn at the origin. The option

    centre allows its origin to set to set to the specified +3-vector, or the string “ee” ensures it is drawn at the +end-effector position.

    +
    +
    +
  • +
+
+ +
+
+fkine(q, end=None, start=None)
+
+ +
+
+fkine_all(q)
+

Compute the pose of every link frame

+

T = robot.fkine_all(q) is an SE3 instance with robot.nlinks + +1 values:

+
    +
  • T[0] is the base transform

  • +
  • T[i] is the pose of link whose number is i

  • +
+
+
Parameters:
+

q (Union[ndarray, List[float], Tuple[float], Set[float]]) – The joint configuration

+
+
Returns:
+

Pose of all links

+
+
Return type:
+

fkine_all

+
+
+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
+
+ +
+
+friction(qd)
+

Manipulator joint friction (Robot superclass)

+

robot.friction(qd) is a vector of joint friction +forces/torques for the robot moving with joint velocities qd.

+

The friction model includes:

+
    +
  • Viscous friction which is a linear function of velocity.

  • +
  • Coulomb friction which is proportional to sign(qd).

  • +
+
+\[\begin{split}\tau_j = G^2 B \dot{q}_j + |G_j| \left\{ \begin{array}{ll} + \tau_{C,j}^+ & \mbox{if $\dot{q}_j > 0$} \\ + \tau_{C,j}^- & \mbox{if $\dot{q}_j < 0$} \end{array} \right.\end{split}\]
+
+
Parameters:
+

qd (ndarray) – The joint velocities of the robot

+
+
Returns:
+

The joint friction forces/torques for the robot

+
+
Return type:
+

friction

+
+
+

Notes

+
    +
  • +
    The friction value should be added to the motor output torque to

    determine the nett torque. It has a negative value when qd > 0.

    +
    +
    +
  • +
  • +
    The returned friction value is referred to the output of the

    gearbox.

    +
    +
    +
  • +
  • +
    The friction parameters in the Link object are referred to the

    motor.

    +
    +
    +
  • +
  • Motor viscous friction is scaled up by \(G^2\).

  • +
  • Motor Coulomb friction is scaled up by math:G.

  • +
  • +
    The appropriate Coulomb friction value to use in the

    non-symmetric case depends on the sign of the joint velocity, +not the motor velocity.

    +
    +
    +
  • +
  • +
    Coulomb friction is zero for zero joint velocity, stiction is

    not modeled.

    +
    +
    +
  • +
  • +
    The absolute value of the gear ratio is used. Negative gear

    ratios are tricky: the Puma560 robot has negative gear ratio for +joints 1 and 3.

    +
    +
    +
  • +
  • +
    The absolute value of the gear ratio is used. Negative gear

    ratios are tricky: the Puma560 has negative gear ratio for +joints 1 and 3.

    +
    +
    +
  • +
+
+

See also

+

Robot.nofriction(), Link.friction()

+
+
+ +
+
+get_path(end=None, start=None)
+

Find a path from start to end

+
+
Parameters:
+
+
+
Raises:
+

ValueError – link not known or ambiguous

+
+
Return type:
+

Tuple[List[TypeVar(LinkType, bound= BaseLink)], int, SE3]

+
+
Returns:
+

    +
  • path – the path from start to end

  • +
  • n – the number of joints in the path

  • +
  • T – the tool transform present after end

  • +
+

+
+
+
+ +
+
+property gravity: ndarray
+

Get/set default gravitational acceleration (Robot superclass)

+
    +
  • robot.name is the default gravitational acceleration

  • +
  • +
    robot.name = ... checks and sets default gravitational

    acceleration

    +
    +
    +
  • +
+
+
Parameters:
+

gravity – the new gravitational acceleration for this robot

+
+
Returns:
+

gravitational acceleration

+
+
Return type:
+

gravity

+
+
+

Notes

+

If the z-axis is upward, out of the Earth, this should be +a positive number.

+
+ +
+
+gravload(q=None, gravity=None)
+

Compute gravity load

+

robot.gravload(q) calculates the joint gravity loading (n) for +the robot in the joint configuration q and using the default +gravitational acceleration specified in the DHRobot object.

+

robot.gravload(q, gravity=g) as above except the gravitational +acceleration is explicitly specified as g.

+

Trajectory operation

+

If q is a matrix (nxm) each column is interpreted as a joint +configuration vector, and the result is a matrix (nxm) each column +being the corresponding joint torques.

+
+
Parameters:
+
+
+
Returns:
+

The generalised joint force/torques due to gravity

+
+
Return type:
+

gravload

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> puma.gravload(puma.qz)
+array([ 0.    , 37.4837,  0.2489,  0.    ,  0.    ,  0.    ])
+
+
+
+ +
+
+gravload_x(q=None, gravity=None, pinv=False, representation='rpy/xyz', Ji=None)
+

Operational space gravity load

+

robot.gravload_x(q) calculates the gravity wrench for +the robot in the joint configuration q and using the default +gravitational acceleration specified in the robot object.

+

robot.gravload_x(q, gravity=g) as above except the gravitational +acceleration is explicitly specified as g.

+
+\[\mathbf{G}_x = \mathbf{J}(q)^{-T} \mathbf{G}(q)\]
+

The transformation to operational space requires an analytical, rather +than geometric, Jacobian. analytical can be one of:

+
+
+ + + + + + + + + + + + + + + + + + + +

Value

Rotational representation

'rpy/xyz'

RPY angular rates in XYZ order (default)

'rpy/zyx'

RPY angular rates in XYZ order

'eul'

Euler angular rates in ZYZ order

'exp'

exponential coordinate rates

+
+

Trajectory operation

+

If q is a matrix (nxm) each column is interpreted as a joint +configuration vector, and the result is a matrix (nxm) each column +being the corresponding joint torques.

+
+
Parameters:
+
    +
  • q – Joint coordinates

  • +
  • gravity – Gravitational acceleration (Optional, if not supplied will +use the gravity attribute of self).

  • +
  • pinv – use pseudo inverse rather than inverse (Default value = False)

  • +
  • analytical – the type of analytical Jacobian to use, default is +‘rpy/xyz’

  • +
  • representation – (Default value = “rpy/xyz”)

  • +
  • Ji

  • +
+
+
Returns:
+

The operational space gravity wrench

+
+
Return type:
+

gravload

+
+
+

Examples

+
    ainv = _umath_linalg.inv(a, signature=signature, extobj=extobj)
+  File "/opt/hostedtoolcache/Python/3.7.17/x64/lib/python3.7/site-packages/numpy/linalg/linalg.py", line 88, in _raise_linalgerror_singular
+    raise LinAlgError("Singular matrix")
+numpy.linalg.LinAlgError: Singular matrix
+
+
+

Notes

+
    +
  • If the robot is not 6 DOF the pinv option is set True.

  • +
  • pinv() is around 5x slower than inv()

  • +
+
+

Warning

+

Assumes that the operational space has 6 DOF.

+
+
+

See also

+

gravload()

+
+
+ +
+
+property grippers: List[Gripper]
+

Grippers attached to the robot

+
+
Returns:
+

A list of grippers

+
+
Return type:
+

grippers

+
+
+
+ +
+
+property hascollision
+

Robot has collision model

+
+
Returns:
+

    +
  • hascollision – Robot has collision model

  • +
  • At least one link has associated collision model.

  • +
+

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.hascollision
+False
+
+
+
+

See also

+

hasgeometry()

+
+
+ +
+
+property hasdynamics
+

Robot has dynamic parameters

+
+
Returns:
+

    +
  • hasdynamics – Robot has dynamic parameters

  • +
  • At least one link has associated dynamic parameters.

  • +
+

+
+
+

Examples

+

+
+
+
+ +
+
+property hasgeometry
+

Robot has geometry model

+

At least one link has associated mesh to describe its shape.

+
+
Returns:
+

Robot has geometry model

+
+
Return type:
+

hasgeometry

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.hasgeometry
+True
+
+
+
+

See also

+

hascollision()

+
+
+ +
+
+hierarchy()
+

Pretty print the robot link hierachy

+
+
Return type:
+

Pretty print of the robot model

+
+
+

Examples

+

Makes a robot and prints the heirachy

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.URDF.Panda()
+>>> robot.hierarchy()
+ panda_link0
+   panda_link1
+     panda_link2
+       panda_link3
+         panda_link4
+           panda_link5
+             panda_link6
+               panda_link7
+                 panda_link8
+                   panda_hand
+                     panda_leftfinger
+                     panda_rightfinger
+
+
+
+ +
+
+inertia(q)
+

Manipulator inertia matrix +inertia(q) is the symmetric joint inertia matrix (n,n) which +relates joint torque to joint acceleration for the robot at joint +configuration q.

+

Trajectory operation

+

If q is a matrix (m,n), each row is interpretted as a joint state +vector, and the result is a 3d-matrix (nxnxk) where each plane +corresponds to the inertia for the corresponding row of q.

+
+
Parameters:
+

q (ndarray) – Joint coordinates

+
+
Returns:
+

The inertia matrix

+
+
Return type:
+

inertia

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> puma.inertia(puma.qz)
+array([[ 3.9611, -0.1627, -0.1389,  0.0016, -0.0004,  0.    ],
+       [-0.1627,  4.4566,  0.3727,  0.    ,  0.0019,  0.    ],
+       [-0.1389,  0.3727,  0.9387,  0.    ,  0.0019,  0.    ],
+       [ 0.0016,  0.    ,  0.    ,  0.1924,  0.    ,  0.    ],
+       [-0.0004,  0.0019,  0.0019,  0.    ,  0.1713,  0.    ],
+       [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.1941]])
+
+
+

Notes

+
    +
  • +
    The diagonal elements M[j,j] are the inertia seen by joint

    actuator j.

    +
    +
    +
  • +
  • +
    The off-diagonal elements M[j,k] are coupling inertias that

    relate acceleration on joint j to force/torque on +joint k.

    +
    +
    +
  • +
  • +
    The diagonal terms include the motor inertia reflected through

    the gear ratio.

    +
    +
    +
  • +
+
+

See also

+

cinertia()

+
+
+ +
+
+inertia_x(q=None, pinv=False, representation='rpy/xyz', Ji=None)
+

Operational space inertia matrix

+

robot.inertia_x(q) is the operational space (Cartesian) inertia +matrix which relates Cartesian force/torque to Cartesian +acceleration at the joint configuration q.

+
+\[\mathbf{M}_x = \mathbf{J}(q)^{-T} \mathbf{M}(q) \mathbf{J}(q)^{-1}\]
+

The transformation to operational space requires an analytical, rather +than geometric, Jacobian. analytical can be one of:

+
+
+ + + + + + + + + + + + + + + + + + + +

Value

Rotational representation

'rpy/xyz'

RPY angular rates in XYZ order (default)

'rpy/zyx'

RPY angular rates in XYZ order

'eul'

Euler angular rates in ZYZ order

'exp'

exponential coordinate rates

+
+

Trajectory operation

+

If q is a matrix (m,n), each row is interpretted as a joint state +vector, and the result is a 3d-matrix (m,n,n) where each plane +corresponds to the Cartesian inertia for the corresponding +row of q.

+
+
Parameters:
+
    +
  • q – Joint coordinates

  • +
  • pinv – use pseudo inverse rather than inverse (Default value = False)

  • +
  • analytical – the type of analytical Jacobian to use, default is +‘rpy/xyz’

  • +
  • representation – (Default value = “rpy/xyz”)

  • +
  • Ji – The inverse analytical Jacobian (base-frame)

  • +
+
+
Returns:
+

The operational space inertia matrix

+
+
Return type:
+

inertia_x

+
+
+

Examples

+
    ainv = _umath_linalg.inv(a, signature=signature, extobj=extobj)
+  File "/opt/hostedtoolcache/Python/3.7.17/x64/lib/python3.7/site-packages/numpy/linalg/linalg.py", line 88, in _raise_linalgerror_singular
+    raise LinAlgError("Singular matrix")
+numpy.linalg.LinAlgError: Singular matrix
+
+
+

Notes

+
    +
  • If the robot is not 6 DOF the pinv option is set True.

  • +
  • pinv() is around 5x slower than inv()

  • +
+
+

Warning

+

Assumes that the operational space has 6 DOF.

+
+
+

See also

+

inertia()

+
+
+ +
+
+isprismatic(j)
+

Check if joint is prismatic

+
+
Returns:
+

True if prismatic

+
+
Return type:
+

j

+
+
+

Examples

+

+
+
+
+

See also

+

Link.isprismatic(), prismaticjoints()

+
+
+ +
+
+isrevolute(j)
+

Check if joint is revolute

+
+
Returns:
+

True if revolute

+
+
Return type:
+

j

+
+
+

Examples

+

+
+
+
+

See also

+

Link.isrevolute(), revolutejoints()

+
+
+ +
+
+itorque(q, qdd)
+

Inertia torque

+

itorque(q, qdd) is the inertia force/torque vector (n) at +the specified joint configuration q (n) and acceleration qdd (n), and +n is the number of robot joints. +It is \(\mathbf{I}(q) \ddot{q}\).

+

Trajectory operation

+

If q and qdd are matrices (m,n), each row is interpretted as a +joint configuration, and the result is a matrix (m,n) where each row is +the corresponding joint torques.

+
+
Parameters:
+
    +
  • q – Joint coordinates

  • +
  • qdd – Joint acceleration

  • +
+
+
Returns:
+

The inertia torque vector

+
+
Return type:
+

itorque

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> puma.itorque(puma.qz, 0.5 * np.ones((6,)))
+array([1.8304, 2.3343, 0.5872, 0.0971, 0.0873, 0.0971])
+
+
+

Notes

+
    +
  • +
    If the robot model contains non-zero motor inertia then this

    will be included in the result.

    +
    +
    +
  • +
+
+

See also

+

inertia()

+
+
+ +
+
+jacob0(q, start=None, end=None)
+
+ +
+
+jacobe(q, start=None, end=None)
+
+ +
+
+jointdynamics(q, qd=None)
+

Transfer function of joint actuator

+

tf = jointdynamics(qd, q) calculates a vector of n +continuous-time transfer functions that represent the transfer +function 1/(Js+B) for each joint based on the dynamic parameters +of the robot and the configuration q (n). n is the number of robot +joints. The result is a list of tuples (J, B) for each joint.

+

tf = jointdynamics(q, qd) as above but include the linearized +effects of Coulomb friction when operating at joint velocity QD +(1xN).

+
+
Parameters:
+
    +
  • q – Joint coordinates

  • +
  • qd – Joint velocity

  • +
+
+
Returns:
+

transfer function denominators

+
+
Return type:
+

list of 2-tuples

+
+
+
+ +
+
+property keywords: List[str]
+
+ +
+ +
+ +
+
+linkcolormap(linkcolors='viridis')
+

Create a colormap for robot joints

+
    +
  • cm = robot.linkcolormap() is an n-element colormap that gives a +unique color for every link. The RGBA colors for link j are +cm(j).

  • +
  • cm = robot.linkcolormap(cmap) as above but cmap is the name +of a valid matplotlib colormap. The default, example above, is the +viridis colormap.

  • +
  • cm = robot.linkcolormap(list of colors) as above but a +colormap is created from a list of n color names given as strings, +tuples or hexstrings.

  • +
+
+
Parameters:
+

linkcolors (Union[List[Any], str]) – list of colors or colormap, defaults to “viridis”

+
+
Returns:
+

the color map

+
+
Return type:
+

color map

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> cm = robot.linkcolormap("inferno")
+>>> print(cm(range(6))) # cm(i) is 3rd color in colormap
+[[0.0015 0.0005 0.0139 1.    ]
+ [0.2582 0.0386 0.4065 1.    ]
+ [0.5783 0.148  0.4044 1.    ]
+ [0.865  0.3168 0.2261 1.    ]
+ [0.9876 0.6453 0.0399 1.    ]
+ [0.9884 0.9984 0.6449 1.    ]]
+>>> cm = robot.linkcolormap(
+...     ['red', 'g', (0,0.5,0), '#0f8040', 'yellow', 'cyan'])
+>>> print(cm(range(6)))
+[[1.     0.     0.     1.    ]
+ [0.     0.5    0.     1.    ]
+ [0.     0.5    0.     1.    ]
+ [0.0588 0.502  0.251  1.    ]
+ [1.     1.     0.     1.    ]
+ [0.     1.     1.     1.    ]]
+
+
+

Notes

+ +
+ +
+ +

Robot links

+
+
Returns:
+

A list of link objects

+
+
Return type:
+

links

+
+
+

Notes

+

It is probably more concise to index the robot object rather +than the list of links, ie. the following are equivalent: +- robot.links[i] +- robot[i]

+
+ +
+
+property manufacturer
+

Get/set robot manufacturer’s name

+
    +
  • robot.manufacturer is the robot manufacturer’s name

  • +
  • robot.manufacturer = ... checks and sets the manufacturer’s name

  • +
+
+
Returns:
+

robot manufacturer’s name

+
+
Return type:
+

manufacturer

+
+
+
+ +
+
+property n: int
+

Number of joints

+
+
Returns:
+

Number of joints

+
+
Return type:
+

n

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.n
+6
+
+
+
+

See also

+

nlinks(), nbranches()

+
+
+ +
+
+property name: str
+

Get/set robot name

+
    +
  • robot.name is the robot name

  • +
  • robot.name = ... checks and sets the robot name

  • +
+
+
Parameters:
+

name – the new robot name

+
+
Returns:
+

the current robot name

+
+
Return type:
+

name

+
+
+
+ +
+
+property nbranches: int
+

Number of branches

+

Number of branches in this robot. Computed as the number of links with +zero children

+
+
Returns:
+

number of branches in the robot’s kinematic tree

+
+
Return type:
+

nbranches

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.ETS.Panda()
+>>> robot.nbranches
+1
+
+
+
+

See also

+

n(), nlinks()

+
+
+ +
+ +

Number of links

+

The returned number is the total of both variable joints and +static links

+
+
Returns:
+

Number of links

+
+
Return type:
+

nlinks

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.nlinks
+6
+
+
+
+

See also

+

n(), nbranches()

+
+
+ +
+
+nofriction(coulomb=True, viscous=False)
+

Remove manipulator joint friction

+

nofriction() copies the robot and returns +a robot with the same link parameters except the Coulomb and/or viscous +friction parameter are set to zero.

+
+
Parameters:
+
    +
  • coulomb (bool) – set the Coulomb friction to 0

  • +
  • viscous (bool) – set the viscous friction to 0

  • +
+
+
Returns:
+

A copy of the robot with dynamic parameters perturbed

+
+
Return type:
+

robot

+
+
+
+

See also

+

Robot.friction(), Link.nofriction()

+
+
+ +
+
+pay(W, q=None, J=None, frame=1)
+

Generalised joint force/torque due to a payload wrench

+

tau = pay(W, J) Returns the generalised joint force/torques due to a +payload wrench W applied to the end-effector. Where the manipulator +Jacobian is J (6xn), and n is the number of robot joints.

+

tau = pay(W, q, frame) as above but the Jacobian is calculated at pose +q in the frame given by frame which is 0 for base frame, 1 for +end-effector frame.

+

Uses the formula tau = J’W, where W is a wrench vector applied at the +end effector, W = [Fx Fy Fz Mx My Mz]’.

+
+
Trajectory operation:

In the case q is nxm or J is 6xnxm then tau is nxm where each row +is the generalised force/torque at the pose given by corresponding +row of q.

+
+
+
+
Parameters:
+
    +
  • W (Union[ndarray, List[float], Tuple[float], Set[float]]) – A wrench vector applied at the end effector, +W = [Fx Fy Fz Mx My Mz]

  • +
  • q (Optional[ndarray]) – Joint coordinates

  • +
  • J (Optional[ndarray]) – The manipulator Jacobian (Optional, if not supplied will +use the q value).

  • +
  • frame (int) – The frame in which to torques are expressed in when J +is not supplied. 0 means base frame of the robot, 1 means end- +effector frame

  • +
+
+
Returns:
+

Joint forces/torques due to w

+
+
Return type:
+

t

+
+
+

Notes

+
    +
  • +
    Wrench vector and Jacobian must be from the same reference

    frame.

    +
    +
    +
  • +
  • Tool transforms are taken into consideration when frame=1.

  • +
  • +
    Must have a constant wrench - no trajectory support for this

    yet.

    +
    +
    +
  • +
+
+ +
+
+paycap(w, tauR, frame=1, q=None)
+

Static payload capacity of a robot

+

wmax, joint = paycap(q, w, f, tauR) returns the maximum permissible +payload wrench wmax (6) applied at the end-effector, and the index +of the joint (zero indexed) which hits its force/torque limit at that +wrench. q (n) is the manipulator pose, w the payload wrench +(6), f the wrench reference frame and tauR (nx2) is a matrix of +joint forces/torques (first col is maximum, second col minimum).

+

Trajectory operation:

+

In the case q is nxm then wmax is Mx6 and J is Mx1 where the rows are +the results at the pose given by corresponding row of q.

+
+
Parameters:
+
    +
  • w (ndarray) – The payload wrench

  • +
  • tauR (ndarray) – Joint torque matrix minimum and maximums

  • +
  • frame (int) – The frame in which to torques are expressed in when J +is not supplied. ‘base’ means base frame of the robot, ‘ee’ means +end-effector frame

  • +
  • q (Union[ndarray, List[float], Tuple[float], Set[float], None]) – Joint coordinates

  • +
+
+
Returns:
+

The maximum permissible payload wrench

+
+
Return type:
+

ndarray(6)

+
+
+

Notes

+
    +
  • Wrench vector and Jacobian must be from the same reference frame

  • +
  • Tool transforms are taken into consideration for frame=1.

  • +
+
+ +
+
+payload(m, p=array([0.0, 0.0, 0.0]))
+

Add a payload to the end-effector

+

payload(m, p) adds payload mass adds a payload with point mass m at +position p in the end-effector coordinate frame.

+

payload(m) adds payload mass adds a payload with point mass m at +in the end-effector coordinate frame.

+

payload(0) removes added payload.

+
+
Parameters:
+
    +
  • m (float) – mass (kg)

  • +
  • p – position in end-effector frame

  • +
+
+
+
+ +
+
+perturb(p=0.1)
+

Perturb robot parameters

+

rp = perturb(p) is a new robot object in which the dynamic parameters +(link mass and inertia) have been perturbed. The perturbation is +multiplicative so that values are multiplied by random numbers in the +interval (1-p) to (1+p). The name string of the perturbed robot is +prefixed by ‘P/’.

+

Useful for investigating the robustness of various model-based control +schemes. For example to vary parameters in the range +/- 10 percent +is: r2 = puma.perturb(0.1)

+
+
Parameters:
+

p – The percent (+/-) to be perturbed. Default 10%

+
+
Returns:
+

A copy of the robot with dynamic parameters perturbed

+
+
Return type:
+

DHRobot

+
+
+
+ +
+
+plot(q, backend=None, block=False, dt=0.05, limits=None, vellipse=False, fellipse=False, fig=None, movie=None, loop=False, **kwargs)
+

Graphical display and animation

+

robot.plot(q, 'pyplot') displays a graphical view of a robot +based on the kinematic model and the joint configuration q. +This is a stick figure polyline which joins the origins of the +link coordinate frames. The plot will autoscale with an aspect +ratio of 1.

+

If q (m,n) representing a joint-space trajectory it will create an +animation with a pause of dt seconds between each frame.

+
+
+q
+

The joint configuration of the robot.

+
+ +
+
+backend
+

The graphical backend to use, currently ‘swift’ +and ‘pyplot’ are implemented. Defaults to ‘swift’ of a Robot +and ‘pyplot` for a DHRobot

+
+ +
+
+block
+

Block operation of the code and keep the figure open

+
+ +
+
+dt
+

if q is a trajectory, this describes the delay in +seconds between frames

+
+ +
+
+limits
+

Custom view limits for the plot. If not supplied will +autoscale, [x1, x2, y1, y2, z1, z2] +(this option is for ‘pyplot’ only)

+
+ +
+
+vellipse
+

(Plot Option) Plot the velocity ellipse at the +end-effector (this option is for ‘pyplot’ only)

+
+ +
+
+fellipse
+

(Plot Option) Plot the force ellipse at the +end-effector (this option is for ‘pyplot’ only)

+
+ +
+
+fig
+

(Plot Option) The figure label to plot in (this option is for +‘pyplot’ only)

+
+ +
+
+movie
+

(Plot Option) The filename to save the movie to (this option is for +‘pyplot’ only)

+
+ +
+
+loop
+

(Plot Option) Loop the movie (this option is for +‘pyplot’ only)

+
+ +
+
+jointaxes
+

(Plot Option) Plot an arrow indicating the axes in +which the joint revolves around(revolute joint) or translates +along (prosmatic joint) (this option is for ‘pyplot’ only)

+
+ +
+
+eeframe
+

(Plot Option) Plot the end-effector coordinate frame +at the location of the end-effector. Uses three arrows, red, +green and blue to indicate the x, y, and z-axes. +(this option is for ‘pyplot’ only)

+
+ +
+
+shadow
+

(Plot Option) Plot a shadow of the robot in the x-y +plane. (this option is for ‘pyplot’ only)

+
+ +
+
+name
+

(Plot Option) Plot the name of the robot near its base +(this option is for ‘pyplot’ only)

+
+ +
+
Returns:
+

A reference to the environment object which controls the +figure

+
+
Return type:
+

env

+
+
+

Notes

+
    +
  • +
    By default this method will block until the figure is dismissed.

    To avoid this set block=False.

    +
    +
    +
  • +
  • +
    For PyPlot, the polyline joins the origins of the link frames,

    but for some Denavit-Hartenberg models those frames may not +actually be on the robot, ie. the lines to not neccessarily +represent the links of the robot.

    +
    +
    +
  • +
+
+

See also

+

teach()

+
+
+ +
+
+plot_ellipse(ellipse, block=True, limits=None, jointaxes=True, eeframe=True, shadow=True, name=True)
+

Plot the an ellipsoid

+

robot.plot_ellipse(ellipsoid) displays the ellipsoid.

+
+
+ellipse
+

the ellipsoid to plot

+
+ +
+
+block
+

Block operation of the code and keep the figure open

+
+ +
+
+limits
+

Custom view limits for the plot. If not supplied will +autoscale, [x1, x2, y1, y2, z1, z2]

+
+ +
+
+jointaxes
+

(Plot Option) Plot an arrow indicating the axes in +which the joint revolves around(revolute joint) or translates +along (prosmatic joint)

+
+ +
+
+eeframe
+

(Plot Option) Plot the end-effector coordinate frame +at the location of the end-effector. Uses three arrows, red, +green and blue to indicate the x, y, and z-axes.

+
+ +
+
+shadow
+

(Plot Option) Plot a shadow of the robot in the x-y +plane

+
+ +
+
+name
+

(Plot Option) Plot the name of the robot near its base

+
+ +
+
Returns:
+

A reference to the PyPlot object which controls the +matplotlib figure

+
+
Return type:
+

env

+
+
+
+ +
+
+plot_fellipse(q, block=True, fellipse=None, limits=None, opt='trans', centre=[0, 0, 0], jointaxes=True, eeframe=True, shadow=True, name=True)
+

Plot the force ellipsoid for manipulator

+

robot.plot_fellipse(q) displays the velocity ellipsoid for the +robot at pose q. The plot will autoscale with an aspect ratio +of 1.

+

robot.plot_fellipse(vellipse) specifies a custon ellipse to plot.

+
+
+q
+

The joint configuration of the robot

+
+ +
+
+block
+

Block operation of the code and keep the figure open

+
+ +
+
+fellipse
+

the vellocity ellipsoid to plot

+
+ +
+
+limits
+

Custom view limits for the plot. If not supplied will +autoscale, [x1, x2, y1, y2, z1, z2]

+
+ +
+
+opt
+

‘trans’ or ‘rot’ will plot either the translational or +rotational force ellipsoid

+
+ +
+
+centre
+

The coordinates to plot the fellipse [x, y, z] or “ee” +to plot at the end-effector location

+
+ +
+
+jointaxes
+

(Plot Option) Plot an arrow indicating the axes in +which the joint revolves around(revolute joint) or translates +along (prosmatic joint)

+
+ +
+
+eeframe
+

(Plot Option) Plot the end-effector coordinate frame +at the location of the end-effector. Uses three arrows, red, +green and blue to indicate the x, y, and z-axes.

+
+ +
+
+shadow
+

(Plot Option) Plot a shadow of the robot in the x-y +plane

+
+ +
+
+name
+

(Plot Option) Plot the name of the robot near its base

+
+ +
+
Raises:
+

ValueError – q or fellipse must be supplied

+
+
Returns:
+

A reference to the PyPlot object which controls the +matplotlib figure

+
+
Return type:
+

env

+
+
+

Notes

+
    +
  • +
    By default the ellipsoid related to translational motion is

    drawn. Use opt='rot' to draw the rotational velocity +ellipsoid.

    +
    +
    +
  • +
  • +
    By default the ellipsoid is drawn at the origin. The option

    centre allows its origin to set to set to the specified +3-vector, or the string “ee” ensures it is drawn at the +end-effector position.

    +
    +
    +
  • +
+
+ +
+
+plot_vellipse(q, block=True, vellipse=None, limits=None, opt='trans', centre=[0, 0, 0], jointaxes=True, eeframe=True, shadow=True, name=True)
+

Plot the velocity ellipsoid for manipulator

+

robot.plot_vellipse(q) displays the velocity ellipsoid for the +robot at pose q. The plot will autoscale with an aspect ratio +of 1.

+

robot.plot_vellipse(vellipse) specifies a custon ellipse to plot.

+
+
block

Block operation of the code and keep the figure open

+
+
q

The joint configuration of the robot

+
+
vellipse

the vellocity ellipsoid to plot

+
+
limits

Custom view limits for the plot. If not supplied will +autoscale, [x1, x2, y1, y2, z1, z2]

+
+
opt

‘trans’ or ‘rot’ will plot either the translational or +rotational velocity ellipsoid

+
+
centre

The coordinates to plot the vellipse [x, y, z] or “ee” +to plot at the end-effector location

+
+
jointaxes

(Plot Option) Plot an arrow indicating the axes in +which the joint revolves around(revolute joint) or translates +along (prosmatic joint)

+
+
eeframe

(Plot Option) Plot the end-effector coordinate frame +at the location of the end-effector. Uses three arrows, red, +green and blue to indicate the x, y, and z-axes.

+
+
shadow

(Plot Option) Plot a shadow of the robot in the x-y +plane

+
+
name

(Plot Option) Plot the name of the robot near its base

+
+
+
+
Raises:
+

ValueError – q or fellipse must be supplied

+
+
Returns:
+

A reference to the PyPlot object which controls the +matplotlib figure

+
+
Return type:
+

env

+
+
+

Notes

+
    +
  • +
    By default the ellipsoid related to translational motion is

    drawn. Use opt='rot' to draw the rotational velocity +ellipsoid.

    +
    +
    +
  • +
  • +
    By default the ellipsoid is drawn at the origin. The option

    centre allows its origin to set to set to the specified +3-vector, or the string “ee” ensures it is drawn at the +end-effector position.

    +
    +
    +
  • +
+
+ +
+
+property prismaticjoints: List[bool]
+

Revolute joints as bool array

+
+
Returns:
+

array of joint type, True if prismatic

+
+
Return type:
+

prismaticjoints

+
+
+

Examples

+

+
+
+

Notes

+

Fixed joints, that maintain a constant link relative pose, +are not included.

+
+

See also

+

Link.isprismatic(), revolutejoints()

+
+
+ +
+
+property q: ndarray
+

Get/set robot joint configuration

+
    +
  • robot.q is the robot joint configuration

  • +
  • robot.q = ... checks and sets the joint configuration

  • +
+
+
Parameters:
+

q – the new robot joint configuration

+
+
Returns:
+

robot joint configuration

+
+
Return type:
+

q

+
+
+
+ +
+
+property qd: ndarray
+

Get/set robot joint velocity

+
    +
  • robot.qd is the robot joint velocity

  • +
  • robot.qd = ... checks and sets the joint velocity

  • +
+
+
Returns:
+

robot joint velocity

+
+
Return type:
+

qd

+
+
+
+ +
+
+property qdd: ndarray
+

Get/set robot joint acceleration

+
    +
  • robot.qdd is the robot joint acceleration

  • +
  • robot.qdd = ... checks and sets the robot joint acceleration

  • +
+
+
Returns:
+

robot joint acceleration

+
+
Return type:
+

qdd

+
+
+
+ +
+
+property qlim: ndarray
+

Joint limits

+

Limits are extracted from the link objects. If joints limits are +not set for:

+
    +
  • a revolute joint [-𝜋. 𝜋] is returned

  • +
  • a prismatic joint an exception is raised

  • +
+
+
+qlim
+

An array of joints limits (2, n)

+
+ +
+
Raises:
+

ValueError – unset limits for a prismatic joint

+
+
Returns:
+

Array of joint limit values

+
+
Return type:
+

qlim

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.qlim
+array([[-2.7925, -1.9199, -2.3562, -4.6426, -1.7453, -4.6426],
+       [ 2.7925,  1.9199,  2.3562,  4.6426,  1.7453,  4.6426]])
+
+
+
+ +
+
+random_q()
+

Return a random joint configuration

+

The value for each joint is uniform randomly distributed between the +limits set for the robot.

+
+

Note

+

The joint limit for all joints must be set.

+
+
+
Returns:
+

Random joint configuration :rtype: ndarray(n)

+
+
Return type:
+

q

+
+
+
+

See also

+

Robot.qlim(), Link.qlim()

+
+
+ +
+
+property reach: float
+

Reach of the robot

+

A conservative estimate of the reach of the robot. It is computed as +the sum of the translational ETs that define the link transform.

+
+

Note

+

Computed on the first access. If kinematic parameters +subsequently change this will not be reflected.

+
+
+
Returns:
+

Maximum reach of the robot

+
+
Return type:
+

reach

+
+
+

Notes

+
    +
  • Probably an overestimate of reach

  • +
  • Used by numerical inverse kinematics to scale translational +error.

  • +
  • For a prismatic joint, uses qlim if it is set

  • +
+
+ +
+
+property revolutejoints: List[bool]
+

Revolute joints as bool array

+
+
Returns:
+

array of joint type, True if revolute

+
+
Return type:
+

revolutejoints

+
+
+

Examples

+

+
+
+

Notes

+

Fixed joints, that maintain a constant link relative pose, +are not included.

+
+

See also

+

Link.isrevolute(), prismaticjoints()

+
+
+ +
+
+property scene_children: List[SceneNode]
+

Returns the child nodes of this object

+
+ +
+
+property scene_parent: Type[SceneNode]
+

Returns the parent node of this object

+
+ +
+
+segments()
+

Segments of branched robot

+

For a single-chain robot with structure:

+
L1 - L2 - L3
+
+
+

the return is [[None, L1, L2, L3]]

+

For a robot with structure:

+
L1 - L2 +-  L3 - L4
+        +- L5 - L6
+
+
+

the return is [[None, L1, L2], [L2, L3, L4], [L2, L5, L6]]

+
+
Returns:
+

Segment list

+
+
Return type:
+

segments

+
+
+

Notes

+
    +
  • the length of the list is the number of segments in the robot

  • +
  • +
    the first segment always starts with None which represents

    the base transform (since there is no base link)

    +
    +
    +
  • +
  • +
    the last link of one segment is also the first link of subsequent

    segments

    +
    +
    +
  • +
+
+ +
+
+showgraph(display_graph=True, **kwargs)
+

Display a link transform graph in browser

+

robot.showgraph() displays a graph of the robot’s link frames +and the ETS between them. It uses GraphViz dot.

+
+
The nodes are:
    +
  • Base is shown as a grey square. This is the world frame origin, +but can be changed using the base attribute of the robot.

  • +
  • Link frames are indicated by circles

  • +
  • ETS transforms are indicated by rounded boxes

  • +
+
+
The edges are:
    +
  • an arrow if jtype is False or the joint is fixed

  • +
  • an arrow with a round head if jtype is True and the joint is +revolute

  • +
  • an arrow with a box head if jtype is True and the joint is +prismatic

  • +
+
+
+

Edge labels or nodes in blue have a fixed transformation to the +preceding link.

+
+
Parameters:
+
    +
  • display_graph (bool) – Open the graph in a browser if True. Otherwise will return the +file path

  • +
  • etsbox – Put the link ETS in a box, otherwise an edge label

  • +
  • jtype – Arrowhead to node indicates revolute or prismatic type

  • +
  • static – Show static joints in blue and bold

  • +
+
+
Return type:
+

Optional[str]

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.URDF.Panda()
+>>> panda.showgraph()
+
+
+_images/panda-graph.svg +
+

See also

+

dotfile()

+
+
+ +
+
+property structure: str
+

Return the joint structure string

+

A string with one letter per joint: R for a revolute +joint, and P for a prismatic joint.

+
+
Returns:
+

joint configuration string

+
+
Return type:
+

structure

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> puma.structure
+'RRRRRR'
+>>> stanford = rtb.models.DH.Stanford()
+>>> stanford.structure
+'RRPRRR'
+
+
+

Notes

+

Fixed joints, that maintain a constant link relative pose, +are not included. +len(self.structure) == self.n.

+
+ +
+
+property symbolic: bool
+
+ +
+
+teach(q, block=True, limits=None, vellipse=False, fellipse=False, backend=None)
+

Graphical teach pendant

+

robot.teach(q) creates a matplotlib plot which allows the user to +“drive” a graphical robot using a graphical slider panel. The robot’s +inital joint configuration is q. The plot will autoscale with an +aspect ratio of 1.

+

robot.teach() as above except the robot’s stored value of q +is used.

+
+
q

The joint configuration of the robot (Optional, +if not supplied will use the stored q values).

+
+
block

Block operation of the code and keep the figure open

+
+
limits

Custom view limits for the plot. If not supplied will +autoscale, [x1, x2, y1, y2, z1, z2]

+
+
vellipse

(Plot Option) Plot the velocity ellipse at the +end-effector (this option is for ‘pyplot’ only)

+
+
fellipse

(Plot Option) Plot the force ellipse at the +end-effector (this option is for ‘pyplot’ only)

+
+
+
+
Returns:
+

A reference to the PyPlot object which controls the +matplotlib figure

+
+
Return type:
+

env

+
+
+

Notes

+
    +
  • +
    Program execution is blocked until the teach window is

    dismissed. If block=False the method is non-blocking but +you need to poll the window manager to ensure that the window +remains responsive.

    +
    +
    +
  • +
  • +
    The slider limits are derived from the joint limit properties.

    If not set then: +- For revolute joints they are assumed to be [-pi, +pi] +- For prismatic joint they are assumed unknown and an error

    +
    +

    occurs.

    +
    +
    +
    +
  • +
+
+ +
+
+todegrees(q)
+

Convert joint angles to degrees

+
+
Parameters:
+

q – The joint configuration of the robot

+
+
Return type:
+

ndarray

+
+
Returns:
+

    +
  • q – a vector of joint coordinates in degrees and metres

  • +
  • robot.todegrees(q) converts joint coordinates q to degrees

  • +
  • taking into account whether elements of q correspond to revolute

  • +
  • or prismatic joints, ie. prismatic joint values are not converted.

  • +
  • If q is a matrix, with one column per joint, the conversion is

  • +
  • performed columnwise.

  • +
+

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> from math import pi
+>>> stanford = rtb.models.DH.Stanford()
+>>> stanford.todegrees([pi/4, pi/8, 2, -pi/4, pi/6, pi/3])
+array([ 45. ,  22.5,   2. , -45. ,  30. ,  60. ])
+
+
+
+ +
+
+property tool: SE3
+

Get/set robot tool transform

+
    +
  • robot.tool is the robot tool transform as an SE3 object

  • +
  • robot._tool is the robot tool transform as a numpy array

  • +
  • robot.tool = ... checks and sets the robot tool transform

  • +
+
+
Parameters:
+

tool – the new robot tool transform (as an SE(3))

+
+
Returns:
+

robot tool transform

+
+
Return type:
+

tool

+
+
+
+ +
+
+toradians(q)
+

Convert joint angles to radians

+

robot.toradians(q) converts joint coordinates q to radians +taking into account whether elements of q correspond to revolute +or prismatic joints, ie. prismatic joint values are not converted.

+

If q is a matrix, with one column per joint, the conversion is +performed columnwise.

+
+
Parameters:
+

q – The joint configuration of the robot

+
+
Returns:
+

a vector of joint coordinates in radians and metres

+
+
Return type:
+

q

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> stanford = rtb.models.DH.Stanford()
+>>> stanford.toradians([10, 20, 2, 30, 40, 50])
+array([0.1745, 0.3491, 2.    , 0.5236, 0.6981, 0.8727])
+
+
+
+ +
+
+property urdf_filepath: str
+
+ +
+
+property urdf_string: str
+
+ +
+
+vellipse(q, opt='trans', unit='rad', centre=[0, 0, 0], scale=0.1)
+

Create a velocity ellipsoid object for plotting with PyPlot

+

robot.vellipse(q) creates a force ellipsoid for the robot at +pose q. The ellipsoid is centered at the origin.

+
+
+q
+

The joint configuration of the robot.

+
+ +
+
+opt
+

‘trans’ or ‘rot’ will plot either the translational or +rotational force ellipsoid

+
+ +
+
+unit
+

‘rad’ or ‘deg’ will plot the ellipsoid in radians or +degrees

+
+ +
+
+centre
+

The centre of the ellipsoid, either ‘ee’ for the end-effector +or a 3-vector [x, y, z] in the world frame

+
+ +
+
+scale
+

The scale factor for the ellipsoid

+
+ +
+
Returns:
+

An EllipsePlot object

+
+
Return type:
+

env

+
+
+

Notes

+
    +
  • +
    By default the ellipsoid related to translational motion is

    drawn. Use opt='rot' to draw the rotational velocity +ellipsoid.

    +
    +
    +
  • +
  • +
    By default the ellipsoid is drawn at the origin. The option

    centre allows its origin to set to set to the specified +3-vector, or the string “ee” ensures it is drawn at the +end-effector position.

    +
    +
    +
  • +
+
+ +
+ +
+ +
+

ERobot models

+
+

Defined using ETS

+

A number of models are defined in terms of elementary transform sequences. +They can be listed by:

+

+
+
+
+
+class roboticstoolbox.models.ETS.Panda[source]
+

Bases: Robot

+

Create model of Franka-Emika Panda manipulator

+

panda = Panda() creates a robot object representing the Franka-Emika +Panda robot arm. This robot is represented using the elementary +transform sequence (ETS).

+

ETS taken from [1] based on +https://frankaemika.github.io/docs/control_parameters.html

+
+
References:
+
    +
  • Kinematic Derivatives using the Elementary Transform +Sequence, J. Haviland and P. Corke

  • +
+
+
+
+ +
+
+class roboticstoolbox.models.ETS.Frankie[source]
+

Bases: Robot

+

A class representing the Franka Emika Panda robot arm. ETS taken from [1] +based on https://frankaemika.github.io/docs/control_parameters.html

+
+
Parameters:
+
    +
  • et_list (list of etb.robot.et) – List of elementary transforms which represent the robot +kinematics

  • +
  • q_idx (list of int) – List of indexes within the ets_list which correspond to +joints

  • +
  • name (str, optional) – Name of the robot

  • +
  • manufacturer (str, optional) – Manufacturer of the robot

  • +
  • base (float np.ndarray(4,4), optional) – Location of the base is the world frame

  • +
  • tool (float np.ndarray(4,4), optional) – Offset of the flange of the robot to the end-effector

  • +
  • qz (float np.ndarray(7,)) – The zero joint angle configuration of the robot

  • +
  • qr (float np.ndarray(7,)) – The ready state joint angle configuration of the robot

  • +
+
+
+
+
References: [1] Kinematic Derivatives using the Elementary Transform

Sequence, J. Haviland and P. Corke

+
+
+
+ +
+
+class roboticstoolbox.models.ETS.Puma560[source]
+

Bases: Robot

+

Create model of Franka-Emika Panda manipulator

+

puma = Puma560() creates a robot object representing the classic +Unimation Puma560 robot arm. This robot is represented using the elementary +transform sequence (ETS).

+
+

Note

+
    +
  • The model has different joint offset conventions compared to +DH.Puma560(). For this robot:

    +
    +
      +
    • Zero joint angles qz is the vertical configuration, +corresponding to qr with DH.Puma560()

    • +
    • qbent is the bent configuration, corresponding to +qz with DH.Puma560()

    • +
    +
    +
  • +
+
+
+
References:
+
+
+
+
+ +
+
+class roboticstoolbox.models.ETS.Planar_Y[source]
+

Bases: Robot

+

Create model of a branched planar manipulator:

+
L0 -- L1 -+- L2a -- L3a -- EEa
+        |
+        +- L2b -- L3b -- EEb
+
+
+

Planar_Y() creates a planar branched manipulator model.

+
+
References:
+
    +
  • Kinematic Derivatives using the Elementary Transform +Sequence, J. Haviland and P. Corke

  • +
+
+
+
+ +
+
+class roboticstoolbox.models.ETS.Planar2[source]
+

Bases: Robot2

+

Create model of a branched planar manipulator:

+
L0 -- L1 -+- L2a -- L3a -- EEa
+        |
+        +- L2b -- L3b -- EEb
+
+
+

Planar_Y() creates a planar branched manipulator model.

+
+
References:
+
    +
  • Kinematic Derivatives using the Elementary Transform +Sequence, J. Haviland and P. Corke

  • +
+
+
+
+ +
+
+class roboticstoolbox.models.ETS.GenericSeven[source]
+

Bases: Robot

+

Create model of a generic seven degree-of-freedom robot

+

robot = GenericSeven() creates a robot object. This robot is represented +using the elementary transform sequence (ETS).

+
+ +
+
+class roboticstoolbox.models.ETS.XYPanda(workspace=5)[source]
+

Bases: Robot

+

Create model of Franka-Emika Panda manipulator on an XY platform

+

xypanda = XYPanda() creates a robot object representing the Franka-Emika +Panda robot arm mounted on an XY platform. This robot is represented using the elementary +transform sequence (ETS).

+

ETS taken from [1] based on +https://frankaemika.github.io/docs/control_parameters.html

+
+
References:
+
    +
  • Kinematic Derivatives using the Elementary Transform +Sequence, J. Haviland and P. Corke

  • +
+
+
+
+ +
+
+

Defined from URDF

+

A number of models are defined in terms of Denavit-Hartenberg parameters, either +standard or modified. They can be listed by:

+

+
+
+
+
+class roboticstoolbox.models.URDF.Panda[source]
+

Bases: Robot

+

Class that imports a Panda URDF model

+

Panda() is a class which imports a Franka-Emika Panda robot definition +from a URDF file. The model describes its kinematic and graphical +characteristics.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.URDF.Panda()
+>>> print(robot)
+ERobot: panda (by Franka Emika), 7 joints (RRRRRRR), 1 gripper, geometry, collision
+┌─────┬──────────────┬───────┬─────────────┬────────────────────────────────────────────────┐
+│link │     link     │ joint │   parent    │              ETS: parent to link               │
+├─────┼──────────────┼───────┼─────────────┼────────────────────────────────────────────────┤
+│   0 │ panda_link0  │       │ BASE        │ SE3()                                          │
+│   1 │ panda_link1  │     0 │ panda_link0 │ SE3(0, 0, 0.333) ⊕ Rz(q0)                      │
+│   2 │ panda_link2  │     1 │ panda_link1 │ SE3(-90°, -0°, 0°) ⊕ Rz(q1)                    │
+│   3 │ panda_link3  │     2 │ panda_link2 │ SE3(0, -0.316, 0; 90°, -0°, 0°) ⊕ Rz(q2)       │
+│   4 │ panda_link4  │     3 │ panda_link3 │ SE3(0.0825, 0, 0; 90°, -0°, 0°) ⊕ Rz(q3)       │
+│   5 │ panda_link5  │     4 │ panda_link4 │ SE3(-0.0825, 0.384, 0; -90°, -0°, 0°) ⊕ Rz(q4) │
+│   6 │ panda_link6  │     5 │ panda_link5 │ SE3(90°, -0°, 0°) ⊕ Rz(q5)                     │
+│   7 │ panda_link7  │     6 │ panda_link6 │ SE3(0.088, 0, 0; 90°, -0°, 0°) ⊕ Rz(q6)        │
+│   8 │ @panda_link8 │       │ panda_link7 │ SE3(0, 0, 0.107)                               │
+└─────┴──────────────┴───────┴─────────────┴────────────────────────────────────────────────┘
+
+┌─────┬─────┬────────┬─────┬───────┬─────┬───────┬──────┐
+│name │ q0  │ q1     │ q2  │ q3    │ q4  │ q5    │ q6   │
+├─────┼─────┼────────┼─────┼───────┼─────┼───────┼──────┤
+│  qr │  0° │ -17.2° │  0° │ -126° │  0° │  115° │  45° │
+│  qz │  0° │  0°    │  0° │  0°   │  0° │  0°   │  0°  │
+└─────┴─────┴────────┴─────┴───────┴─────┴───────┴──────┘
+
+
+
+

Defined joint configurations are:

+
    +
  • qz, zero joint angle configuration, ‘L’ shaped configuration

  • +
  • qr, vertical ‘READY’ configuration

  • +
  • qs, arm is stretched out in the x-direction

  • +
  • qn, arm is at a nominal non-singular configuration

  • +
+

Code author: Jesse Haviland

+

Section author: Peter Corke

+
+ +
+
+class roboticstoolbox.models.URDF.Frankie[source]
+

Bases: Robot

+

Class that imports a Frankie URDF model

+

Frankie() is a class which imports a Frankie robot definition +from a URDF file. The model describes its kinematic and graphical +characteristics.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.URDF.Frankie()
+>>> print(robot)
+ERobot: frankie (by Franka Emika), 9 joints (RPRRRRRRR), 1 gripper, geometry, collision
+┌─────┬────────────────┬───────┬────────────────┬────────────────────────────────────────────────┐
+│link │      link      │ joint │     parent     │              ETS: parent to link               │
+├─────┼────────────────┼───────┼────────────────┼────────────────────────────────────────────────┤
+│   0 │ panda_base0    │       │ BASE           │ SE3()                                          │
+│   1 │ panda_base1    │     0 │ panda_base0    │ SE3() ⊕ Rz(q0)                                 │
+│   2 │ panda_base_arm │     1 │ panda_base1    │ SE3() ⊕ tx(q1)                                 │
+│   3 │ panda_link0    │       │ panda_base_arm │ SE3(0.15, 0, 0.38)                             │
+│   4 │ panda_link1    │     2 │ panda_link0    │ SE3(0, 0, 0.333) ⊕ Rz(q2)                      │
+│   5 │ panda_link2    │     3 │ panda_link1    │ SE3(-90°, -0°, 0°) ⊕ Rz(q3)                    │
+│   6 │ panda_link3    │     4 │ panda_link2    │ SE3(0, -0.316, 0; 90°, -0°, 0°) ⊕ Rz(q4)       │
+│   7 │ panda_link4    │     5 │ panda_link3    │ SE3(0.0825, 0, 0; 90°, -0°, 0°) ⊕ Rz(q5)       │
+│   8 │ panda_link5    │     6 │ panda_link4    │ SE3(-0.0825, 0.384, 0; -90°, -0°, 0°) ⊕ Rz(q6) │
+│   9 │ panda_link6    │     7 │ panda_link5    │ SE3(90°, -0°, 0°) ⊕ Rz(q7)                     │
+│  10 │ panda_link7    │     8 │ panda_link6    │ SE3(0.088, 0, 0; 90°, -0°, 0°) ⊕ Rz(q8)        │
+│  11 │ @panda_link8   │       │ panda_link7    │ SE3(0, 0, 0.107)                               │
+└─────┴────────────────┴───────┴────────────────┴────────────────────────────────────────────────┘
+
+┌─────┬─────┬────┬─────┬────────┬─────┬───────┬─────┬───────┬──────┐
+│name │ q0  │ q1 │ q2  │ q3     │ q4  │ q5    │ q6  │ q7    │ q8   │
+├─────┼─────┼────┼─────┼────────┼─────┼───────┼─────┼───────┼──────┤
+│  qr │  0° │  0 │  0° │ -17.2° │  0° │ -126° │  0° │  115° │  45° │
+│  qz │  0° │  0 │  0° │  0°    │  0° │  0°   │  0° │  0°   │  0°  │
+└─────┴─────┴────┴─────┴────────┴─────┴───────┴─────┴───────┴──────┘
+
+
+
+

Defined joint configurations are:

+
    +
  • qz, zero joint angle configuration, ‘L’ shaped configuration

  • +
  • qr, vertical ‘READY’ configuration

  • +
  • qs, arm is stretched out in the x-direction

  • +
  • qn, arm is at a nominal non-singular configuration

  • +
+

Code author: Jesse Haviland

+

Section author: Peter Corke

+
+ +
+
+class roboticstoolbox.models.URDF.FrankieOmni[source]
+

Bases: Robot

+

Class that imports an Omnidirectional Frankie URDF model

+

FrankieOmni() is a class which imports a FrankieOmni robot definition +from a URDF file. The model describes its kinematic and graphical +characteristics.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.URDF.FrankieOmni()
+>>> print(robot)
+ERobot: FrankieOmni (by Custom), 10 joints (PPRRRRRRRR), 1 gripper, 12 branches, dynamics, geometry, collision
+┌─────┬────────────────────────┬───────┬───────────────────┬────────────────────────────────────────────────┐
+│link │          link          │ joint │      parent       │              ETS: parent to link               │
+├─────┼────────────────────────┼───────┼───────────────────┼────────────────────────────────────────────────┤
+│   0 │ virtual0               │       │ BASE              │ SE3()                                          │
+│   1 │ virtual1               │     0 │ virtual0          │ SE3() ⊕ tx(q0)                                 │
+│   2 │ virtual2               │     1 │ virtual1          │ SE3() ⊕ ty(q1)                                 │
+│   3 │ base_link              │     2 │ virtual2          │ SE3() ⊕ Rz(q2)                                 │
+│   4 │ chassis_link           │       │ base_link         │ SE3()                                          │
+│   5 │ left_side_cover_link   │       │ chassis_link      │ SE3()                                          │
+│   6 │ right_side_cover_link  │       │ chassis_link      │ SE3()                                          │
+│   7 │ front_cover_link       │       │ chassis_link      │ SE3()                                          │
+│   8 │ rear_cover_link        │       │ chassis_link      │ SE3()                                          │
+│   9 │ front_lights_link      │       │ chassis_link      │ SE3()                                          │
+│  10 │ rear_lights_link       │       │ chassis_link      │ SE3()                                          │
+│  11 │ top_link               │       │ chassis_link      │ SE3()                                          │
+│  12 │ axle_link              │       │ chassis_link      │ SE3(0, 0, 0.05)                                │
+│  13 │ front_rocker_link      │       │ axle_link         │ SE3(0.319, 0, 0)                               │
+│  14 │ front_left_wheel_link  │       │ front_rocker_link │ SE3(0, 0.2755, 0)                              │
+│  15 │ front_right_wheel_link │       │ front_rocker_link │ SE3(0, -0.2755, 0)                             │
+│  16 │ rear_rocker_link       │       │ axle_link         │ SE3(-0.319, 0, 0)                              │
+│  17 │ rear_left_wheel_link   │       │ rear_rocker_link  │ SE3(0, 0.2755, 0)                              │
+│  18 │ rear_right_wheel_link  │       │ rear_rocker_link  │ SE3(0, -0.2755, 0)                             │
+│  19 │ base_arm               │       │ base_link         │ tz(0.28)                                       │
+│  20 │ panda_link0            │       │ base_arm          │ SE3()                                          │
+│  21 │ panda_link1            │     3 │ panda_link0       │ SE3(0, 0, 0.333) ⊕ Rz(q3)                      │
+│  22 │ panda_link2            │     4 │ panda_link1       │ SE3(-90°, -0°, 0°) ⊕ Rz(q4)                    │
+│  23 │ panda_link3            │     5 │ panda_link2       │ SE3(0, -0.316, 0; 90°, -0°, 0°) ⊕ Rz(q5)       │
+│  24 │ panda_link4            │     6 │ panda_link3       │ SE3(0.0825, 0, 0; 90°, -0°, 0°) ⊕ Rz(q6)       │
+│  25 │ panda_link5            │     7 │ panda_link4       │ SE3(-0.0825, 0.384, 0; -90°, -0°, 0°) ⊕ Rz(q7) │
+│  26 │ panda_link6            │     8 │ panda_link5       │ SE3(90°, -0°, 0°) ⊕ Rz(q8)                     │
+│  27 │ panda_link7            │     9 │ panda_link6       │ SE3(0.088, 0, 0; 90°, -0°, 0°) ⊕ Rz(q9)        │
+│  28 │ @panda_link8           │       │ panda_link7       │ SE3(0, 0, 0.107)                               │
+└─────┴────────────────────────┴───────┴───────────────────┴────────────────────────────────────────────────┘
+
+┌─────┬────┬────┬─────┬─────┬────────┬─────┬───────┬─────┬───────┬──────┐
+│name │ q0 │ q1 │ q2  │ q3  │ q4     │ q5  │ q6    │ q7  │ q8    │ q9   │
+├─────┼────┼────┼─────┼─────┼────────┼─────┼───────┼─────┼───────┼──────┤
+│  qr │  0 │  0 │  0° │  0° │ -17.2° │  0° │ -126° │  0° │  115° │  45° │
+│  qz │  0 │  0 │  0° │  0° │  0°    │  0° │  0°   │  0° │  0°   │  0°  │
+└─────┴────┴────┴─────┴─────┴────────┴─────┴───────┴─────┴───────┴──────┘
+
+
+
+

Defined joint configurations are:

+
    +
  • qz, zero joint angle configuration, ‘L’ shaped configuration

  • +
  • qr, vertical ‘READY’ configuration

  • +
  • qs, arm is stretched out in the x-direction

  • +
  • qn, arm is at a nominal non-singular configuration

  • +
+

Code author: Jesse Haviland

+

Section author: Peter Corke

+
+ +
+
+class roboticstoolbox.models.URDF.UR3[source]
+

Bases: Robot

+

Class that imports a UR3 URDF model

+

UR3() is a class which imports a Universal Robotics UR3 robot +definition from a URDF file. The model describes its kinematic and +graphical characteristics.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.URDF.UR3()
+>>> print(robot)
+ERobot: UR3 (by Universal Robotics), 6 joints (RRRRRR), 3 branches, dynamics, geometry, collision
+┌─────┬────────────────┬───────┬────────────────┬──────────────────────────────────────────┐
+│link │      link      │ joint │     parent     │           ETS: parent to link            │
+├─────┼────────────────┼───────┼────────────────┼──────────────────────────────────────────┤
+│   0 │ world          │       │ BASE           │ SE3()                                    │
+│   1 │ base_link      │       │ world          │ SE3()                                    │
+│   2 │ shoulder_link  │     0 │ base_link      │ SE3(0, 0, 0.1519) ⊕ Rz(q0)               │
+│   3 │ upper_arm_link │     1 │ shoulder_link  │ SE3(0, 0.1198, 0; 0°, 90°, -0°) ⊕ Ry(q1) │
+│   4 │ forearm_link   │     2 │ upper_arm_link │ SE3(0, -0.0925, 0.2437) ⊕ Ry(q2)         │
+│   5 │ wrist_1_link   │     3 │ forearm_link   │ SE3(0, 0, 0.2132; 0°, 90°, -0°) ⊕ Ry(q3) │
+│   6 │ wrist_2_link   │     4 │ wrist_1_link   │ SE3(0, 0.08505, 0) ⊕ Rz(q4)              │
+│   7 │ wrist_3_link   │     5 │ wrist_2_link   │ SE3(0, 0, 0.08535) ⊕ Ry(q5)              │
+│   8 │ @ee_link       │       │ wrist_3_link   │ SE3(0, 0.0819, 0; 0°, -0°, 90°)          │
+│   9 │ @tool0         │       │ wrist_3_link   │ SE3(0, 0.0819, 0; -90°, -0°, 0°)         │
+│  10 │ @base          │       │ base_link      │ SE3(0°, -0°, -180°)                      │
+└─────┴────────────────┴───────┴────────────────┴──────────────────────────────────────────┘
+
+┌─────┬───────┬─────┬─────┬─────┬──────┬─────┐
+│name │ q0    │ q1  │ q2  │ q3  │ q4   │ q5  │
+├─────┼───────┼─────┼─────┼─────┼──────┼─────┤
+│  qr │  180° │  0° │  0° │  0° │  90° │  0° │
+│  qz │  0°   │  0° │  0° │  0° │  0°  │  0° │
+└─────┴───────┴─────┴─────┴─────┴──────┴─────┘
+
+
+
+

Defined joint configurations are:

+
    +
  • qz, zero joint angle configuration, ‘L’ shaped configuration

  • +
  • qr, vertical ‘READY’ configuration

  • +
+

Code author: Jesse Haviland

+

Section author: Peter Corke

+
+ +
+
+class roboticstoolbox.models.URDF.UR5[source]
+

Bases: Robot

+

Class that imports a UR5 URDF model

+

UR3() is a class which imports a Universal Robotics UR5 robot +definition from a URDF file. The model describes its kinematic and +graphical characteristics.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.URDF.UR5()
+>>> print(robot)
+ERobot: UR5 (by Universal Robotics), 6 joints (RRRRRR), 1 gripper, 3 branches, dynamics, geometry, collision
+┌─────┬────────────────┬───────┬────────────────┬──────────────────────────────────────────┐
+│link │      link      │ joint │     parent     │           ETS: parent to link            │
+├─────┼────────────────┼───────┼────────────────┼──────────────────────────────────────────┤
+│   0 │ world          │       │ BASE           │ SE3()                                    │
+│   1 │ base_link      │       │ world          │ SE3()                                    │
+│   2 │ shoulder_link  │     0 │ base_link      │ SE3(0, 0, 0.08916) ⊕ Rz(q0)              │
+│   3 │ upper_arm_link │     1 │ shoulder_link  │ SE3(0, 0.1358, 0; 0°, 90°, -0°) ⊕ Ry(q1) │
+│   4 │ forearm_link   │     2 │ upper_arm_link │ SE3(0, -0.1197, 0.425) ⊕ Ry(q2)          │
+│   5 │ wrist_1_link   │     3 │ forearm_link   │ SE3(0, 0, 0.3922; 0°, 90°, -0°) ⊕ Ry(q3) │
+│   6 │ wrist_2_link   │     4 │ wrist_1_link   │ SE3(0, 0.093, 0) ⊕ Rz(q4)                │
+│   7 │ @wrist_3_link  │     5 │ wrist_2_link   │ SE3(0, 0, 0.09465) ⊕ Ry(q5)              │
+│   8 │ tool0          │       │ wrist_3_link   │ SE3(0, 0.0823, 0; -90°, -0°, 0°)         │
+│   9 │ base           │       │ base_link      │ SE3(0°, -0°, -180°)                      │
+└─────┴────────────────┴───────┴────────────────┴──────────────────────────────────────────┘
+
+┌─────┬────────┬────────┬────────┬──────┬────────┬─────┐
+│name │ q0     │ q1     │ q2     │ q3   │ q4     │ q5  │
+├─────┼────────┼────────┼────────┼──────┼────────┼─────┤
+│  qr │  180°  │  0°    │  0°    │  0°  │  90°   │  0° │
+│  qz │  0°    │  0°    │  0°    │  0°  │  0°    │  0° │
+│  qn │ -40.4° │  20.7° │ -85.6° │  65° │ -40.4° │  0° │
+│  q1 │  0°    │ -90°   │  90°   │  0°  │  90°   │  0° │
+└─────┴────────┴────────┴────────┴──────┴────────┴─────┘
+
+
+
+

Defined joint configurations are:

+
    +
  • qz, zero joint angle configuration, ‘L’ shaped configuration

  • +
  • qr, vertical ‘READY’ configuration

  • +
+

Code author: Jesse Haviland

+

Section author: Peter Corke

+
+ +
+
+class roboticstoolbox.models.URDF.UR10[source]
+

Bases: Robot

+

Class that imports a UR10 URDF model

+

UR3() is a class which imports a Universal Robotics UR310 robot +definition from a URDF file. The model describes its kinematic and +graphical characteristics.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.URDF.UR10()
+>>> print(robot)
+ERobot: UR10 (by Universal Robotics), 6 joints (RRRRRR), 3 branches, dynamics, geometry, collision
+┌─────┬────────────────┬───────┬────────────────┬──────────────────────────────────────────┐
+│link │      link      │ joint │     parent     │           ETS: parent to link            │
+├─────┼────────────────┼───────┼────────────────┼──────────────────────────────────────────┤
+│   0 │ world          │       │ BASE           │ SE3()                                    │
+│   1 │ base_link      │       │ world          │ SE3()                                    │
+│   2 │ shoulder_link  │     0 │ base_link      │ SE3(0, 0, 0.1273) ⊕ Rz(q0)               │
+│   3 │ upper_arm_link │     1 │ shoulder_link  │ SE3(0, 0.2209, 0; 0°, 90°, -0°) ⊕ Ry(q1) │
+│   4 │ forearm_link   │     2 │ upper_arm_link │ SE3(0, -0.1719, 0.612) ⊕ Ry(q2)          │
+│   5 │ wrist_1_link   │     3 │ forearm_link   │ SE3(0, 0, 0.5723; 0°, 90°, -0°) ⊕ Ry(q3) │
+│   6 │ wrist_2_link   │     4 │ wrist_1_link   │ SE3(0, 0.1149, 0) ⊕ Rz(q4)               │
+│   7 │ wrist_3_link   │     5 │ wrist_2_link   │ SE3(0, 0, 0.1157) ⊕ Ry(q5)               │
+│   8 │ @ee_link       │       │ wrist_3_link   │ SE3(0, 0.0922, 0; 0°, -0°, 90°)          │
+│   9 │ @tool0         │       │ wrist_3_link   │ SE3(0, 0.0922, 0; -90°, -0°, 0°)         │
+│  10 │ @base          │       │ base_link      │ SE3(0°, -0°, -180°)                      │
+└─────┴────────────────┴───────┴────────────────┴──────────────────────────────────────────┘
+
+┌─────┬───────┬─────┬─────┬─────┬──────┬─────┐
+│name │ q0    │ q1  │ q2  │ q3  │ q4   │ q5  │
+├─────┼───────┼─────┼─────┼─────┼──────┼─────┤
+│  qr │  180° │  0° │  0° │  0° │  90° │  0° │
+│  qz │  0°   │  0° │  0° │  0° │  0°  │  0° │
+└─────┴───────┴─────┴─────┴─────┴──────┴─────┘
+
+
+
+

Defined joint configurations are:

+
    +
  • qz, zero joint angle configuration, ‘L’ shaped configuration

  • +
  • qr, vertical ‘READY’ configuration

  • +
+

Code author: Jesse Haviland

+

Section author: Peter Corke

+
+ +
+
+class roboticstoolbox.models.URDF.Puma560[source]
+

Bases: Robot

+

Class that imports a Puma 560 URDF model

+

Puma560() is a class which imports a Unimation Puma560 robot definition +from a URDF file. The model describes its kinematic and graphical +characteristics.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.URDF.Puma560()
+>>> print(robot)
+ERobot: Puma560 (by Unimation), 6 joints (RRRRRR), geometry, collision
+┌─────┬────────┬───────┬────────┬───────────────────────────────────────────────────┐
+│link │  link  │ joint │ parent │                ETS: parent to link                │
+├─────┼────────┼───────┼────────┼───────────────────────────────────────────────────┤
+│   0 │ link1  │       │ BASE   │ SE3()                                             │
+│   1 │ link2  │     0 │ link1  │ SE3(0, 0, 0.5486; 90°, -0°, 0°) ⊕ Ry(q0)          │
+│   2 │ link3  │     1 │ link2  │ SE3(0, 0.07493, 0.1422) ⊕ Rz(q1)                  │
+│   3 │ link4  │     2 │ link3  │ SE3(0.4318, 0, 0.0254; 0°, -0°, 90°) ⊕ Rz(q2)     │
+│   4 │ link5  │     3 │ link4  │ SE3(0.3518, 0, -0.0381; -90°, -0°, -90°) ⊕ Rz(q3) │
+│   5 │ link6  │     4 │ link5  │ SE3(0, 0, 0.0803; 90°, -0°, 0°) ⊕ Rz(q4)          │
+│   6 │ @link7 │     5 │ link6  │ SE3(0, 0.05334, 0; -90°, -0°, 0°) ⊕ Rz(q5)        │
+└─────┴────────┴───────┴────────┴───────────────────────────────────────────────────┘
+
+┌─────┬───────┬────────┬────────┬────────┬────────┬────────┐
+│name │ q0    │ q1     │ q2     │ q3     │ q4     │ q5     │
+├─────┼───────┼────────┼────────┼────────┼────────┼────────┤
+│  qr │  0°   │  90°   │ -90°   │  0°    │  0°    │  0°    │
+│  qz │  0°   │  0°    │  0°    │  0°    │  0°    │  0°    │
+│  ru │ -0°   │  45°   │  180°  │ -0°    │  45°   │  0°    │
+│  rd │ -0°   │ -47.8° │  5.39° │ -180°  │  47.6° │  180°  │
+│  lu │  152° │ -225°  │  5.39° │  145°  │  55.8° │  21.4° │
+│  ld │  152° │ -132°  │  180°  │  38.6° │  49.3° │  152°  │
+│  qs │  0°   │  0°    │ -90°   │  0°    │  0°    │  0°    │
+│  qn │  0°   │  45°   │  180°  │  0°    │  45°   │  0°    │
+└─────┴───────┴────────┴────────┴────────┴────────┴────────┘
+
+
+
+

Defined joint configurations are:

+
    +
  • qz, zero joint angle configuration, ‘L’ shaped configuration

  • +
  • qr, vertical ‘READY’ configuration

  • +
  • qs, arm is stretched out in the x-direction

  • +
  • qn, arm is at a nominal non-singular configuration

  • +
+
+

Warning

+

This file has been modified so that the zero-angle pose is the +same as the DH model in the toolbox. j3 rotation is changed from +-𝜋/2 to 𝜋/2. Dimensions are also slightly different. Both models +include the pedestal height.

+
+ +

Code author: Jesse Haviland

+

Section author: Peter Corke

+
+ +
+
+class roboticstoolbox.models.URDF.px100[source]
+

Bases: Robot

+

Class that imports a PX100 URDF model

+

px100() is a class which imports an Interbotix px100 robot definition +from a URDF file. The model describes its kinematic and graphical +characteristics.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.URDF.px100()
+>>> print(robot)
+ERobot: px100 (by Interbotix), 7 joints (RRRRRPP), 4 branches, dynamics, geometry, collision
+┌─────┬─────────────────────┬───────┬───────────────────┬────────────────────────────────────────────┐
+│link │        link         │ joint │      parent       │            ETS: parent to link             │
+├─────┼─────────────────────┼───────┼───────────────────┼────────────────────────────────────────────┤
+│   0 │ /base_link          │       │ BASE              │ SE3()                                      │
+│   1 │ /shoulder_link      │     0 │ /base_link        │ SE3(0, 0, 0.0508) ⊕ Rz(q0)                 │
+│   2 │ /upper_arm_link     │     1 │ /shoulder_link    │ SE3(0, 0, 0.04225) ⊕ Ry(q1)                │
+│   3 │ /forearm_link       │     2 │ /upper_arm_link   │ SE3(0.035, 0, 0.1; 180°, -0°, 0°) ⊕ Ry(q2) │
+│   4 │ /gripper_link       │     3 │ /forearm_link     │ SE3(0.1, 0, 0) ⊕ Ry(q3)                    │
+│   5 │ /ee_arm_link        │       │ /gripper_link     │ SE3(0.063, 0, 0; -180°, -0°, 0°)           │
+│   6 │ @/gripper_prop_link │     4 │ /ee_arm_link      │ SE3(0.0055, 0, 0) ⊕ Rx(q4)                 │
+│   7 │ /gripper_bar_link   │       │ /ee_arm_link      │ SE3()                                      │
+│   8 │ /fingers_link       │       │ /gripper_bar_link │ SE3(0.023, 0, 0)                           │
+│   9 │ @/left_finger_link  │     5 │ /fingers_link     │ SE3() ⊕ ty(q5)                             │
+│  10 │ @/right_finger_link │     6 │ /fingers_link     │ SE3() ⊕ ty(q6)                             │
+│  11 │ @/ee_gripper_link   │       │ /fingers_link     │ SE3(0.02757, 0, 0)                         │
+└─────┴─────────────────────┴───────┴───────────────────┴────────────────────────────────────────────┘
+
+┌─────┬─────┬────────┬─────┬───────┬─────┬────┬────────┐
+│name │ q0  │ q1     │ q2  │ q3    │ q4  │ q5 │ q6     │
+├─────┼─────┼────────┼─────┼───────┼─────┼────┼────────┤
+│  qr │  0° │ -17.2° │  0° │ -126° │  0° │  2 │  0.785 │
+│  qz │  0° │  0°    │  0° │  0°   │  0° │  0 │  0     │
+└─────┴─────┴────────┴─────┴───────┴─────┴────┴────────┘
+
+
+
+

Defined joint configurations are:

+
    +
  • qz, zero joint angle configuration, ‘L’ shaped configuration

  • +
  • qr, vertical ‘READY’ configuration

  • +
+
+
Reference:
+
+
+
+

Code author: Jesse Haviland

+

Section author: Peter Corke

+
+ +
+
+class roboticstoolbox.models.URDF.px150[source]
+

Bases: Robot

+

Class that imports a PX150 URDF model

+

px150() is a class which imports an Interbotix px150 robot definition +from a URDF file. The model describes its kinematic and graphical +characteristics.

+
IndexError: index 7 is out of bounds for axis 0 with size 7
+
+
+

Defined joint configurations are:

+
    +
  • qz, zero joint angle configuration, ‘L’ shaped configuration

  • +
  • qr, vertical ‘READY’ configuration

  • +
+
+
Reference:
+
+
+
+

Code author: Jesse Haviland

+

Section author: Peter Corke

+
+ +
+
+class roboticstoolbox.models.URDF.rx150[source]
+

Bases: Robot

+

Class that imports a RX150 URDF model

+

rx150() is a class which imports an Interbotix rx150 robot definition +from a URDF file. The model describes its kinematic and graphical +characteristics.

+
IndexError: index 7 is out of bounds for axis 0 with size 7
+
+
+

Defined joint configurations are:

+
    +
  • qz, zero joint angle configuration, ‘L’ shaped configuration

  • +
  • qr, vertical ‘READY’ configuration

  • +
+
+
Reference:
+
+
+
+

Code author: Jesse Haviland

+

Section author: Peter Corke

+
+ +
+
+class roboticstoolbox.models.URDF.rx200[source]
+

Bases: Robot

+

Class that imports a RX200 URDF model

+

rx200() is a class which imports an Interbotix rx200 robot definition +from a URDF file. The model describes its kinematic and graphical +characteristics.

+
IndexError: index 7 is out of bounds for axis 0 with size 7
+
+
+

Defined joint configurations are:

+
    +
  • qz, zero joint angle configuration, ‘L’ shaped configuration

  • +
  • qr, vertical ‘READY’ configuration

  • +
+
+
Reference:
+
+
+
+

Code author: Jesse Haviland

+

Section author: Peter Corke

+
+ +
+
+class roboticstoolbox.models.URDF.vx300[source]
+

Bases: Robot

+

Class that imports a VX300 URDF model

+

vx300() is a class which imports an Interbotix vx300 robot definition +from a URDF file. The model describes its kinematic and graphical +characteristics.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.URDF.vx300()
+>>> print(robot)
+ERobot: vx300 (by Interbotix), 8 joints (RRRRRRPP), 4 branches, dynamics, geometry, collision
+┌─────┬─────────────────────┬───────┬───────────────────┬──────────────────────────────────────────────┐
+│link │        link         │ joint │      parent       │             ETS: parent to link              │
+├─────┼─────────────────────┼───────┼───────────────────┼──────────────────────────────────────────────┤
+│   0 │ /base_link          │       │ BASE              │ SE3()                                        │
+│   1 │ /shoulder_link      │     0 │ /base_link        │ SE3(0, 0, 0.0787) ⊕ Rz(q0)                   │
+│   2 │ /upper_arm_link     │     1 │ /shoulder_link    │ SE3(0, 0, 0.04805) ⊕ Ry(q1)                  │
+│   3 │ /forearm_link       │     2 │ /upper_arm_link   │ SE3(0.05955, 0, 0.3; 180°, -0°, 0°) ⊕ Ry(q2) │
+│   4 │ /wrist_link         │     3 │ /forearm_link     │ SE3(0.3, 0, 0) ⊕ Ry(q3)                      │
+│   5 │ /gripper_link       │     4 │ /wrist_link       │ SE3(0.06974, 0, 0; -180°, -0°, 0°) ⊕ Rx(q4)  │
+│   6 │ /ee_arm_link        │       │ /gripper_link     │ SE3(0.04283, 0, 0)                           │
+│   7 │ @/gripper_prop_link │     5 │ /ee_arm_link      │ SE3(0.005675, 0, 0) ⊕ Rx(q5)                 │
+│   8 │ /gripper_bar_link   │       │ /ee_arm_link      │ SE3()                                        │
+│   9 │ /fingers_link       │       │ /gripper_bar_link │ SE3(0.02587, 0, 0)                           │
+│  10 │ @/left_finger_link  │     6 │ /fingers_link     │ SE3() ⊕ ty(q6)                               │
+│  11 │ @/right_finger_link │     7 │ /fingers_link     │ SE3() ⊕ ty(q7)                               │
+│  12 │ @/ee_gripper_link   │       │ /fingers_link     │ SE3(0.0385, 0, 0)                            │
+└─────┴─────────────────────┴───────┴───────────────────┴──────────────────────────────────────────────┘
+
+┌─────┬─────┬────────┬─────┬───────┬─────┬───────┬────────┬────┐
+│name │ q0  │ q1     │ q2  │ q3    │ q4  │ q5    │ q6     │ q7 │
+├─────┼─────┼────────┼─────┼───────┼─────┼───────┼────────┼────┤
+│  qr │  0° │ -17.2° │  0° │ -126° │  0° │  115° │  0.785 │  0 │
+│  qz │  0° │  0°    │  0° │  0°   │  0° │  0°   │  0     │  0 │
+└─────┴─────┴────────┴─────┴───────┴─────┴───────┴────────┴────┘
+
+
+
+

Defined joint configurations are:

+
    +
  • qz, zero joint angle configuration, ‘L’ shaped configuration

  • +
  • qr, vertical ‘READY’ configuration

  • +
+
+
Reference:
+
+
+
+

Code author: Jesse Haviland

+

Section author: Peter Corke

+
+ +
+
+class roboticstoolbox.models.URDF.vx300s[source]
+

Bases: Robot

+

Class that imports a VX300s URDF model

+

vx300s() is a class which imports an Interbotix vx300s robot definition +from a URDF file. The model describes its kinematic and graphical +characteristics.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.URDF.vx300s()
+>>> print(robot)
+ERobot: vx300s (by Interbotix), 9 joints (RRRRRRRPP), 4 branches, dynamics, geometry, collision
+┌─────┬─────────────────────┬───────┬─────────────────────┬──────────────────────────────────────────────┐
+│link │        link         │ joint │       parent        │             ETS: parent to link              │
+├─────┼─────────────────────┼───────┼─────────────────────┼──────────────────────────────────────────────┤
+│   0 │ /base_link          │       │ BASE                │ SE3()                                        │
+│   1 │ /shoulder_link      │     0 │ /base_link          │ SE3(0, 0, 0.0787) ⊕ Rz(q0)                   │
+│   2 │ /upper_arm_link     │     1 │ /shoulder_link      │ SE3(0, 0, 0.04805) ⊕ Ry(q1)                  │
+│   3 │ /upper_forearm_link │     2 │ /upper_arm_link     │ SE3(0.05955, 0, 0.3; 180°, -0°, 0°) ⊕ Ry(q2) │
+│   4 │ /lower_forearm_link │     3 │ /upper_forearm_link │ SE3(0.2, 0, 0) ⊕ Rx(q3)                      │
+│   5 │ /wrist_link         │     4 │ /lower_forearm_link │ SE3(0.1, 0, 0) ⊕ Ry(q4)                      │
+│   6 │ /gripper_link       │     5 │ /wrist_link         │ SE3(0.06974, 0, 0; -180°, -0°, 0°) ⊕ Rx(q5)  │
+│   7 │ /ee_arm_link        │       │ /gripper_link       │ SE3(0.04283, 0, 0)                           │
+│   8 │ @/gripper_prop_link │     6 │ /ee_arm_link        │ SE3(0.005675, 0, 0) ⊕ Rx(q6)                 │
+│   9 │ /gripper_bar_link   │       │ /ee_arm_link        │ SE3()                                        │
+│  10 │ /fingers_link       │       │ /gripper_bar_link   │ SE3(0.02587, 0, 0)                           │
+│  11 │ @/left_finger_link  │     7 │ /fingers_link       │ SE3() ⊕ ty(q7)                               │
+│  12 │ @/right_finger_link │     8 │ /fingers_link       │ SE3() ⊕ ty(q8)                               │
+│  13 │ @/ee_gripper_link   │       │ /fingers_link       │ SE3(0.0385, 0, 0)                            │
+└─────┴─────────────────────┴───────┴─────────────────────┴──────────────────────────────────────────────┘
+
+┌─────┬─────┬────────┬─────┬───────┬─────┬───────┬──────┬────┬────┐
+│name │ q0  │ q1     │ q2  │ q3    │ q4  │ q5    │ q6   │ q7 │ q8 │
+├─────┼─────┼────────┼─────┼───────┼─────┼───────┼──────┼────┼────┤
+│  qr │  0° │ -17.2° │  0° │ -126° │  0° │  115° │  45° │  0 │  0 │
+│  qz │  0° │  0°    │  0° │  0°   │  0° │  0°   │  0°  │  0 │  0 │
+└─────┴─────┴────────┴─────┴───────┴─────┴───────┴──────┴────┴────┘
+
+
+
+

Defined joint configurations are:

+
    +
  • qz, zero joint angle configuration, ‘L’ shaped configuration

  • +
  • qr, vertical ‘READY’ configuration

  • +
+
+
Reference:
+
+
+
+

Code author: Jesse Haviland

+

Section author: Peter Corke

+
+ +
+
+class roboticstoolbox.models.URDF.wx200[source]
+

Bases: Robot

+

Class that imports a WX200 URDF model

+

wx200() is a class which imports an Interbotix wx200 robot definition +from a URDF file. The model describes its kinematic and graphical +characteristics.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.URDF.wx200()
+>>> print(robot)
+ERobot: wx200 (by Interbotix), 8 joints (RRRRRRPP), 4 branches, dynamics, geometry, collision
+┌─────┬─────────────────────┬───────┬───────────────────┬───────────────────────────────────────────┐
+│link │        link         │ joint │      parent       │            ETS: parent to link            │
+├─────┼─────────────────────┼───────┼───────────────────┼───────────────────────────────────────────┤
+│   0 │ /base_link          │       │ BASE              │ SE3()                                     │
+│   1 │ /shoulder_link      │     0 │ /base_link        │ SE3(0, 0, 0.0716) ⊕ Rz(q0)                │
+│   2 │ /upper_arm_link     │     1 │ /shoulder_link    │ SE3(0, 0, 0.03865) ⊕ Ry(q1)               │
+│   3 │ /forearm_link       │     2 │ /upper_arm_link   │ SE3(0.05, 0, 0.2; 180°, -0°, 0°) ⊕ Ry(q2) │
+│   4 │ /wrist_link         │     3 │ /forearm_link     │ SE3(0.2, 0, 0) ⊕ Ry(q3)                   │
+│   5 │ /gripper_link       │     4 │ /wrist_link       │ SE3(0.065, 0, 0; -180°, -0°, 0°) ⊕ Rx(q4) │
+│   6 │ /ee_arm_link        │       │ /gripper_link     │ SE3(0.043, 0, 0)                          │
+│   7 │ @/gripper_prop_link │     5 │ /ee_arm_link      │ SE3(0.0055, 0, 0) ⊕ Rx(q5)                │
+│   8 │ /gripper_bar_link   │       │ /ee_arm_link      │ SE3()                                     │
+│   9 │ /fingers_link       │       │ /gripper_bar_link │ SE3(0.023, 0, 0)                          │
+│  10 │ @/left_finger_link  │     6 │ /fingers_link     │ SE3() ⊕ ty(q6)                            │
+│  11 │ @/right_finger_link │     7 │ /fingers_link     │ SE3() ⊕ ty(q7)                            │
+│  12 │ @/ee_gripper_link   │       │ /fingers_link     │ SE3(0.02757, 0, 0)                        │
+└─────┴─────────────────────┴───────┴───────────────────┴───────────────────────────────────────────┘
+
+┌─────┬─────┬────────┬─────┬───────┬─────┬───────┬────────┬────┐
+│name │ q0  │ q1     │ q2  │ q3    │ q4  │ q5    │ q6     │ q7 │
+├─────┼─────┼────────┼─────┼───────┼─────┼───────┼────────┼────┤
+│  qr │  0° │ -17.2° │  0° │ -126° │  0° │  115° │  0.785 │  0 │
+│  qz │  0° │  0°    │  0° │  0°   │  0° │  0°   │  0     │  0 │
+└─────┴─────┴────────┴─────┴───────┴─────┴───────┴────────┴────┘
+
+
+
+

Defined joint configurations are:

+
    +
  • qz, zero joint angle configuration, ‘L’ shaped configuration

  • +
  • qr, vertical ‘READY’ configuration

  • +
+
+
Reference:
+
+
+
+

Code author: Jesse Haviland

+

Section author: Peter Corke

+
+ +
+
+class roboticstoolbox.models.URDF.wx250[source]
+

Bases: Robot

+

Class that imports a WX250 URDF model

+

wx250() is a class which imports an Interbotix wx250 robot definition +from a URDF file. The model describes its kinematic and graphical +characteristics.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.URDF.wx250()
+>>> print(robot)
+ERobot: wx250 (by Interbotix), 8 joints (RRRRRRPP), 4 branches, dynamics, geometry, collision
+┌─────┬─────────────────────┬───────┬───────────────────┬───────────────────────────────────────────────┐
+│link │        link         │ joint │      parent       │              ETS: parent to link              │
+├─────┼─────────────────────┼───────┼───────────────────┼───────────────────────────────────────────────┤
+│   0 │ /base_link          │       │ BASE              │ SE3()                                         │
+│   1 │ /shoulder_link      │     0 │ /base_link        │ SE3(0, 0, 0.0716) ⊕ Rz(q0)                    │
+│   2 │ /upper_arm_link     │     1 │ /shoulder_link    │ SE3(0, 0, 0.03865) ⊕ Ry(q1)                   │
+│   3 │ /forearm_link       │     2 │ /upper_arm_link   │ SE3(0.04975, 0, 0.25; 180°, -0°, 0°) ⊕ Ry(q2) │
+│   4 │ /wrist_link         │     3 │ /forearm_link     │ SE3(0.25, 0, 0) ⊕ Ry(q3)                      │
+│   5 │ /gripper_link       │     4 │ /wrist_link       │ SE3(0.065, 0, 0; -180°, -0°, 0°) ⊕ Rx(q4)     │
+│   6 │ /ee_arm_link        │       │ /gripper_link     │ SE3(0.043, 0, 0)                              │
+│   7 │ @/gripper_prop_link │     5 │ /ee_arm_link      │ SE3(0.0055, 0, 0) ⊕ Rx(q5)                    │
+│   8 │ /gripper_bar_link   │       │ /ee_arm_link      │ SE3()                                         │
+│   9 │ /fingers_link       │       │ /gripper_bar_link │ SE3(0.023, 0, 0)                              │
+│  10 │ @/left_finger_link  │     6 │ /fingers_link     │ SE3() ⊕ ty(q6)                                │
+│  11 │ @/right_finger_link │     7 │ /fingers_link     │ SE3() ⊕ ty(q7)                                │
+│  12 │ @/ee_gripper_link   │       │ /fingers_link     │ SE3(0.02757, 0, 0)                            │
+└─────┴─────────────────────┴───────┴───────────────────┴───────────────────────────────────────────────┘
+
+┌─────┬─────┬────────┬─────┬───────┬─────┬───────┬────────┬────┐
+│name │ q0  │ q1     │ q2  │ q3    │ q4  │ q5    │ q6     │ q7 │
+├─────┼─────┼────────┼─────┼───────┼─────┼───────┼────────┼────┤
+│  qr │  0° │ -17.2° │  0° │ -126° │  0° │  115° │  0.785 │  0 │
+│  qz │  0° │  0°    │  0° │  0°   │  0° │  0°   │  0     │  0 │
+└─────┴─────┴────────┴─────┴───────┴─────┴───────┴────────┴────┘
+
+
+
+

Defined joint configurations are:

+
    +
  • qz, zero joint angle configuration, ‘L’ shaped configuration

  • +
  • qr, vertical ‘READY’ configuration

  • +
+
+
Reference:
+
+
+
+

Code author: Jesse Haviland

+

Section author: Peter Corke

+
+ +
+
+class roboticstoolbox.models.URDF.wx250s[source]
+

Bases: Robot

+

Class that imports a wx250s URDF model

+

wx250s() is a class which imports an Interbotix wx250s robot definition +from a URDF file. The model describes its kinematic and graphical +characteristics.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.URDF.wx250s()
+>>> print(robot)
+ERobot: wx250s (by Interbotix), 9 joints (RRRRRRRPP), 4 branches, dynamics, geometry, collision
+┌─────┬─────────────────────┬───────┬─────────────────────┬───────────────────────────────────────────────┐
+│link │        link         │ joint │       parent        │              ETS: parent to link              │
+├─────┼─────────────────────┼───────┼─────────────────────┼───────────────────────────────────────────────┤
+│   0 │ /base_link          │       │ BASE                │ SE3()                                         │
+│   1 │ /shoulder_link      │     0 │ /base_link          │ SE3(0, 0, 0.0716) ⊕ Rz(q0)                    │
+│   2 │ /upper_arm_link     │     1 │ /shoulder_link      │ SE3(0, 0, 0.03865) ⊕ Ry(q1)                   │
+│   3 │ /upper_forearm_link │     2 │ /upper_arm_link     │ SE3(0.04975, 0, 0.25; 180°, -0°, 0°) ⊕ Ry(q2) │
+│   4 │ /lower_forearm_link │     3 │ /upper_forearm_link │ SE3(0.175, 0, 0) ⊕ Rx(q3)                     │
+│   5 │ /wrist_link         │     4 │ /lower_forearm_link │ SE3(0.075, 0, 0) ⊕ Ry(q4)                     │
+│   6 │ /gripper_link       │     5 │ /wrist_link         │ SE3(0.065, 0, 0; -180°, -0°, 0°) ⊕ Rx(q5)     │
+│   7 │ /ee_arm_link        │       │ /gripper_link       │ SE3(0.043, 0, 0)                              │
+│   8 │ @/gripper_prop_link │     6 │ /ee_arm_link        │ SE3(0.0055, 0, 0) ⊕ Rx(q6)                    │
+│   9 │ /gripper_bar_link   │       │ /ee_arm_link        │ SE3()                                         │
+│  10 │ /fingers_link       │       │ /gripper_bar_link   │ SE3(0.023, 0, 0)                              │
+│  11 │ @/left_finger_link  │     7 │ /fingers_link       │ SE3() ⊕ ty(q7)                                │
+│  12 │ @/right_finger_link │     8 │ /fingers_link       │ SE3() ⊕ ty(q8)                                │
+│  13 │ @/ee_gripper_link   │       │ /fingers_link       │ SE3(0.02757, 0, 0)                            │
+└─────┴─────────────────────┴───────┴─────────────────────┴───────────────────────────────────────────────┘
+
+┌─────┬─────┬────────┬─────┬───────┬─────┬───────┬──────┬────┬────┐
+│name │ q0  │ q1     │ q2  │ q3    │ q4  │ q5    │ q6   │ q7 │ q8 │
+├─────┼─────┼────────┼─────┼───────┼─────┼───────┼──────┼────┼────┤
+│  qr │  0° │ -17.2° │  0° │ -126° │  0° │  115° │  45° │  0 │  0 │
+│  qz │  0° │  0°    │  0° │  0°   │  0° │  0°   │  0°  │  0 │  0 │
+└─────┴─────┴────────┴─────┴───────┴─────┴───────┴──────┴────┴────┘
+
+
+
+

Defined joint configurations are:

+
    +
  • qz, zero joint angle configuration, ‘L’ shaped configuration

  • +
  • qr, vertical ‘READY’ configuration

  • +
+
+
Reference:
+
+
+
+

Code author: Jesse Haviland

+

Section author: Peter Corke

+
+ +
+
+class roboticstoolbox.models.URDF.Mico[source]
+

Bases: Robot

+

Class that imports a Mico URDF model

+

Panda() is a class which imports a Kinova Mico robot definition +from a URDF file. The model describes its kinematic and graphical +characteristics.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.URDF.Mico()
+>>> print(robot)
+ERobot: j2n4s300 (by Kinova), 4 joints (RRRR), 1 gripper, dynamics, geometry, collision
+┌─────┬────────────────────┬───────┬────────────────────┬────────────────────────────────────────────────────┐
+│link │        link        │ joint │       parent       │                ETS: parent to link                 │
+├─────┼────────────────────┼───────┼────────────────────┼────────────────────────────────────────────────────┤
+│   0 │ world              │       │ BASE               │ SE3()                                              │
+│   1 │ root               │       │ world              │ SE3()                                              │
+│   2 │ j2n4s300_link_base │       │ root               │ SE3()                                              │
+│   3 │ j2n4s300_link_1    │     0 │ j2n4s300_link_base │ SE3(0, 0, 0.1568; 180°, 7.017e-15°, 180°) ⊕ Rz(q0) │
+│   4 │ j2n4s300_link_2    │     1 │ j2n4s300_link_1    │ SE3(0, 0.0016, -0.1187; -90°, -0°, 180°) ⊕ Rz(q1)  │
+│   5 │ j2n4s300_link_3    │     2 │ j2n4s300_link_2    │ SE3(0, -0.41, 0; 180°, 7.017e-15°, 180°) ⊕ Rz(q2)  │
+│   6 │ @j2n4s300_link_4   │     3 │ j2n4s300_link_3    │ SE3(0, 0.2073, -0.0114; -90°, -0°, 180°) ⊕ Rz(q3)  │
+└─────┴────────────────────┴───────┴────────────────────┴────────────────────────────────────────────────────┘
+
+┌─────┬─────┬──────┬──────┬─────┐
+│name │ q0  │ q1   │ q2   │ q3  │
+├─────┼─────┼──────┼──────┼─────┤
+│  qr │  0° │  45° │  60° │  0° │
+│  qz │  0° │  0°  │  0°  │  0° │
+└─────┴─────┴──────┴──────┴─────┘
+
+
+
+

Defined joint configurations are:

+
    +
  • qz, zero joint angle configuration, ‘L’ shaped configuration

  • +
  • qr, vertical ‘READY’ configuration

  • +
+

Code author: Jesse Haviland

+

Section author: Peter Corke

+
+ +
+
+class roboticstoolbox.models.URDF.PR2[source]
+

Bases: Robot

+
+ +
+
+class roboticstoolbox.models.URDF.LBR[source]
+

Bases: Robot

+

Class that imports a LBR URDF model

+

LBR() is a class which imports a Franka-Emika LBR robot definition +from a URDF file. The model describes its kinematic and graphical +characteristics.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.URDF.LBR()
+>>> print(robot)
+ERobot: kuka_lbr_iiwa_14_r820 (by Kuka), 7 joints (RRRRRRR), 2 branches, geometry, collision
+┌─────┬───────────┬───────┬───────────┬───────────────────────────────────┐
+│link │   link    │ joint │  parent   │        ETS: parent to link        │
+├─────┼───────────┼───────┼───────────┼───────────────────────────────────┤
+│   0 │ base_link │       │ BASE      │ SE3()                             │
+│   1 │ link_1    │     0 │ base_link │ SE3() ⊕ Rz(q0)                    │
+│   2 │ link_2    │     1 │ link_1    │ SE3(-0.0004362, 0, 0.36) ⊕ Ry(q1) │
+│   3 │ link_3    │     2 │ link_2    │ SE3() ⊕ Rz(q2)                    │
+│   4 │ link_4    │     3 │ link_3    │ SE3(0.0004362, 0, 0.42) ⊕ Ry(-q3) │
+│   5 │ link_5    │     4 │ link_4    │ SE3() ⊕ Rz(q4)                    │
+│   6 │ link_6    │     5 │ link_5    │ SE3(0, 0, 0.4) ⊕ Ry(q5)           │
+│   7 │ link_7    │     6 │ link_6    │ SE3() ⊕ Rz(q6)                    │
+│   8 │ @tool0    │       │ link_7    │ SE3(0, 0, 0.126)                  │
+│   9 │ @base     │       │ base_link │ SE3()                             │
+└─────┴───────────┴───────┴───────────┴───────────────────────────────────┘
+
+┌─────┬─────┬────────┬─────┬───────┬─────┬────────┬──────┐
+│name │ q0  │ q1     │ q2  │ q3    │ q4  │ q5     │ q6   │
+├─────┼─────┼────────┼─────┼───────┼─────┼────────┼──────┤
+│  qr │  0° │ -17.2° │  0° │ -109° │  0° │  85.9° │  45° │
+│  qz │  0° │  0°    │  0° │  0°   │  0° │  0°    │  0°  │
+└─────┴─────┴────────┴─────┴───────┴─────┴────────┴──────┘
+
+
+
+

Defined joint configurations are:

+
    +
  • qz, zero joint angle configuration, ‘L’ shaped configuration

  • +
  • qr, vertical ‘READY’ configuration

  • +
  • qs, arm is stretched out in the x-direction

  • +
  • qn, arm is at a nominal non-singular configuration

  • +
+

Code author: Jesse Haviland

+

Section author: Peter Corke

+
+ +
+
+class roboticstoolbox.models.URDF.KinovaGen3[source]
+

Bases: Robot

+

Class that imports a KinovaGen3 URDF model

+

KinovaGen3() is a class which imports a KinovaGen3 robot definition +from a URDF file. The model describes its kinematic and graphical +characteristics.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.URDF.KinovaGen3()
+>>> print(robot)
+ERobot: gen3 (by Kinova), 7 joints (RRRRRRR), dynamics, geometry, collision
+┌─────┬────────────────────────┬───────┬────────────────────────┬────────────────────────────────────────────────────────────────────┐
+│link │          link          │ joint │         parent         │                        ETS: parent to link                         │
+├─────┼────────────────────────┼───────┼────────────────────────┼────────────────────────────────────────────────────────────────────┤
+│   0 │ base_link              │       │ BASE                   │ SE3()                                                              │
+│   1 │ shoulder_link          │     0 │ base_link              │ SE3(0, 0, 0.1564; -180°, 1.583e-16°, -2.825e-34°) ⊕ Rz(q0)         │
+│   2 │ half_arm_1_link        │     1 │ shoulder_link          │ SE3(0, 0.005375, -0.1284; 90°, 1.223e-15°, -6.361e-15°) ⊕ Rz(q1)   │
+│   3 │ half_arm_2_link        │     2 │ half_arm_1_link        │ SE3(0, -0.2104, -0.006375; -90°, 7.062e-31°, -1.669e-14°) ⊕ Rz(q2) │
+│   4 │ forearm_link           │     3 │ half_arm_2_link        │ SE3(0, 0.006375, -0.2104; 90°, -3.836e-15°, -9.541e-15°) ⊕ Rz(q3)  │
+│   5 │ spherical_wrist_1_link │     4 │ forearm_link           │ SE3(0, -0.2084, -0.006375; -90°, 1.272e-14°, -3.651e-15°) ⊕ Rz(q4) │
+│   6 │ spherical_wrist_2_link │     5 │ spherical_wrist_1_link │ SE3(0, 0.000175, -0.1059; 90°, 5.276e-26°, -4.707e-13°) ⊕ Rz(q5)   │
+│   7 │ bracelet_link          │     6 │ spherical_wrist_2_link │ SE3(0, -0.1059, -0.000175; -90°, -3.181e-15°, 5.523e-15°) ⊕ Rz(q6) │
+│   8 │ end_effector_link      │       │ bracelet_link          │ SE3(0, 0, -0.06153; 180°, 6.299e-31°, 0°)                          │
+│   9 │ @tool_frame            │       │ end_effector_link      │ SE3()                                                              │
+└─────┴────────────────────────┴───────┴────────────────────────┴────────────────────────────────────────────────────────────────────┘
+
+┌─────┬───────┬────────┬─────┬────────┬─────┬────────┬──────┐
+│name │ q0    │ q1     │ q2  │ q3     │ q4  │ q5     │ q6   │
+├─────┼───────┼────────┼─────┼────────┼─────┼────────┼──────┤
+│  qr │  180° │ -17.2° │  0° │ -91.7° │  0° │ -57.3° │  90° │
+│  qz │  0°   │  0°    │  0° │  0°    │  0° │  0°    │  0°  │
+└─────┴───────┴────────┴─────┴────────┴─────┴────────┴──────┘
+
+
+
+

Defined joint configurations are:

+
    +
  • qz, zero joint angle configuration, ‘L’ shaped configuration

  • +
  • qr, vertical ‘READY’ configuration

  • +
  • qs, arm is stretched out in the x-direction

  • +
  • qn, arm is at a nominal non-singular configuration

  • +
+

Code author: Jesse Haviland

+

Section author: Peter Corke

+
+ +
+
+class roboticstoolbox.models.URDF.YuMi[source]
+

Bases: Robot

+

Class that imports an ABB YuMi URDF model

+

YuMi() is a class which imports an ABB YuMi (IRB14000) robot definition +from a URDF file. The model describes its kinematic and graphical +characteristics.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.URDF.YuMi()
+>>> print(robot)
+ERobot: yumi (by ABB), 14 joints (RRRRRRRRRRRRRR), 2 grippers, 2 branches, dynamics, geometry, collision
+┌─────┬─────────────────┬───────┬────────────────┬───────────────────────────────────────────────────────────────────┐
+│link │      link       │ joint │     parent     │                        ETS: parent to link                        │
+├─────┼─────────────────┼───────┼────────────────┼───────────────────────────────────────────────────────────────────┤
+│   0 │ world           │       │ BASE           │ SE3()                                                             │
+│   1 │ yumi_base_link  │       │ world          │ SE3(0, 0, 0.1)                                                    │
+│   2 │ yumi_body       │       │ yumi_base_link │ SE3()                                                             │
+│   3 │ yumi_link_1_r   │     0 │ yumi_body      │ SE3(0.05355, -0.0725, 0.4149; -56.12°, -32.56°, -132.7°) ⊕ Rz(q0) │
+│   4 │ yumi_link_2_r   │     1 │ yumi_link_1_r  │ SE3(0.03, 0, 0.1; 90°, -0°, 0°) ⊕ Rz(q1)                          │
+│   5 │ yumi_link_3_r   │     2 │ yumi_link_2_r  │ SE3(-0.03, 0.1728, 0; -90°, -0°, 0°) ⊕ Rz(q2)                     │
+│   6 │ yumi_link_4_r   │     3 │ yumi_link_3_r  │ SE3(-0.04188, 0, 0.07873; 0°, -90°, 90°) ⊕ Rz(q3)                 │
+│   7 │ yumi_link_5_r   │     4 │ yumi_link_4_r  │ SE3(0.0405, 0.1646, 0; -90°, -0°, 0°) ⊕ Rz(q4)                    │
+│   8 │ yumi_link_6_r   │     5 │ yumi_link_5_r  │ SE3(-0.027, 0, 0.1004; 90°, -0°, 0°) ⊕ Rz(q5)                     │
+│   9 │ yumi_link_7_r   │     6 │ yumi_link_6_r  │ SE3(0.027, 0.029, 0; -90°, -0°, 0°) ⊕ Rz(q6)                      │
+│  10 │ @gripper_r_base │       │ yumi_link_7_r  │ SE3(0, 0, 0.007; 0°, -0°, -180°)                                  │
+│  11 │ yumi_link_1_l   │     7 │ yumi_body      │ SE3(0.05355, 0.0725, 0.4149; 56.04°, -32.75°, 132.8°) ⊕ Rz(q7)    │
+│  12 │ yumi_link_2_l   │     8 │ yumi_link_1_l  │ SE3(0.03, 0, 0.1; 90°, -0°, 0°) ⊕ Rz(q8)                          │
+│  13 │ yumi_link_3_l   │     9 │ yumi_link_2_l  │ SE3(-0.03, 0.1728, 0; -90°, -0°, 0°) ⊕ Rz(q9)                     │
+│  14 │ yumi_link_4_l   │    10 │ yumi_link_3_l  │ SE3(-0.04188, 0, 0.07873; 0°, -90°, 90°) ⊕ Rz(q10)                │
+│  15 │ yumi_link_5_l   │    11 │ yumi_link_4_l  │ SE3(0.0405, 0.1646, 0; -90°, -0°, 0°) ⊕ Rz(q11)                   │
+│  16 │ yumi_link_6_l   │    12 │ yumi_link_5_l  │ SE3(-0.027, 0, 0.1004; 90°, -0°, 0°) ⊕ Rz(q12)                    │
+│  17 │ yumi_link_7_l   │    13 │ yumi_link_6_l  │ SE3(0.027, 0.029, 0; -90°, -0°, 0°) ⊕ Rz(q13)                     │
+│  18 │ @gripper_l_base │       │ yumi_link_7_l  │ SE3(0, 0, 0.007; 0°, -0°, -180°)                                  │
+└─────┴─────────────────┴───────┴────────────────┴───────────────────────────────────────────────────────────────────┘
+
+┌─────┬─────┬────────┬─────┬───────┬─────┬───────┬──────┬─────┬────────┬─────┬───────┬─────┬───────┬──────┐
+│name │ q0  │ q1     │ q2  │ q3    │ q4  │ q5    │ q6   │ q7  │ q8     │ q9  │ q10   │ q11 │ q12   │ q13  │
+├─────┼─────┼────────┼─────┼───────┼─────┼───────┼──────┼─────┼────────┼─────┼───────┼─────┼───────┼──────┤
+│  qr │  0° │ -17.2° │  0° │ -126° │  0° │  115° │  45° │  0° │ -17.2° │  0° │ -126° │  0° │  115° │  45° │
+│  qz │  0° │  0°    │  0° │  0°   │  0° │  0°   │  0°  │  0° │  0°    │  0° │  0°   │  0° │  0°   │  0°  │
+│  q1 │  0° │ -22.9° │  0° │  0°   │  0° │  0°   │  0°  │  0° │ -22.9° │  0° │  0°   │  0° │  0°   │  0°  │
+└─────┴─────┴────────┴─────┴───────┴─────┴───────┴──────┴─────┴────────┴─────┴───────┴─────┴───────┴──────┘
+
+
+
+

Defined joint configurations are:

+
    +
  • qz, zero joint angle configuration, ‘L’ shaped configuration

  • +
  • qr, vertical ‘READY’ configuration

  • +
+
+
Reference:
+
+
+
+

Code author: Jesse Haviland

+

Section author: Peter Corke

+
+ +
+
+class roboticstoolbox.models.URDF.Fetch[source]
+

Bases: Robot

+

Class that imports a Fetch URDF model

+

Fetch() is a class which imports a Fetch robot definition +from a URDF file. The model describes its kinematic and graphical +characteristics.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.URDF.Fetch()
+>>> print(robot)
+ERobot: fetch (by Fetch), 10 joints (RPPRRRRRRR), 1 gripper, 6 branches, dynamics, geometry, collision
+┌─────┬────────────────────┬───────┬────────────────────┬─────────────────────────────────────────────────────────┐
+│link │        link        │ joint │       parent       │                   ETS: parent to link                   │
+├─────┼────────────────────┼───────┼────────────────────┼─────────────────────────────────────────────────────────┤
+│   0 │ base0              │       │ BASE               │ SE3()                                                   │
+│   1 │ base1              │     0 │ base0              │ SE3() ⊕ Rz(q0)                                          │
+│   2 │ base_link          │     1 │ base1              │ SE3() ⊕ tx(q1)                                          │
+│   3 │ torso_lift_link    │     2 │ base_link          │ SE3(-0.08687, 0, 0.3774; -3.508e-15°, -0°, 0°) ⊕ tz(q2) │
+│   4 │ shoulder_pan_link  │     3 │ torso_lift_link    │ SE3(0.1195, 0, 0.3486) ⊕ Rz(q3)                         │
+│   5 │ shoulder_lift_link │     4 │ shoulder_pan_link  │ SE3(0.117, 0, 0.06) ⊕ Ry(q4)                            │
+│   6 │ upperarm_roll_link │     5 │ shoulder_lift_link │ SE3(0.219, 0, 0) ⊕ Rx(q5)                               │
+│   7 │ elbow_flex_link    │     6 │ upperarm_roll_link │ SE3(0.133, 0, 0) ⊕ Ry(q6)                               │
+│   8 │ forearm_roll_link  │     7 │ elbow_flex_link    │ SE3(0.197, 0, 0) ⊕ Rx(q7)                               │
+│   9 │ wrist_flex_link    │     8 │ forearm_roll_link  │ SE3(0.1245, 0, 0) ⊕ Ry(q8)                              │
+│  10 │ @wrist_roll_link   │     9 │ wrist_flex_link    │ SE3(0.1385, 0, 0) ⊕ Rx(q9)                              │
+│  11 │ bellows_link       │       │ torso_lift_link    │ SE3()                                                   │
+│  12 │ bellows_link2      │       │ torso_lift_link    │ SE3()                                                   │
+│  13 │ estop_link         │       │ base_link          │ SE3(-0.1246, 0.2389, 0.3113; 90°, -0°, 0°)              │
+│  14 │ laser_link         │       │ base_link          │ SE3(0.235, 0, 0.2878; -180°, -0°, 0°)                   │
+│  15 │ torso_fixed_link   │       │ base_link          │ SE3(-0.08687, 0, 0.3774; -3.508e-15°, -0°, 0°)          │
+└─────┴────────────────────┴───────┴────────────────────┴─────────────────────────────────────────────────────────┘
+
+┌─────┬─────┬────┬───────┬────────┬────────┬────────┬────────┬─────┬────────┬─────┐
+│name │ q0  │ q1 │ q2    │ q3     │ q4     │ q5     │ q6     │ q7  │ q8     │ q9  │
+├─────┼─────┼────┼───────┼────────┼────────┼────────┼────────┼─────┼────────┼─────┤
+│  qr │  0° │  0 │  0.05 │  75.6° │  80.2° │ -11.5° │  98.5° │  0° │  95.1° │  0° │
+│  qz │  0° │  0 │  0    │  0°    │  0°    │  0°    │  0°    │  0° │  0°    │  0° │
+└─────┴─────┴────┴───────┴────────┴────────┴────────┴────────┴─────┴────────┴─────┘
+
+
+
+

Defined joint configurations are:

+
    +
  • qz, zero joint angle configuration, arm is stretched out in the x-direction

  • +
  • qr, tucked arm configuration

  • +
+

Code author: Kerry He

+

Section author: Peter Corke

+
+ +
+
+class roboticstoolbox.models.URDF.FetchCamera[source]
+

Bases: Robot

+

Class that imports a FetchCamera URDF model

+

FetchCamera() is a class which imports a FetchCamera robot definition +from a URDF file. The model describes its kinematic and graphical +characteristics.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.URDF.Fetch()
+>>> print(robot)
+ERobot: fetch (by Fetch), 10 joints (RPPRRRRRRR), 1 gripper, 6 branches, dynamics, geometry, collision
+┌─────┬────────────────────┬───────┬────────────────────┬─────────────────────────────────────────────────────────┐
+│link │        link        │ joint │       parent       │                   ETS: parent to link                   │
+├─────┼────────────────────┼───────┼────────────────────┼─────────────────────────────────────────────────────────┤
+│   0 │ base0              │       │ BASE               │ SE3()                                                   │
+│   1 │ base1              │     0 │ base0              │ SE3() ⊕ Rz(q0)                                          │
+│   2 │ base_link          │     1 │ base1              │ SE3() ⊕ tx(q1)                                          │
+│   3 │ torso_lift_link    │     2 │ base_link          │ SE3(-0.08687, 0, 0.3774; -3.508e-15°, -0°, 0°) ⊕ tz(q2) │
+│   4 │ shoulder_pan_link  │     3 │ torso_lift_link    │ SE3(0.1195, 0, 0.3486) ⊕ Rz(q3)                         │
+│   5 │ shoulder_lift_link │     4 │ shoulder_pan_link  │ SE3(0.117, 0, 0.06) ⊕ Ry(q4)                            │
+│   6 │ upperarm_roll_link │     5 │ shoulder_lift_link │ SE3(0.219, 0, 0) ⊕ Rx(q5)                               │
+│   7 │ elbow_flex_link    │     6 │ upperarm_roll_link │ SE3(0.133, 0, 0) ⊕ Ry(q6)                               │
+│   8 │ forearm_roll_link  │     7 │ elbow_flex_link    │ SE3(0.197, 0, 0) ⊕ Rx(q7)                               │
+│   9 │ wrist_flex_link    │     8 │ forearm_roll_link  │ SE3(0.1245, 0, 0) ⊕ Ry(q8)                              │
+│  10 │ @wrist_roll_link   │     9 │ wrist_flex_link    │ SE3(0.1385, 0, 0) ⊕ Rx(q9)                              │
+│  11 │ bellows_link       │       │ torso_lift_link    │ SE3()                                                   │
+│  12 │ bellows_link2      │       │ torso_lift_link    │ SE3()                                                   │
+│  13 │ estop_link         │       │ base_link          │ SE3(-0.1246, 0.2389, 0.3113; 90°, -0°, 0°)              │
+│  14 │ laser_link         │       │ base_link          │ SE3(0.235, 0, 0.2878; -180°, -0°, 0°)                   │
+│  15 │ torso_fixed_link   │       │ base_link          │ SE3(-0.08687, 0, 0.3774; -3.508e-15°, -0°, 0°)          │
+└─────┴────────────────────┴───────┴────────────────────┴─────────────────────────────────────────────────────────┘
+
+┌─────┬─────┬────┬───────┬────────┬────────┬────────┬────────┬─────┬────────┬─────┐
+│name │ q0  │ q1 │ q2    │ q3     │ q4     │ q5     │ q6     │ q7  │ q8     │ q9  │
+├─────┼─────┼────┼───────┼────────┼────────┼────────┼────────┼─────┼────────┼─────┤
+│  qr │  0° │  0 │  0.05 │  75.6° │  80.2° │ -11.5° │  98.5° │  0° │  95.1° │  0° │
+│  qz │  0° │  0 │  0    │  0°    │  0°    │  0°    │  0°    │  0° │  0°    │  0° │
+└─────┴─────┴────┴───────┴────────┴────────┴────────┴────────┴─────┴────────┴─────┘
+
+
+
+

Defined joint configurations are:

+
    +
  • qz, zero joint angle configuration

  • +
  • qr, zero joint angle configuration

  • +
+

Code author: Kerry He

+

Section author: Peter Corke

+
+ +
+
+class roboticstoolbox.models.URDF.Valkyrie(variant='A')[source]
+

Bases: Robot

+

Class that imports a NASA Valkyrie URDF model

+

Valkyrie() is a class which imports a NASA Valkyrie robot definition +from a URDF file. The model describes its kinematic and graphical +characteristics.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.URDF.Valkyrie()
+>>> print(robot)
+ERobot: valkyrie (by NASA), 58 joints (RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR), 18 branches, dynamics, geometry, collision
+┌─────┬──────────────────────────────┬───────┬─────────────────────────────┬──────────────────────────────────────────────────────────┐
+│link │             link             │ joint │           parent            │                   ETS: parent to link                    │
+├─────┼──────────────────────────────┼───────┼─────────────────────────────┼──────────────────────────────────────────────────────────┤
+│   0 │ pelvis                       │       │ BASE                        │ SE3()                                                    │
+│   1 │ torsoYawLink                 │     0 │ pelvis                      │ SE3() ⊕ Rz(q0)                                           │
+│   2 │ torsoPitchLink               │     1 │ torsoYawLink                │ SE3(0.04191, 0, 0) ⊕ Ry(q1)                              │
+│   3 │ torso                        │     2 │ torsoPitchLink              │ SE3(0, 0, 0.0203) ⊕ Rx(q2)                               │
+│   4 │ lowerNeckPitchLink           │     3 │ torso                       │ SE3(0.02035, 0, 0.3384) ⊕ Ry(q3)                         │
+│   5 │ neckYawLink                  │     4 │ lowerNeckPitchLink          │ SE3(-0.05192, 0, 0) ⊕ Rz(q4)                             │
+│   6 │ upperNeckPitchLink           │     5 │ neckYawLink                 │ SE3(-0.06, 0, 0.196) ⊕ Ry(q5)                            │
+│   7 │ @multisense_root_link        │       │ upperNeckPitchLink          │ SE3(0.1836, 0, 0.07535; -180°, 7.5°, 0°)                 │
+│   8 │ rightShoulderPitchLink       │     6 │ torso                       │ SE3(-0.0316, 0, 0.2984) ⊕ Ry(q6)                         │
+│   9 │ rightShoulderRollLink        │     7 │ rightShoulderPitchLink      │ SE3(0, -0.2499, 0) ⊕ Rx(q7)                              │
+│  10 │ rightShoulderYawLink         │     8 │ rightShoulderRollLink       │ SE3() ⊕ Ry(q8)                                           │
+│  11 │ rightElbowPitchLink          │     9 │ rightShoulderYawLink        │ SE3(0.0254, -0.33, 0) ⊕ Rz(q9)                           │
+│  12 │ rightForearmLink             │    10 │ rightElbowPitchLink         │ SE3(-0.0254, 0, 0) ⊕ Ry(q10)                             │
+│  13 │ rightWristRollLink           │    11 │ rightForearmLink            │ SE3(0, -0.2871, 0) ⊕ Rx(q11)                             │
+│  14 │ rightPalm                    │    12 │ rightWristRollLink          │ SE3() ⊕ Rz(q12)                                          │
+│  15 │ rightThumbRollLink           │    13 │ rightPalm                   │ SE3(0.0049, -0.0351, 0.0228; 20°, -0°, 0°) ⊕ Ry(q13)     │
+│  16 │ rightThumbPitch1Link         │    14 │ rightThumbRollLink          │ SE3(0, 0, 0.0229; -20°, -0°, 0°) ⊕ Rx(q14)               │
+│  17 │ rightThumbPitch2Link         │    15 │ rightThumbPitch1Link        │ SE3(0, -0.0066, 0.0375) ⊕ Rx(q15)                        │
+│  18 │ @rightThumbPitch3Link        │    16 │ rightThumbPitch2Link        │ SE3(0, -0.0049, 0.0275) ⊕ Rx(q16)                        │
+│  19 │ rightIndexFingerPitch1Link   │    17 │ rightPalm                   │ SE3(0.0022, -0.0976, 0.0235; -9.998°, -0°, 0°) ⊕ Rz(q17) │
+│  20 │ rightIndexFingerPitch2Link   │    18 │ rightIndexFingerPitch1Link  │ SE3(0, -0.0381, 0) ⊕ Rz(q18)                             │
+│  21 │ @rightIndexFingerPitch3Link  │    19 │ rightIndexFingerPitch2Link  │ SE3(0, -0.0229, 0) ⊕ Rz(q19)                             │
+│  22 │ rightMiddleFingerPitch1Link  │    20 │ rightPalm                   │ SE3(0.0022, -0.097, -0.0119; 7.002°, -0°, 0°) ⊕ Rz(q20)  │
+│  23 │ rightMiddleFingerPitch2Link  │    21 │ rightMiddleFingerPitch1Link │ SE3(0, -0.0381, 0) ⊕ Rz(q21)                             │
+│  24 │ @rightMiddleFingerPitch3Link │    22 │ rightMiddleFingerPitch2Link │ SE3(0, -0.0229, 0) ⊕ Rz(q22)                             │
+│  25 │ rightPinkyPitch1Link         │    23 │ rightPalm                   │ SE3(0.0022, -0.0838, -0.041; 7.002°, -0°, 0°) ⊕ Rz(q23)  │
+│  26 │ rightPinkyPitch2Link         │    24 │ rightPinkyPitch1Link        │ SE3(0, -0.0381, 0) ⊕ Rz(q24)                             │
+│  27 │ @rightPinkyPitch3Link        │    25 │ rightPinkyPitch2Link        │ SE3(0, -0.0229, 0) ⊕ Rz(q25)                             │
+│  28 │ leftShoulderPitchLink        │    26 │ torso                       │ SE3(-0.0316, 0, 0.2984) ⊕ Ry(q26)                        │
+│  29 │ leftShoulderRollLink         │    27 │ leftShoulderPitchLink       │ SE3(0, 0.2499, 0) ⊕ Rx(q27)                              │
+│  30 │ leftShoulderYawLink          │    28 │ leftShoulderRollLink        │ SE3() ⊕ Ry(q28)                                          │
+│  31 │ leftElbowPitchLink           │    29 │ leftShoulderYawLink         │ SE3(0.0254, 0.33, 0) ⊕ Rz(q29)                           │
+│  32 │ leftForearmLink              │    30 │ leftElbowPitchLink          │ SE3(-0.0254, 0, 0) ⊕ Ry(q30)                             │
+│  33 │ leftWristRollLink            │    31 │ leftForearmLink             │ SE3(0, 0.2871, 0) ⊕ Rx(q31)                              │
+│  34 │ leftPalm                     │    32 │ leftWristRollLink           │ SE3() ⊕ Rz(q32)                                          │
+│  35 │ leftThumbRollLink            │    33 │ leftPalm                    │ SE3(0.0049, 0.0351, 0.0228; -20°, -0°, 0°) ⊕ Ry(q33)     │
+│  36 │ leftThumbPitch1Link          │    34 │ leftThumbRollLink           │ SE3(0, 0, 0.0229; 20°, -0°, 0°) ⊕ Rx(q34)                │
+│  37 │ leftThumbPitch2Link          │    35 │ leftThumbPitch1Link         │ SE3(0, 0.0066, 0.0375) ⊕ Rx(q35)                         │
+│  38 │ @leftThumbPitch3Link         │    36 │ leftThumbPitch2Link         │ SE3(0, 0.0049, 0.0275) ⊕ Rx(q36)                         │
+│  39 │ leftIndexFingerPitch1Link    │    37 │ leftPalm                    │ SE3(0.0022, 0.0976, 0.0235; 9.998°, -0°, 0°) ⊕ Rz(q37)   │
+│  40 │ leftIndexFingerPitch2Link    │    38 │ leftIndexFingerPitch1Link   │ SE3(0, 0.0381, 0) ⊕ Rz(q38)                              │
+│  41 │ @leftIndexFingerPitch3Link   │    39 │ leftIndexFingerPitch2Link   │ SE3(0, 0.0229, 0) ⊕ Rz(q39)                              │
+│  42 │ leftMiddleFingerPitch1Link   │    40 │ leftPalm                    │ SE3(0.0022, 0.097, -0.0119; -7.002°, -0°, 0°) ⊕ Rz(q40)  │
+│  43 │ leftMiddleFingerPitch2Link   │    41 │ leftMiddleFingerPitch1Link  │ SE3(0, 0.0381, 0) ⊕ Rz(q41)                              │
+│  44 │ @leftMiddleFingerPitch3Link  │    42 │ leftMiddleFingerPitch2Link  │ SE3(0, 0.0229, 0) ⊕ Rz(q42)                              │
+│  45 │ leftPinkyPitch1Link          │    43 │ leftPalm                    │ SE3(0.0022, 0.0838, -0.041; -7.002°, -0°, 0°) ⊕ Rz(q43)  │
+│  46 │ leftPinkyPitch2Link          │    44 │ leftPinkyPitch1Link         │ SE3(0, 0.0381, 0) ⊕ Rz(q44)                              │
+│  47 │ @leftPinkyPitch3Link         │    45 │ leftPinkyPitch2Link         │ SE3(0, 0.0229, 0) ⊕ Rz(q45)                              │
+│  48 │ @leftTorsoImu_Frame          │       │ torso                       │ SE3(-0.06276, 0.1342, 0.3631; -89.99°, -0°, 0°)          │
+│  49 │ @rightHazardCamera_Frame     │       │ torso                       │ SE3(0.0345, -0.0406, 0.1135; 89.95°, -0°, 89.95°)        │
+│  50 │ @leftHazardCamera_Frame      │       │ torso                       │ SE3(0.0345, 0.0406, 0.1135; 89.95°, -0°, 89.95°)         │
+│  51 │ rightHipYawLink              │    46 │ pelvis                      │ SE3(0, -0.1016, -0.1853) ⊕ Rz(q46)                       │
+│  52 │ rightHipRollLink             │    47 │ rightHipYawLink             │ SE3() ⊕ Rx(q47)                                          │
+│  53 │ rightHipPitchLink            │    48 │ rightHipRollLink            │ SE3(0, 0, -0.0609) ⊕ Ry(q48)                             │
+│  54 │ rightKneePitchLink           │    49 │ rightHipPitchLink           │ SE3(0.0001122, -0.0361, -0.431) ⊕ Ry(q49)                │
+│  55 │ rightAnklePitchLink          │    50 │ rightKneePitchLink          │ SE3(-0.01024, 0, -0.4063) ⊕ Ry(q50)                      │
+│  56 │ rightFoot                    │    51 │ rightAnklePitchLink         │ SE3() ⊕ Rx(q51)                                          │
+│  57 │ @rightCOP_Frame              │       │ rightFoot                   │ SE3(0.0189, 0, -0.0783)                                  │
+│  58 │ @rightFootSixAxis_Frame      │       │ rightFoot                   │ SE3(0.02156, 0, -0.05105; 179.9°, -0°, 0°)               │
+│  59 │ leftHipYawLink               │    52 │ pelvis                      │ SE3(0, 0.1016, -0.1853) ⊕ Rz(q52)                        │
+│  60 │ leftHipRollLink              │    53 │ leftHipYawLink              │ SE3() ⊕ Rx(q53)                                          │
+│  61 │ leftHipPitchLink             │    54 │ leftHipRollLink             │ SE3(0, 0, -0.0609) ⊕ Ry(q54)                             │
+│  62 │ leftKneePitchLink            │    55 │ leftHipPitchLink            │ SE3(0.0001122, 0.0361, -0.431) ⊕ Ry(q55)                 │
+│  63 │ leftAnklePitchLink           │    56 │ leftKneePitchLink           │ SE3(-0.01024, 0, -0.4063) ⊕ Ry(q56)                      │
+│  64 │ leftFoot                     │    57 │ leftAnklePitchLink          │ SE3() ⊕ Rx(q57)                                          │
+│  65 │ @leftCOP_Frame               │       │ leftFoot                    │ SE3(0.0189, 0, -0.0783)                                  │
+│  66 │ @leftFootSixAxis_Frame       │       │ leftFoot                    │ SE3(0.02156, 0, -0.05105; 179.9°, -0°, 0°)               │
+│  67 │ @pelvisRearImu_Frame         │       │ pelvis                      │ SE3(-0.07584, 0, -0.1111; -180°, 27.3°, -180°)           │
+│  68 │ @pelvisMiddleImu_Frame       │       │ pelvis                      │ SE3(0, 0, -0.1082; 180°, -0°, 0°)                        │
+└─────┴──────────────────────────────┴───────┴─────────────────────────────┴──────────────────────────────────────────────────────────┘
+
+
+
+

Defined joint configurations are:

+
    +
  • qz, zero joint angle configuration, ‘L’ shaped configuration

  • +
  • qr, vertical ‘READY’ configuration

  • +
+
+
Reference:
+
+
+
+

Code author: Peter Corke

+
+ +
+
+class roboticstoolbox.models.URDF.AL5D[source]
+

Bases: Robot

+

Class that imports a AL5D URDF model

+

AL5D() is a class which imports a Lynxmotion AL5D robot definition +from a URDF file. The model describes its kinematic and graphical +characteristics.

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.URDF.AL5D()
+>>> print(robot)
+ERobot: AL5D (by Lynxmotion), 4 joints (RRRR), dynamics, geometry
+┌─────┬────────┬───────┬────────┬─────────────────────────────────────────────────────┐
+│link │  link  │ joint │ parent │                 ETS: parent to link                 │
+├─────┼────────┼───────┼────────┼─────────────────────────────────────────────────────┤
+│   0 │ base   │       │ BASE   │ SE3()                                               │
+│   1 │ link1  │     0 │ base   │ SE3(0, 0, 0.06858; 180°, 3.379e-08°, 180°) ⊕ Rz(q0) │
+│   2 │ link2  │     1 │ link1  │ SE3(0.002, 0, 0; 0°, 90°, -90°) ⊕ Rz(q1)            │
+│   3 │ link3  │     2 │ link2  │ SE3(0.1468, 0, 0; 180°, 3.379e-08°, -90°) ⊕ Rz(q2)  │
+│   4 │ @link4 │     3 │ link3  │ SE3(0.1775, 0, 0; 180°, -0°, 90°) ⊕ Rz(q3)          │
+└─────┴────────┴───────┴────────┴─────────────────────────────────────────────────────┘
+
+┌─────┬─────┬─────┬──────┬─────┐
+│name │ q0  │ q1  │ q2   │ q3  │
+├─────┼─────┼─────┼──────┼─────┤
+│  qz │  0° │  0° │  0°  │  0° │
+│  up │  0° │  0° │  90° │  0° │
+└─────┴─────┴─────┴──────┴─────┘
+
+
+
+

Defined joint configurations are:

+
    +
  • qz, zero joint angle configuration, ‘L’ shaped configuration

  • +
  • up, robot poiting upwards

  • +
+

Code author: Tassos Natsakis

+
+ +
+
+
+ + +
+
+
+ +
+ +
+

© Copyright 2023, Jesse Haviland and Peter Corke. + Last updated on 11-May-2024. +

+
+ + + +
+
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/arm_ets.html b/arm_ets.html new file mode 100644 index 00000000..c104a40f --- /dev/null +++ b/arm_ets.html @@ -0,0 +1,1460 @@ + + + + + + + + + + + Elementary transform sequence (ETS) models — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

Elementary transform sequence (ETS) models

+

Code author: Jesse Haviland

+

Elementary transforms are canonic rotations or translations about, or along, +the x-, y- or z-axes. The amount of rotation or translation can be a constant +or a variable. A variable amount corresponds to a joint.

+

Consider the example:

+
 1>>> from roboticstoolbox import ET
+ 2>>> ET.Rx(45, 'deg')
+ 3ET.Rx(eta=0.7853981633974483)
+ 4>>> ET.tz(0.75)
+ 5ET.tz(eta=0.75)
+ 6>>> e = ET.Rx(0.3) * ET.tz(0.75)
+ 7>>> print(e)
+ 8Rx(17.19°) ⊕ tz(0.75)
+ 9>>> e.fkine([])
+10SE3(array([[ 1.    ,  0.    ,  0.    ,  0.    ],
+11           [ 0.    ,  0.9553, -0.2955, -0.2216],
+12           [ 0.    ,  0.2955,  0.9553,  0.7165],
+13           [ 0.    ,  0.    ,  0.    ,  1.    ]]))
+
+
+

In lines 2-5 we defined two elementary transforms. Line 2 defines a rotation +of 45° about the x-axis, and line 4 defines a translation of 0.75m along the z-axis. +Compounding them in line 6, the result is the two elementary transforms in a +sequence - an elementary transform sequence (ETS). An ETS can be arbitrarily +long.

+

Line 8 evaluates the forward kinematics of the sequence, substituting in values, +and the result is an SE(3) matrix encapsulated in an 4x4 numpy array.

+

The real power comes from having variable arguments to the elementary transforms +as shown in this example where we define a simple two link planar manipulator.

+
 1>>> from roboticstoolbox import ET
+ 2>>> e = ET.Rz() * ET.tx(1) * ET.Rz() * ET.tx(1)
+ 3>>> print(e)
+ 4Rz(q0) ⊕ tx(1) ⊕ Rz(q1) ⊕ tx(1)
+ 5>>> len(e)
+ 64
+ 7>>> e[1:3]
+ 8[ET.tx(eta=1.0), ET.Rz(jindex=1)]
+ 9>>> e.fkine([0, 0])
+10SE3(array([[1., 0., 0., 2.],
+11           [0., 1., 0., 0.],
+12           [0., 0., 1., 0.],
+13           [0., 0., 0., 1.]]))
+14>>> e.fkine([1.57, -1.57])
+15SE3(array([[1.    , 0.    , 0.    , 1.0008],
+16           [0.    , 1.    , 0.    , 1.    ],
+17           [0.    , 0.    , 1.    , 0.    ],
+18           [0.    , 0.    , 0.    , 1.    ]]))
+
+
+

Line 2 succinctly describes the kinematics in terms of elementary transforms: a +rotation around the z-axis by the first joint angle, then a translation in the +x-direction, then a rotation around the z-axis by the second joint angle, and +finally a translation in the x-direction.

+

Line 3 creates the elementary transform sequence, with variable transforms. +e is a single object that encapsulates a list of elementary transforms, and list like +methods such as len as well as indexing and slicing as shown in lines 5-8.

+

Lines 9-18 evaluate the sequence, and substitutes elements from the passed +arrays as the joint variables.

+

This approach is general enough to be able to describe any serial-link robot +manipulator. For a branched manipulator we can use ETS to describe the +connections between every parent and child link pair.

+

The ETS inherits list-like properties and has methods like reverse and pop.

+

Reference:

+
+
+
+
+

ETS - 3D

+
+
+class roboticstoolbox.robot.ETS.ETS(arg=None)[source]
+

Bases: BaseETS

+

This class implements an elementary transform sequence (ETS) for 3D

+

An instance can contain an elementary transform (ET) or an elementary +transform sequence (ETS). It has list-like properties by subclassing +UserList, which means we can perform indexing, slicing pop, insert, as well +as using it as an iterator over its values.

+
    +
  • ETS() an empty ETS list

  • +
  • ETS(et) an ETS containing a single ET

  • +
  • ETS([et0, et1, et2]) an ETS consisting of three ET’s

  • +
+
+
Parameters:
+

arg (Union[List[Union[ETS, ET]], List[ET], List[ETS], ET, ETS, None]) – Function to compute ET value

+
+
+

Examples

+
>>> from roboticstoolbox import ETS, ET
+>>> e = ET.Rz(0.3) # a single ET, rotation about z
+>>> ets1 = ETS(e)
+>>> len(ets1)
+1
+>>> ets2 = ET.Rz(0.3) * ET.tx(2) # an ETS
+>>> len(ets2)                    # of length 2
+2
+>>> ets2[1]                      # an ET sliced from the ETS
+ET.tx(eta=2.0)
+
+
+

References

+
    +
    1. +
    2. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I:

    3. +
    +
    +

    Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

    +
    +
  • +
    1. +
    2. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II:

    3. +
    +
    +

    Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

    +
    +
  • +
+
+

See also

+

rx(), ry(), rz(), tx(), ty(), tz()

+
+
+
+__mul__(other)[source]
+
+
Return type:
+

ETS

+
+
+
+ +
+
+compile()[source]
+

Compile an ETS

+

Perform constant folding for faster evaluation. Consecutive constant +ETs are compounded, leading to a constant ET which is denoted by +SE3 when displayed.

+
+
Returns:
+

optimised ETS

+
+
Return type:
+

compile

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.ETS.Panda()
+>>> ets = robot.ets()
+>>> ets
+[ET.tz(eta=0.333), ET.Rz(jindex=0), ET.Rx(eta=-1.5707963267948966), ET.Rz(jindex=1), ET.Rx(eta=1.5707963267948966), ET.tz(eta=0.316), ET.Rz(jindex=2), ET.tx(eta=0.0825), ET.Rx(eta=1.5707963267948966), ET.Rz(jindex=3), ET.tx(eta=-0.0825), ET.Rx(eta=-1.5707963267948966), ET.tz(eta=0.384), ET.Rz(jindex=4), ET.Rx(eta=1.5707963267948966), ET.Rz(jindex=5), ET.tx(eta=0.088), ET.Rx(eta=1.5707963267948966), ET.tz(eta=0.107), ET.Rz(jindex=6), ET.tz(eta=0.10300000000000001), ET.Rz(eta=-0.7853981633974483)]
+>>> ets.compile()
+[ET.SE3(T=array([[1.   , 0.   , 0.   , 0.   ],
+       [0.   , 1.   , 0.   , 0.   ],
+       [0.   , 0.   , 1.   , 0.333],
+       [0.   , 0.   , 0.   , 1.   ]])), ET.Rz(jindex=0), ET.SE3(T=array([[ 1.,  0.,  0.,  0.],
+       [ 0.,  0.,  1.,  0.],
+       [ 0., -1.,  0.,  0.],
+       [ 0.,  0.,  0.,  1.]])), ET.Rz(jindex=1), ET.SE3(T=array([[ 1.   ,  0.   ,  0.   ,  0.   ],
+       [ 0.   ,  0.   , -1.   , -0.316],
+       [ 0.   ,  1.   ,  0.   ,  0.   ],
+       [ 0.   ,  0.   ,  0.   ,  1.   ]])), ET.Rz(jindex=2), ET.SE3(T=array([[ 1.    ,  0.    ,  0.    ,  0.0825],
+       [ 0.    ,  0.    , -1.    ,  0.    ],
+       [ 0.    ,  1.    ,  0.    ,  0.    ],
+       [ 0.    ,  0.    ,  0.    ,  1.    ]])), ET.Rz(jindex=3), ET.SE3(T=array([[ 1.    ,  0.    ,  0.    , -0.0825],
+       [ 0.    ,  0.    ,  1.    ,  0.384 ],
+       [ 0.    , -1.    ,  0.    ,  0.    ],
+       [ 0.    ,  0.    ,  0.    ,  1.    ]])), ET.Rz(jindex=4), ET.SE3(T=array([[ 1.,  0.,  0.,  0.],
+       [ 0.,  0., -1.,  0.],
+       [ 0.,  1.,  0.,  0.],
+       [ 0.,  0.,  0.,  1.]])), ET.Rz(jindex=5), ET.SE3(T=array([[ 1.   ,  0.   ,  0.   ,  0.088],
+       [ 0.   ,  0.   , -1.   , -0.107],
+       [ 0.   ,  1.   ,  0.   ,  0.   ],
+       [ 0.   ,  0.   ,  0.   ,  1.   ]])), ET.Rz(jindex=6), ET.SE3(T=array([[ 0.7071,  0.7071,  0.    ,  0.    ],
+       [-0.7071,  0.7071,  0.    ,  0.    ],
+       [ 0.    ,  0.    ,  1.    ,  0.103 ],
+       [ 0.    ,  0.    ,  0.    ,  1.    ]]))]
+
+
+
+

See also

+

isconstant()

+
+
+ +
+
+insert(arg, i=-1)[source]
+

Insert value

+

Inserts an ET or ETS into the ET sequence. The inserted value is at position +i.

+
+
Parameters:
+
    +
  • i (int) – insert an ET or ETS into the ETS, default is at the end

  • +
  • arg (Union[ET, ETS]) – the elementary transform or sequence to insert

  • +
+
+
Return type:
+

None

+
+
+

Examples

+
>>> from roboticstoolbox import ET
+>>> e = ET.Rz() * ET.tx(1) * ET.Rz() * ET.tx(1)
+>>> f = ET.Ry()
+>>> e.insert(f, 2)
+>>> e
+[ET.Rz(jindex=0), ET.tx(eta=1.0), ET.Ry(), ET.Rz(jindex=1), ET.tx(eta=1.0)]
+
+
+
+ +
+
+fkine(q, base=None, tool=None, include_base=True)[source]
+

Forward kinematics

+

T = ets.fkine(q) evaluates forward kinematics for the ets at +joint configuration q.

+

Trajectory operation: +If q has multiple rows (mxn), it is considered a trajectory and the +result is an SE3 instance with m values.

+
+
+q
+

Joint coordinates

+
+ +
+
+base
+

A base transform applied before the ETS

+
+ +
+
+tool
+

tool transform, optional

+
+ +
+
+include_base
+

set to True if the base transform should be considered

+
+ +
+
Return type:
+

SE3

+
+
Returns:
+

The transformation matrix representing the pose of the +end-effector

+
+
+

Examples

+

The following example makes a panda robot object, gets the ets, and +solves for the forward kinematics at the listed configuration.

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda().ets()
+>>> panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+SE3(array([[ 0.995 , -0.    ,  0.0998,  0.484 ],
+           [-0.    , -1.    ,  0.    ,  0.    ],
+           [ 0.0998, -0.    , -0.995 ,  0.4126],
+           [ 0.    ,  0.    ,  0.    ,  1.    ]]))
+
+
+

Notes

+
    +
  • A tool transform, if provided, is incorporated into the result.

  • +
  • Works from the end-effector link to the base

  • +
+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
+
+ +
+
+jacob0(q, tool=None)[source]
+

Manipulator geometric Jacobian in the base frame

+

robot.jacobo(q) is the manipulator Jacobian matrix which maps +joint velocity to end-effector spatial velocity expressed in the +base frame.

+

End-effector spatial velocity \(\nu = (v_x, v_y, v_z, \omega_x, \omega_y, \omega_z)^T\) +is related to joint velocity by \({}^{E}\!\nu = \mathbf{J}_m(q) \dot{q}\).

+
+
Parameters:
+
+
+
Returns:
+

Manipulator Jacobian in the base frame

+
+
Return type:
+

J0

+
+
+

Examples

+

The following example makes a Puma560 robot object, and solves for the +base-frame Jacobian at the zero joint angle configuration

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.Puma560().ets()
+>>> puma.jacob0([0, 0, 0, 0, 0, 0])
+array([[ 0.1295, -0.4854, -0.4854, -0.    , -0.0533,  0.    ],
+       [ 0.4318,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+       [-0.    ,  0.4318,  0.    ,  0.    ,  0.    ,  0.    ],
+       [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+       [ 0.    , -1.    , -1.    ,  0.    , -1.    ,  0.    ],
+       [ 1.    ,  0.    ,  0.    ,  1.    ,  0.    ,  1.    ]])
+
+
+

Notes

+
    +
  • +
    This is the geometric Jacobian as described in texts by

    Corke, Spong etal., Siciliano etal. The end-effector velocity is +described in terms of translational and angular velocity, not a +velocity twist as per the text by Lynch & Park.

    +
    +
    +
  • +
+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
+
+ +
+
+jacobe(q, tool=None)[source]
+

Manipulator geometric Jacobian in the end-effector frame

+

robot.jacobe(q) is the manipulator Jacobian matrix which maps +joint velocity to end-effector spatial velocity expressed in the +end frame.

+

End-effector spatial velocity \(\nu = (v_x, v_y, v_z, \omega_x, \omega_y, \omega_z)^T\) +is related to joint velocity by \({}^{E}\!\nu = \mathbf{J}_m(q) \dot{q}\).

+
+
Parameters:
+
    +
  • q (Union[ndarray, List[float], Tuple[float], Set[float]]) – Joint coordinate vector

  • +
  • end – the particular link or gripper whose velocity the Jacobian +describes, defaults to the end-effector if only one is present

  • +
  • start – the link considered as the base frame, defaults to the robots’s base frame

  • +
  • tool (Union[ndarray, SE3, None]) – a static tool transformation matrix to apply to the +end of end, defaults to None

  • +
+
+
Returns:
+

Manipulator Jacobian in the end frame

+
+
Return type:
+

Je

+
+
+

Examples

+

The following example makes a Puma560 robot object, and solves for the +end-effector frame Jacobian at the zero joint angle configuration

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.Puma560().ets()
+>>> puma.jacobe([0, 0, 0, 0, 0, 0])
+array([[ 0.1295, -0.4854, -0.4854, -0.    , -0.0533,  0.    ],
+       [ 0.4318,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+       [-0.    ,  0.4318,  0.    ,  0.    ,  0.    ,  0.    ],
+       [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+       [ 0.    , -1.    , -1.    ,  0.    , -1.    ,  0.    ],
+       [ 1.    ,  0.    ,  0.    ,  1.    ,  0.    ,  1.    ]])
+
+
+

Notes

+
    +
  • +
    This is the geometric Jacobian as described in texts by

    Corke, Spong etal., Siciliano etal. The end-effector velocity is +described in terms of translational and angular velocity, not a +velocity twist as per the text by Lynch & Park.

    +
    +
    +
  • +
+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
+
+ +
+
+hessian0(q=None, J0=None, tool=None)[source]
+

Manipulator Hessian

+

The manipulator Hessian tensor maps joint acceleration to end-effector +spatial acceleration, expressed in the base frame. This +function calulcates this based on the ETS of the robot. One of J0 or q +is required. Supply J0 if already calculated to save computation time

+
+
Parameters:
+
+
+
Returns:
+

The manipulator Hessian in the base frame

+
+
Return type:
+

h0

+
+
+

Synopsis

+

This method computes the manipulator Hessian in the base frame. If +we take the time derivative of the differential kinematic relationship +.. math:

+
\nu    &= \mat{J}(\vec{q}) \dvec{q} \\
+\alpha &= \dmat{J} \dvec{q} + \mat{J} \ddvec{q}
+
+
+

where +.. math:

+
\dmat{J} = \mat{H} \dvec{q}
+
+
+

and \(\mat{H} \in \mathbb{R}^{6\times n \times n}\) is the +Hessian tensor.

+

The elements of the Hessian are +.. math:

+
\mat{H}_{i,j,k} =  \frac{d^2 u_i}{d q_j d q_k}
+
+
+

where \(u = \{t_x, t_y, t_z, r_x, r_y, r_z\}\) are the elements +of the spatial velocity vector.

+

Similarly, we can write +.. math:

+
\mat{J}_{i,j} = \frac{d u_i}{d q_j}
+
+
+

Examples

+

The following example makes a Panda robot object, and solves for the +base frame Hessian at the given joint angle configuration

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda().ets()
+>>> panda.hessian0([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+array([[[-0.484 , -0.    , -0.486 , -0.    , -0.1547,  0.    , -0.    ],
+        [ 0.    ,  0.0796,  0.    ,  0.2466,  0.    ,  0.2006,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+        [ 0.    , -1.    , -0.    ,  1.    ,  0.    ,  1.    ,  0.    ],
+        [ 0.    , -0.    , -0.2955, -0.    ,  0.9463, -0.    ,  0.0998],
+        [ 0.    ,  0.    , -0.    , -0.    ,  0.    , -0.    ,  0.    ]],
+
+       [[-0.    , -0.484 , -0.    ,  0.4986, -0.    ,  0.1086,  0.    ],
+        [ 0.0796,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+        [ 0.    , -0.0796, -0.    , -0.2466, -0.    , -0.2006, -0.    ],
+        [ 0.    ,  0.    ,  0.9553,  0.    , -0.3233,  0.    , -0.995 ],
+        [ 0.    ,  0.    , -0.    , -0.    ,  0.    , -0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.2955,  0.    , -0.9463,  0.    , -0.0998]],
+
+       [[-0.486 , -0.    , -0.4643, -0.    , -0.1478,  0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.383 ,  0.    ,  0.2237,  0.    ],
+        [ 0.    , -0.    , -0.1436, -0.    , -0.0457, -0.    , -0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.9553,  0.    ,  0.9553,  0.    ],
+        [ 0.    ,  0.    ,  0.    , -0.    ,  0.8085, -0.    , -0.1987],
+        [ 0.    ,  0.    ,  0.    ,  0.2955,  0.    ,  0.2955,  0.    ]],
+
+       [[-0.    ,  0.4986, -0.    , -0.4986,  0.    , -0.1086, -0.    ],
+        [ 0.2466,  0.    ,  0.383 ,  0.    ,  0.    ,  0.    ,  0.    ],
+        [ 0.    , -0.2466, -0.    ,  0.2466,  0.    ,  0.2006,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.3233,  0.    ,  0.995 ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    , -0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.9463,  0.    ,  0.0998]],
+
+       [[-0.1547, -0.    , -0.1478,  0.    ,  0.05  , -0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    , -0.1676, -0.    ],
+        [ 0.    , -0.    , -0.0457,  0.    ,  0.1464,  0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    , -0.3233,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    , -0.    ,  0.9093],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    , -0.9463, -0.    ]],
+
+       [[ 0.    ,  0.1086,  0.    , -0.1086, -0.    , -0.1086, -0.    ],
+        [ 0.2006,  0.    ,  0.2237,  0.    , -0.1676,  0.    ,  0.    ],
+        [ 0.    , -0.2006, -0.    ,  0.2006,  0.    ,  0.2006,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.995 ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    , -0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.0998]],
+
+       [[-0.    ,  0.    ,  0.    , -0.    ,  0.    , -0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    , -0.    ,  0.    , -0.    ],
+        [ 0.    , -0.    , -0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ]]])
+
+
+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+ +
+
+hessiane(q=None, Je=None, tool=None)[source]
+

Manipulator Hessian

+

The manipulator Hessian tensor maps joint acceleration to end-effector +spatial acceleration, expressed in the end-effector coordinate frame. This +function calulcates this based on the ETS of the robot. One of J0 or q +is required. Supply J0 if already calculated to save computation time

+
+
Parameters:
+
    +
  • q (Union[ndarray, List[float], Tuple[float], Set[float], None]) – The joint angles/configuration of the robot (Optional, +if not supplied will use the stored q values).

  • +
  • J0 – The manipulator Jacobian in the end-effector frame

  • +
  • tool (Union[ndarray, SE3, None]) – a static tool transformation matrix to apply to the +end of end, defaults to None

  • +
+
+
Returns:
+

The manipulator Hessian in end-effector frame

+
+
Return type:
+

he

+
+
+

Synopsis

+

This method computes the manipulator Hessian in the end-effector frame. If +we take the time derivative of the differential kinematic relationship +.. math:

+
\nu    &= \mat{J}(\vec{q}) \dvec{q} \\
+\alpha &= \dmat{J} \dvec{q} + \mat{J} \ddvec{q}
+
+
+

where +.. math:

+
\dmat{J} = \mat{H} \dvec{q}
+
+
+

and \(\mat{H} \in \mathbb{R}^{6\times n \times n}\) is the +Hessian tensor.

+

The elements of the Hessian are +.. math:

+
\mat{H}_{i,j,k} =  \frac{d^2 u_i}{d q_j d q_k}
+
+
+

where \(u = \{t_x, t_y, t_z, r_x, r_y, r_z\}\) are the elements +of the spatial velocity vector.

+

Similarly, we can write +.. math:

+
\mat{J}_{i,j} = \frac{d u_i}{d q_j}
+
+
+

Examples

+

The following example makes a Panda robot object, and solves for the +end-effector frame Hessian at the given joint angle configuration

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda().ets()
+>>> panda.hessiane([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+array([[[-0.4816, -0.    , -0.4835, -0.    , -0.1539, -0.    ,  0.    ],
+        [ 0.    , -0.0796,  0.    , -0.2466,  0.    , -0.2006,  0.    ],
+        [-0.0483, -0.    , -0.0485, -0.    , -0.0154, -0.    ,  0.    ],
+        [ 0.    , -0.995 ,  0.    ,  0.995 , -0.    ,  0.995 , -0.    ],
+        [ 0.    ,  0.    ,  0.2955, -0.    , -0.9463, -0.    , -0.0998],
+        [ 0.    , -0.0998,  0.    ,  0.0998, -0.    ,  0.0998,  0.    ]],
+
+       [[-0.    , -0.4896, -0.    ,  0.4715, -0.    ,  0.088 ,  0.    ],
+        [-0.0796,  0.    ,  0.    , -0.    ,  0.    , -0.    ,  0.    ],
+        [-0.    ,  0.0309,  0.    ,  0.2952,  0.    ,  0.2104, -0.    ],
+        [ 0.    ,  0.    ,  0.9801,  0.    , -0.4161,  0.    , -1.    ],
+        [ 0.    ,  0.    , -0.    , -0.    ,  0.    , -0.    ,  0.    ],
+        [ 0.    ,  0.    , -0.1987, -0.    ,  0.9093, -0.    ,  0.    ]],
+
+       [[-0.4835, -0.    , -0.4763, -0.    , -0.1516, -0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    , -0.383 ,  0.    , -0.2237,  0.    ],
+        [-0.0485,  0.    ,  0.0965, -0.    ,  0.0307, -0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.9801, -0.    ,  0.9801,  0.    ],
+        [ 0.    ,  0.    ,  0.    , -0.    , -0.8085, -0.    ,  0.1987],
+        [ 0.    ,  0.    ,  0.    , -0.1987, -0.    , -0.1987, -0.    ]],
+
+       [[-0.    ,  0.4715, -0.    , -0.4715,  0.    , -0.088 ,  0.    ],
+        [-0.2466, -0.    , -0.383 ,  0.    , -0.    ,  0.    , -0.    ],
+        [-0.    ,  0.2952, -0.    , -0.2952, -0.    , -0.2104,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.4161,  0.    ,  1.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    , -0.    ,  0.    , -0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    , -0.9093,  0.    ,  0.    ]],
+
+       [[-0.1539, -0.    , -0.1516,  0.    ,  0.0644,  0.    , -0.    ],
+        [ 0.    ,  0.    ,  0.    , -0.    , -0.    ,  0.1676, -0.    ],
+        [-0.0154,  0.    ,  0.0307, -0.    , -0.1407, -0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    , -0.4161, -0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    , -0.9093],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.9093,  0.    ]],
+
+       [[-0.    ,  0.088 , -0.    , -0.088 ,  0.    , -0.088 ,  0.    ],
+        [-0.2006, -0.    , -0.2237,  0.    ,  0.1676,  0.    , -0.    ],
+        [-0.    ,  0.2104, -0.    , -0.2104, -0.    , -0.2104,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  1.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    , -0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ]],
+
+       [[ 0.    ,  0.    ,  0.    ,  0.    , -0.    ,  0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    , -0.    , -0.    , -0.    , -0.    ],
+        [ 0.    , -0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ]]])
+
+
+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+ +
+
+__str__(q=None)
+

Pretty prints the ETS

+

q controls how the joint variables are displayed:

+
    +
  • +
    None, format depends on number of joint variables
      +
    • one, display joint variable as q

    • +
    • more, display joint variables as q0, q1, …

    • +
    • if a joint index was provided, use this value

    • +
    +
    +
    +
  • +
  • “”, display all joint variables as empty parentheses ()

  • +
  • “θ”, display all joint variables as (θ)

  • +
  • format string with passed joint variables (j, j+1), so “θ{0}” +would display joint variables as θ0, θ1, … while “θ{1}” would +display joint variables as θ1, θ2, … j is either the joint +index, if provided, otherwise a sequential value.

  • +
+
+
Parameters:
+

q (Optional[str]) – control how joint variables are displayed

+
+
Returns:
+

Pretty printed ETS

+
+
Return type:
+

str

+
+
+

Examples

+
>>> from roboticstoolbox import ET
+>>> e = ET.Rz() * ET.tx(1) * ET.Rz()
+>>> print(e[:2])
+[ET.Rz(jindex=0), ET.tx(eta=1.0)]
+>>> print(e)
+Rz(q0) ⊕ tx(1) ⊕ Rz(q1)
+>>> print(e.__str__(""))
+Rz() ⊕ tx(1) ⊕ Rz()
+>>> print(e.__str__({0}"))  # numbering from 0
+Rz(θ0) ⊕ tx(1) ⊕ Rz(θ1)
+>>> print(e.__str__({1}"))  # numbering from 1
+Rz(θ1) ⊕ tx(1) ⊕ Rz(θ2)
+>>> # explicit joint indices
+>>> e = ET.Rz(jindex=3) * ET.tx(1) * ET.Rz(jindex=4)
+>>> print(e)
+Rz(q3) ⊕ tx(1) ⊕ Rz(q4)
+>>> print(e.__str__({0}"))
+Rz(θ3) ⊕ tx(1) ⊕ Rz(θ4)
+
+
+

Angular parameters are converted to degrees, except if they +are symbolic.

+
>>> from roboticstoolbox import ET
+>>> from spatialmath.base import symbol
+>>> theta, d = symbol('theta, d')
+>>> e = ET.Rx(theta) * ET.tx(2) * ET.Rx(45, 'deg') * ET.Ry(0.2) * ET.ty(d)
+>>> str(e)
+'Rx(theta) ⊕ tx(2) ⊕ Rx(45°) ⊕ Ry(11.46°) ⊕ ty(d)'
+
+
+
+ +
+
+__repr__()
+

Return repr(self).

+
+ +
+
+__getitem__(i)
+

Index or slice an ETS

+
+
Parameters:
+

i – the index or slince

+
+
Returns:
+

Elementary transform

+
+
Return type:
+

et

+
+
+

Examples

+
>>> from roboticstoolbox import ET
+>>> e = ET.Rz() * ET.tx(1) * ET.Rz() * ET.tx(1)
+>>> e[0]
+ET.Rz(jindex=0)
+>>> e[1]
+ET.tx(eta=1.0)
+>>> e[1:3]
+[ET.tx(eta=1.0), ET.Rz(jindex=1)]
+
+
+
+ +
+
+property n: int
+

Number of joints

+

Counts the number of joints in the ETS.

+
+
Returns:
+

the number of joints in the ETS

+
+
Return type:
+

n

+
+
+

Examples

+
>>> from roboticstoolbox import ET
+>>> e = ET.Rx() * ET.tx(1) * ET.tz()
+>>> e.n
+2
+
+
+
+

See also

+

joints()

+
+
+ +
+
+property m: int
+

Number of transforms

+

Counts the number of transforms in the ETS.

+
+
Returns:
+

the number of transforms in the ETS

+
+
Return type:
+

m

+
+
+

Examples

+
>>> from roboticstoolbox import ET
+>>> e = ET.Rx() * ET.tx(1) * ET.tz()
+>>> e.m
+3
+
+
+
+ +
+
+property structure: str
+

Joint structure string

+

A string comprising the characters ‘R’ or ‘P’ which indicate the types +of joints in order from left to right.

+
+
Returns:
+

A string indicating the joint types

+
+
Return type:
+

structure

+
+
+

Examples

+
>>> from roboticstoolbox import ET
+>>> e = ET.tz() * ET.tx(1) * ET.Rz() * ET.tx(1)
+>>> e.structure
+'PR'
+
+
+
+ +
+
+joints()
+

Get a list of the variable ETs with this ETS

+
+
Returns:
+

list of ETs that are joints

+
+
Return type:
+

joints

+
+
+

Examples

+
>>> from roboticstoolbox import ET
+>>> e = ET.Rz() * ET.tx(1) * ET.Rz() * ET.tx(1)
+>>> e.joints()
+[ET.Rz(jindex=0), ET.Rz(jindex=1)]
+
+
+
+ +
+
+split()
+

Split ETS into link segments

+
+
Returns:
+

a list of ETS, each one, apart from the last, +ends with a variable ET.

+
+
Return type:
+

split

+
+
+
+ +
+
+inv()
+

Inverse of ETS

+

The inverse of a given ETS. It is computed as the inverse of the +individual ETs in the reverse order.

+
+\[(\mathbf{E}_0, \mathbf{E}_1 \cdots \mathbf{E}_{n-1} )^{-1} = (\mathbf{E}_{n-1}^{-1}, \mathbf{E}_{n-2}^{-1} \cdots \mathbf{E}_0^{-1}{n-1} )\]
+
+
Returns:
+

Inverse of the ETS

+
+
Return type:
+

inv

+
+
+

Examples

+
>>> from roboticstoolbox import ET
+>>> e = ET.Rz(jindex=2) * ET.tx(1) * ET.Rx(jindex=3,flip=True) * ET.tx(1)
+>>> print(e)
+Rz(q2) ⊕ tx(1) ⊕ Rx(-q3) ⊕ tx(1)
+>>> print(e.inv())
+tx(-1) ⊕ Rx(q3) ⊕ tx(-1) ⊕ Rz(-q2)
+
+
+

Notes

+
    +
  • +
    It is essential to use explicit joint indices to account for

    the reversed order of the transforms.

    +
    +
    +
  • +
+
+ +
+ +
+
+

ETS - 2D

+
+
+class roboticstoolbox.robot.ETS.ETS2(arg=None)[source]
+

Bases: BaseETS

+

This class implements an elementary transform sequence (ETS) for 2D

+
+
Parameters:
+

arg (Union[List[Union[ETS2, ET2]], List[ET2], List[ETS2], ET2, ETS2, None]) – Function to compute ET value

+
+
+

An instance can contain an elementary transform (ET) or an elementary +transform sequence (ETS). It has list-like properties by subclassing +UserList, which means we can perform indexing, slicing pop, insert, as well +as using it as an iterator over its values.

+
    +
  • ETS() an empty ETS list

  • +
  • ET2.XY(η) is a constant elementary transform

  • +
  • ET2.XY(η, 'deg') as above but the angle is expressed in degrees

  • +
  • ET2.XY() is a joint variable, the value is left free until evaluation +time

  • +
  • ET2.XY(j=J) as above but the joint index is explicitly given, this +might correspond to the joint number of a multi-joint robot.

  • +
  • ET2.XY(flip=True) as above but the joint moves in the opposite sense

  • +
+

where XY is one of R, tx, ty.

+

Example

+
Traceback (most recent call last):
+  File "<input>", line 1, in <module>
+AttributeError: type object 'ETS2' has no attribute 'R'
+Traceback (most recent call last):
+  File "<input>", line 1, in <module>
+NameError: name 'e' is not defined
+Traceback (most recent call last):
+  File "<input>", line 1, in <module>
+NameError: name 'e' is not defined
+
+
+
+
References:
+
    +
  • Kinematic Derivatives using the Elementary Transform Sequence, +J. Haviland and P. Corke

  • +
+
+
Seealso:
+

r(), tx(), ty()

+
+
+
+
+__str__(q=None)
+

Pretty prints the ETS

+

q controls how the joint variables are displayed:

+
    +
  • +
    None, format depends on number of joint variables
      +
    • one, display joint variable as q

    • +
    • more, display joint variables as q0, q1, …

    • +
    • if a joint index was provided, use this value

    • +
    +
    +
    +
  • +
  • “”, display all joint variables as empty parentheses ()

  • +
  • “θ”, display all joint variables as (θ)

  • +
  • format string with passed joint variables (j, j+1), so “θ{0}” +would display joint variables as θ0, θ1, … while “θ{1}” would +display joint variables as θ1, θ2, … j is either the joint +index, if provided, otherwise a sequential value.

  • +
+
+
Parameters:
+

q (Optional[str]) – control how joint variables are displayed

+
+
Returns:
+

Pretty printed ETS

+
+
Return type:
+

str

+
+
+

Examples

+
>>> from roboticstoolbox import ET
+>>> e = ET.Rz() * ET.tx(1) * ET.Rz()
+>>> print(e[:2])
+[ET.Rz(jindex=0), ET.tx(eta=1.0)]
+>>> print(e)
+Rz(q0) ⊕ tx(1) ⊕ Rz(q1)
+>>> print(e.__str__(""))
+Rz() ⊕ tx(1) ⊕ Rz()
+>>> print(e.__str__({0}"))  # numbering from 0
+Rz(θ0) ⊕ tx(1) ⊕ Rz(θ1)
+>>> print(e.__str__({1}"))  # numbering from 1
+Rz(θ1) ⊕ tx(1) ⊕ Rz(θ2)
+>>> # explicit joint indices
+>>> e = ET.Rz(jindex=3) * ET.tx(1) * ET.Rz(jindex=4)
+>>> print(e)
+Rz(q3) ⊕ tx(1) ⊕ Rz(q4)
+>>> print(e.__str__({0}"))
+Rz(θ3) ⊕ tx(1) ⊕ Rz(θ4)
+
+
+

Angular parameters are converted to degrees, except if they +are symbolic.

+
>>> from roboticstoolbox import ET
+>>> from spatialmath.base import symbol
+>>> theta, d = symbol('theta, d')
+>>> e = ET.Rx(theta) * ET.tx(2) * ET.Rx(45, 'deg') * ET.Ry(0.2) * ET.ty(d)
+>>> str(e)
+'Rx(theta) ⊕ tx(2) ⊕ Rx(45°) ⊕ Ry(11.46°) ⊕ ty(d)'
+
+
+
+ +
+
+__repr__()
+

Return repr(self).

+
+ +
+
+__getitem__(i)
+

Index or slice an ETS

+
+
Parameters:
+

i – the index or slince

+
+
Returns:
+

Elementary transform

+
+
Return type:
+

et

+
+
+

Examples

+
>>> from roboticstoolbox import ET
+>>> e = ET.Rz() * ET.tx(1) * ET.Rz() * ET.tx(1)
+>>> e[0]
+ET.Rz(jindex=0)
+>>> e[1]
+ET.tx(eta=1.0)
+>>> e[1:3]
+[ET.tx(eta=1.0), ET.Rz(jindex=1)]
+
+
+
+ +
+
+property n: int
+

Number of joints

+

Counts the number of joints in the ETS.

+
+
Returns:
+

the number of joints in the ETS

+
+
Return type:
+

n

+
+
+

Examples

+
>>> from roboticstoolbox import ET
+>>> e = ET.Rx() * ET.tx(1) * ET.tz()
+>>> e.n
+2
+
+
+
+

See also

+

joints()

+
+
+ +
+
+property m: int
+

Number of transforms

+

Counts the number of transforms in the ETS.

+
+
Returns:
+

the number of transforms in the ETS

+
+
Return type:
+

m

+
+
+

Examples

+
>>> from roboticstoolbox import ET
+>>> e = ET.Rx() * ET.tx(1) * ET.tz()
+>>> e.m
+3
+
+
+
+ +
+
+property structure: str
+

Joint structure string

+

A string comprising the characters ‘R’ or ‘P’ which indicate the types +of joints in order from left to right.

+
+
Returns:
+

A string indicating the joint types

+
+
Return type:
+

structure

+
+
+

Examples

+
>>> from roboticstoolbox import ET
+>>> e = ET.tz() * ET.tx(1) * ET.Rz() * ET.tx(1)
+>>> e.structure
+'PR'
+
+
+
+ +
+
+joints()
+

Get a list of the variable ETs with this ETS

+
+
Returns:
+

list of ETs that are joints

+
+
Return type:
+

joints

+
+
+

Examples

+
>>> from roboticstoolbox import ET
+>>> e = ET.Rz() * ET.tx(1) * ET.Rz() * ET.tx(1)
+>>> e.joints()
+[ET.Rz(jindex=0), ET.Rz(jindex=1)]
+
+
+
+ +
+
+split()
+

Split ETS into link segments

+
+
Returns:
+

a list of ETS, each one, apart from the last, +ends with a variable ET.

+
+
Return type:
+

split

+
+
+
+ +
+
+inv()
+

Inverse of ETS

+

The inverse of a given ETS. It is computed as the inverse of the +individual ETs in the reverse order.

+
+\[(\mathbf{E}_0, \mathbf{E}_1 \cdots \mathbf{E}_{n-1} )^{-1} = (\mathbf{E}_{n-1}^{-1}, \mathbf{E}_{n-2}^{-1} \cdots \mathbf{E}_0^{-1}{n-1} )\]
+
+
Returns:
+

Inverse of the ETS

+
+
Return type:
+

inv

+
+
+

Examples

+
>>> from roboticstoolbox import ET
+>>> e = ET.Rz(jindex=2) * ET.tx(1) * ET.Rx(jindex=3,flip=True) * ET.tx(1)
+>>> print(e)
+Rz(q2) ⊕ tx(1) ⊕ Rx(-q3) ⊕ tx(1)
+>>> print(e.inv())
+tx(-1) ⊕ Rx(q3) ⊕ tx(-1) ⊕ Rz(-q2)
+
+
+

Notes

+
    +
  • +
    It is essential to use explicit joint indices to account for

    the reversed order of the transforms.

    +
    +
    +
  • +
+
+ +
+
+__mul__(other)[source]
+
+
Return type:
+

ETS2

+
+
+
+ +
+
+compile()[source]
+

Compile an ETS2

+
+
Return type:
+

ETS2

+
+
Returns:
+

optimised ETS2

+
+
+

Perform constant folding for faster evaluation. Consecutive constant +ETs are compounded, leading to a constant ET which is denoted by +SE3 when displayed.

+
+
Seealso:
+

isconstant()

+
+
+
+ +
+
+insert(arg, i=-1)[source]
+

Insert value

+
+
Parameters:
+
    +
  • i (int) – insert an ET or ETS into the ETS, default is at the end

  • +
  • arg (Union[ET2, ETS2]) – the elementary transform or sequence to insert

  • +
+
+
Return type:
+

None

+
+
+

Inserts an ET or ETS into the ET sequence. The inserted value is at position +i.

+

Example:

+
>>> from roboticstoolbox import ET2
+>>> e = ET2.R() * ET2.tx(1) * ET2.R() * ET2.tx(1)
+>>> f = ET2.R()
+>>> e.insert(f, 2)
+>>> e
+[ET2.R(jindex=0), ET2.tx(eta=1.0), ET2.R(), ET2.R(jindex=1), ET2.tx(eta=1.0)]
+
+
+
+ +
+
+fkine(q, base=None, tool=None, include_base=True)[source]
+

Forward kinematics +:type q: Union[ndarray, List[float], Tuple[float], Set[float]] +:param q: Joint coordinates +:type q: ArrayLike +:type base: Union[ndarray, SE2, None] +:param base: base transform, optional +:type tool: Union[ndarray, SE2, None] +:param tool: tool transform, optional

+
+
Return type:
+

SE2

+
+
Returns:
+

The transformation matrix representing the pose of the +end-effector

+
+
+
    +
  • T = ets.fkine(q) evaluates forward kinematics for the robot at +joint configuration q.

  • +
+

Trajectory operation: +If q has multiple rows (mxn), it is considered a trajectory and the +result is an SE2 instance with m values. +.. note:

+
- The robot's base tool transform, if set, is incorporated
+  into the result.
+- A tool transform, if provided, is incorporated into the result.
+- Works from the end-effector link to the base
+
+
+
+
References:
+
    +
  • Kinematic Derivatives using the Elementary Transform +Sequence, J. Haviland and P. Corke

  • +
+
+
+
+ +
+
+jacob0(q)[source]
+
+
Return type:
+

ndarray

+
+
+
+ +
+
+jacobe(q)[source]
+

Jacobian in base frame

+
+
Parameters:
+

q (ArrayLike) – joint coordinates

+
+
Returns:
+

Jacobian matrix

+
+
+

jacobe(q) is the manipulator Jacobian matrix which maps joint +velocity to end-effector spatial velocity.

+

End-effector spatial velocity \(\nu = (v_x, v_y, v_z, \omega_x, \omega_y, \omega_z)^T\) +is related to joint velocity by \({}^{e}\nu = {}^{e}\mathbf{J}_0(q) \dot{q}\).

+
+
Seealso:
+

jacob(), hessian0()

+
+
+
+ +
+ +
+
+ + +
+
+
+ +
+ +
+

© Copyright 2023, Jesse Haviland and Peter Corke. + Last updated on 11-May-2024. +

+
+ + + +
+
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/arm_models.html b/arm_models.html new file mode 100644 index 00000000..3b938026 --- /dev/null +++ b/arm_models.html @@ -0,0 +1,167 @@ + + + + + + + + + + + Manipulator models — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+ +
+ +
+ +
+

© Copyright 2023, Jesse Haviland and Peter Corke. + Last updated on 11-May-2024. +

+
+ + + +
+
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/arm_superclass.html b/arm_superclass.html new file mode 100644 index 00000000..bc00eb38 --- /dev/null +++ b/arm_superclass.html @@ -0,0 +1,12086 @@ + + + + + + + + + + + Superclasses — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

Superclasses

+
+

Robot

+
Inheritance diagram of roboticstoolbox.ERobot
+ + + +

The various models ERobot models all subclass this class.

+

@author: Jesse Haviland +@author: Peter Corke

+
+
+class roboticstoolbox.robot.Robot.Robot(arg, gripper_links=None, name='', manufacturer='', comment='', base=None, tool=None, gravity=[0, 0, -9.81], keywords=[], symbolic=False, configs=None, check_jindex=True, urdf_string=None, urdf_filepath=None)[source]
+

Bases: BaseRobot[Link], RobotKinematicsMixin

+
+
+static URDF_read(file_path, tld=None, xacro_tld=None)[source]
+

Read a URDF file as Links

+

File should be specified relative to RTBDATA/URDF/xacro

+
+
Parameters:
+
    +
  • file_path – File path relative to the xacro folder

  • +
  • tld – A custom top-level directory which holds the xacro data, +defaults to None

  • +
  • xacro_tld – A custom top-level within the xacro data, +defaults to None

  • +
+
+
Return type:
+

Tuple[List[Link], str, str, Union[Path, PurePosixPath]]

+
+
Returns:
+

    +
  • links – a list of links

  • +
  • name – the name of the robot

  • +
  • urdf – a string representing the URDF

  • +
  • file_path – a path to the original file

  • +
+

+
+
+

Notes

+

If tld is not supplied, filepath pointing to xacro data should +be directly under RTBDATA/URDF/xacro OR under ./xacro relative +to the model file calling this method. If tld is supplied, then +`file_path` needs to be relative to tld

+
+ +
+
+classmethod URDF(file_path, gripper=None)[source]
+

Construct a Robot object from URDF file

+
+
Parameters:
+
    +
  • file_path (str) – the path to the URDF

  • +
  • gripper (Union[int, str, None]) – index or name of the gripper link(s)

  • +
+
+
Returns:
+

    +
  • If gripper is specified, links from that link outward are removed

  • +
  • from the rigid-body tree and folded into a Gripper object.

  • +
+

+
+
+
+ +
+
+property reach: float
+

Reach of the robot

+

A conservative estimate of the reach of the robot. It is computed as +the sum of the translational ETs that define the link transform.

+
+

Note

+

Computed on the first access. If kinematic parameters +subsequently change this will not be reflected.

+
+
+
Returns:
+

Maximum reach of the robot

+
+
Return type:
+

reach

+
+
+

Notes

+
    +
  • Probably an overestimate of reach

  • +
  • Used by numerical inverse kinematics to scale translational +error.

  • +
  • For a prismatic joint, uses qlim if it is set

  • +
+
+ +
+
+fkine_all(q)[source]
+

Compute the pose of every link frame

+

T = robot.fkine_all(q) is an SE3 instance with robot.nlinks + +1 values:

+
    +
  • T[0] is the base transform

  • +
  • T[i] is the pose of link whose number is i

  • +
+
+
Parameters:
+

q (Union[ndarray, List[float], Tuple[float], Set[float]]) – The joint configuration

+
+
Returns:
+

Pose of all links

+
+
Return type:
+

fkine_all

+
+
+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
+
+ +
+
+manipulability(q=None, J=None, end=None, start=None, method='yoshikawa', axes='all', **kwargs)[source]
+

Manipulability measure

+

manipulability(q) is the scalar manipulability index +for the robot at the joint configuration q. It indicates +dexterity, that is, how well conditioned the robot is for motion +with respect to the 6 degrees of Cartesian motion. The values is +zero if the robot is at a singularity.

+
+
Parameters:
+
    +
  • q – Joint coordinates, one of J or q required

  • +
  • J – Jacobian in base frame if already computed, one of J or +q required

  • +
  • method (Literal['yoshikawa', 'asada', 'minsingular', 'invcondition']) – method to use, “yoshikawa” (default), “invcondition”, +“minsingular” or “asada”

  • +
  • axes (Union[Literal['all', 'trans', 'rot'], List[bool]]) – Task space axes to consider: “all” [default], +“trans”, or “rot”

  • +
+
+
Return type:
+

manipulability

+
+
+

Synopsis

+

Various measures are supported:

+
+
Measure | Description |
+
+

|-------------------|————————————————-| +| "yoshikawa" | Volume of the velocity ellipsoid, distance | +| | from singularity [Yoshikawa85] | +| "invcondition"``| Inverse condition number of Jacobian, isotropy  | +|                   | of the velocity ellipsoid [Klein87]_            | +| ``"minsingular" | Minimum singular value of the Jacobian, | +| | distance from singularity [Klein87] | +| "asada" | Isotropy of the task-space acceleration | +| | ellipsoid which is a function of the Cartesian | +| | inertia matrix which depends on the inertial | +| | parameters [Asada83] |

+

Trajectory operation:

+

If q is a matrix (m,n) then the result (m,) is a vector of +manipulability indices for each joint configuration specified by a row +of q.

+

Notes

+
    +
  • Invokes the jacob0 method of the robot if J is not passed

  • +
  • +
    The “all” option includes rotational and translational

    dexterity, but this involves adding different units. It can be +more useful to look at the translational and rotational +manipulability separately.

    +
    +
    +
  • +
  • +
    Examples in the RVC book (1st edition) can be replicated by

    using the “all” option

    +
    +
    +
  • +
  • +
    Asada’s measure requires inertial a robot model with inertial

    parameters.

    +
    +
    +
  • +
+

References

+
+
+[Yoshikawa85] +

Manipulability of Robotic Mechanisms. Yoshikawa T., +The International Journal of Robotics Research. +1985;4(2):3-9. doi:10.1177/027836498500400201

+
+
+[Asada83] +

A geometrical representation of manipulator dynamics and +its application to arm design, H. Asada, +Journal of Dynamic Systems, Measurement, and Control, +vol. 105, p. 131, 1983.

+
+
+[Klein87] +

Dexterity Measures for the Design and Control of +Kinematically Redundant Manipulators. Klein CA, Blaho BE. +The International Journal of Robotics Research. +1987;6(2):72-83. doi:10.1177/027836498700600206

+
+
+
    +
  • Robotics, Vision & Control, Chap 8, P. Corke, Springer 2011.

  • +
+
+

Changed in version 1.0.3: Removed ‘both’ option for axes, added a custom list option.

+
+
+ +
+
+jtraj(T1, T2, t, **kwargs)[source]
+

Joint-space trajectory between SE(3) poses

+

The initial and final poses are mapped to joint space using inverse +kinematics:

+
    +
  • if the object has an analytic solution ikine_a that will be used,

  • +
  • otherwise the general numerical algorithm ikine_lm will be used.

  • +
+

traj = obot.jtraj(T1, T2, t) is a trajectory object whose +attribute traj.q is a row-wise joint-space trajectory.

+
+
Parameters:
+
    +
  • T1 (Union[ndarray, SE3]) – initial end-effector pose

  • +
  • T2 (Union[ndarray, SE3]) – final end-effector pose

  • +
  • t (Union[ndarray, int]) – time vector or number of steps

  • +
  • kwargs – arguments passed to the IK solver

  • +
+
+
Return type:
+

trajectory

+
+
+
+ +
+
+jacob0_dot(q, qd, J0=None, representation=None)[source]
+

Derivative of Jacobian

+

robot.jacob_dot(q, qd) computes the rate of change of the +Jacobian elements

+
+\[\dmat{J} = \frac{d \mat{J}}{d \vec{q}} \frac{d \vec{q}}{dt}\]
+

where the first term is the rank-3 Hessian.

+
+
Parameters:
+
    +
  • q

  • +
  • robot (The joint configuration of the) –

  • +
  • qd (Union[ndarray, List[float], Tuple[float], Set[float]]) – The joint velocity of the robot

  • +
  • J0 – Jacobian in {0} frame

  • +
  • representation (Optional[Literal['rpy/xyz', 'rpy/zyx', 'eul', 'exp']]) – angular representation

  • +
+
+
Returns:
+

The derivative of the manipulator Jacobian

+
+
Return type:
+

jdot

+
+
+

Synopsis

+

If J0 is already calculated for the joint +coordinates q it can be passed in to to save computation time.

+

It is computed as the mode-3 product of the Hessian tensor and the +velocity vector.

+

The derivative of an analytical Jacobian can be obtained by setting +representation as

+

|``representation`` | Rotational representation | +|---------------------|————————————-| +|'rpy/xyz' | RPY angular rates in XYZ order | +|'rpy/zyx' | RPY angular rates in XYZ order | +|'eul' | Euler angular rates in ZYZ order | +|'exp' | exponential coordinate rates |

+

References

+
    +
  • +
    Kinematic Derivatives using the Elementary Transform

    Sequence, J. Haviland and P. Corke

    +
    +
    +
  • +
+
+

See also

+

jacob0(), hessian0()

+
+
+ +
+
+jacobm(q=None, J=None, H=None, end=None, start=None, axes='all')[source]
+

The manipulability Jacobian

+

This measure relates the rate of change of the manipulability to the +joint velocities of the robot. One of J or q is required. Supply J +and H if already calculated to save computation time

+
+
Parameters:
+
    +
  • q – The joint angles/configuration of the robot (Optional, +if not supplied will use the stored q values).

  • +
  • J – The manipulator Jacobian in any frame

  • +
  • H – The manipulator Hessian in any frame

  • +
  • end (Union[str, Link, Gripper, None]) – the final link or Gripper which the Hessian represents

  • +
  • start (Union[str, Link, Gripper, None]) – the first link which the Hessian represents

  • +
+
+
Returns:
+

The manipulability Jacobian

+
+
Return type:
+

jacobm

+
+
+

Synopsis

+

Yoshikawa’s manipulability measure

+
+\[m(\vec{q}) = \sqrt{\mat{J}(\vec{q}) \mat{J}(\vec{q})^T}\]
+

This method returns its Jacobian with respect to configuration

+
+\[\frac{\partial m(\vec{q})}{\partial \vec{q}}\]
+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+ +
+
+closest_point(q, shape, inf_dist=1.0, skip=False)[source]
+

Find the closest point between robot and shape

+

closest_point(shape, inf_dist) returns the minimum euclidean +distance between this robot and shape, provided it is less than +inf_dist. It will also return the points on self and shape in the +world frame which connect the line of length distance between the +shapes. If the distance is negative then the shapes are collided.

+
+
+shape
+

The shape to compare distance to

+
+ +
+
+inf_dist
+

The minimum distance within which to consider +the shape

+
+ +
+
+skip
+

Skip setting all shape transforms based on q, use this +option if using this method in conjuction with Swift to save time

+
+ +
+
Return type:
+

Tuple[Optional[int], Optional[ndarray], Optional[ndarray]]

+
+
Returns:
+

    +
  • d – distance between the robot and shape

  • +
  • p1 – [x, y, z] point on the robot (in the world frame)

  • +
  • p2 – [x, y, z] point on the shape (in the world frame)

  • +
+

+
+
+
+ +
+
+iscollided(q, shape, skip=False)[source]
+

Check if the robot is in collision with a shape

+

iscollided(shape) checks if this robot and shape have collided

+
+
+shape
+

The shape to compare distance to

+
+ +
+
+skip
+

Skip setting all shape transforms based on q, use this +option if using this method in conjuction with Swift to save time

+
+ +
+
Returns:
+

True if shapes have collided

+
+
Return type:
+

iscollided

+
+
+
+ +
+
+collided(q, shape, skip=False)[source]
+

Check if the robot is in collision with a shape

+

collided(shape) checks if this robot and shape have collided

+
+
+shape
+

The shape to compare distance to

+
+ +
+
+skip
+

Skip setting all shape transforms based on q, use this +option if using this method in conjuction with Swift to save time

+
+ +
+
Returns:
+

True if shapes have collided

+
+
Return type:
+

collided

+
+
+
+ +
+
+joint_velocity_damper(ps=0.05, pi=0.1, n=None, gain=1.0)[source]
+

Compute the joint velocity damper for QP motion control

+

Formulates an inequality contraint which, when optimised for will +make it impossible for the robot to run into joint limits. Requires +the joint limits of the robot to be specified. See examples/mmc.py +for use case

+
+
+ps
+

The minimum angle (in radians) in which the joint is +allowed to approach to its limit

+
+ +
+
+pi
+

The influence angle (in radians) in which the velocity +damper becomes active

+
+ +
+
+n
+

The number of joints to consider. Defaults to all joints

+
+ +
+
+gain
+

The gain for the velocity damper

+
+ +
+
Return type:
+

Tuple[ndarray, ndarray]

+
+
Returns:
+

    +
  • Ain – A (6,) vector inequality contraint for an optisator

  • +
  • Bin – b (6,) vector inequality contraint for an optisator

  • +
+

+
+
+
+ +
+ +

Compute a collision constrain for QP motion control

+

Formulates an inequality contraint which, when optimised for will +make it impossible for the robot to run into a collision. Requires +See examples/neo.py for use case

+
+
+ds
+

The minimum distance in which a joint is allowed to +approach the collision object shape

+
+ +
+
+di
+

The influence distance in which the velocity +damper becomes active

+
+ +
+
+xi
+

The gain for the velocity damper

+
+ +
+
+end
+

The end link of the robot to consider

+
+ +
+
+start
+

The start link of the robot to consider

+
+ +
+
+collision_list
+

A list of shapes to consider for collision

+
+ +
+
Returns:
+

    +
  • Ain – A (6,) vector inequality contraint for an optisator

  • +
  • Bin – b (6,) vector inequality contraint for an optisator

  • +
+

+
+
+
+ +
+
+vision_collision_damper(shape, camera=None, camera_n=0, q=None, di=0.3, ds=0.05, xi=1.0, end=None, start=None, collision_list=None)[source]
+

Compute a vision collision constrain for QP motion control

+

Formulates an inequality contraint which, when optimised for will +make it impossible for the robot to run into a line of sight. +See examples/fetch_vision.py for use case

+
+
+camera
+

The camera link, either as a robotic link or SE3 +pose

+
+ +
+
+camera_n
+

Degrees of freedom of the camera link

+
+ +
+
+ds
+

The minimum distance in which a joint is allowed to +approach the collision object shape

+
+ +
+
+di
+

The influence distance in which the velocity +damper becomes active

+
+ +
+
+xi
+

The gain for the velocity damper

+
+ +
+
+end
+

The end link of the robot to consider

+
+ +
+
+start
+

The start link of the robot to consider

+
+ +
+
+collision_list
+

A list of shapes to consider for collision

+
+ +
+
Returns:
+

    +
  • Ain – A (6,) vector inequality contraint for an optisator

  • +
  • Bin – b (6,) vector inequality contraint for an optisator

  • +
+

+
+
+
+ +
+
+rne(q, qd, qdd, symbolic=False, gravity=None)[source]
+

Compute inverse dynamics via recursive Newton-Euler formulation

+

rne_dh(q, qd, qdd) where the arguments have shape (n,) where n is +the number of robot joints. The result has shape (n,).

+

rne_dh(q, qd, qdd) where the arguments have shape (m,n) where n +is the number of robot joints and where m is the number of steps in +the joint trajectory. The result has shape (m,n).

+

rne_dh(p) where the input is a 1D array p = [q, qd, qdd] with +shape (3n,), and the result has shape (n,).

+

rne_dh(p) where the input is a 2D array p = [q, qd, qdd] with +shape (m,3n) and the result has shape (m,n).

+
+
Parameters:
+
+
+
Returns:
+

Joint force/torques

+
+
Return type:
+

tau

+
+
+

Notes

+
    +
  • This version supports symbolic model parameters

  • +
  • Verified against MATLAB code

  • +
+
+ +
+
+__getitem__(i)
+

Get link

+

This also supports iterating over each link in the robot object, +from the base to the tool.

+
+
Parameters:
+

i (Union[int, str]) – link number or name

+
+
Returns:
+

i’th link or named link

+
+
Return type:
+

link

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> print(robot[1]) # print the 2nd link
+RevoluteDH:   θ=q,  d=0,  a=0.4318,  ⍺=0.0
+>>> print([link.a for link in robot])  # print all the a_j values
+[0, 0.4318, 0.0203, 0, 0, 0]
+
+
+

Notes

+
+
Robot supports link lookup by name,

eg. robot['link1']

+
+
+
+ +
+
+__str__()
+

Pretty prints the ETS Model of the robot.

+
+
Returns:
+

Pretty print of the robot model

+
+
Return type:
+

str

+
+
+

Notes

+
    +
  • Constant links are shown in blue.

  • +
  • End-effector links are prefixed with an @

  • +
  • Angles in degrees

  • +
  • +
    The robot base frame is denoted as BASE and is equal to the

    robot’s base attribute.

    +
    +
    +
  • +
+
+ +
+
+accel(q, qd, torque, gravity=None)
+

Compute acceleration due to applied torque

+

qdd = accel(q, qd, torque) calculates a vector (n) of joint +accelerations that result from applying the actuator force/torque (n) +to the manipulator in state q (n) and qd (n), and n is +the number of robot joints.

+
+\[\ddot{q} = \mathbf{M}^{-1} \left(\tau - \mathbf{C}(q)\dot{q} - \mathbf{g}(q)\right)\]
+

Trajectory operation

+

If q, qd, torque are matrices (m,n) then qdd is a matrix (m,n) +where each row is the acceleration corresponding to the equivalent rows +of q, qd, torque.

+
+
Parameters:
+
    +
  • q – Joint coordinates

  • +
  • qd – Joint velocity

  • +
  • torque – Joint torques of the robot

  • +
  • gravity – Gravitational acceleration (Optional, if not supplied will +use the gravity attribute of self).

  • +
+
+
Return type:
+

ndarray(n)

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> puma.accel(puma.qz, 0.5 * np.ones(6), np.zeros(6))
+array([ -7.5544, -12.22  ,  -6.4022,  -5.4303,  -4.9518,  -2.1178])
+
+
+

Notes

+
    +
  • +
    Useful for simulation of manipulator dynamics, in

    conjunction with a numerical integration function.

    +
    +
    +
  • +
  • +
    Uses the method 1 of Walker and Orin to compute the forward

    dynamics.

    +
    +
    +
  • +
  • +
    Featherstone’s method is more efficient for robots with large

    numbers of joints.

    +
    +
    +
  • +
  • Joint friction is considered.

  • +
+

References

+
    +
  • +
    Efficient dynamic computer simulation of robotic mechanisms,

    M. W. Walker and D. E. Orin, +ASME Journa of Dynamic Systems, Measurement and Control, vol. +104, no. 3, pp. 205-211, 1982.

    +
    +
    +
  • +
+
+ +
+
+accel_x(q, xd, wrench, gravity=None, pinv=False, representation='rpy/xyz')
+

Operational space acceleration due to applied wrench

+

xdd = accel_x(q, qd, wrench) is the operational space acceleration +due to wrench applied to the end-effector of a robot in joint +configuration q and joint velocity qd.

+
+\[\ddot{x} = \mathbf{J}(q) \mathbf{M}(q)^{-1} \left( + \mathbf{J}(q)^T w - \mathbf{C}(q)\dot{q} - \mathbf{g}(q) + \right)\]
+

Trajectory operation

+

If q, qd, torque are matrices (m,n) then qdd is a matrix (m,n) +where each row is the acceleration corresponding to the equivalent rows +of q, qd, wrench.

+
+
Parameters:
+
    +
  • q – Joint coordinates

  • +
  • qd – Joint velocity

  • +
  • wrench – Wrench applied to the end-effector

  • +
  • gravity – Gravitational acceleration (Optional, if not supplied will +use the gravity attribute of self).

  • +
  • pinv – use pseudo inverse rather than inverse

  • +
  • analytical – the type of analytical Jacobian to use, default is +‘rpy/xyz’

  • +
  • xd

  • +
  • representation – (Default value = “rpy/xyz”)

  • +
+
+
Returns:
+

Operational space accelerations of the end-effector

+
+
Return type:
+

accel

+
+
+

Examples

+
    ainv = _umath_linalg.inv(a, signature=signature, extobj=extobj)
+  File "/opt/hostedtoolcache/Python/3.7.17/x64/lib/python3.7/site-packages/numpy/linalg/linalg.py", line 88, in _raise_linalgerror_singular
+    raise LinAlgError("Singular matrix")
+numpy.linalg.LinAlgError: Singular matrix
+
+
+

Notes

+
    +
  • +
    Useful for simulation of manipulator dynamics, in

    conjunction with a numerical integration function.

    +
    +
    +
  • +
  • +
    Uses the method 1 of Walker and Orin to compute the forward

    dynamics.

    +
    +
    +
  • +
  • +
    Featherstone’s method is more efficient for robots with large

    numbers of joints.

    +
    +
    +
  • +
  • Joint friction is considered.

  • +
+
+

See also

+

accel()

+
+
+ +
+
+addconfiguration(name, q)
+

Add a named joint configuration

+

Add a named configuration to the robot instance’s dictionary of named +configurations.

+
+
Parameters:
+
    +
  • name (str) – Name of the joint configuration

  • +
  • q (ndarray) – Joint configuration

  • +
+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.addconfiguration_attr("mypos", [0.1, 0.2, 0.3, 0.4, 0.5, 0.6])
+>>> robot.configs["mypos"]
+array([0.1, 0.2, 0.3, 0.4, 0.5, 0.6])
+
+
+
+

See also

+

addconfiguration()

+
+
+ +
+
+addconfiguration_attr(name, q, unit='rad')
+

Add a named joint configuration as an attribute

+
+
Parameters:
+
+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.addconfiguration_attr("mypos", [0.1, 0.2, 0.3, 0.4, 0.5, 0.6])
+>>> robot.mypos
+array([0.1, 0.2, 0.3, 0.4, 0.5, 0.6])
+>>> robot.configs["mypos"]
+array([0.1, 0.2, 0.3, 0.4, 0.5, 0.6])
+
+
+

Notes

+
    +
  • Used in robot model init method to store the qr configuration

  • +
  • +
    Dynamically adding attributes to objects can cause issues with

    Python type checking.

    +
    +
    +
  • +
  • +
    Configuration is also added to the robot instance’s dictionary of

    named configurations.

    +
    +
    +
  • +
+
+

See also

+

addconfiguration()

+
+
+ +
+
+attach(object)
+
+ +
+
+attach_to(object)
+
+ +
+
+property base: SE3
+

Get/set robot base transform

+
    +
  • robot.base is the robot base transform

  • +
  • robot.base = ... checks and sets the robot base transform

  • +
+
+
Parameters:
+

base – the new robot base transform

+
+
Returns:
+

the current robot base transform

+
+
Return type:
+

base

+
+
+
+ +
+ +

Get the robot base link

+
    +
  • robot.base_link is the robot base link

  • +
+
+
Returns:
+

the first link in the robot tree

+
+
Return type:
+

base_link

+
+
+
+ +
+
+cinertia(q)
+

Deprecated, use inertia_x

+
+ +
+
+property comment: str
+

Get/set robot comment

+
    +
  • robot.comment is the robot comment

  • +
  • robot.comment = ... checks and sets the robot comment

  • +
+
+
Parameters:
+

name – the new robot comment

+
+
Returns:
+

robot comment

+
+
Return type:
+

comment

+
+
+
+ +
+
+property configs: Dict[str, ndarray]
+
+ +
+
+configurations_str(border='thin')
+
+ +
+
+property control_mode: str
+

Get/set robot control mode

+
    +
  • robot.control_type is the robot control mode

  • +
  • robot.control_type = ... checks and sets the robot control mode

  • +
+
+
Parameters:
+

control_mode – the new robot control mode

+
+
Returns:
+

the current robot control mode

+
+
Return type:
+

control_mode

+
+
+
+ +
+
+copy()
+
+ +
+
+coriolis(q, qd)
+

Coriolis and centripetal term

+

coriolis(q, qd) calculates the Coriolis/centripetal matrix (n,n) +for the robot in configuration q and velocity qd, where n +is the number of joints.

+

The product \(\mathbf{C} \dot{q}\) is the vector of joint +force/torque due to velocity coupling. The diagonal elements are due to +centripetal effects and the off-diagonal elements are due to Coriolis +effects. This matrix is also known as the velocity coupling matrix, +since it describes the disturbance forces on any joint due to +velocity of all other joints.

+

Trajectory operation

+

If q and qd are matrices (m,n), each row is interpretted as a +joint configuration, and the result (n,n,m) is a 3d-matrix where +each plane corresponds to a row of q and qd.

+
+
Parameters:
+
    +
  • q – Joint coordinates

  • +
  • qd – Joint velocity

  • +
+
+
Returns:
+

Velocity matrix

+
+
Return type:
+

coriolis

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> puma.coriolis(puma.qz, 0.5 * np.ones((6,)))
+array([[-0.4017, -0.5513, -0.2025, -0.0007, -0.0013,  0.    ],
+       [ 0.2023, -0.1937, -0.3868, -0.    , -0.002 ,  0.    ],
+       [ 0.1987,  0.193 , -0.    ,  0.    , -0.0001,  0.    ],
+       [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+       [ 0.0007,  0.0007,  0.0001,  0.    ,  0.    ,  0.    ],
+       [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ]])
+
+
+

Notes

+
    +
  • +
    Joint viscous friction is also a joint force proportional to

    velocity but it is eliminated in the computation of this value.

    +
    +
    +
  • +
  • Computationally slow, involves \(n^2/2\) invocations of RNE.

  • +
+
+ +
+
+coriolis_x(q, qd, pinv=False, representation='rpy/xyz', J=None, Ji=None, Jd=None, C=None, Mx=None)
+

Operational space Coriolis and centripetal term

+

coriolis_x(q, qd) is the Coriolis/centripetal matrix (m,m) +in operational space for the robot in configuration q and velocity +qd, where n is the number of joints.

+
+\[\mathbf{C}_x = \mathbf{J}(q)^{-T} \left( + \mathbf{C}(q) - \mathbf{M}_x(q) \mathbf{J})(q) + \right) \mathbf{J}(q)^{-1}\]
+

The product \(\mathbf{C} \dot{x}\) is the operational space wrench +due to joint velocity coupling. This matrix is also known as the +velocity coupling matrix, since it describes the disturbance forces on +any joint due to velocity of all other joints.

+

The transformation to operational space requires an analytical, rather +than geometric, Jacobian. analytical can be one of:

+
+
+ + + + + + + + + + + + + + + + + + + +

Value

Rotational representation

'rpy/xyz'

RPY angular rates in XYZ order (default)

'rpy/zyx'

RPY angular rates in XYZ order

'eul'

Euler angular rates in ZYZ order

'exp'

exponential coordinate rates

+
+

Trajectory operation

+

If q and qd are matrices (m,n), each row is interpretted as a +joint configuration, and the result (n,n,m) is a 3d-matrix where +each plane corresponds to a row of q and qd.

+
+
Parameters:
+
    +
  • q – Joint coordinates

  • +
  • qd – Joint velocity

  • +
  • pinv – use pseudo inverse rather than inverse (Default value = False)

  • +
  • analytical – the type of analytical Jacobian to use, default is +‘rpy/xyz’

  • +
  • representation – (Default value = “rpy/xyz”)

  • +
  • J

  • +
  • Ji

  • +
  • Jd

  • +
  • C

  • +
  • Mx

  • +
+
+
Returns:
+

Operational space velocity matrix

+
+
Return type:
+

ndarray(6,6) or ndarray(m,6,6)

+
+
+

Examples

+
    ainv = _umath_linalg.inv(a, signature=signature, extobj=extobj)
+  File "/opt/hostedtoolcache/Python/3.7.17/x64/lib/python3.7/site-packages/numpy/linalg/linalg.py", line 88, in _raise_linalgerror_singular
+    raise LinAlgError("Singular matrix")
+numpy.linalg.LinAlgError: Singular matrix
+
+
+

Notes

+
    +
  • +
    Joint viscous friction is also a joint force proportional to

    velocity but it is eliminated in the computation of this value.

    +
    +
    +
  • +
  • Computationally slow, involves \(n^2/2\) invocations of RNE.

  • +
  • If the robot is not 6 DOF the pinv option is set True.

  • +
  • pinv() is around 5x slower than inv()

  • +
+
+

Warning

+

Assumes that the operational space has 6 DOF.

+
+ +
+ +
+
+property default_backend
+

Get default graphical backend

+
    +
  • +
    robot.default_backend Get the default graphical backend, used when

    no explicit backend is passed to Robot.plot.

    +
    +
    +
  • +
  • +
    robot.default_backend = ... Set the default graphical backend, used when

    no explicit backend is passed to Robot.plot. The default set here will +be overridden if the particular Robot subclass cannot support it.

    +
    +
    +
  • +
+
+
Returns:
+

backend name

+
+
Return type:
+

default_backend

+
+
+
+ +
+ +

A link search method

+

Visit all links from start in depth-first order and will apply +func to each visited link

+
+
Parameters:
+
+
+
Returns:
+

A list of links

+
+
Return type:
+

links

+
+
+
+ +
+
+dotfile(filename, etsbox=False, ets='full', jtype=False, static=True)
+

Write a link transform graph as a GraphViz dot file

+
+
The file can be processed using dot:

% dot -Tpng -o out.png dotfile.dot

+
+
The nodes are:
    +
  • Base is shown as a grey square. This is the world frame origin, +but can be changed using the base attribute of the robot.

  • +
  • Link frames are indicated by circles

  • +
  • ETS transforms are indicated by rounded boxes

  • +
+
+
The edges are:
    +
  • an arrow if jtype is False or the joint is fixed

  • +
  • an arrow with a round head if jtype is True and the joint is +revolute

  • +
  • an arrow with a box head if jtype is True and the joint is +prismatic

  • +
+
+
+

Edge labels or nodes in blue have a fixed transformation to the +preceding link.

+
+

Note

+
+
If filename is a file object then the file will not

be closed after the GraphViz model is written.

+
+
+
+
+
Parameters:
+
    +
  • file – Name of file to write to

  • +
  • etsbox (bool) – Put the link ETS in a box, otherwise an edge label

  • +
  • ets (Literal['full', 'brief']) – Display the full ets with “full” or a brief version with “brief”

  • +
  • jtype (bool) – Arrowhead to node indicates revolute or prismatic type

  • +
  • static (bool) – Show static joints in blue and bold

  • +
+
+
+
+

See also

+

showgraph()

+
+
+ +
+
+dynamics()
+

Pretty print the dynamic parameters (Robot superclass)

+

The dynamic parameters (inertial and friction) are printed in a table, +with one row per link.

+

Examples

+

+
+
+
+ +
+
+dynamics_list()
+

Print dynamic parameters (Robot superclass)

+

Display the kinematic and dynamic parameters to the console in +reable format

+
+ +
+
+dynchanged(what=None)
+

Dynamic parameters have changed

+

Called from a property setter to inform the robot that the cache of +dynamic parameters is invalid.

+
+

See also

+

roboticstoolbox.Link._listen_dyn()

+
+
+ +
+ +
+ +
+
+ets(start=None, end=None)
+

Robot to ETS

+

robot.ets() is an ETS representing the kinematics from base to +end-effector.

+

robot.ets(end=link) is an ETS representing the kinematics from +base to the link link specified as a Link reference or a name.

+

robot.ets(start=l1, end=l2) is an ETS representing the kinematics +from link l1 to link l2.

+

:param : +:type : param start: start of path, defaults to base_link +:param : +:type : param end: end of path, defaults to end-effector

+
+
Raises:
+
+
+
Returns:
+

elementary transform sequence

+
+
Return type:
+

ets

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.ETS.Panda()
+>>> panda.ets()
+[ET.tz(eta=0.333), ET.Rz(jindex=0), ET.Rx(eta=-1.5707963267948966), ET.Rz(jindex=1), ET.Rx(eta=1.5707963267948966), ET.tz(eta=0.316), ET.Rz(jindex=2), ET.tx(eta=0.0825), ET.Rx(eta=1.5707963267948966), ET.Rz(jindex=3), ET.tx(eta=-0.0825), ET.Rx(eta=-1.5707963267948966), ET.tz(eta=0.384), ET.Rz(jindex=4), ET.Rx(eta=1.5707963267948966), ET.Rz(jindex=5), ET.tx(eta=0.088), ET.Rx(eta=1.5707963267948966), ET.tz(eta=0.107), ET.Rz(jindex=6), ET.tz(eta=0.10300000000000001), ET.Rz(eta=-0.7853981633974483)]
+
+
+
+ +
+
+fdyn(T, q0, Q=None, Q_args={}, qd0=None, solver='RK45', solver_args={}, dt=None, progress=False)
+

Integrate forward dynamics

+

tg = R.fdyn(T, q) integrates the dynamics of the robot with zero +input torques over the time interval 0 to T and returns the +trajectory as a namedtuple with elements:

+
    +
  • t the time vector (M,)

  • +
  • q the joint coordinates (M,n)

  • +
  • qd the joint velocities (M,n)

  • +
+

tg = R.fdyn(T, q, torqfun) as above but the torque applied to the +joints is given by the provided function:

+
tau = function(robot, t, q, qd, **args)
+
+
+

where the inputs are:

+
+
    +
  • the robot object

  • +
  • current time

  • +
  • current joint coordinates (n,)

  • +
  • current joint velocity (n,)

  • +
  • args, optional keyword arguments can be specified, these are

  • +
+

passed in from the targs kewyword argument.

+
+

The function must return a Numpy array (n,) of joint forces/torques.

+
+
Parameters:
+
+
+
Returns:
+

robot trajectory

+
+
Return type:
+

trajectory

+
+
+

Examples

+

To apply zero joint torque to the robot without Coulomb +friction:

+
>>> def myfunc(robot, t, q, qd):
+>>>     return np.zeros((robot.n,))
+
+
+
>>> tg = robot.nofriction().fdyn(5, q0, myfunc)
+
+
+
>>> plt.figure()
+>>> plt.plot(tg.t, tg.q)
+>>> plt.show()
+
+
+

We could also use a lambda function:

+
>>> tg = robot.nofriction().fdyn(
+>>>     5, q0, lambda r, t, q, qd: np.zeros((r.n,)))
+
+
+

The robot is controlled by a PD controller. We first define a +function to compute the control which has additional parameters for +the setpoint and control gains (qstar, P, D):

+
>>> def myfunc(robot, t, q, qd, qstar, P, D):
+>>>     return (qstar - q) * P + qd * D  # P, D are (6,)
+
+
+
>>> tg = robot.fdyn(10, q0, myfunc, torque_args=(qstar, P, D)) )
+
+
+

Many integrators have variable step length which is problematic if we +want to animate the result. If dt is specified then the solver +results are interpolated in time steps of dt.

+

Notes

+ +
+

See also

+

DHRobot.accel(), DHRobot.nofriction(), DHRobot.rne()

+
+
+ +
+
+fellipse(q, opt='trans', unit='rad', centre=[0, 0, 0])
+

Create a force ellipsoid object for plotting with PyPlot

+

robot.fellipse(q) creates a force ellipsoid for the robot at +pose q. The ellipsoid is centered at the origin.

+
+
+q
+

The joint configuration of the robot.

+
+ +
+
+opt
+

‘trans’ or ‘rot’ will plot either the translational or +rotational force ellipsoid

+
+ +
+
+unit
+

‘rad’ or ‘deg’ will plot the ellipsoid in radians or +degrees

+
+ +
+
+centre
+

The centre of the ellipsoid, either ‘ee’ for the end-effector +or a 3-vector [x, y, z] in the world frame

+
+ +
+
Returns:
+

An EllipsePlot object

+
+
Return type:
+

env

+
+
+

Notes

+
    +
  • +
    By default the ellipsoid related to translational motion is

    drawn. Use opt='rot' to draw the rotational velocity +ellipsoid.

    +
    +
    +
  • +
  • +
    By default the ellipsoid is drawn at the origin. The option

    centre allows its origin to set to set to the specified +3-vector, or the string “ee” ensures it is drawn at the +end-effector position.

    +
    +
    +
  • +
+
+ +
+
+fkine(q, end=None, start=None, tool=None, include_base=True)
+

Forward kinematics

+

T = robot.fkine(q) evaluates forward kinematics for the robot at +joint configuration q.

+

Trajectory operation: +If q has multiple rows (mxn), it is considered a trajectory and the +result is an SE3 instance with m values.

+
+
+q
+

Joint coordinates

+
+ +
+
+end
+

end-effector or gripper to compute forward kinematics to

+
+ +
+
+start
+

the link to compute forward kinematics from

+
+ +
+
+tool
+

tool transform, optional

+
+ +
+
Return type:
+

SE3

+
+
Returns:
+

The transformation matrix representing the pose of the +end-effector

+
+
+

Examples

+

The following example makes a panda robot object, and solves for the +forward kinematics at the listed configuration.

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda()
+>>> panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+SE3(array([[ 0.995 , -0.    ,  0.0998,  0.484 ],
+           [-0.    , -1.    ,  0.    ,  0.    ],
+           [ 0.0998, -0.    , -0.995 ,  0.4126],
+           [ 0.    ,  0.    ,  0.    ,  1.    ]]))
+
+
+

Notes

+
    +
  • +
    For a robot with a single end-effector there is no need to

    specify end

    +
    +
    +
  • +
  • +
    For a robot with multiple end-effectors, the end must

    be specified.

    +
    +
    +
  • +
  • +
    The robot’s base tool transform, if set, is incorporated

    into the result.

    +
    +
    +
  • +
  • A tool transform, if provided, is incorporated into the result.

  • +
  • Works from the end-effector link to the base

  • +
+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+ +
+
+friction(qd)
+

Manipulator joint friction (Robot superclass)

+

robot.friction(qd) is a vector of joint friction +forces/torques for the robot moving with joint velocities qd.

+

The friction model includes:

+
    +
  • Viscous friction which is a linear function of velocity.

  • +
  • Coulomb friction which is proportional to sign(qd).

  • +
+
+\[\begin{split}\tau_j = G^2 B \dot{q}_j + |G_j| \left\{ \begin{array}{ll} + \tau_{C,j}^+ & \mbox{if $\dot{q}_j > 0$} \\ + \tau_{C,j}^- & \mbox{if $\dot{q}_j < 0$} \end{array} \right.\end{split}\]
+
+
Parameters:
+

qd (ndarray) – The joint velocities of the robot

+
+
Returns:
+

The joint friction forces/torques for the robot

+
+
Return type:
+

friction

+
+
+

Notes

+
    +
  • +
    The friction value should be added to the motor output torque to

    determine the nett torque. It has a negative value when qd > 0.

    +
    +
    +
  • +
  • +
    The returned friction value is referred to the output of the

    gearbox.

    +
    +
    +
  • +
  • +
    The friction parameters in the Link object are referred to the

    motor.

    +
    +
    +
  • +
  • Motor viscous friction is scaled up by \(G^2\).

  • +
  • Motor Coulomb friction is scaled up by math:G.

  • +
  • +
    The appropriate Coulomb friction value to use in the

    non-symmetric case depends on the sign of the joint velocity, +not the motor velocity.

    +
    +
    +
  • +
  • +
    Coulomb friction is zero for zero joint velocity, stiction is

    not modeled.

    +
    +
    +
  • +
  • +
    The absolute value of the gear ratio is used. Negative gear

    ratios are tricky: the Puma560 robot has negative gear ratio for +joints 1 and 3.

    +
    +
    +
  • +
  • +
    The absolute value of the gear ratio is used. Negative gear

    ratios are tricky: the Puma560 has negative gear ratio for +joints 1 and 3.

    +
    +
    +
  • +
+
+

See also

+

Robot.nofriction(), Link.friction()

+
+
+ +
+
+get_path(end=None, start=None)
+

Find a path from start to end

+
+
Parameters:
+
+
+
Raises:
+

ValueError – link not known or ambiguous

+
+
Return type:
+

Tuple[List[TypeVar(LinkType, bound= BaseLink)], int, SE3]

+
+
Returns:
+

    +
  • path – the path from start to end

  • +
  • n – the number of joints in the path

  • +
  • T – the tool transform present after end

  • +
+

+
+
+
+ +
+
+property gravity: ndarray
+

Get/set default gravitational acceleration (Robot superclass)

+
    +
  • robot.name is the default gravitational acceleration

  • +
  • +
    robot.name = ... checks and sets default gravitational

    acceleration

    +
    +
    +
  • +
+
+
Parameters:
+

gravity – the new gravitational acceleration for this robot

+
+
Returns:
+

gravitational acceleration

+
+
Return type:
+

gravity

+
+
+

Notes

+

If the z-axis is upward, out of the Earth, this should be +a positive number.

+
+ +
+
+gravload(q=None, gravity=None)
+

Compute gravity load

+

robot.gravload(q) calculates the joint gravity loading (n) for +the robot in the joint configuration q and using the default +gravitational acceleration specified in the DHRobot object.

+

robot.gravload(q, gravity=g) as above except the gravitational +acceleration is explicitly specified as g.

+

Trajectory operation

+

If q is a matrix (nxm) each column is interpreted as a joint +configuration vector, and the result is a matrix (nxm) each column +being the corresponding joint torques.

+
+
Parameters:
+
+
+
Returns:
+

The generalised joint force/torques due to gravity

+
+
Return type:
+

gravload

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> puma.gravload(puma.qz)
+array([ 0.    , 37.4837,  0.2489,  0.    ,  0.    ,  0.    ])
+
+
+
+ +
+
+gravload_x(q=None, gravity=None, pinv=False, representation='rpy/xyz', Ji=None)
+

Operational space gravity load

+

robot.gravload_x(q) calculates the gravity wrench for +the robot in the joint configuration q and using the default +gravitational acceleration specified in the robot object.

+

robot.gravload_x(q, gravity=g) as above except the gravitational +acceleration is explicitly specified as g.

+
+\[\mathbf{G}_x = \mathbf{J}(q)^{-T} \mathbf{G}(q)\]
+

The transformation to operational space requires an analytical, rather +than geometric, Jacobian. analytical can be one of:

+
+
+ + + + + + + + + + + + + + + + + + + +

Value

Rotational representation

'rpy/xyz'

RPY angular rates in XYZ order (default)

'rpy/zyx'

RPY angular rates in XYZ order

'eul'

Euler angular rates in ZYZ order

'exp'

exponential coordinate rates

+
+

Trajectory operation

+

If q is a matrix (nxm) each column is interpreted as a joint +configuration vector, and the result is a matrix (nxm) each column +being the corresponding joint torques.

+
+
Parameters:
+
    +
  • q – Joint coordinates

  • +
  • gravity – Gravitational acceleration (Optional, if not supplied will +use the gravity attribute of self).

  • +
  • pinv – use pseudo inverse rather than inverse (Default value = False)

  • +
  • analytical – the type of analytical Jacobian to use, default is +‘rpy/xyz’

  • +
  • representation – (Default value = “rpy/xyz”)

  • +
  • Ji

  • +
+
+
Returns:
+

The operational space gravity wrench

+
+
Return type:
+

gravload

+
+
+

Examples

+
    ainv = _umath_linalg.inv(a, signature=signature, extobj=extobj)
+  File "/opt/hostedtoolcache/Python/3.7.17/x64/lib/python3.7/site-packages/numpy/linalg/linalg.py", line 88, in _raise_linalgerror_singular
+    raise LinAlgError("Singular matrix")
+numpy.linalg.LinAlgError: Singular matrix
+
+
+

Notes

+
    +
  • If the robot is not 6 DOF the pinv option is set True.

  • +
  • pinv() is around 5x slower than inv()

  • +
+
+

Warning

+

Assumes that the operational space has 6 DOF.

+
+
+

See also

+

gravload()

+
+
+ +
+
+property grippers: List[Gripper]
+

Grippers attached to the robot

+
+
Returns:
+

A list of grippers

+
+
Return type:
+

grippers

+
+
+
+ +
+
+property hascollision
+

Robot has collision model

+
+
Returns:
+

    +
  • hascollision – Robot has collision model

  • +
  • At least one link has associated collision model.

  • +
+

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.hascollision
+False
+
+
+
+

See also

+

hasgeometry()

+
+
+ +
+
+property hasdynamics
+

Robot has dynamic parameters

+
+
Returns:
+

    +
  • hasdynamics – Robot has dynamic parameters

  • +
  • At least one link has associated dynamic parameters.

  • +
+

+
+
+

Examples

+

+
+
+
+ +
+
+property hasgeometry
+

Robot has geometry model

+

At least one link has associated mesh to describe its shape.

+
+
Returns:
+

Robot has geometry model

+
+
Return type:
+

hasgeometry

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.hasgeometry
+True
+
+
+
+

See also

+

hascollision()

+
+
+ +
+
+hessian0(q=None, end=None, start=None, J0=None, tool=None)
+

Manipulator Hessian

+

The manipulator Hessian tensor maps joint acceleration to end-effector +spatial acceleration, expressed in the start frame. This +function calulcates this based on the ETS of the robot. One of J0 or q +is required. Supply J0 if already calculated to save computation time

+
+
Parameters:
+
    +
  • q – The joint angles/configuration of the robot (Optional, +if not supplied will use the stored q values).

  • +
  • end (Union[str, Link, Gripper, None]) – the final link/Gripper which the Hessian represents

  • +
  • start (Union[str, Link, Gripper, None]) – the first link which the Hessian represents

  • +
  • J0 – The manipulator Jacobian in the start frame

  • +
  • tool (Union[ndarray, SE3, None]) – a static tool transformation matrix to apply to the +end of end, defaults to None

  • +
+
+
Returns:
+

The manipulator Hessian in the start frame

+
+
Return type:
+

h0

+
+
+

Synopsis

+

This method computes the manipulator Hessian in the start frame. If +we take the time derivative of the differential kinematic relationship +.. math:

+
\nu    &= \mat{J}(\vec{q}) \dvec{q} \\
+\alpha &= \dmat{J} \dvec{q} + \mat{J} \ddvec{q}
+
+
+

where +.. math:

+
\dmat{J} = \mat{H} \dvec{q}
+
+
+

and \(\mat{H} \in \mathbb{R}^{6\times n \times n}\) is the +Hessian tensor.

+

The elements of the Hessian are +.. math:

+
\mat{H}_{i,j,k} =  \frac{d^2 u_i}{d q_j d q_k}
+
+
+

where \(u = \{t_x, t_y, t_z, r_x, r_y, r_z\}\) are the elements +of the spatial velocity vector.

+

Similarly, we can write +.. math:

+
\mat{J}_{i,j} = \frac{d u_i}{d q_j}
+
+
+

Examples

+

The following example makes a Panda robot object, and solves for the +base frame Hessian at the given joint angle configuration

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda()
+>>> panda.hessian0([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+array([[[-0.484 , -0.    , -0.486 , -0.    , -0.1547,  0.    , -0.    ],
+        [ 0.    ,  0.0796,  0.    ,  0.2466,  0.    ,  0.2006,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+        [ 0.    , -1.    , -0.    ,  1.    ,  0.    ,  1.    ,  0.    ],
+        [ 0.    , -0.    , -0.2955, -0.    ,  0.9463, -0.    ,  0.0998],
+        [ 0.    ,  0.    , -0.    , -0.    ,  0.    , -0.    ,  0.    ]],
+
+       [[-0.    , -0.484 , -0.    ,  0.4986, -0.    ,  0.1086,  0.    ],
+        [ 0.0796,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+        [ 0.    , -0.0796, -0.    , -0.2466, -0.    , -0.2006, -0.    ],
+        [ 0.    ,  0.    ,  0.9553,  0.    , -0.3233,  0.    , -0.995 ],
+        [ 0.    ,  0.    , -0.    , -0.    ,  0.    , -0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.2955,  0.    , -0.9463,  0.    , -0.0998]],
+
+       [[-0.486 , -0.    , -0.4643, -0.    , -0.1478,  0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.383 ,  0.    ,  0.2237,  0.    ],
+        [ 0.    , -0.    , -0.1436, -0.    , -0.0457, -0.    , -0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.9553,  0.    ,  0.9553,  0.    ],
+        [ 0.    ,  0.    ,  0.    , -0.    ,  0.8085, -0.    , -0.1987],
+        [ 0.    ,  0.    ,  0.    ,  0.2955,  0.    ,  0.2955,  0.    ]],
+
+       [[-0.    ,  0.4986, -0.    , -0.4986,  0.    , -0.1086, -0.    ],
+        [ 0.2466,  0.    ,  0.383 ,  0.    ,  0.    ,  0.    ,  0.    ],
+        [ 0.    , -0.2466, -0.    ,  0.2466,  0.    ,  0.2006,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.3233,  0.    ,  0.995 ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    , -0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.9463,  0.    ,  0.0998]],
+
+       [[-0.1547, -0.    , -0.1478,  0.    ,  0.05  , -0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    , -0.1676, -0.    ],
+        [ 0.    , -0.    , -0.0457,  0.    ,  0.1464,  0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    , -0.3233,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    , -0.    ,  0.9093],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    , -0.9463, -0.    ]],
+
+       [[ 0.    ,  0.1086,  0.    , -0.1086, -0.    , -0.1086, -0.    ],
+        [ 0.2006,  0.    ,  0.2237,  0.    , -0.1676,  0.    ,  0.    ],
+        [ 0.    , -0.2006, -0.    ,  0.2006,  0.    ,  0.2006,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.995 ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    , -0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.0998]],
+
+       [[-0.    ,  0.    ,  0.    , -0.    ,  0.    , -0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    , -0.    ,  0.    , -0.    ],
+        [ 0.    , -0.    , -0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ]]])
+
+
+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+ +
+
+hessiane(q=None, end=None, start=None, Je=None, tool=None)
+

Manipulator Hessian

+

The manipulator Hessian tensor maps joint acceleration to end-effector +spatial acceleration, expressed in the end coordinate frame. This +function calulcates this based on the ETS of the robot. One of J0 or q +is required. Supply J0 if already calculated to save computation time

+
+
Parameters:
+
    +
  • q – The joint angles/configuration of the robot (Optional, +if not supplied will use the stored q values).

  • +
  • end (Union[str, Link, Gripper, None]) – the final link/Gripper which the Hessian represents

  • +
  • start (Union[str, Link, Gripper, None]) – the first link which the Hessian represents

  • +
  • J0 – The manipulator Jacobian in the end frame

  • +
  • tool (Union[ndarray, SE3, None]) – a static tool transformation matrix to apply to the +end of end, defaults to None

  • +
+
+
Returns:
+

The manipulator Hessian in end frame

+
+
Return type:
+

he

+
+
+

Synopsis

+

This method computes the manipulator Hessian in the end frame. If +we take the time derivative of the differential kinematic relationship +.. math:

+
\nu    &= \mat{J}(\vec{q}) \dvec{q} \\
+\alpha &= \dmat{J} \dvec{q} + \mat{J} \ddvec{q}
+
+
+

where +.. math:

+
\dmat{J} = \mat{H} \dvec{q}
+
+
+

and \(\mat{H} \in \mathbb{R}^{6\times n \times n}\) is the +Hessian tensor.

+

The elements of the Hessian are +.. math:

+
\mat{H}_{i,j,k} =  \frac{d^2 u_i}{d q_j d q_k}
+
+
+

where \(u = \{t_x, t_y, t_z, r_x, r_y, r_z\}\) are the elements +of the spatial velocity vector.

+

Similarly, we can write +.. math:

+
\mat{J}_{i,j} = \frac{d u_i}{d q_j}
+
+
+

Examples

+

The following example makes a Panda robot object, and solves for the +end-effector frame Hessian at the given joint angle configuration

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda()
+>>> panda.hessiane([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+array([[[-0.4816, -0.    , -0.4835, -0.    , -0.1539, -0.    ,  0.    ],
+        [ 0.    , -0.0796,  0.    , -0.2466,  0.    , -0.2006,  0.    ],
+        [-0.0483, -0.    , -0.0485, -0.    , -0.0154, -0.    ,  0.    ],
+        [ 0.    , -0.995 ,  0.    ,  0.995 , -0.    ,  0.995 , -0.    ],
+        [ 0.    ,  0.    ,  0.2955, -0.    , -0.9463, -0.    , -0.0998],
+        [ 0.    , -0.0998,  0.    ,  0.0998, -0.    ,  0.0998,  0.    ]],
+
+       [[-0.    , -0.4896, -0.    ,  0.4715, -0.    ,  0.088 ,  0.    ],
+        [-0.0796,  0.    ,  0.    , -0.    ,  0.    , -0.    ,  0.    ],
+        [-0.    ,  0.0309,  0.    ,  0.2952,  0.    ,  0.2104, -0.    ],
+        [ 0.    ,  0.    ,  0.9801,  0.    , -0.4161,  0.    , -1.    ],
+        [ 0.    ,  0.    , -0.    , -0.    ,  0.    , -0.    ,  0.    ],
+        [ 0.    ,  0.    , -0.1987, -0.    ,  0.9093, -0.    ,  0.    ]],
+
+       [[-0.4835, -0.    , -0.4763, -0.    , -0.1516, -0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    , -0.383 ,  0.    , -0.2237,  0.    ],
+        [-0.0485,  0.    ,  0.0965, -0.    ,  0.0307, -0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.9801, -0.    ,  0.9801,  0.    ],
+        [ 0.    ,  0.    ,  0.    , -0.    , -0.8085, -0.    ,  0.1987],
+        [ 0.    ,  0.    ,  0.    , -0.1987, -0.    , -0.1987, -0.    ]],
+
+       [[-0.    ,  0.4715, -0.    , -0.4715,  0.    , -0.088 ,  0.    ],
+        [-0.2466, -0.    , -0.383 ,  0.    , -0.    ,  0.    , -0.    ],
+        [-0.    ,  0.2952, -0.    , -0.2952, -0.    , -0.2104,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.4161,  0.    ,  1.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    , -0.    ,  0.    , -0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    , -0.9093,  0.    ,  0.    ]],
+
+       [[-0.1539, -0.    , -0.1516,  0.    ,  0.0644,  0.    , -0.    ],
+        [ 0.    ,  0.    ,  0.    , -0.    , -0.    ,  0.1676, -0.    ],
+        [-0.0154,  0.    ,  0.0307, -0.    , -0.1407, -0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    , -0.4161, -0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    , -0.9093],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.9093,  0.    ]],
+
+       [[-0.    ,  0.088 , -0.    , -0.088 ,  0.    , -0.088 ,  0.    ],
+        [-0.2006, -0.    , -0.2237,  0.    ,  0.1676,  0.    , -0.    ],
+        [-0.    ,  0.2104, -0.    , -0.2104, -0.    , -0.2104,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  1.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    , -0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ]],
+
+       [[ 0.    ,  0.    ,  0.    ,  0.    , -0.    ,  0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    , -0.    , -0.    , -0.    , -0.    ],
+        [ 0.    , -0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+        [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ]]])
+
+
+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+ +
+
+hierarchy()
+

Pretty print the robot link hierachy

+
+
Return type:
+

Pretty print of the robot model

+
+
+

Examples

+

Makes a robot and prints the heirachy

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.URDF.Panda()
+>>> robot.hierarchy()
+ panda_link0
+   panda_link1
+     panda_link2
+       panda_link3
+         panda_link4
+           panda_link5
+             panda_link6
+               panda_link7
+                 panda_link8
+                   panda_hand
+                     panda_leftfinger
+                     panda_rightfinger
+
+
+
+ +
+
+ik_GN(Tep, end=None, start=None, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, pinv=True, pinv_damping=0.0)
+

Fast numerical inverse kinematics by Gauss-Newton optimization

+

sol = ets.ik_GN(Tep) are the joint coordinates (n) corresponding +to the robot end-effector pose Tep which is an SE3 or ndarray object. +This method can be used for robots with any number of degrees of freedom. This +is a fast solver implemented in C++.

+

See the Inverse Kinematics Docs Page for more details and for a +tutorial on numerical IK, see here.

+
+

Note

+

When using this method with redundant robots (>6 DoF), pinv must be set to True

+
+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose or pose trajectory

  • +
  • end (Union[str, Link, Gripper, None]) – the link considered as the end-effector

  • +
  • start (Union[str, Link, Gripper, None]) – the link considered as the base frame, defaults to the robots’s base frame

  • +
  • q0 (Optional[ndarray]) – initial joint configuration (default to random valid joint +configuration contrained by the joint limits of the robot)

  • +
  • ilimit (int) – maximum number of iterations per search

  • +
  • slimit (int) – maximum number of search attempts

  • +
  • tol (float) – final error tolerance

  • +
  • mask (Optional[ndarray]) – a mask vector which weights the end-effector error priority. +Corresponds to translation in X, Y and Z and rotation about X, Y and Z +respectively

  • +
  • joint_limits (bool) – constrain the solution to being within the joint limits of +the robot (reject solution with invalid joint configurations and perfrom +another search up to the slimit)

  • +
  • pinv (int) – Use the psuedo-inverse instad of the normal matrix inverse

  • +
  • pinv_damping (float) – Damping factor for the psuedo-inverse

  • +
+
+
Return type:
+

Tuple[ndarray, int, int, int, float]

+
+
Returns:
+

    +
  • sol – tuple (q, success, iterations, searches, residual)

  • +
  • The return value sol is a tuple with elements

  • +
  • ============ ========== ===============================================

  • +
  • Element Type Description

  • +
  • ============ ========== ===============================================

  • +
  • q ndarray(n) joint coordinates in units of radians or metres

  • +
  • success int whether a solution was found

  • +
  • iterations int total number of iterations

  • +
  • searches int total number of searches

  • +
  • residual float final value of cost function

  • +
  • ============ ========== ===============================================

  • +
  • If success == 0 the q values will be valid numbers, but the

  • +
  • solution will be in error. The amount of error is indicated by

  • +
  • the residual.

  • +
+

+
+
+

Synopsis

+

Each iteration uses the Gauss-Newton optimisation method

+
+\[\begin{split}\vec{q}_{k+1} &= \vec{q}_k + +\left( +{\mat{J}(\vec{q}_k)}^\top +\mat{W}_e \ +{\mat{J}(\vec{q}_k)} +\right)^{-1} +\bf{g}_k \\ +\bf{g}_k &= +{\mat{J}(\vec{q}_k)}^\top +\mat{W}_e +\vec{e}_k\end{split}\]
+

where \(\mat{J} = {^0\mat{J}}\) is the base-frame manipulator Jacobian. If +\(\mat{J}(\vec{q}_k)\) is non-singular, and \(\mat{W}_e = \mat{1}_n\), then +the above provides the pseudoinverse solution. However, if \(\mat{J}(\vec{q}_k)\) +is singular, the above can not be computed and the GN solution is infeasible.

+

Examples

+

The following example gets a panda robot object, makes a goal +pose Tep, and then solves for the joint coordinates which result in the pose +Tep using the ikine_GN method.

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda()
+>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> panda.ik_GN(Tep)
+(array([-1.0805, -0.5328,  0.9086, -2.1781,  0.4612,  1.9018,  0.4239]), 1, 510, 32, 2.803306327008683e-09)
+
+
+

Notes

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
ik_NR

A fast numerical inverse kinematics solver using Newton-Raphson optimisation

+
+
ik_GN

A fast numerical inverse kinematics solver using Gauss-Newton optimisation

+
+
+
+
+ +
+
+ik_LM(Tep, end=None, start=None, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, k=1.0, method='chan')
+

Fast levenberg-Marquadt Numerical Inverse Kinematics Solver

+

A method which provides functionality to perform numerical inverse kinematics (IK) +using the Levemberg-Marquadt method. This +is a fast solver implemented in C++.

+

See the Inverse Kinematics Docs Page for more details and for a +tutorial on numerical IK, see here.

+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose

  • +
  • end (Union[str, Link, Gripper, None]) – the link considered as the end-effector

  • +
  • start (Union[str, Link, Gripper, None]) – the link considered as the base frame, defaults to the robots’s base frame

  • +
  • q0 (Optional[ndarray]) – The initial joint coordinate vector

  • +
  • ilimit (int) – How many iterations are allowed within a search before a new search +is started

  • +
  • slimit (int) – How many searches are allowed before being deemed unsuccessful

  • +
  • tol (float) – Maximum allowed residual error E

  • +
  • mask (Optional[ndarray]) – A 6 vector which assigns weights to Cartesian degrees-of-freedom +error priority

  • +
  • joint_limits (bool) – Reject solutions with joint limit violations

  • +
  • seed – A seed for the private RNG used to generate random joint coordinate +vectors

  • +
  • k (float) – Sets the gain value for the damping matrix Wn in the next iteration. See +synopsis

  • +
  • method (Literal['chan', 'wampler', 'sugihara']) – One of “chan”, “sugihara” or “wampler”. Defines which method is used +to calculate the damping matrix Wn in the step method

  • +
+
+
Return type:
+

Tuple[ndarray, int, int, int, float]

+
+
+

Synopsis

+

The operation is defined by the choice of the method kwarg.

+

The step is deined as

+
+\[\begin{split}\vec{q}_{k+1} +&= +\vec{q}_k + +\left( + \mat{A}_k +\right)^{-1} +\bf{g}_k \\ +% +\mat{A}_k +&= +{\mat{J}(\vec{q}_k)}^\top +\mat{W}_e \ +{\mat{J}(\vec{q}_k)} ++ +\mat{W}_n\end{split}\]
+

where \(\mat{W}_n = \text{diag}(\vec{w_n})(\vec{w_n} \in \mathbb{R}^n_{>0})\) is a +diagonal damping matrix. The damping matrix ensures that \(\mat{A}_k\) is +non-singular and positive definite. The performance of the LM method largely depends +on the choice of \(\mat{W}_n\).

+

Chan’s Method

+

Chan proposed

+
+\[\mat{W}_n += +λ E_k \mat{1}_n\]
+

where λ is a constant which reportedly does not have much influence on performance. +Use the kwarg k to adjust the weighting term λ.

+

Sugihara’s Method

+

Sugihara proposed

+
+\[\mat{W}_n += +E_k \mat{1}_n + \text{diag}(\hat{\vec{w}}_n)\]
+

where \(\hat{\vec{w}}_n \in \mathbb{R}^n\), \(\hat{w}_{n_i} = l^2 \sim 0.01 l^2\), +and \(l\) is the length of a typical link within the manipulator. We provide the +variable k as a kwarg to adjust the value of \(w_n\).

+

Wampler’s Method

+

Wampler proposed \(\vec{w_n}\) to be a constant. This is set through the k kwarg.

+

Examples

+

The following example makes a panda robot object, makes a goal +pose Tep, and then solves for the joint coordinates which result in the pose +Tep using the ikine_LM method.

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda()
+>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> panda.ikine_LM(Tep)
+IKSolution(q=array([-0.3834, -0.3202,  0.3427, -2.1984,  0.1219,  1.9934,  0.6892]), success=True, iterations=27, searches=2, residual=3.829062192451652e-09, reason='Success')
+
+
+

Notes

+

The value for the k kwarg will depend on the method chosen and the arm you are +using. Use the following as a rough guide chan, k = 1.0 - 0.01, +wampler, k = 0.01 - 0.0001, and sugihara, k = 0.1 - 0.0001

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian.

+

This class supports null-space motion to assist with maximising manipulability and +avoiding joint limits. These are enabled by setting kq and km to non-zero values.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
ik_NR

A fast numerical inverse kinematics solver using Newton-Raphson optimisation

+
+
ik_GN

A fast numerical inverse kinematics solver using Gauss-Newton optimisation

+
+
+
+
+

Changed in version 1.0.4: Merged the Levemberg-Marquadt IK solvers into the ik_LM method

+
+
+ +
+
+ik_NR(Tep, end=None, start=None, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, pinv=True, pinv_damping=0.0)
+

Fast numerical inverse kinematics using Newton-Raphson optimization

+

sol = ets.ik_NR(Tep) are the joint coordinates (n) corresponding +to the robot end-effector pose Tep which is an SE3 or ndarray object. +This method can be used for robots with any number of degrees of freedom. This +is a fast solver implemented in C++.

+

See the Inverse Kinematics Docs Page for more details and for a +tutorial on numerical IK, see here.

+
+

Note

+

When using this method with redundant robots (>6 DoF), pinv must be set to True

+
+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose or pose trajectory

  • +
  • end (Union[str, Link, Gripper, None]) – the link considered as the end-effector

  • +
  • start (Union[str, Link, Gripper, None]) – the link considered as the base frame, defaults to the robots’s base frame

  • +
  • q0 (Optional[ndarray]) – initial joint configuration (default to random valid joint +configuration contrained by the joint limits of the robot)

  • +
  • ilimit (int) – maximum number of iterations per search

  • +
  • slimit (int) – maximum number of search attempts

  • +
  • tol (float) – final error tolerance

  • +
  • mask (Optional[ndarray]) – a mask vector which weights the end-effector error priority. +Corresponds to translation in X, Y and Z and rotation about X, Y and Z +respectively

  • +
  • joint_limits (bool) – constrain the solution to being within the joint limits of +the robot (reject solution with invalid joint configurations and perfrom +another search up to the slimit)

  • +
  • pinv (int) – Use the psuedo-inverse instad of the normal matrix inverse

  • +
  • pinv_damping (float) – Damping factor for the psuedo-inverse

  • +
+
+
Return type:
+

Tuple[ndarray, int, int, int, float]

+
+
Returns:
+

    +
  • sol – tuple (q, success, iterations, searches, residual)

  • +
  • The return value sol is a tuple with elements

  • +
  • ============ ========== ===============================================

  • +
  • Element Type Description

  • +
  • ============ ========== ===============================================

  • +
  • q ndarray(n) joint coordinates in units of radians or metres

  • +
  • success int whether a solution was found

  • +
  • iterations int total number of iterations

  • +
  • searches int total number of searches

  • +
  • residual float final value of cost function

  • +
  • ============ ========== ===============================================

  • +
  • If success == 0 the q values will be valid numbers, but the

  • +
  • solution will be in error. The amount of error is indicated by

  • +
  • the residual.

  • +
+

+
+
+

Synopsis

+

Each iteration uses the Newton-Raphson optimisation method

+
+\[\vec{q}_{k+1} = \vec{q}_k + {^0\mat{J}(\vec{q}_k)}^{-1} \vec{e}_k\]
+

Examples

+

The following example gets a panda robot object, makes a goal +pose Tep, and then solves for the joint coordinates which result in the pose +Tep using the ikine_GN method.

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda()
+>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> panda.ik_NR(Tep)
+(array([-1.0805, -0.5328,  0.9086, -2.1781,  0.4612,  1.9018,  0.4239]), 1, 489, 32, 2.8033063270234757e-09)
+
+
+

Notes

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
ik_LM

A fast numerical inverse kinematics solver using Levenberg-Marquadt optimisation

+
+
ik_GN

A fast numerical inverse kinematics solver using Gauss-Newton optimisation

+
+
+
+
+ +
+
+ikine_GN(Tep, end=None, start=None, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, seed=None, pinv=False, kq=0.0, km=0.0, ps=0.0, pi=0.3, **kwargs)
+

Gauss-Newton Numerical Inverse Kinematics Solver

+

A method which provides functionality to perform numerical inverse kinematics (IK) +using the Gauss-Newton method.

+

See the Inverse Kinematics Docs Page for more details and for a +tutorial on numerical IK, see here.

+
+

Note

+

When using this method with redundant robots (>6 DoF), pinv must be set to True

+
+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose

  • +
  • end (Union[str, Link, Gripper, None]) – the link considered as the end-effector

  • +
  • start (Union[str, Link, Gripper, None]) – the link considered as the base frame, defaults to the robots’s base frame

  • +
  • q0 (Union[ndarray, List[float], Tuple[float], Set[float], None]) – The initial joint coordinate vector

  • +
  • ilimit (int) – How many iterations are allowed within a search before a new search +is started

  • +
  • slimit (int) – How many searches are allowed before being deemed unsuccessful

  • +
  • tol (float) – Maximum allowed residual error E

  • +
  • mask (Union[ndarray, List[float], Tuple[float], Set[float], None]) – A 6 vector which assigns weights to Cartesian degrees-of-freedom +error priority

  • +
  • joint_limits (bool) – Reject solutions with joint limit violations

  • +
  • seed (Optional[int]) – A seed for the private RNG used to generate random joint coordinate +vectors

  • +
  • pinv (bool) – If True, will use the psuedoinverse in the step method instead of +the normal inverse

  • +
  • kq (float) – The gain for joint limit avoidance. Setting to 0.0 will remove this +completely from the solution

  • +
  • km (float) – The gain for maximisation. Setting to 0.0 will remove this completely +from the solution

  • +
  • ps (float) – The minimum angle/distance (in radians or metres) in which the joint is +allowed to approach to its limit

  • +
  • pi (Union[ndarray, float]) – The influence angle/distance (in radians or metres) in null space motion +becomes active

  • +
+
+
+

Synopsis

+

Each iteration uses the Gauss-Newton optimisation method

+
+\[\begin{split}\vec{q}_{k+1} &= \vec{q}_k + +\left( +{\mat{J}(\vec{q}_k)}^\top +\mat{W}_e \ +{\mat{J}(\vec{q}_k)} +\right)^{-1} +\bf{g}_k \\ +\bf{g}_k &= +{\mat{J}(\vec{q}_k)}^\top +\mat{W}_e +\vec{e}_k\end{split}\]
+

where \(\mat{J} = {^0\mat{J}}\) is the base-frame manipulator Jacobian. If +\(\mat{J}(\vec{q}_k)\) is non-singular, and \(\mat{W}_e = \mat{1}_n\), then +the above provides the pseudoinverse solution. However, if \(\mat{J}(\vec{q}_k)\) +is singular, the above can not be computed and the GN solution is infeasible.

+

Examples

+

The following example gets a panda robot object, makes a goal +pose Tep, and then solves for the joint coordinates which result in the pose +Tep using the ikine_GN method.

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda()
+>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> panda.ikine_GN(Tep)
+IKSolution(q=array([ 2.0377, -0.5789, -1.7111, -2.1663, -0.9973,  1.8724,  2.0736]), success=False, iterations=100, searches=100, residual=0.0, reason='iteration and search limit reached, 100 numpy.LinAlgError encountered')
+
+
+

Notes

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian.

+

This class supports null-space motion to assist with maximising manipulability and +avoiding joint limits. These are enabled by setting kq and km to non-zero values.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
IK_NR

An IK Solver class which implements the Newton-Raphson optimisation technique

+
+
ikine_LM

Implements the IK_LM class as a method within the ETS class

+
+
ikine_NR

Implements the IK_NR class as a method within the ETS class

+
+
ikine_QP

Implements the IK_QP class as a method within the ETS class

+
+
+
+
+

Changed in version 1.0.4: Added the Gauss-Newton IK solver method on the Robot class

+
+
+ +
+
+ikine_LM(Tep, end=None, start=None, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, seed=None, k=1.0, method='chan', kq=0.0, km=0.0, ps=0.0, pi=0.3, **kwargs)
+

Levenberg-Marquadt Numerical Inverse Kinematics Solver

+

A method which provides functionality to perform numerical inverse kinematics (IK) +using the Levemberg-Marquadt method.

+

See the Inverse Kinematics Docs Page for more details and for a +tutorial on numerical IK, see here.

+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose

  • +
  • end (Union[str, Link, Gripper, None]) – the link considered as the end-effector

  • +
  • start (Union[str, Link, Gripper, None]) – the link considered as the base frame, defaults to the robots’s base frame

  • +
  • q0 (Union[ndarray, List[float], Tuple[float], Set[float], None]) – The initial joint coordinate vector

  • +
  • ilimit (int) – How many iterations are allowed within a search before a new search +is started

  • +
  • slimit (int) – How many searches are allowed before being deemed unsuccessful

  • +
  • tol (float) – Maximum allowed residual error E

  • +
  • mask (Union[ndarray, List[float], Tuple[float], Set[float], None]) – A 6 vector which assigns weights to Cartesian degrees-of-freedom +error priority

  • +
  • joint_limits (bool) – Reject solutions with joint limit violations

  • +
  • seed (Optional[int]) – A seed for the private RNG used to generate random joint coordinate +vectors

  • +
  • k (float) – Sets the gain value for the damping matrix Wn in the next iteration. See +synopsis

  • +
  • method (Literal['chan', 'wampler', 'sugihara']) – One of “chan”, “sugihara” or “wampler”. Defines which method is used +to calculate the damping matrix Wn in the step method

  • +
  • kq (float) – The gain for joint limit avoidance. Setting to 0.0 will remove this +completely from the solution

  • +
  • km (float) – The gain for maximisation. Setting to 0.0 will remove this completely +from the solution

  • +
  • ps (float) – The minimum angle/distance (in radians or metres) in which the joint is +allowed to approach to its limit

  • +
  • pi (Union[ndarray, float]) – The influence angle/distance (in radians or metres) in null space motion +becomes active

  • +
+
+
+

Synopsis

+

The operation is defined by the choice of the method kwarg.

+

The step is deined as

+
+\[\begin{split}\vec{q}_{k+1} +&= +\vec{q}_k + +\left( + \mat{A}_k +\right)^{-1} +\bf{g}_k \\ +% +\mat{A}_k +&= +{\mat{J}(\vec{q}_k)}^\top +\mat{W}_e \ +{\mat{J}(\vec{q}_k)} ++ +\mat{W}_n\end{split}\]
+

where \(\mat{W}_n = \text{diag}(\vec{w_n})(\vec{w_n} \in \mathbb{R}^n_{>0})\) is a +diagonal damping matrix. The damping matrix ensures that \(\mat{A}_k\) is +non-singular and positive definite. The performance of the LM method largely depends +on the choice of \(\mat{W}_n\).

+

Chan’s Method

+

Chan proposed

+
+\[\mat{W}_n += +λ E_k \mat{1}_n\]
+

where λ is a constant which reportedly does not have much influence on performance. +Use the kwarg k to adjust the weighting term λ.

+

Sugihara’s Method

+

Sugihara proposed

+
+\[\mat{W}_n += +E_k \mat{1}_n + \text{diag}(\hat{\vec{w}}_n)\]
+

where \(\hat{\vec{w}}_n \in \mathbb{R}^n\), \(\hat{w}_{n_i} = l^2 \sim 0.01 l^2\), +and \(l\) is the length of a typical link within the manipulator. We provide the +variable k as a kwarg to adjust the value of \(w_n\).

+

Wampler’s Method

+

Wampler proposed \(\vec{w_n}\) to be a constant. This is set through the k kwarg.

+

Examples

+

The following example makes a panda robot object, makes a goal +pose Tep, and then solves for the joint coordinates which result in the pose +Tep using the ikine_LM method.

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda()
+>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> panda.ikine_LM(Tep)
+IKSolution(q=array([ 0.076 , -0.3008, -0.0683, -2.1999, -0.0233,  1.9998,  0.8038]), success=True, iterations=31, searches=2, residual=6.103979291931972e-11, reason='Success')
+
+
+

Notes

+

The value for the k kwarg will depend on the method chosen and the arm you are +using. Use the following as a rough guide chan, k = 1.0 - 0.01, +wampler, k = 0.01 - 0.0001, and sugihara, k = 0.1 - 0.0001

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian.

+

This class supports null-space motion to assist with maximising manipulability and +avoiding joint limits. These are enabled by setting kq and km to non-zero values.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
IK_LM

An IK Solver class which implements the Levemberg Marquadt optimisation technique

+
+
ikine_NR

Implements the IK_NR class as a method within the Robot class

+
+
ikine_GN

Implements the IK_GN class as a method within the Robot class

+
+
ikine_QP

Implements the IK_QP class as a method within the Robot class

+
+
+
+
+

Changed in version 1.0.4: Added the Levemberg-Marquadt IK solver method on the Robot class

+
+
+ +
+
+ikine_NR(Tep, end=None, start=None, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, seed=None, pinv=False, kq=0.0, km=0.0, ps=0.0, pi=0.3, **kwargs)
+

Newton-Raphson Numerical Inverse Kinematics Solver

+

A method which provides functionality to perform numerical inverse kinematics (IK) +using the Newton-Raphson method.

+

See the Inverse Kinematics Docs Page for more details and for a +tutorial on numerical IK, see here.

+
+

Note

+

When using this method with redundant robots (>6 DoF), pinv must be set to True

+
+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose

  • +
  • end (Union[str, Link, Gripper, None]) – the link considered as the end-effector

  • +
  • start (Union[str, Link, Gripper, None]) – the link considered as the base frame, defaults to the robots’s base frame

  • +
  • q0 (Union[ndarray, List[float], Tuple[float], Set[float], None]) – The initial joint coordinate vector

  • +
  • ilimit (int) – How many iterations are allowed within a search before a new search +is started

  • +
  • slimit (int) – How many searches are allowed before being deemed unsuccessful

  • +
  • tol (float) – Maximum allowed residual error E

  • +
  • mask (Union[ndarray, List[float], Tuple[float], Set[float], None]) – A 6 vector which assigns weights to Cartesian degrees-of-freedom +error priority

  • +
  • joint_limits (bool) – Reject solutions with joint limit violations

  • +
  • seed (Optional[int]) – A seed for the private RNG used to generate random joint coordinate +vectors

  • +
  • pinv (bool) – If True, will use the psuedoinverse in the step method instead of +the normal inverse

  • +
  • kq (float) – The gain for joint limit avoidance. Setting to 0.0 will remove this +completely from the solution

  • +
  • km (float) – The gain for maximisation. Setting to 0.0 will remove this completely +from the solution

  • +
  • ps (float) – The minimum angle/distance (in radians or metres) in which the joint is +allowed to approach to its limit

  • +
  • pi (Union[ndarray, float]) – The influence angle/distance (in radians or metres) in null space motion +becomes active

  • +
+
+
+

Synopsis

+

Each iteration uses the Newton-Raphson optimisation method

+
+\[\vec{q}_{k+1} = \vec{q}_k + {^0\mat{J}(\vec{q}_k)}^{-1} \vec{e}_k\]
+

Examples

+

The following example gets a panda robot object, makes a goal +pose Tep, and then solves for the joint coordinates which result in the pose +Tep using the ikine_NR method.

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda()
+>>> Tep = panda.fkine([0, -0.3, 0, -2.2, 0, 2, 0.7854])
+>>> panda.ikine_NR(Tep)
+IKSolution(q=array([-0.9478, -1.036 , -0.5051, -2.1583, -0.0992,  1.8519,  0.0562]), success=False, iterations=100, searches=100, residual=0.0, reason='iteration and search limit reached, 100 numpy.LinAlgError encountered')
+
+
+

Notes

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian.

+

This class supports null-space motion to assist with maximising manipulability and +avoiding joint limits. These are enabled by setting kq and km to non-zero values.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
IK_NR

An IK Solver class which implements the Newton-Raphson optimisation technique

+
+
ikine_LM

Implements the IK_LM class as a method within the ETS class

+
+
ikine_GN

Implements the IK_GN class as a method within the ETS class

+
+
ikine_QP

Implements the IK_QP class as a method within the ETS class

+
+
+
+
+

Changed in version 1.0.4: Added the Newton-Raphson IK solver method on the Robot class

+
+
+ +
+
+ikine_QP(Tep, end=None, start=None, q0=None, ilimit=30, slimit=100, tol=1e-06, mask=None, joint_limits=True, seed=None, kj=1.0, ks=1.0, kq=0.0, km=0.0, ps=0.0, pi=0.3, **kwargs)
+

Quadratic Programming Numerical Inverse Kinematics Solver

+

A method that provides functionality to perform numerical inverse kinematics +(IK) using a quadratic progamming approach.

+

See the Inverse Kinematics Docs Page for more details and for a +tutorial on numerical IK, see here.

+
+
Parameters:
+
    +
  • Tep (Union[ndarray, SE3]) – The desired end-effector pose

  • +
  • end (Union[str, Link, Gripper, None]) – the link considered as the end-effector

  • +
  • start (Union[str, Link, Gripper, None]) – the link considered as the base frame, defaults to the robots’s base frame

  • +
  • q0 (Union[ndarray, List[float], Tuple[float], Set[float], None]) – The initial joint coordinate vector

  • +
  • ilimit (int) – How many iterations are allowed within a search before a new search +is started

  • +
  • slimit (int) – How many searches are allowed before being deemed unsuccessful

  • +
  • tol (float) – Maximum allowed residual error E

  • +
  • mask (Union[ndarray, List[float], Tuple[float], Set[float], None]) – A 6 vector which assigns weights to Cartesian degrees-of-freedom +error priority

  • +
  • joint_limits (bool) – Reject solutions with joint limit violations

  • +
  • seed (Optional[int]) – A seed for the private RNG used to generate random joint coordinate +vectors

  • +
  • kj – A gain for joint velocity norm minimisation

  • +
  • ks – A gain which adjusts the cost of slack (intentional error)

  • +
  • kq (float) – The gain for joint limit avoidance. Setting to 0.0 will remove this +completely from the solution

  • +
  • km (float) – The gain for maximisation. Setting to 0.0 will remove this completely +from the solution

  • +
  • ps (float) – The minimum angle/distance (in radians or metres) in which the joint is +allowed to approach to its limit

  • +
  • pi (Union[ndarray, float]) – The influence angle/distance (in radians or metres) in null space motion +becomes active

  • +
+
+
Raises:
+

ImportError – If the package qpsolvers is not installed

+
+
+

Synopsis

+

Each iteration uses the following approach

+
+\[\vec{q}_{k+1} = \vec{q}_{k} + \dot{\vec{q}}.\]
+

where the QP is defined as

+
+\[\begin{split}\min_x \quad f_o(\vec{x}) &= \frac{1}{2} \vec{x}^\top \mathcal{Q} \vec{x}+ \mathcal{C}^\top \vec{x}, \\ +\text{subject to} \quad \mathcal{J} \vec{x} &= \vec{\nu}, \\ +\mathcal{A} \vec{x} &\leq \mathcal{B}, \\ +\vec{x}^- &\leq \vec{x} \leq \vec{x}^+\end{split}\]
+

with

+
+\[\begin{split}\vec{x} &= +\begin{pmatrix} + \dvec{q} \\ \vec{\delta} +\end{pmatrix} \in \mathbb{R}^{(n+6)} \\ +\mathcal{Q} &= +\begin{pmatrix} + \lambda_q \mat{1}_{n} & \mathbf{0}_{6 \times 6} \\ \mathbf{0}_{n \times n} & \lambda_\delta \mat{1}_{6} +\end{pmatrix} \in \mathbb{R}^{(n+6) \times (n+6)} \\ +\mathcal{J} &= +\begin{pmatrix} + \mat{J}(\vec{q}) & \mat{1}_{6} +\end{pmatrix} \in \mathbb{R}^{6 \times (n+6)} \\ +\mathcal{C} &= +\begin{pmatrix} + \mat{J}_m \\ \bf{0}_{6 \times 1} +\end{pmatrix} \in \mathbb{R}^{(n + 6)} \\ +\mathcal{A} &= +\begin{pmatrix} + \mat{1}_{n \times n + 6} \\ +\end{pmatrix} \in \mathbb{R}^{(l + n) \times (n + 6)} \\ +\mathcal{B} &= +\eta +\begin{pmatrix} + \frac{\rho_0 - \rho_s} + {\rho_i - \rho_s} \\ + \vdots \\ + \frac{\rho_n - \rho_s} + {\rho_i - \rho_s} +\end{pmatrix} \in \mathbb{R}^{n} \\ +\vec{x}^{-, +} &= +\begin{pmatrix} + \dvec{q}^{-, +} \\ + \vec{\delta}^{-, +} +\end{pmatrix} \in \mathbb{R}^{(n+6)},\end{split}\]
+

where \(\vec{\delta} \in \mathbb{R}^6\) is the slack vector, +\(\lambda_\delta \in \mathbb{R}^+\) is a gain term which adjusts the +cost of the norm of the slack vector in the optimiser, +\(\dvec{q}^{-,+}\) are the minimum and maximum joint velocities, and +\(\dvec{\delta}^{-,+}\) are the minimum and maximum slack velocities.

+

Examples

+

The following example gets a panda robot object, makes a goal +pose Tep, and then solves for the joint coordinates which result in the pose +Tep using the ikine_QP method.

+
  File "/opt/hostedtoolcache/Python/3.7.17/x64/lib/python3.7/site-packages/roboticstoolbox/robot/IK.py", line 1331, in __init__
+    "the package qpsolvers is required for this class. \nInstall using 'pip"
+ImportError: the package qpsolvers is required for this class. 
+Install using 'pip install qpsolvers'
+
+
+

Notes

+

When using the this method, the initial joint coordinates \(q_0\), should correspond +to a non-singular manipulator pose, since it uses the manipulator Jacobian.

+

This class supports null-space motion to assist with maximising manipulability and +avoiding joint limits. These are enabled by setting kq and km to non-zero values.

+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+

See also

+
+
IK_NR

An IK Solver class which implements the Newton-Raphson optimisation technique

+
+
ikine_LM

Implements the IK_LM class as a method within the ETS class

+
+
ikine_GN

Implements the IK_GN class as a method within the ETS class

+
+
ikine_NR

Implements the IK_NR class as a method within the ETS class

+
+
+
+
+

Changed in version 1.0.4: Added the Quadratic Programming IK solver method on the Robot class

+
+
+ +
+
+inertia(q)
+

Manipulator inertia matrix +inertia(q) is the symmetric joint inertia matrix (n,n) which +relates joint torque to joint acceleration for the robot at joint +configuration q.

+

Trajectory operation

+

If q is a matrix (m,n), each row is interpretted as a joint state +vector, and the result is a 3d-matrix (nxnxk) where each plane +corresponds to the inertia for the corresponding row of q.

+
+
Parameters:
+

q (ndarray) – Joint coordinates

+
+
Returns:
+

The inertia matrix

+
+
Return type:
+

inertia

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> puma.inertia(puma.qz)
+array([[ 3.9611, -0.1627, -0.1389,  0.0016, -0.0004,  0.    ],
+       [-0.1627,  4.4566,  0.3727,  0.    ,  0.0019,  0.    ],
+       [-0.1389,  0.3727,  0.9387,  0.    ,  0.0019,  0.    ],
+       [ 0.0016,  0.    ,  0.    ,  0.1924,  0.    ,  0.    ],
+       [-0.0004,  0.0019,  0.0019,  0.    ,  0.1713,  0.    ],
+       [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.1941]])
+
+
+

Notes

+
    +
  • +
    The diagonal elements M[j,j] are the inertia seen by joint

    actuator j.

    +
    +
    +
  • +
  • +
    The off-diagonal elements M[j,k] are coupling inertias that

    relate acceleration on joint j to force/torque on +joint k.

    +
    +
    +
  • +
  • +
    The diagonal terms include the motor inertia reflected through

    the gear ratio.

    +
    +
    +
  • +
+
+

See also

+

cinertia()

+
+
+ +
+
+inertia_x(q=None, pinv=False, representation='rpy/xyz', Ji=None)
+

Operational space inertia matrix

+

robot.inertia_x(q) is the operational space (Cartesian) inertia +matrix which relates Cartesian force/torque to Cartesian +acceleration at the joint configuration q.

+
+\[\mathbf{M}_x = \mathbf{J}(q)^{-T} \mathbf{M}(q) \mathbf{J}(q)^{-1}\]
+

The transformation to operational space requires an analytical, rather +than geometric, Jacobian. analytical can be one of:

+
+
+ + + + + + + + + + + + + + + + + + + +

Value

Rotational representation

'rpy/xyz'

RPY angular rates in XYZ order (default)

'rpy/zyx'

RPY angular rates in XYZ order

'eul'

Euler angular rates in ZYZ order

'exp'

exponential coordinate rates

+
+

Trajectory operation

+

If q is a matrix (m,n), each row is interpretted as a joint state +vector, and the result is a 3d-matrix (m,n,n) where each plane +corresponds to the Cartesian inertia for the corresponding +row of q.

+
+
Parameters:
+
    +
  • q – Joint coordinates

  • +
  • pinv – use pseudo inverse rather than inverse (Default value = False)

  • +
  • analytical – the type of analytical Jacobian to use, default is +‘rpy/xyz’

  • +
  • representation – (Default value = “rpy/xyz”)

  • +
  • Ji – The inverse analytical Jacobian (base-frame)

  • +
+
+
Returns:
+

The operational space inertia matrix

+
+
Return type:
+

inertia_x

+
+
+

Examples

+
    ainv = _umath_linalg.inv(a, signature=signature, extobj=extobj)
+  File "/opt/hostedtoolcache/Python/3.7.17/x64/lib/python3.7/site-packages/numpy/linalg/linalg.py", line 88, in _raise_linalgerror_singular
+    raise LinAlgError("Singular matrix")
+numpy.linalg.LinAlgError: Singular matrix
+
+
+

Notes

+
    +
  • If the robot is not 6 DOF the pinv option is set True.

  • +
  • pinv() is around 5x slower than inv()

  • +
+
+

Warning

+

Assumes that the operational space has 6 DOF.

+
+
+

See also

+

inertia()

+
+
+ +
+
+isprismatic(j)
+

Check if joint is prismatic

+
+
Returns:
+

True if prismatic

+
+
Return type:
+

j

+
+
+

Examples

+

+
+
+
+

See also

+

Link.isprismatic(), prismaticjoints()

+
+
+ +
+
+isrevolute(j)
+

Check if joint is revolute

+
+
Returns:
+

True if revolute

+
+
Return type:
+

j

+
+
+

Examples

+

+
+
+
+

See also

+

Link.isrevolute(), revolutejoints()

+
+
+ +
+
+itorque(q, qdd)
+

Inertia torque

+

itorque(q, qdd) is the inertia force/torque vector (n) at +the specified joint configuration q (n) and acceleration qdd (n), and +n is the number of robot joints. +It is \(\mathbf{I}(q) \ddot{q}\).

+

Trajectory operation

+

If q and qdd are matrices (m,n), each row is interpretted as a +joint configuration, and the result is a matrix (m,n) where each row is +the corresponding joint torques.

+
+
Parameters:
+
    +
  • q – Joint coordinates

  • +
  • qdd – Joint acceleration

  • +
+
+
Returns:
+

The inertia torque vector

+
+
Return type:
+

itorque

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> puma.itorque(puma.qz, 0.5 * np.ones((6,)))
+array([1.8304, 2.3343, 0.5872, 0.0971, 0.0873, 0.0971])
+
+
+

Notes

+
    +
  • +
    If the robot model contains non-zero motor inertia then this

    will be included in the result.

    +
    +
    +
  • +
+
+

See also

+

inertia()

+
+
+ +
+
+jacob0(q, end=None, start=None, tool=None)
+

Manipulator geometric Jacobian in the start frame

+

robot.jacobo(q) is the manipulator Jacobian matrix which maps +joint velocity to end-effector spatial velocity expressed in the +base frame.

+

End-effector spatial velocity \(\nu = (v_x, v_y, v_z, \omega_x, \omega_y, \omega_z)^T\) +is related to joint velocity by \({}^{E}\!\nu = \mathbf{J}_m(q) \dot{q}\).

+
+
Parameters:
+
    +
  • q (Union[ndarray, List[float], Tuple[float], Set[float]]) – Joint coordinate vector

  • +
  • end (Union[str, Link, Gripper, None]) – the particular link or gripper whose velocity the Jacobian +describes, defaults to the end-effector if only one is present

  • +
  • start (Union[str, Link, Gripper, None]) – the link considered as the base frame, defaults to the robots’s base frame

  • +
  • tool (Union[ndarray, SE3, None]) – a static tool transformation matrix to apply to the +end of end, defaults to None

  • +
+
+
Returns:
+

Manipulator Jacobian in the start frame

+
+
Return type:
+

J0

+
+
+

Examples

+

The following example makes a Puma560 robot object, and solves for the +base-frame Jacobian at the zero joint angle configuration

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.Puma560()
+>>> puma.jacob0([0, 0, 0, 0, 0, 0])
+array([[ 0.1295, -0.4854, -0.4854, -0.    , -0.0533,  0.    ],
+       [ 0.4318,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+       [-0.    ,  0.4318,  0.    ,  0.    ,  0.    ,  0.    ],
+       [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+       [ 0.    , -1.    , -1.    ,  0.    , -1.    ,  0.    ],
+       [ 1.    ,  0.    ,  0.    ,  1.    ,  0.    ,  1.    ]])
+
+
+

Notes

+
    +
  • +
    This is the geometric Jacobian as described in texts by

    Corke, Spong etal., Siciliano etal. The end-effector velocity is +described in terms of translational and angular velocity, not a +velocity twist as per the text by Lynch & Park.

    +
    +
    +
  • +
+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
+
+ +
+
+jacob0_analytical(q, representation='rpy/xyz', end=None, start=None, tool=None)
+

Manipulator analytical Jacobian in the start frame

+

robot.jacob0_analytical(q) is the manipulator Jacobian matrix which maps +joint velocity to end-effector spatial velocity expressed in the +start frame.

+
+
Parameters:
+
    +
  • q (Union[ndarray, List[float], Tuple[float], Set[float]]) – Joint coordinate vector

  • +
  • representation (Literal['rpy/xyz', 'rpy/zyx', 'eul', 'exp']) – angular representation

  • +
  • end (Union[str, Link, Gripper, None]) – the particular link or gripper whose velocity the Jacobian +describes, defaults to the base link

  • +
  • start (Union[str, Link, Gripper, None]) – the link considered as the end-effector, defaults to the robots’s end-effector

  • +
  • tool (Union[ndarray, SE3, None]) – a static tool transformation matrix to apply to the +end of end, defaults to None

  • +
+
+
Returns:
+

Manipulator Jacobian in the start frame

+
+
Return type:
+

jacob0

+
+
+

Synopsis

+

End-effector spatial velocity \(\nu = (v_x, v_y, v_z, \omega_x, \omega_y, \omega_z)^T\) +is related to joint velocity by \({}^{E}\!\nu = \mathbf{J}_m(q) \dot{q}\).

+

|``representation`` | Rotational representation | +|---------------------|————————————-| +|'rpy/xyz' | RPY angular rates in XYZ order | +|'rpy/zyx' | RPY angular rates in XYZ order | +|'eul' | Euler angular rates in ZYZ order | +|'exp' | exponential coordinate rates |

+

Examples

+

Makes a robot object and computes the analytic Jacobian for the given +joint configuration

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.ETS.Puma560()
+>>> puma.jacob0_analytical([0, 0, 0, 0, 0, 0])
+array([[ 0.15  ,  0.8636,  0.4318,  0.    ,  0.    ,  0.    ],
+       [ 0.2203,  0.    ,  0.    ,  0.2   ,  0.    ,  0.2   ],
+       [ 0.    , -0.2203, -0.2   ,  0.    , -0.2   ,  0.    ],
+       [ 1.    ,  0.    ,  0.    ,  1.    ,  0.    ,  1.    ],
+       [ 0.    ,  1.    ,  1.    ,  0.    ,  1.    ,  0.    ],
+       [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ]])
+
+
+
+ +
+
+jacobe(q, end=None, start=None, tool=None)
+

Manipulator geometric Jacobian in the end-effector frame

+

robot.jacobe(q) is the manipulator Jacobian matrix which maps +joint velocity to end-effector spatial velocity expressed in the +end frame.

+

End-effector spatial velocity \(\nu = (v_x, v_y, v_z, \omega_x, \omega_y, \omega_z)^T\) +is related to joint velocity by \({}^{E}\!\nu = \mathbf{J}_m(q) \dot{q}\).

+
+
Parameters:
+
    +
  • q (Union[ndarray, List[float], Tuple[float], Set[float]]) – Joint coordinate vector

  • +
  • end (Union[str, Link, Gripper, None]) – the particular link or gripper whose velocity the Jacobian +describes, defaults to the end-effector if only one is present

  • +
  • start (Union[str, Link, Gripper, None]) – the link considered as the base frame, defaults to the robots’s base frame

  • +
  • tool (Union[ndarray, SE3, None]) – a static tool transformation matrix to apply to the +end of end, defaults to None

  • +
+
+
Returns:
+

Manipulator Jacobian in the end frame

+
+
Return type:
+

Je

+
+
+

Examples

+

The following example makes a Puma560 robot object, and solves for the +end-effector frame Jacobian at the zero joint angle configuration

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.Puma560()
+>>> puma.jacobe([0, 0, 0, 0, 0, 0])
+array([[ 0.1295, -0.4854, -0.4854, -0.    , -0.0533,  0.    ],
+       [ 0.4318,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+       [-0.    ,  0.4318,  0.    ,  0.    ,  0.    ,  0.    ],
+       [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+       [ 0.    , -1.    , -1.    ,  0.    , -1.    ,  0.    ],
+       [ 1.    ,  0.    ,  0.    ,  1.    ,  0.    ,  1.    ]])
+
+
+

Notes

+
    +
  • +
    This is the geometric Jacobian as described in texts by

    Corke, Spong etal., Siciliano etal. The end-effector velocity is +described in terms of translational and angular velocity, not a +velocity twist as per the text by Lynch & Park.

    +
    +
    +
  • +
+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
+
+ +
+
+jointdynamics(q, qd=None)
+

Transfer function of joint actuator

+

tf = jointdynamics(qd, q) calculates a vector of n +continuous-time transfer functions that represent the transfer +function 1/(Js+B) for each joint based on the dynamic parameters +of the robot and the configuration q (n). n is the number of robot +joints. The result is a list of tuples (J, B) for each joint.

+

tf = jointdynamics(q, qd) as above but include the linearized +effects of Coulomb friction when operating at joint velocity QD +(1xN).

+
+
Parameters:
+
    +
  • q – Joint coordinates

  • +
  • qd – Joint velocity

  • +
+
+
Returns:
+

transfer function denominators

+
+
Return type:
+

list of 2-tuples

+
+
+
+ +
+
+property keywords: List[str]
+
+ +
+ +
+ +
+
+linkcolormap(linkcolors='viridis')
+

Create a colormap for robot joints

+
    +
  • cm = robot.linkcolormap() is an n-element colormap that gives a +unique color for every link. The RGBA colors for link j are +cm(j).

  • +
  • cm = robot.linkcolormap(cmap) as above but cmap is the name +of a valid matplotlib colormap. The default, example above, is the +viridis colormap.

  • +
  • cm = robot.linkcolormap(list of colors) as above but a +colormap is created from a list of n color names given as strings, +tuples or hexstrings.

  • +
+
+
Parameters:
+

linkcolors (Union[List[Any], str]) – list of colors or colormap, defaults to “viridis”

+
+
Returns:
+

the color map

+
+
Return type:
+

color map

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> cm = robot.linkcolormap("inferno")
+>>> print(cm(range(6))) # cm(i) is 3rd color in colormap
+[[0.0015 0.0005 0.0139 1.    ]
+ [0.2582 0.0386 0.4065 1.    ]
+ [0.5783 0.148  0.4044 1.    ]
+ [0.865  0.3168 0.2261 1.    ]
+ [0.9876 0.6453 0.0399 1.    ]
+ [0.9884 0.9984 0.6449 1.    ]]
+>>> cm = robot.linkcolormap(
+...     ['red', 'g', (0,0.5,0), '#0f8040', 'yellow', 'cyan'])
+>>> print(cm(range(6)))
+[[1.     0.     0.     1.    ]
+ [0.     0.5    0.     1.    ]
+ [0.     0.5    0.     1.    ]
+ [0.0588 0.502  0.251  1.    ]
+ [1.     1.     0.     1.    ]
+ [0.     1.     1.     1.    ]]
+
+
+

Notes

+ +
+ +
+ +

Robot links

+
+
Returns:
+

A list of link objects

+
+
Return type:
+

links

+
+
+

Notes

+

It is probably more concise to index the robot object rather +than the list of links, ie. the following are equivalent: +- robot.links[i] +- robot[i]

+
+ +
+
+property manufacturer
+

Get/set robot manufacturer’s name

+
    +
  • robot.manufacturer is the robot manufacturer’s name

  • +
  • robot.manufacturer = ... checks and sets the manufacturer’s name

  • +
+
+
Returns:
+

robot manufacturer’s name

+
+
Return type:
+

manufacturer

+
+
+
+ +
+
+property n: int
+

Number of joints

+
+
Returns:
+

Number of joints

+
+
Return type:
+

n

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.n
+6
+
+
+
+

See also

+

nlinks(), nbranches()

+
+
+ +
+
+property name: str
+

Get/set robot name

+
    +
  • robot.name is the robot name

  • +
  • robot.name = ... checks and sets the robot name

  • +
+
+
Parameters:
+

name – the new robot name

+
+
Returns:
+

the current robot name

+
+
Return type:
+

name

+
+
+
+ +
+
+property nbranches: int
+

Number of branches

+

Number of branches in this robot. Computed as the number of links with +zero children

+
+
Returns:
+

number of branches in the robot’s kinematic tree

+
+
Return type:
+

nbranches

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.ETS.Panda()
+>>> robot.nbranches
+1
+
+
+
+

See also

+

n(), nlinks()

+
+
+ +
+ +

Number of links

+

The returned number is the total of both variable joints and +static links

+
+
Returns:
+

Number of links

+
+
Return type:
+

nlinks

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.nlinks
+6
+
+
+
+

See also

+

n(), nbranches()

+
+
+ +
+
+nofriction(coulomb=True, viscous=False)
+

Remove manipulator joint friction

+

nofriction() copies the robot and returns +a robot with the same link parameters except the Coulomb and/or viscous +friction parameter are set to zero.

+
+
Parameters:
+
    +
  • coulomb (bool) – set the Coulomb friction to 0

  • +
  • viscous (bool) – set the viscous friction to 0

  • +
+
+
Returns:
+

A copy of the robot with dynamic parameters perturbed

+
+
Return type:
+

robot

+
+
+
+

See also

+

Robot.friction(), Link.nofriction()

+
+
+ +
+
+partial_fkine0(q, n=3, end=None, start=None)
+

Manipulator Forward Kinematics nth Partial Derivative

+

This method computes the nth derivative of the forward kinematics where n is +greater than or equal to 3. This is an extension of the differential kinematics +where the Jacobian is the first partial derivative and the Hessian is the +second.

+
+
Parameters:
+
    +
  • q (Union[ndarray, List[float], Tuple[float], Set[float]]) – The joint angles/configuration of the robot (Optional, +if not supplied will use the stored q values).

  • +
  • end (Union[str, Link, Gripper, None]) – the final link/Gripper which the Hessian represents

  • +
  • start (Union[str, Link, Gripper, None]) – the first link which the Hessian represents

  • +
  • tool – a static tool transformation matrix to apply to the +end of end, defaults to None

  • +
+
+
Returns:
+

The nth Partial Derivative of the forward kinematics

+
+
Return type:
+

A

+
+
+

Examples

+

The following example makes a Panda robot object, and solves for the +base-effector frame 4th defivative of the forward kinematics at the given +joint angle configuration

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.Panda()
+>>> panda.partial_fkine0([0, -0.3, 0, -2.2, 0, 2, 0.7854], n=4)
+array([[[[[ 0.484 ,  0.    ,  0.486 , ...,  0.1547, -0.    ,  0.    ],
+          [-0.    , -0.0796, -0.    , ..., -0.    , -0.2006,  0.    ],
+          [-0.    , -0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
+          [ 0.    ,  1.    ,  0.    , ..., -0.    , -1.    , -0.    ],
+          [ 0.    ,  0.    ,  0.2955, ..., -0.9463,  0.    , -0.0998],
+          [ 0.    , -0.    ,  0.    , ..., -0.    ,  0.    , -0.    ]],
+
+         [[ 0.    ,  0.484 ,  0.    , ...,  0.    , -0.1086,  0.    ],
+          [-0.    , -0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
+          [ 0.    , -0.    , -0.    , ..., -0.    ,  0.    ,  0.    ],
+          [-1.    ,  0.    , -0.9553, ...,  0.3233, -0.    ,  0.995 ],
+          [-0.    ,  0.    ,  0.    , ..., -0.    ,  0.    , -0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.    ]],
+
+         [[ 0.4624,  0.    ,  0.4643, ...,  0.1478, -0.    ,  0.    ],
+          [-0.    ,  0.067 , -0.    , ..., -0.    , -0.2237,  0.    ],
+          [-0.    ,  0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
+          [-0.    ,  0.9553,  0.    , ..., -0.    , -0.9553, -0.    ],
+          [-0.2955, -0.    ,  0.    , ..., -0.8085,  0.    ,  0.1987],
+          [-0.    , -0.    ,  0.    , ..., -0.    ,  0.    ,  0.    ]],
+
+         ...,
+
+         [[-0.1565, -0.    , -0.1571, ..., -0.05  ,  0.    ,  0.    ],
+          [ 0.    , -0.4323,  0.    , ..., -0.    ,  0.1676,  0.    ],
+          [ 0.    , -0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    , -0.3233,  0.    , ...,  0.    ,  0.3233, -0.    ],
+          [ 0.9463,  0.    ,  0.8085, ...,  0.    ,  0.    , -0.9093],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    , -0.    ]],
+
+         [[ 0.    , -0.484 ,  0.    , ..., -0.    ,  0.1086,  0.    ],
+          [-0.    ,  0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
+          [ 0.    ,  0.    , -0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 1.    ,  0.    ,  0.9553, ..., -0.3233,  0.    , -0.995 ],
+          [-0.    , -0.    , -0.    , ..., -0.    ,  0.    ,  0.    ],
+          [-0.    , -0.    , -0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[-0.4816, -0.    , -0.4835, ..., -0.1539,  0.    ,  0.    ],
+          [ 0.    ,  0.0309,  0.    , ...,  0.    ,  0.2104,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    , -0.995 ,  0.    , ...,  0.    ,  0.995 ,  0.    ],
+          [ 0.0998,  0.    , -0.1987, ...,  0.9093, -0.    ,  0.    ],
+          [ 0.    ,  0.    , -0.    , ...,  0.    , -0.    ,  0.    ]]],
+
+
+        [[[ 0.    ,  0.    ,  0.    , ...,  0.    , -0.    ,  0.    ],
+          [-0.    , -0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
+          [-0.    , -0.0796, -0.    , ..., -0.    , -0.2006,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ..., -0.    , -0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.    , -0.    ],
+          [ 0.    , -0.    ,  0.2955, ..., -0.9463,  0.    , -0.0998]],
+
+         [[-0.    ,  0.    ,  0.    , ...,  0.    , -0.    ,  0.    ],
+          [ 0.    , -0.0796, -0.    , ..., -0.    , -0.2006,  0.    ],
+          [-0.0796, -0.    ,  0.    , ...,  0.    , -0.    ,  0.    ],
+          [-0.    ,  0.    , -0.    , ...,  0.    , -0.    ,  0.    ],
+          [-0.    ,  0.    ,  0.2955, ..., -0.9463,  0.    , -0.0998],
+          [ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.    , -0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [-0.143 , -0.    , -0.1436, ..., -0.0457, -0.    ,  0.    ],
+          [-0.    ,  0.    , -0.    , ..., -0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    , -0.    ,  0.    ],
+          [-0.    , -0.2955,  0.    , ..., -0.    ,  0.2955,  0.    ],
+          [-0.2955, -0.    ,  0.    , ..., -0.    ,  0.    ,  0.    ]],
+
+         ...,
+
+         [[ 0.    ,  0.    , -0.    , ..., -0.    ,  0.    ,  0.    ],
+          [ 0.4581,  0.    ,  0.4599, ...,  0.1464,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    , -0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.9463,  0.    , ...,  0.    , -0.9463, -0.    ],
+          [ 0.9463,  0.    ,  0.    , ...,  0.    , -0.    ,  0.    ]],
+
+         [[ 0.    , -0.    ,  0.    , ..., -0.    ,  0.    ,  0.    ],
+          [-0.    ,  0.0796,  0.    , ...,  0.    ,  0.2006,  0.    ],
+          [ 0.0796,  0.    , -0.    , ..., -0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ..., -0.    ,  0.    , -0.    ],
+          [-0.    , -0.    , -0.2955, ...,  0.9463,  0.    ,  0.0998],
+          [-0.    , -0.    , -0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[-0.    ,  0.    , -0.    , ..., -0.    , -0.    ,  0.    ],
+          [ 0.0483,  0.    ,  0.0485, ...,  0.0154,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    , -0.    ,  0.    ],
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+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         ...,
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]]],
+
+
+        ...,
+
+
+        [[[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         ...,
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]]],
+
+
+        [[[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         ...,
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]]],
+
+
+        [[[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         ...,
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]],
+
+         [[ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ],
+          [ 0.    ,  0.    ,  0.    , ...,  0.    ,  0.    ,  0.    ]]]]])
+
+
+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part II: +Acceleration and Advanced Applications.” arXiv preprint arXiv:2207.01794 (2022).

  • +
+
+ +
+
+pay(W, q=None, J=None, frame=1)
+

Generalised joint force/torque due to a payload wrench

+

tau = pay(W, J) Returns the generalised joint force/torques due to a +payload wrench W applied to the end-effector. Where the manipulator +Jacobian is J (6xn), and n is the number of robot joints.

+

tau = pay(W, q, frame) as above but the Jacobian is calculated at pose +q in the frame given by frame which is 0 for base frame, 1 for +end-effector frame.

+

Uses the formula tau = J’W, where W is a wrench vector applied at the +end effector, W = [Fx Fy Fz Mx My Mz]’.

+
+
Trajectory operation:

In the case q is nxm or J is 6xnxm then tau is nxm where each row +is the generalised force/torque at the pose given by corresponding +row of q.

+
+
+
+
Parameters:
+
    +
  • W (Union[ndarray, List[float], Tuple[float], Set[float]]) – A wrench vector applied at the end effector, +W = [Fx Fy Fz Mx My Mz]

  • +
  • q (Optional[ndarray]) – Joint coordinates

  • +
  • J (Optional[ndarray]) – The manipulator Jacobian (Optional, if not supplied will +use the q value).

  • +
  • frame (int) – The frame in which to torques are expressed in when J +is not supplied. 0 means base frame of the robot, 1 means end- +effector frame

  • +
+
+
Returns:
+

Joint forces/torques due to w

+
+
Return type:
+

t

+
+
+

Notes

+
    +
  • +
    Wrench vector and Jacobian must be from the same reference

    frame.

    +
    +
    +
  • +
  • Tool transforms are taken into consideration when frame=1.

  • +
  • +
    Must have a constant wrench - no trajectory support for this

    yet.

    +
    +
    +
  • +
+
+ +
+
+paycap(w, tauR, frame=1, q=None)
+

Static payload capacity of a robot

+

wmax, joint = paycap(q, w, f, tauR) returns the maximum permissible +payload wrench wmax (6) applied at the end-effector, and the index +of the joint (zero indexed) which hits its force/torque limit at that +wrench. q (n) is the manipulator pose, w the payload wrench +(6), f the wrench reference frame and tauR (nx2) is a matrix of +joint forces/torques (first col is maximum, second col minimum).

+

Trajectory operation:

+

In the case q is nxm then wmax is Mx6 and J is Mx1 where the rows are +the results at the pose given by corresponding row of q.

+
+
Parameters:
+
    +
  • w (ndarray) – The payload wrench

  • +
  • tauR (ndarray) – Joint torque matrix minimum and maximums

  • +
  • frame (int) – The frame in which to torques are expressed in when J +is not supplied. ‘base’ means base frame of the robot, ‘ee’ means +end-effector frame

  • +
  • q (Union[ndarray, List[float], Tuple[float], Set[float], None]) – Joint coordinates

  • +
+
+
Returns:
+

The maximum permissible payload wrench

+
+
Return type:
+

ndarray(6)

+
+
+

Notes

+
    +
  • Wrench vector and Jacobian must be from the same reference frame

  • +
  • Tool transforms are taken into consideration for frame=1.

  • +
+
+ +
+
+payload(m, p=array([0.0, 0.0, 0.0]))
+

Add a payload to the end-effector

+

payload(m, p) adds payload mass adds a payload with point mass m at +position p in the end-effector coordinate frame.

+

payload(m) adds payload mass adds a payload with point mass m at +in the end-effector coordinate frame.

+

payload(0) removes added payload.

+
+
Parameters:
+
    +
  • m (float) – mass (kg)

  • +
  • p – position in end-effector frame

  • +
+
+
+
+ +
+
+perturb(p=0.1)
+

Perturb robot parameters

+

rp = perturb(p) is a new robot object in which the dynamic parameters +(link mass and inertia) have been perturbed. The perturbation is +multiplicative so that values are multiplied by random numbers in the +interval (1-p) to (1+p). The name string of the perturbed robot is +prefixed by ‘P/’.

+

Useful for investigating the robustness of various model-based control +schemes. For example to vary parameters in the range +/- 10 percent +is: r2 = puma.perturb(0.1)

+
+
Parameters:
+

p – The percent (+/-) to be perturbed. Default 10%

+
+
Returns:
+

A copy of the robot with dynamic parameters perturbed

+
+
Return type:
+

DHRobot

+
+
+
+ +
+
+plot(q, backend=None, block=False, dt=0.05, limits=None, vellipse=False, fellipse=False, fig=None, movie=None, loop=False, **kwargs)
+

Graphical display and animation

+

robot.plot(q, 'pyplot') displays a graphical view of a robot +based on the kinematic model and the joint configuration q. +This is a stick figure polyline which joins the origins of the +link coordinate frames. The plot will autoscale with an aspect +ratio of 1.

+

If q (m,n) representing a joint-space trajectory it will create an +animation with a pause of dt seconds between each frame.

+
+
+q
+

The joint configuration of the robot.

+
+ +
+
+backend
+

The graphical backend to use, currently ‘swift’ +and ‘pyplot’ are implemented. Defaults to ‘swift’ of a Robot +and ‘pyplot` for a DHRobot

+
+ +
+
+block
+

Block operation of the code and keep the figure open

+
+ +
+
+dt
+

if q is a trajectory, this describes the delay in +seconds between frames

+
+ +
+
+limits
+

Custom view limits for the plot. If not supplied will +autoscale, [x1, x2, y1, y2, z1, z2] +(this option is for ‘pyplot’ only)

+
+ +
+
+vellipse
+

(Plot Option) Plot the velocity ellipse at the +end-effector (this option is for ‘pyplot’ only)

+
+ +
+
+fellipse
+

(Plot Option) Plot the force ellipse at the +end-effector (this option is for ‘pyplot’ only)

+
+ +
+
+fig
+

(Plot Option) The figure label to plot in (this option is for +‘pyplot’ only)

+
+ +
+
+movie
+

(Plot Option) The filename to save the movie to (this option is for +‘pyplot’ only)

+
+ +
+
+loop
+

(Plot Option) Loop the movie (this option is for +‘pyplot’ only)

+
+ +
+
+jointaxes
+

(Plot Option) Plot an arrow indicating the axes in +which the joint revolves around(revolute joint) or translates +along (prosmatic joint) (this option is for ‘pyplot’ only)

+
+ +
+
+eeframe
+

(Plot Option) Plot the end-effector coordinate frame +at the location of the end-effector. Uses three arrows, red, +green and blue to indicate the x, y, and z-axes. +(this option is for ‘pyplot’ only)

+
+ +
+
+shadow
+

(Plot Option) Plot a shadow of the robot in the x-y +plane. (this option is for ‘pyplot’ only)

+
+ +
+
+name
+

(Plot Option) Plot the name of the robot near its base +(this option is for ‘pyplot’ only)

+
+ +
+
Returns:
+

A reference to the environment object which controls the +figure

+
+
Return type:
+

env

+
+
+

Notes

+
    +
  • +
    By default this method will block until the figure is dismissed.

    To avoid this set block=False.

    +
    +
    +
  • +
  • +
    For PyPlot, the polyline joins the origins of the link frames,

    but for some Denavit-Hartenberg models those frames may not +actually be on the robot, ie. the lines to not neccessarily +represent the links of the robot.

    +
    +
    +
  • +
+
+

See also

+

teach()

+
+
+ +
+
+plot_ellipse(ellipse, block=True, limits=None, jointaxes=True, eeframe=True, shadow=True, name=True)
+

Plot the an ellipsoid

+

robot.plot_ellipse(ellipsoid) displays the ellipsoid.

+
+
+ellipse
+

the ellipsoid to plot

+
+ +
+
+block
+

Block operation of the code and keep the figure open

+
+ +
+
+limits
+

Custom view limits for the plot. If not supplied will +autoscale, [x1, x2, y1, y2, z1, z2]

+
+ +
+
+jointaxes
+

(Plot Option) Plot an arrow indicating the axes in +which the joint revolves around(revolute joint) or translates +along (prosmatic joint)

+
+ +
+
+eeframe
+

(Plot Option) Plot the end-effector coordinate frame +at the location of the end-effector. Uses three arrows, red, +green and blue to indicate the x, y, and z-axes.

+
+ +
+
+shadow
+

(Plot Option) Plot a shadow of the robot in the x-y +plane

+
+ +
+
+name
+

(Plot Option) Plot the name of the robot near its base

+
+ +
+
Returns:
+

A reference to the PyPlot object which controls the +matplotlib figure

+
+
Return type:
+

env

+
+
+
+ +
+
+plot_fellipse(q, block=True, fellipse=None, limits=None, opt='trans', centre=[0, 0, 0], jointaxes=True, eeframe=True, shadow=True, name=True)
+

Plot the force ellipsoid for manipulator

+

robot.plot_fellipse(q) displays the velocity ellipsoid for the +robot at pose q. The plot will autoscale with an aspect ratio +of 1.

+

robot.plot_fellipse(vellipse) specifies a custon ellipse to plot.

+
+
+q
+

The joint configuration of the robot

+
+ +
+
+block
+

Block operation of the code and keep the figure open

+
+ +
+
+fellipse
+

the vellocity ellipsoid to plot

+
+ +
+
+limits
+

Custom view limits for the plot. If not supplied will +autoscale, [x1, x2, y1, y2, z1, z2]

+
+ +
+
+opt
+

‘trans’ or ‘rot’ will plot either the translational or +rotational force ellipsoid

+
+ +
+
+centre
+

The coordinates to plot the fellipse [x, y, z] or “ee” +to plot at the end-effector location

+
+ +
+
+jointaxes
+

(Plot Option) Plot an arrow indicating the axes in +which the joint revolves around(revolute joint) or translates +along (prosmatic joint)

+
+ +
+
+eeframe
+

(Plot Option) Plot the end-effector coordinate frame +at the location of the end-effector. Uses three arrows, red, +green and blue to indicate the x, y, and z-axes.

+
+ +
+
+shadow
+

(Plot Option) Plot a shadow of the robot in the x-y +plane

+
+ +
+
+name
+

(Plot Option) Plot the name of the robot near its base

+
+ +
+
Raises:
+

ValueError – q or fellipse must be supplied

+
+
Returns:
+

A reference to the PyPlot object which controls the +matplotlib figure

+
+
Return type:
+

env

+
+
+

Notes

+
    +
  • +
    By default the ellipsoid related to translational motion is

    drawn. Use opt='rot' to draw the rotational velocity +ellipsoid.

    +
    +
    +
  • +
  • +
    By default the ellipsoid is drawn at the origin. The option

    centre allows its origin to set to set to the specified +3-vector, or the string “ee” ensures it is drawn at the +end-effector position.

    +
    +
    +
  • +
+
+ +
+
+plot_vellipse(q, block=True, vellipse=None, limits=None, opt='trans', centre=[0, 0, 0], jointaxes=True, eeframe=True, shadow=True, name=True)
+

Plot the velocity ellipsoid for manipulator

+

robot.plot_vellipse(q) displays the velocity ellipsoid for the +robot at pose q. The plot will autoscale with an aspect ratio +of 1.

+

robot.plot_vellipse(vellipse) specifies a custon ellipse to plot.

+
+
block

Block operation of the code and keep the figure open

+
+
q

The joint configuration of the robot

+
+
vellipse

the vellocity ellipsoid to plot

+
+
limits

Custom view limits for the plot. If not supplied will +autoscale, [x1, x2, y1, y2, z1, z2]

+
+
opt

‘trans’ or ‘rot’ will plot either the translational or +rotational velocity ellipsoid

+
+
centre

The coordinates to plot the vellipse [x, y, z] or “ee” +to plot at the end-effector location

+
+
jointaxes

(Plot Option) Plot an arrow indicating the axes in +which the joint revolves around(revolute joint) or translates +along (prosmatic joint)

+
+
eeframe

(Plot Option) Plot the end-effector coordinate frame +at the location of the end-effector. Uses three arrows, red, +green and blue to indicate the x, y, and z-axes.

+
+
shadow

(Plot Option) Plot a shadow of the robot in the x-y +plane

+
+
name

(Plot Option) Plot the name of the robot near its base

+
+
+
+
Raises:
+

ValueError – q or fellipse must be supplied

+
+
Returns:
+

A reference to the PyPlot object which controls the +matplotlib figure

+
+
Return type:
+

env

+
+
+

Notes

+
    +
  • +
    By default the ellipsoid related to translational motion is

    drawn. Use opt='rot' to draw the rotational velocity +ellipsoid.

    +
    +
    +
  • +
  • +
    By default the ellipsoid is drawn at the origin. The option

    centre allows its origin to set to set to the specified +3-vector, or the string “ee” ensures it is drawn at the +end-effector position.

    +
    +
    +
  • +
+
+ +
+
+property prismaticjoints: List[bool]
+

Revolute joints as bool array

+
+
Returns:
+

array of joint type, True if prismatic

+
+
Return type:
+

prismaticjoints

+
+
+

Examples

+

+
+
+

Notes

+

Fixed joints, that maintain a constant link relative pose, +are not included.

+
+

See also

+

Link.isprismatic(), revolutejoints()

+
+
+ +
+
+property q: ndarray
+

Get/set robot joint configuration

+
    +
  • robot.q is the robot joint configuration

  • +
  • robot.q = ... checks and sets the joint configuration

  • +
+
+
Parameters:
+

q – the new robot joint configuration

+
+
Returns:
+

robot joint configuration

+
+
Return type:
+

q

+
+
+
+ +
+
+property qd: ndarray
+

Get/set robot joint velocity

+
    +
  • robot.qd is the robot joint velocity

  • +
  • robot.qd = ... checks and sets the joint velocity

  • +
+
+
Returns:
+

robot joint velocity

+
+
Return type:
+

qd

+
+
+
+ +
+
+property qdd: ndarray
+

Get/set robot joint acceleration

+
    +
  • robot.qdd is the robot joint acceleration

  • +
  • robot.qdd = ... checks and sets the robot joint acceleration

  • +
+
+
Returns:
+

robot joint acceleration

+
+
Return type:
+

qdd

+
+
+
+ +
+
+property qlim: ndarray
+

Joint limits

+

Limits are extracted from the link objects. If joints limits are +not set for:

+
    +
  • a revolute joint [-𝜋. 𝜋] is returned

  • +
  • a prismatic joint an exception is raised

  • +
+
+
+qlim
+

An array of joints limits (2, n)

+
+ +
+
Raises:
+

ValueError – unset limits for a prismatic joint

+
+
Returns:
+

Array of joint limit values

+
+
Return type:
+

qlim

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.qlim
+array([[-2.7925, -1.9199, -2.3562, -4.6426, -1.7453, -4.6426],
+       [ 2.7925,  1.9199,  2.3562,  4.6426,  1.7453,  4.6426]])
+
+
+
+ +
+
+random_q()
+

Return a random joint configuration

+

The value for each joint is uniform randomly distributed between the +limits set for the robot.

+
+

Note

+

The joint limit for all joints must be set.

+
+
+
Returns:
+

Random joint configuration :rtype: ndarray(n)

+
+
Return type:
+

q

+
+
+
+

See also

+

Robot.qlim(), Link.qlim()

+
+
+ +
+
+property revolutejoints: List[bool]
+

Revolute joints as bool array

+
+
Returns:
+

array of joint type, True if revolute

+
+
Return type:
+

revolutejoints

+
+
+

Examples

+

+
+
+

Notes

+

Fixed joints, that maintain a constant link relative pose, +are not included.

+
+

See also

+

Link.isrevolute(), prismaticjoints()

+
+
+ +
+
+property scene_children: List[SceneNode]
+

Returns the child nodes of this object

+
+ +
+
+property scene_parent: Type[SceneNode]
+

Returns the parent node of this object

+
+ +
+
+segments()
+

Segments of branched robot

+

For a single-chain robot with structure:

+
L1 - L2 - L3
+
+
+

the return is [[None, L1, L2, L3]]

+

For a robot with structure:

+
L1 - L2 +-  L3 - L4
+        +- L5 - L6
+
+
+

the return is [[None, L1, L2], [L2, L3, L4], [L2, L5, L6]]

+
+
Returns:
+

Segment list

+
+
Return type:
+

segments

+
+
+

Notes

+
    +
  • the length of the list is the number of segments in the robot

  • +
  • +
    the first segment always starts with None which represents

    the base transform (since there is no base link)

    +
    +
    +
  • +
  • +
    the last link of one segment is also the first link of subsequent

    segments

    +
    +
    +
  • +
+
+ +
+
+showgraph(display_graph=True, **kwargs)
+

Display a link transform graph in browser

+

robot.showgraph() displays a graph of the robot’s link frames +and the ETS between them. It uses GraphViz dot.

+
+
The nodes are:
    +
  • Base is shown as a grey square. This is the world frame origin, +but can be changed using the base attribute of the robot.

  • +
  • Link frames are indicated by circles

  • +
  • ETS transforms are indicated by rounded boxes

  • +
+
+
The edges are:
    +
  • an arrow if jtype is False or the joint is fixed

  • +
  • an arrow with a round head if jtype is True and the joint is +revolute

  • +
  • an arrow with a box head if jtype is True and the joint is +prismatic

  • +
+
+
+

Edge labels or nodes in blue have a fixed transformation to the +preceding link.

+
+
Parameters:
+
    +
  • display_graph (bool) – Open the graph in a browser if True. Otherwise will return the +file path

  • +
  • etsbox – Put the link ETS in a box, otherwise an edge label

  • +
  • jtype – Arrowhead to node indicates revolute or prismatic type

  • +
  • static – Show static joints in blue and bold

  • +
+
+
Return type:
+

Optional[str]

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.URDF.Panda()
+>>> panda.showgraph()
+
+
+_images/panda-graph.svg +
+

See also

+

dotfile()

+
+
+ +
+
+property structure: str
+

Return the joint structure string

+

A string with one letter per joint: R for a revolute +joint, and P for a prismatic joint.

+
+
Returns:
+

joint configuration string

+
+
Return type:
+

structure

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> puma.structure
+'RRRRRR'
+>>> stanford = rtb.models.DH.Stanford()
+>>> stanford.structure
+'RRPRRR'
+
+
+

Notes

+

Fixed joints, that maintain a constant link relative pose, +are not included. +len(self.structure) == self.n.

+
+ +
+
+property symbolic: bool
+
+ +
+
+teach(q, block=True, limits=None, vellipse=False, fellipse=False, backend=None)
+

Graphical teach pendant

+

robot.teach(q) creates a matplotlib plot which allows the user to +“drive” a graphical robot using a graphical slider panel. The robot’s +inital joint configuration is q. The plot will autoscale with an +aspect ratio of 1.

+

robot.teach() as above except the robot’s stored value of q +is used.

+
+
q

The joint configuration of the robot (Optional, +if not supplied will use the stored q values).

+
+
block

Block operation of the code and keep the figure open

+
+
limits

Custom view limits for the plot. If not supplied will +autoscale, [x1, x2, y1, y2, z1, z2]

+
+
vellipse

(Plot Option) Plot the velocity ellipse at the +end-effector (this option is for ‘pyplot’ only)

+
+
fellipse

(Plot Option) Plot the force ellipse at the +end-effector (this option is for ‘pyplot’ only)

+
+
+
+
Returns:
+

A reference to the PyPlot object which controls the +matplotlib figure

+
+
Return type:
+

env

+
+
+

Notes

+
    +
  • +
    Program execution is blocked until the teach window is

    dismissed. If block=False the method is non-blocking but +you need to poll the window manager to ensure that the window +remains responsive.

    +
    +
    +
  • +
  • +
    The slider limits are derived from the joint limit properties.

    If not set then: +- For revolute joints they are assumed to be [-pi, +pi] +- For prismatic joint they are assumed unknown and an error

    +
    +

    occurs.

    +
    +
    +
    +
  • +
+
+ +
+
+todegrees(q)
+

Convert joint angles to degrees

+
+
Parameters:
+

q – The joint configuration of the robot

+
+
Return type:
+

ndarray

+
+
Returns:
+

    +
  • q – a vector of joint coordinates in degrees and metres

  • +
  • robot.todegrees(q) converts joint coordinates q to degrees

  • +
  • taking into account whether elements of q correspond to revolute

  • +
  • or prismatic joints, ie. prismatic joint values are not converted.

  • +
  • If q is a matrix, with one column per joint, the conversion is

  • +
  • performed columnwise.

  • +
+

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> from math import pi
+>>> stanford = rtb.models.DH.Stanford()
+>>> stanford.todegrees([pi/4, pi/8, 2, -pi/4, pi/6, pi/3])
+array([ 45. ,  22.5,   2. , -45. ,  30. ,  60. ])
+
+
+
+ +
+
+property tool: SE3
+

Get/set robot tool transform

+
    +
  • robot.tool is the robot tool transform as an SE3 object

  • +
  • robot._tool is the robot tool transform as a numpy array

  • +
  • robot.tool = ... checks and sets the robot tool transform

  • +
+
+
Parameters:
+

tool – the new robot tool transform (as an SE(3))

+
+
Returns:
+

robot tool transform

+
+
Return type:
+

tool

+
+
+
+ +
+
+toradians(q)
+

Convert joint angles to radians

+

robot.toradians(q) converts joint coordinates q to radians +taking into account whether elements of q correspond to revolute +or prismatic joints, ie. prismatic joint values are not converted.

+

If q is a matrix, with one column per joint, the conversion is +performed columnwise.

+
+
Parameters:
+

q – The joint configuration of the robot

+
+
Returns:
+

a vector of joint coordinates in radians and metres

+
+
Return type:
+

q

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> stanford = rtb.models.DH.Stanford()
+>>> stanford.toradians([10, 20, 2, 30, 40, 50])
+array([0.1745, 0.3491, 2.    , 0.5236, 0.6981, 0.8727])
+
+
+
+ +
+
+property urdf_filepath: str
+
+ +
+
+property urdf_string: str
+
+ +
+
+vellipse(q, opt='trans', unit='rad', centre=[0, 0, 0], scale=0.1)
+

Create a velocity ellipsoid object for plotting with PyPlot

+

robot.vellipse(q) creates a force ellipsoid for the robot at +pose q. The ellipsoid is centered at the origin.

+
+
+q
+

The joint configuration of the robot.

+
+ +
+
+opt
+

‘trans’ or ‘rot’ will plot either the translational or +rotational force ellipsoid

+
+ +
+
+unit
+

‘rad’ or ‘deg’ will plot the ellipsoid in radians or +degrees

+
+ +
+
+centre
+

The centre of the ellipsoid, either ‘ee’ for the end-effector +or a 3-vector [x, y, z] in the world frame

+
+ +
+
+scale
+

The scale factor for the ellipsoid

+
+ +
+
Returns:
+

An EllipsePlot object

+
+
Return type:
+

env

+
+
+

Notes

+
    +
  • +
    By default the ellipsoid related to translational motion is

    drawn. Use opt='rot' to draw the rotational velocity +ellipsoid.

    +
    +
    +
  • +
  • +
    By default the ellipsoid is drawn at the origin. The option

    centre allows its origin to set to set to the specified +3-vector, or the string “ee” ensures it is drawn at the +end-effector position.

    +
    +
    +
  • +
+
+ +
+ +
+
+class roboticstoolbox.robot.Robot.Robot2(arg, **kwargs)[source]
+

Bases: BaseRobot[Link2]

+
+
+__getitem__(i)
+

Get link

+

This also supports iterating over each link in the robot object, +from the base to the tool.

+
+
Parameters:
+

i (Union[int, str]) – link number or name

+
+
Returns:
+

i’th link or named link

+
+
Return type:
+

link

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> print(robot[1]) # print the 2nd link
+RevoluteDH:   θ=q,  d=0,  a=0.4318,  ⍺=0.0
+>>> print([link.a for link in robot])  # print all the a_j values
+[0, 0.4318, 0.0203, 0, 0, 0]
+
+
+

Notes

+
+
Robot supports link lookup by name,

eg. robot['link1']

+
+
+
+ +
+
+__str__()
+

Pretty prints the ETS Model of the robot.

+
+
Returns:
+

Pretty print of the robot model

+
+
Return type:
+

str

+
+
+

Notes

+
    +
  • Constant links are shown in blue.

  • +
  • End-effector links are prefixed with an @

  • +
  • Angles in degrees

  • +
  • +
    The robot base frame is denoted as BASE and is equal to the

    robot’s base attribute.

    +
    +
    +
  • +
+
+ +
+
+accel(q, qd, torque, gravity=None)
+

Compute acceleration due to applied torque

+

qdd = accel(q, qd, torque) calculates a vector (n) of joint +accelerations that result from applying the actuator force/torque (n) +to the manipulator in state q (n) and qd (n), and n is +the number of robot joints.

+
+\[\ddot{q} = \mathbf{M}^{-1} \left(\tau - \mathbf{C}(q)\dot{q} - \mathbf{g}(q)\right)\]
+

Trajectory operation

+

If q, qd, torque are matrices (m,n) then qdd is a matrix (m,n) +where each row is the acceleration corresponding to the equivalent rows +of q, qd, torque.

+
+
Parameters:
+
    +
  • q – Joint coordinates

  • +
  • qd – Joint velocity

  • +
  • torque – Joint torques of the robot

  • +
  • gravity – Gravitational acceleration (Optional, if not supplied will +use the gravity attribute of self).

  • +
+
+
Return type:
+

ndarray(n)

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> puma.accel(puma.qz, 0.5 * np.ones(6), np.zeros(6))
+array([ -7.5544, -12.22  ,  -6.4022,  -5.4303,  -4.9518,  -2.1178])
+
+
+

Notes

+
    +
  • +
    Useful for simulation of manipulator dynamics, in

    conjunction with a numerical integration function.

    +
    +
    +
  • +
  • +
    Uses the method 1 of Walker and Orin to compute the forward

    dynamics.

    +
    +
    +
  • +
  • +
    Featherstone’s method is more efficient for robots with large

    numbers of joints.

    +
    +
    +
  • +
  • Joint friction is considered.

  • +
+

References

+
    +
  • +
    Efficient dynamic computer simulation of robotic mechanisms,

    M. W. Walker and D. E. Orin, +ASME Journa of Dynamic Systems, Measurement and Control, vol. +104, no. 3, pp. 205-211, 1982.

    +
    +
    +
  • +
+
+ +
+
+accel_x(q, xd, wrench, gravity=None, pinv=False, representation='rpy/xyz')
+

Operational space acceleration due to applied wrench

+

xdd = accel_x(q, qd, wrench) is the operational space acceleration +due to wrench applied to the end-effector of a robot in joint +configuration q and joint velocity qd.

+
+\[\ddot{x} = \mathbf{J}(q) \mathbf{M}(q)^{-1} \left( + \mathbf{J}(q)^T w - \mathbf{C}(q)\dot{q} - \mathbf{g}(q) + \right)\]
+

Trajectory operation

+

If q, qd, torque are matrices (m,n) then qdd is a matrix (m,n) +where each row is the acceleration corresponding to the equivalent rows +of q, qd, wrench.

+
+
Parameters:
+
    +
  • q – Joint coordinates

  • +
  • qd – Joint velocity

  • +
  • wrench – Wrench applied to the end-effector

  • +
  • gravity – Gravitational acceleration (Optional, if not supplied will +use the gravity attribute of self).

  • +
  • pinv – use pseudo inverse rather than inverse

  • +
  • analytical – the type of analytical Jacobian to use, default is +‘rpy/xyz’

  • +
  • xd

  • +
  • representation – (Default value = “rpy/xyz”)

  • +
+
+
Returns:
+

Operational space accelerations of the end-effector

+
+
Return type:
+

accel

+
+
+

Examples

+
    ainv = _umath_linalg.inv(a, signature=signature, extobj=extobj)
+  File "/opt/hostedtoolcache/Python/3.7.17/x64/lib/python3.7/site-packages/numpy/linalg/linalg.py", line 88, in _raise_linalgerror_singular
+    raise LinAlgError("Singular matrix")
+numpy.linalg.LinAlgError: Singular matrix
+
+
+

Notes

+
    +
  • +
    Useful for simulation of manipulator dynamics, in

    conjunction with a numerical integration function.

    +
    +
    +
  • +
  • +
    Uses the method 1 of Walker and Orin to compute the forward

    dynamics.

    +
    +
    +
  • +
  • +
    Featherstone’s method is more efficient for robots with large

    numbers of joints.

    +
    +
    +
  • +
  • Joint friction is considered.

  • +
+
+

See also

+

accel()

+
+
+ +
+
+addconfiguration(name, q)
+

Add a named joint configuration

+

Add a named configuration to the robot instance’s dictionary of named +configurations.

+
+
Parameters:
+
    +
  • name (str) – Name of the joint configuration

  • +
  • q (ndarray) – Joint configuration

  • +
+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.addconfiguration_attr("mypos", [0.1, 0.2, 0.3, 0.4, 0.5, 0.6])
+>>> robot.configs["mypos"]
+array([0.1, 0.2, 0.3, 0.4, 0.5, 0.6])
+
+
+
+

See also

+

addconfiguration()

+
+
+ +
+
+addconfiguration_attr(name, q, unit='rad')
+

Add a named joint configuration as an attribute

+
+
Parameters:
+
+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.addconfiguration_attr("mypos", [0.1, 0.2, 0.3, 0.4, 0.5, 0.6])
+>>> robot.mypos
+array([0.1, 0.2, 0.3, 0.4, 0.5, 0.6])
+>>> robot.configs["mypos"]
+array([0.1, 0.2, 0.3, 0.4, 0.5, 0.6])
+
+
+

Notes

+
    +
  • Used in robot model init method to store the qr configuration

  • +
  • +
    Dynamically adding attributes to objects can cause issues with

    Python type checking.

    +
    +
    +
  • +
  • +
    Configuration is also added to the robot instance’s dictionary of

    named configurations.

    +
    +
    +
  • +
+
+

See also

+

addconfiguration()

+
+
+ +
+
+attach(object)
+
+ +
+
+attach_to(object)
+
+ +
+ +

Get the robot base link

+
    +
  • robot.base_link is the robot base link

  • +
+
+
Returns:
+

the first link in the robot tree

+
+
Return type:
+

base_link

+
+
+
+ +
+
+cinertia(q)
+

Deprecated, use inertia_x

+
+ +
+
+property comment: str
+

Get/set robot comment

+
    +
  • robot.comment is the robot comment

  • +
  • robot.comment = ... checks and sets the robot comment

  • +
+
+
Parameters:
+

name – the new robot comment

+
+
Returns:
+

robot comment

+
+
Return type:
+

comment

+
+
+
+ +
+
+property configs: Dict[str, ndarray]
+
+ +
+
+configurations_str(border='thin')
+
+ +
+
+property control_mode: str
+

Get/set robot control mode

+
    +
  • robot.control_type is the robot control mode

  • +
  • robot.control_type = ... checks and sets the robot control mode

  • +
+
+
Parameters:
+

control_mode – the new robot control mode

+
+
Returns:
+

the current robot control mode

+
+
Return type:
+

control_mode

+
+
+
+ +
+
+copy()
+
+ +
+
+coriolis(q, qd)
+

Coriolis and centripetal term

+

coriolis(q, qd) calculates the Coriolis/centripetal matrix (n,n) +for the robot in configuration q and velocity qd, where n +is the number of joints.

+

The product \(\mathbf{C} \dot{q}\) is the vector of joint +force/torque due to velocity coupling. The diagonal elements are due to +centripetal effects and the off-diagonal elements are due to Coriolis +effects. This matrix is also known as the velocity coupling matrix, +since it describes the disturbance forces on any joint due to +velocity of all other joints.

+

Trajectory operation

+

If q and qd are matrices (m,n), each row is interpretted as a +joint configuration, and the result (n,n,m) is a 3d-matrix where +each plane corresponds to a row of q and qd.

+
+
Parameters:
+
    +
  • q – Joint coordinates

  • +
  • qd – Joint velocity

  • +
+
+
Returns:
+

Velocity matrix

+
+
Return type:
+

coriolis

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> puma.coriolis(puma.qz, 0.5 * np.ones((6,)))
+array([[-0.4017, -0.5513, -0.2025, -0.0007, -0.0013,  0.    ],
+       [ 0.2023, -0.1937, -0.3868, -0.    , -0.002 ,  0.    ],
+       [ 0.1987,  0.193 , -0.    ,  0.    , -0.0001,  0.    ],
+       [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ],
+       [ 0.0007,  0.0007,  0.0001,  0.    ,  0.    ,  0.    ],
+       [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.    ]])
+
+
+

Notes

+
    +
  • +
    Joint viscous friction is also a joint force proportional to

    velocity but it is eliminated in the computation of this value.

    +
    +
    +
  • +
  • Computationally slow, involves \(n^2/2\) invocations of RNE.

  • +
+
+ +
+
+coriolis_x(q, qd, pinv=False, representation='rpy/xyz', J=None, Ji=None, Jd=None, C=None, Mx=None)
+

Operational space Coriolis and centripetal term

+

coriolis_x(q, qd) is the Coriolis/centripetal matrix (m,m) +in operational space for the robot in configuration q and velocity +qd, where n is the number of joints.

+
+\[\mathbf{C}_x = \mathbf{J}(q)^{-T} \left( + \mathbf{C}(q) - \mathbf{M}_x(q) \mathbf{J})(q) + \right) \mathbf{J}(q)^{-1}\]
+

The product \(\mathbf{C} \dot{x}\) is the operational space wrench +due to joint velocity coupling. This matrix is also known as the +velocity coupling matrix, since it describes the disturbance forces on +any joint due to velocity of all other joints.

+

The transformation to operational space requires an analytical, rather +than geometric, Jacobian. analytical can be one of:

+
+
+ + + + + + + + + + + + + + + + + + + +

Value

Rotational representation

'rpy/xyz'

RPY angular rates in XYZ order (default)

'rpy/zyx'

RPY angular rates in XYZ order

'eul'

Euler angular rates in ZYZ order

'exp'

exponential coordinate rates

+
+

Trajectory operation

+

If q and qd are matrices (m,n), each row is interpretted as a +joint configuration, and the result (n,n,m) is a 3d-matrix where +each plane corresponds to a row of q and qd.

+
+
Parameters:
+
    +
  • q – Joint coordinates

  • +
  • qd – Joint velocity

  • +
  • pinv – use pseudo inverse rather than inverse (Default value = False)

  • +
  • analytical – the type of analytical Jacobian to use, default is +‘rpy/xyz’

  • +
  • representation – (Default value = “rpy/xyz”)

  • +
  • J

  • +
  • Ji

  • +
  • Jd

  • +
  • C

  • +
  • Mx

  • +
+
+
Returns:
+

Operational space velocity matrix

+
+
Return type:
+

ndarray(6,6) or ndarray(m,6,6)

+
+
+

Examples

+
    ainv = _umath_linalg.inv(a, signature=signature, extobj=extobj)
+  File "/opt/hostedtoolcache/Python/3.7.17/x64/lib/python3.7/site-packages/numpy/linalg/linalg.py", line 88, in _raise_linalgerror_singular
+    raise LinAlgError("Singular matrix")
+numpy.linalg.LinAlgError: Singular matrix
+
+
+

Notes

+
    +
  • +
    Joint viscous friction is also a joint force proportional to

    velocity but it is eliminated in the computation of this value.

    +
    +
    +
  • +
  • Computationally slow, involves \(n^2/2\) invocations of RNE.

  • +
  • If the robot is not 6 DOF the pinv option is set True.

  • +
  • pinv() is around 5x slower than inv()

  • +
+
+

Warning

+

Assumes that the operational space has 6 DOF.

+
+
+

See also

+

coriolis(), inertia_x(), hessian0()

+
+
+ +
+
+property default_backend
+

Get default graphical backend

+
    +
  • +
    robot.default_backend Get the default graphical backend, used when

    no explicit backend is passed to Robot.plot.

    +
    +
    +
  • +
  • +
    robot.default_backend = ... Set the default graphical backend, used when

    no explicit backend is passed to Robot.plot. The default set here will +be overridden if the particular Robot subclass cannot support it.

    +
    +
    +
  • +
+
+
Returns:
+

backend name

+
+
Return type:
+

default_backend

+
+
+
+ +
+ +

A link search method

+

Visit all links from start in depth-first order and will apply +func to each visited link

+
+
Parameters:
+
+
+
Returns:
+

A list of links

+
+
Return type:
+

links

+
+
+
+ +
+
+dotfile(filename, etsbox=False, ets='full', jtype=False, static=True)
+

Write a link transform graph as a GraphViz dot file

+
+
The file can be processed using dot:

% dot -Tpng -o out.png dotfile.dot

+
+
The nodes are:
    +
  • Base is shown as a grey square. This is the world frame origin, +but can be changed using the base attribute of the robot.

  • +
  • Link frames are indicated by circles

  • +
  • ETS transforms are indicated by rounded boxes

  • +
+
+
The edges are:
    +
  • an arrow if jtype is False or the joint is fixed

  • +
  • an arrow with a round head if jtype is True and the joint is +revolute

  • +
  • an arrow with a box head if jtype is True and the joint is +prismatic

  • +
+
+
+

Edge labels or nodes in blue have a fixed transformation to the +preceding link.

+
+

Note

+
+
If filename is a file object then the file will not

be closed after the GraphViz model is written.

+
+
+
+
+
Parameters:
+
    +
  • file – Name of file to write to

  • +
  • etsbox (bool) – Put the link ETS in a box, otherwise an edge label

  • +
  • ets (Literal['full', 'brief']) – Display the full ets with “full” or a brief version with “brief”

  • +
  • jtype (bool) – Arrowhead to node indicates revolute or prismatic type

  • +
  • static (bool) – Show static joints in blue and bold

  • +
+
+
+
+

See also

+

showgraph()

+
+
+ +
+
+dynamics()
+

Pretty print the dynamic parameters (Robot superclass)

+

The dynamic parameters (inertial and friction) are printed in a table, +with one row per link.

+

Examples

+

+
+
+
+ +
+
+dynamics_list()
+

Print dynamic parameters (Robot superclass)

+

Display the kinematic and dynamic parameters to the console in +reable format

+
+ +
+
+dynchanged(what=None)
+

Dynamic parameters have changed

+

Called from a property setter to inform the robot that the cache of +dynamic parameters is invalid.

+
+

See also

+

roboticstoolbox.Link._listen_dyn()

+
+
+ +
+ +
+ +
+
+ets(start=None, end=None)
+

Robot to ETS

+

robot.ets() is an ETS representing the kinematics from base to +end-effector.

+

robot.ets(end=link) is an ETS representing the kinematics from +base to the link link specified as a Link reference or a name.

+

robot.ets(start=l1, end=l2) is an ETS representing the kinematics +from link l1 to link l2.

+

:param : +:type : param start: start of path, defaults to base_link +:param : +:type : param end: end of path, defaults to end-effector

+
+
Raises:
+
+
+
Returns:
+

elementary transform sequence

+
+
Return type:
+

ets

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.ETS.Panda()
+>>> panda.ets()
+[ET.tz(eta=0.333), ET.Rz(jindex=0), ET.Rx(eta=-1.5707963267948966), ET.Rz(jindex=1), ET.Rx(eta=1.5707963267948966), ET.tz(eta=0.316), ET.Rz(jindex=2), ET.tx(eta=0.0825), ET.Rx(eta=1.5707963267948966), ET.Rz(jindex=3), ET.tx(eta=-0.0825), ET.Rx(eta=-1.5707963267948966), ET.tz(eta=0.384), ET.Rz(jindex=4), ET.Rx(eta=1.5707963267948966), ET.Rz(jindex=5), ET.tx(eta=0.088), ET.Rx(eta=1.5707963267948966), ET.tz(eta=0.107), ET.Rz(jindex=6), ET.tz(eta=0.10300000000000001), ET.Rz(eta=-0.7853981633974483)]
+
+
+
+ +
+
+fdyn(T, q0, Q=None, Q_args={}, qd0=None, solver='RK45', solver_args={}, dt=None, progress=False)
+

Integrate forward dynamics

+

tg = R.fdyn(T, q) integrates the dynamics of the robot with zero +input torques over the time interval 0 to T and returns the +trajectory as a namedtuple with elements:

+
    +
  • t the time vector (M,)

  • +
  • q the joint coordinates (M,n)

  • +
  • qd the joint velocities (M,n)

  • +
+

tg = R.fdyn(T, q, torqfun) as above but the torque applied to the +joints is given by the provided function:

+
tau = function(robot, t, q, qd, **args)
+
+
+

where the inputs are:

+
+
    +
  • the robot object

  • +
  • current time

  • +
  • current joint coordinates (n,)

  • +
  • current joint velocity (n,)

  • +
  • args, optional keyword arguments can be specified, these are

  • +
+

passed in from the targs kewyword argument.

+
+

The function must return a Numpy array (n,) of joint forces/torques.

+
+
Parameters:
+
+
+
Returns:
+

robot trajectory

+
+
Return type:
+

trajectory

+
+
+

Examples

+

To apply zero joint torque to the robot without Coulomb +friction:

+
>>> def myfunc(robot, t, q, qd):
+>>>     return np.zeros((robot.n,))
+
+
+
>>> tg = robot.nofriction().fdyn(5, q0, myfunc)
+
+
+
>>> plt.figure()
+>>> plt.plot(tg.t, tg.q)
+>>> plt.show()
+
+
+

We could also use a lambda function:

+
>>> tg = robot.nofriction().fdyn(
+>>>     5, q0, lambda r, t, q, qd: np.zeros((r.n,)))
+
+
+

The robot is controlled by a PD controller. We first define a +function to compute the control which has additional parameters for +the setpoint and control gains (qstar, P, D):

+
>>> def myfunc(robot, t, q, qd, qstar, P, D):
+>>>     return (qstar - q) * P + qd * D  # P, D are (6,)
+
+
+
>>> tg = robot.fdyn(10, q0, myfunc, torque_args=(qstar, P, D)) )
+
+
+

Many integrators have variable step length which is problematic if we +want to animate the result. If dt is specified then the solver +results are interpolated in time steps of dt.

+

Notes

+ +
+

See also

+

DHRobot.accel(), DHRobot.nofriction(), DHRobot.rne()

+
+
+ +
+
+fellipse(q, opt='trans', unit='rad', centre=[0, 0, 0])
+

Create a force ellipsoid object for plotting with PyPlot

+

robot.fellipse(q) creates a force ellipsoid for the robot at +pose q. The ellipsoid is centered at the origin.

+
+
+q
+

The joint configuration of the robot.

+
+ +
+
+opt
+

‘trans’ or ‘rot’ will plot either the translational or +rotational force ellipsoid

+
+ +
+
+unit
+

‘rad’ or ‘deg’ will plot the ellipsoid in radians or +degrees

+
+ +
+
+centre
+

The centre of the ellipsoid, either ‘ee’ for the end-effector +or a 3-vector [x, y, z] in the world frame

+
+ +
+
Returns:
+

An EllipsePlot object

+
+
Return type:
+

env

+
+
+

Notes

+
    +
  • +
    By default the ellipsoid related to translational motion is

    drawn. Use opt='rot' to draw the rotational velocity +ellipsoid.

    +
    +
    +
  • +
  • +
    By default the ellipsoid is drawn at the origin. The option

    centre allows its origin to set to set to the specified +3-vector, or the string “ee” ensures it is drawn at the +end-effector position.

    +
    +
    +
  • +
+
+ +
+
+friction(qd)
+

Manipulator joint friction (Robot superclass)

+

robot.friction(qd) is a vector of joint friction +forces/torques for the robot moving with joint velocities qd.

+

The friction model includes:

+
    +
  • Viscous friction which is a linear function of velocity.

  • +
  • Coulomb friction which is proportional to sign(qd).

  • +
+
+\[\begin{split}\tau_j = G^2 B \dot{q}_j + |G_j| \left\{ \begin{array}{ll} + \tau_{C,j}^+ & \mbox{if $\dot{q}_j > 0$} \\ + \tau_{C,j}^- & \mbox{if $\dot{q}_j < 0$} \end{array} \right.\end{split}\]
+
+
Parameters:
+

qd (ndarray) – The joint velocities of the robot

+
+
Returns:
+

The joint friction forces/torques for the robot

+
+
Return type:
+

friction

+
+
+

Notes

+
    +
  • +
    The friction value should be added to the motor output torque to

    determine the nett torque. It has a negative value when qd > 0.

    +
    +
    +
  • +
  • +
    The returned friction value is referred to the output of the

    gearbox.

    +
    +
    +
  • +
  • +
    The friction parameters in the Link object are referred to the

    motor.

    +
    +
    +
  • +
  • Motor viscous friction is scaled up by \(G^2\).

  • +
  • Motor Coulomb friction is scaled up by math:G.

  • +
  • +
    The appropriate Coulomb friction value to use in the

    non-symmetric case depends on the sign of the joint velocity, +not the motor velocity.

    +
    +
    +
  • +
  • +
    Coulomb friction is zero for zero joint velocity, stiction is

    not modeled.

    +
    +
    +
  • +
  • +
    The absolute value of the gear ratio is used. Negative gear

    ratios are tricky: the Puma560 robot has negative gear ratio for +joints 1 and 3.

    +
    +
    +
  • +
  • +
    The absolute value of the gear ratio is used. Negative gear

    ratios are tricky: the Puma560 has negative gear ratio for +joints 1 and 3.

    +
    +
    +
  • +
+
+

See also

+

Robot.nofriction(), Link.friction()

+
+
+ +
+
+get_path(end=None, start=None)
+

Find a path from start to end

+
+
Parameters:
+
+
+
Raises:
+

ValueError – link not known or ambiguous

+
+
Return type:
+

Tuple[List[TypeVar(LinkType, bound= BaseLink)], int, SE3]

+
+
Returns:
+

    +
  • path – the path from start to end

  • +
  • n – the number of joints in the path

  • +
  • T – the tool transform present after end

  • +
+

+
+
+
+ +
+
+property gravity: ndarray
+

Get/set default gravitational acceleration (Robot superclass)

+
    +
  • robot.name is the default gravitational acceleration

  • +
  • +
    robot.name = ... checks and sets default gravitational

    acceleration

    +
    +
    +
  • +
+
+
Parameters:
+

gravity – the new gravitational acceleration for this robot

+
+
Returns:
+

gravitational acceleration

+
+
Return type:
+

gravity

+
+
+

Notes

+

If the z-axis is upward, out of the Earth, this should be +a positive number.

+
+ +
+
+gravload(q=None, gravity=None)
+

Compute gravity load

+

robot.gravload(q) calculates the joint gravity loading (n) for +the robot in the joint configuration q and using the default +gravitational acceleration specified in the DHRobot object.

+

robot.gravload(q, gravity=g) as above except the gravitational +acceleration is explicitly specified as g.

+

Trajectory operation

+

If q is a matrix (nxm) each column is interpreted as a joint +configuration vector, and the result is a matrix (nxm) each column +being the corresponding joint torques.

+
+
Parameters:
+
+
+
Returns:
+

The generalised joint force/torques due to gravity

+
+
Return type:
+

gravload

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> puma.gravload(puma.qz)
+array([ 0.    , 37.4837,  0.2489,  0.    ,  0.    ,  0.    ])
+
+
+
+ +
+
+gravload_x(q=None, gravity=None, pinv=False, representation='rpy/xyz', Ji=None)
+

Operational space gravity load

+

robot.gravload_x(q) calculates the gravity wrench for +the robot in the joint configuration q and using the default +gravitational acceleration specified in the robot object.

+

robot.gravload_x(q, gravity=g) as above except the gravitational +acceleration is explicitly specified as g.

+
+\[\mathbf{G}_x = \mathbf{J}(q)^{-T} \mathbf{G}(q)\]
+

The transformation to operational space requires an analytical, rather +than geometric, Jacobian. analytical can be one of:

+
+
+ + + + + + + + + + + + + + + + + + + +

Value

Rotational representation

'rpy/xyz'

RPY angular rates in XYZ order (default)

'rpy/zyx'

RPY angular rates in XYZ order

'eul'

Euler angular rates in ZYZ order

'exp'

exponential coordinate rates

+
+

Trajectory operation

+

If q is a matrix (nxm) each column is interpreted as a joint +configuration vector, and the result is a matrix (nxm) each column +being the corresponding joint torques.

+
+
Parameters:
+
    +
  • q – Joint coordinates

  • +
  • gravity – Gravitational acceleration (Optional, if not supplied will +use the gravity attribute of self).

  • +
  • pinv – use pseudo inverse rather than inverse (Default value = False)

  • +
  • analytical – the type of analytical Jacobian to use, default is +‘rpy/xyz’

  • +
  • representation – (Default value = “rpy/xyz”)

  • +
  • Ji

  • +
+
+
Returns:
+

The operational space gravity wrench

+
+
Return type:
+

gravload

+
+
+

Examples

+
    ainv = _umath_linalg.inv(a, signature=signature, extobj=extobj)
+  File "/opt/hostedtoolcache/Python/3.7.17/x64/lib/python3.7/site-packages/numpy/linalg/linalg.py", line 88, in _raise_linalgerror_singular
+    raise LinAlgError("Singular matrix")
+numpy.linalg.LinAlgError: Singular matrix
+
+
+

Notes

+
    +
  • If the robot is not 6 DOF the pinv option is set True.

  • +
  • pinv() is around 5x slower than inv()

  • +
+
+

Warning

+

Assumes that the operational space has 6 DOF.

+
+
+

See also

+

gravload()

+
+
+ +
+
+property grippers: List[Gripper]
+

Grippers attached to the robot

+
+
Returns:
+

A list of grippers

+
+
Return type:
+

grippers

+
+
+
+ +
+
+property hascollision
+

Robot has collision model

+
+
Returns:
+

    +
  • hascollision – Robot has collision model

  • +
  • At least one link has associated collision model.

  • +
+

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.hascollision
+False
+
+
+
+

See also

+

hasgeometry()

+
+
+ +
+
+property hasdynamics
+

Robot has dynamic parameters

+
+
Returns:
+

    +
  • hasdynamics – Robot has dynamic parameters

  • +
  • At least one link has associated dynamic parameters.

  • +
+

+
+
+

Examples

+

+
+
+
+ +
+
+property hasgeometry
+

Robot has geometry model

+

At least one link has associated mesh to describe its shape.

+
+
Returns:
+

Robot has geometry model

+
+
Return type:
+

hasgeometry

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.hasgeometry
+True
+
+
+
+

See also

+

hascollision()

+
+
+ +
+
+hierarchy()
+

Pretty print the robot link hierachy

+
+
Return type:
+

Pretty print of the robot model

+
+
+

Examples

+

Makes a robot and prints the heirachy

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.URDF.Panda()
+>>> robot.hierarchy()
+ panda_link0
+   panda_link1
+     panda_link2
+       panda_link3
+         panda_link4
+           panda_link5
+             panda_link6
+               panda_link7
+                 panda_link8
+                   panda_hand
+                     panda_leftfinger
+                     panda_rightfinger
+
+
+
+ +
+
+inertia(q)
+

Manipulator inertia matrix +inertia(q) is the symmetric joint inertia matrix (n,n) which +relates joint torque to joint acceleration for the robot at joint +configuration q.

+

Trajectory operation

+

If q is a matrix (m,n), each row is interpretted as a joint state +vector, and the result is a 3d-matrix (nxnxk) where each plane +corresponds to the inertia for the corresponding row of q.

+
+
Parameters:
+

q (ndarray) – Joint coordinates

+
+
Returns:
+

The inertia matrix

+
+
Return type:
+

inertia

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> puma.inertia(puma.qz)
+array([[ 3.9611, -0.1627, -0.1389,  0.0016, -0.0004,  0.    ],
+       [-0.1627,  4.4566,  0.3727,  0.    ,  0.0019,  0.    ],
+       [-0.1389,  0.3727,  0.9387,  0.    ,  0.0019,  0.    ],
+       [ 0.0016,  0.    ,  0.    ,  0.1924,  0.    ,  0.    ],
+       [-0.0004,  0.0019,  0.0019,  0.    ,  0.1713,  0.    ],
+       [ 0.    ,  0.    ,  0.    ,  0.    ,  0.    ,  0.1941]])
+
+
+

Notes

+
    +
  • +
    The diagonal elements M[j,j] are the inertia seen by joint

    actuator j.

    +
    +
    +
  • +
  • +
    The off-diagonal elements M[j,k] are coupling inertias that

    relate acceleration on joint j to force/torque on +joint k.

    +
    +
    +
  • +
  • +
    The diagonal terms include the motor inertia reflected through

    the gear ratio.

    +
    +
    +
  • +
+
+

See also

+

cinertia()

+
+
+ +
+
+inertia_x(q=None, pinv=False, representation='rpy/xyz', Ji=None)
+

Operational space inertia matrix

+

robot.inertia_x(q) is the operational space (Cartesian) inertia +matrix which relates Cartesian force/torque to Cartesian +acceleration at the joint configuration q.

+
+\[\mathbf{M}_x = \mathbf{J}(q)^{-T} \mathbf{M}(q) \mathbf{J}(q)^{-1}\]
+

The transformation to operational space requires an analytical, rather +than geometric, Jacobian. analytical can be one of:

+
+
+ + + + + + + + + + + + + + + + + + + +

Value

Rotational representation

'rpy/xyz'

RPY angular rates in XYZ order (default)

'rpy/zyx'

RPY angular rates in XYZ order

'eul'

Euler angular rates in ZYZ order

'exp'

exponential coordinate rates

+
+

Trajectory operation

+

If q is a matrix (m,n), each row is interpretted as a joint state +vector, and the result is a 3d-matrix (m,n,n) where each plane +corresponds to the Cartesian inertia for the corresponding +row of q.

+
+
Parameters:
+
    +
  • q – Joint coordinates

  • +
  • pinv – use pseudo inverse rather than inverse (Default value = False)

  • +
  • analytical – the type of analytical Jacobian to use, default is +‘rpy/xyz’

  • +
  • representation – (Default value = “rpy/xyz”)

  • +
  • Ji – The inverse analytical Jacobian (base-frame)

  • +
+
+
Returns:
+

The operational space inertia matrix

+
+
Return type:
+

inertia_x

+
+
+

Examples

+
    ainv = _umath_linalg.inv(a, signature=signature, extobj=extobj)
+  File "/opt/hostedtoolcache/Python/3.7.17/x64/lib/python3.7/site-packages/numpy/linalg/linalg.py", line 88, in _raise_linalgerror_singular
+    raise LinAlgError("Singular matrix")
+numpy.linalg.LinAlgError: Singular matrix
+
+
+

Notes

+
    +
  • If the robot is not 6 DOF the pinv option is set True.

  • +
  • pinv() is around 5x slower than inv()

  • +
+
+

Warning

+

Assumes that the operational space has 6 DOF.

+
+
+

See also

+

inertia()

+
+
+ +
+
+isprismatic(j)
+

Check if joint is prismatic

+
+
Returns:
+

True if prismatic

+
+
Return type:
+

j

+
+
+

Examples

+

+
+
+
+

See also

+

Link.isprismatic(), prismaticjoints()

+
+
+ +
+
+isrevolute(j)
+

Check if joint is revolute

+
+
Returns:
+

True if revolute

+
+
Return type:
+

j

+
+
+

Examples

+

+
+
+
+

See also

+

Link.isrevolute(), revolutejoints()

+
+
+ +
+
+itorque(q, qdd)
+

Inertia torque

+

itorque(q, qdd) is the inertia force/torque vector (n) at +the specified joint configuration q (n) and acceleration qdd (n), and +n is the number of robot joints. +It is \(\mathbf{I}(q) \ddot{q}\).

+

Trajectory operation

+

If q and qdd are matrices (m,n), each row is interpretted as a +joint configuration, and the result is a matrix (m,n) where each row is +the corresponding joint torques.

+
+
Parameters:
+
    +
  • q – Joint coordinates

  • +
  • qdd – Joint acceleration

  • +
+
+
Returns:
+

The inertia torque vector

+
+
Return type:
+

itorque

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> puma.itorque(puma.qz, 0.5 * np.ones((6,)))
+array([1.8304, 2.3343, 0.5872, 0.0971, 0.0873, 0.0971])
+
+
+

Notes

+
    +
  • +
    If the robot model contains non-zero motor inertia then this

    will be included in the result.

    +
    +
    +
  • +
+
+

See also

+

inertia()

+
+
+ +
+
+jointdynamics(q, qd=None)
+

Transfer function of joint actuator

+

tf = jointdynamics(qd, q) calculates a vector of n +continuous-time transfer functions that represent the transfer +function 1/(Js+B) for each joint based on the dynamic parameters +of the robot and the configuration q (n). n is the number of robot +joints. The result is a list of tuples (J, B) for each joint.

+

tf = jointdynamics(q, qd) as above but include the linearized +effects of Coulomb friction when operating at joint velocity QD +(1xN).

+
+
Parameters:
+
    +
  • q – Joint coordinates

  • +
  • qd – Joint velocity

  • +
+
+
Returns:
+

transfer function denominators

+
+
Return type:
+

list of 2-tuples

+
+
+
+ +
+
+property keywords: List[str]
+
+ +
+ +
+ +
+
+linkcolormap(linkcolors='viridis')
+

Create a colormap for robot joints

+
    +
  • cm = robot.linkcolormap() is an n-element colormap that gives a +unique color for every link. The RGBA colors for link j are +cm(j).

  • +
  • cm = robot.linkcolormap(cmap) as above but cmap is the name +of a valid matplotlib colormap. The default, example above, is the +viridis colormap.

  • +
  • cm = robot.linkcolormap(list of colors) as above but a +colormap is created from a list of n color names given as strings, +tuples or hexstrings.

  • +
+
+
Parameters:
+

linkcolors (Union[List[Any], str]) – list of colors or colormap, defaults to “viridis”

+
+
Returns:
+

the color map

+
+
Return type:
+

color map

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> cm = robot.linkcolormap("inferno")
+>>> print(cm(range(6))) # cm(i) is 3rd color in colormap
+[[0.0015 0.0005 0.0139 1.    ]
+ [0.2582 0.0386 0.4065 1.    ]
+ [0.5783 0.148  0.4044 1.    ]
+ [0.865  0.3168 0.2261 1.    ]
+ [0.9876 0.6453 0.0399 1.    ]
+ [0.9884 0.9984 0.6449 1.    ]]
+>>> cm = robot.linkcolormap(
+...     ['red', 'g', (0,0.5,0), '#0f8040', 'yellow', 'cyan'])
+>>> print(cm(range(6)))
+[[1.     0.     0.     1.    ]
+ [0.     0.5    0.     1.    ]
+ [0.     0.5    0.     1.    ]
+ [0.0588 0.502  0.251  1.    ]
+ [1.     1.     0.     1.    ]
+ [0.     1.     1.     1.    ]]
+
+
+

Notes

+ +
+ +
+ +

Robot links

+
+
Returns:
+

A list of link objects

+
+
Return type:
+

links

+
+
+

Notes

+

It is probably more concise to index the robot object rather +than the list of links, ie. the following are equivalent: +- robot.links[i] +- robot[i]

+
+ +
+
+property manufacturer
+

Get/set robot manufacturer’s name

+
    +
  • robot.manufacturer is the robot manufacturer’s name

  • +
  • robot.manufacturer = ... checks and sets the manufacturer’s name

  • +
+
+
Returns:
+

robot manufacturer’s name

+
+
Return type:
+

manufacturer

+
+
+
+ +
+
+property n: int
+

Number of joints

+
+
Returns:
+

Number of joints

+
+
Return type:
+

n

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.n
+6
+
+
+
+

See also

+

nlinks(), nbranches()

+
+
+ +
+
+property name: str
+

Get/set robot name

+
    +
  • robot.name is the robot name

  • +
  • robot.name = ... checks and sets the robot name

  • +
+
+
Parameters:
+

name – the new robot name

+
+
Returns:
+

the current robot name

+
+
Return type:
+

name

+
+
+
+ +
+
+property nbranches: int
+

Number of branches

+

Number of branches in this robot. Computed as the number of links with +zero children

+
+
Returns:
+

number of branches in the robot’s kinematic tree

+
+
Return type:
+

nbranches

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.ETS.Panda()
+>>> robot.nbranches
+1
+
+
+
+

See also

+

n(), nlinks()

+
+
+ +
+ +

Number of links

+

The returned number is the total of both variable joints and +static links

+
+
Returns:
+

Number of links

+
+
Return type:
+

nlinks

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.nlinks
+6
+
+
+
+

See also

+

n(), nbranches()

+
+
+ +
+
+nofriction(coulomb=True, viscous=False)
+

Remove manipulator joint friction

+

nofriction() copies the robot and returns +a robot with the same link parameters except the Coulomb and/or viscous +friction parameter are set to zero.

+
+
Parameters:
+
    +
  • coulomb (bool) – set the Coulomb friction to 0

  • +
  • viscous (bool) – set the viscous friction to 0

  • +
+
+
Returns:
+

A copy of the robot with dynamic parameters perturbed

+
+
Return type:
+

robot

+
+
+
+

See also

+

Robot.friction(), Link.nofriction()

+
+
+ +
+
+pay(W, q=None, J=None, frame=1)
+

Generalised joint force/torque due to a payload wrench

+

tau = pay(W, J) Returns the generalised joint force/torques due to a +payload wrench W applied to the end-effector. Where the manipulator +Jacobian is J (6xn), and n is the number of robot joints.

+

tau = pay(W, q, frame) as above but the Jacobian is calculated at pose +q in the frame given by frame which is 0 for base frame, 1 for +end-effector frame.

+

Uses the formula tau = J’W, where W is a wrench vector applied at the +end effector, W = [Fx Fy Fz Mx My Mz]’.

+
+
Trajectory operation:

In the case q is nxm or J is 6xnxm then tau is nxm where each row +is the generalised force/torque at the pose given by corresponding +row of q.

+
+
+
+
Parameters:
+
    +
  • W (Union[ndarray, List[float], Tuple[float], Set[float]]) – A wrench vector applied at the end effector, +W = [Fx Fy Fz Mx My Mz]

  • +
  • q (Optional[ndarray]) – Joint coordinates

  • +
  • J (Optional[ndarray]) – The manipulator Jacobian (Optional, if not supplied will +use the q value).

  • +
  • frame (int) – The frame in which to torques are expressed in when J +is not supplied. 0 means base frame of the robot, 1 means end- +effector frame

  • +
+
+
Returns:
+

Joint forces/torques due to w

+
+
Return type:
+

t

+
+
+

Notes

+
    +
  • +
    Wrench vector and Jacobian must be from the same reference

    frame.

    +
    +
    +
  • +
  • Tool transforms are taken into consideration when frame=1.

  • +
  • +
    Must have a constant wrench - no trajectory support for this

    yet.

    +
    +
    +
  • +
+
+ +
+
+paycap(w, tauR, frame=1, q=None)
+

Static payload capacity of a robot

+

wmax, joint = paycap(q, w, f, tauR) returns the maximum permissible +payload wrench wmax (6) applied at the end-effector, and the index +of the joint (zero indexed) which hits its force/torque limit at that +wrench. q (n) is the manipulator pose, w the payload wrench +(6), f the wrench reference frame and tauR (nx2) is a matrix of +joint forces/torques (first col is maximum, second col minimum).

+

Trajectory operation:

+

In the case q is nxm then wmax is Mx6 and J is Mx1 where the rows are +the results at the pose given by corresponding row of q.

+
+
Parameters:
+
    +
  • w (ndarray) – The payload wrench

  • +
  • tauR (ndarray) – Joint torque matrix minimum and maximums

  • +
  • frame (int) – The frame in which to torques are expressed in when J +is not supplied. ‘base’ means base frame of the robot, ‘ee’ means +end-effector frame

  • +
  • q (Union[ndarray, List[float], Tuple[float], Set[float], None]) – Joint coordinates

  • +
+
+
Returns:
+

The maximum permissible payload wrench

+
+
Return type:
+

ndarray(6)

+
+
+

Notes

+
    +
  • Wrench vector and Jacobian must be from the same reference frame

  • +
  • Tool transforms are taken into consideration for frame=1.

  • +
+
+ +
+
+payload(m, p=array([0.0, 0.0, 0.0]))
+

Add a payload to the end-effector

+

payload(m, p) adds payload mass adds a payload with point mass m at +position p in the end-effector coordinate frame.

+

payload(m) adds payload mass adds a payload with point mass m at +in the end-effector coordinate frame.

+

payload(0) removes added payload.

+
+
Parameters:
+
    +
  • m (float) – mass (kg)

  • +
  • p – position in end-effector frame

  • +
+
+
+
+ +
+
+perturb(p=0.1)
+

Perturb robot parameters

+

rp = perturb(p) is a new robot object in which the dynamic parameters +(link mass and inertia) have been perturbed. The perturbation is +multiplicative so that values are multiplied by random numbers in the +interval (1-p) to (1+p). The name string of the perturbed robot is +prefixed by ‘P/’.

+

Useful for investigating the robustness of various model-based control +schemes. For example to vary parameters in the range +/- 10 percent +is: r2 = puma.perturb(0.1)

+
+
Parameters:
+

p – The percent (+/-) to be perturbed. Default 10%

+
+
Returns:
+

A copy of the robot with dynamic parameters perturbed

+
+
Return type:
+

DHRobot

+
+
+
+ +
+
+plot(q, backend=None, block=False, dt=0.05, limits=None, vellipse=False, fellipse=False, fig=None, movie=None, loop=False, **kwargs)
+

Graphical display and animation

+

robot.plot(q, 'pyplot') displays a graphical view of a robot +based on the kinematic model and the joint configuration q. +This is a stick figure polyline which joins the origins of the +link coordinate frames. The plot will autoscale with an aspect +ratio of 1.

+

If q (m,n) representing a joint-space trajectory it will create an +animation with a pause of dt seconds between each frame.

+
+
+q
+

The joint configuration of the robot.

+
+ +
+
+backend
+

The graphical backend to use, currently ‘swift’ +and ‘pyplot’ are implemented. Defaults to ‘swift’ of a Robot +and ‘pyplot` for a DHRobot

+
+ +
+
+block
+

Block operation of the code and keep the figure open

+
+ +
+
+dt
+

if q is a trajectory, this describes the delay in +seconds between frames

+
+ +
+
+limits
+

Custom view limits for the plot. If not supplied will +autoscale, [x1, x2, y1, y2, z1, z2] +(this option is for ‘pyplot’ only)

+
+ +
+
+vellipse
+

(Plot Option) Plot the velocity ellipse at the +end-effector (this option is for ‘pyplot’ only)

+
+ +
+
+fellipse
+

(Plot Option) Plot the force ellipse at the +end-effector (this option is for ‘pyplot’ only)

+
+ +
+
+fig
+

(Plot Option) The figure label to plot in (this option is for +‘pyplot’ only)

+
+ +
+
+movie
+

(Plot Option) The filename to save the movie to (this option is for +‘pyplot’ only)

+
+ +
+
+loop
+

(Plot Option) Loop the movie (this option is for +‘pyplot’ only)

+
+ +
+
+jointaxes
+

(Plot Option) Plot an arrow indicating the axes in +which the joint revolves around(revolute joint) or translates +along (prosmatic joint) (this option is for ‘pyplot’ only)

+
+ +
+
+eeframe
+

(Plot Option) Plot the end-effector coordinate frame +at the location of the end-effector. Uses three arrows, red, +green and blue to indicate the x, y, and z-axes. +(this option is for ‘pyplot’ only)

+
+ +
+
+shadow
+

(Plot Option) Plot a shadow of the robot in the x-y +plane. (this option is for ‘pyplot’ only)

+
+ +
+
+name
+

(Plot Option) Plot the name of the robot near its base +(this option is for ‘pyplot’ only)

+
+ +
+
Returns:
+

A reference to the environment object which controls the +figure

+
+
Return type:
+

env

+
+
+

Notes

+
    +
  • +
    By default this method will block until the figure is dismissed.

    To avoid this set block=False.

    +
    +
    +
  • +
  • +
    For PyPlot, the polyline joins the origins of the link frames,

    but for some Denavit-Hartenberg models those frames may not +actually be on the robot, ie. the lines to not neccessarily +represent the links of the robot.

    +
    +
    +
  • +
+
+

See also

+

teach()

+
+
+ +
+
+plot_ellipse(ellipse, block=True, limits=None, jointaxes=True, eeframe=True, shadow=True, name=True)
+

Plot the an ellipsoid

+

robot.plot_ellipse(ellipsoid) displays the ellipsoid.

+
+
+ellipse
+

the ellipsoid to plot

+
+ +
+
+block
+

Block operation of the code and keep the figure open

+
+ +
+
+limits
+

Custom view limits for the plot. If not supplied will +autoscale, [x1, x2, y1, y2, z1, z2]

+
+ +
+
+jointaxes
+

(Plot Option) Plot an arrow indicating the axes in +which the joint revolves around(revolute joint) or translates +along (prosmatic joint)

+
+ +
+
+eeframe
+

(Plot Option) Plot the end-effector coordinate frame +at the location of the end-effector. Uses three arrows, red, +green and blue to indicate the x, y, and z-axes.

+
+ +
+
+shadow
+

(Plot Option) Plot a shadow of the robot in the x-y +plane

+
+ +
+
+name
+

(Plot Option) Plot the name of the robot near its base

+
+ +
+
Returns:
+

A reference to the PyPlot object which controls the +matplotlib figure

+
+
Return type:
+

env

+
+
+
+ +
+
+plot_fellipse(q, block=True, fellipse=None, limits=None, opt='trans', centre=[0, 0, 0], jointaxes=True, eeframe=True, shadow=True, name=True)
+

Plot the force ellipsoid for manipulator

+

robot.plot_fellipse(q) displays the velocity ellipsoid for the +robot at pose q. The plot will autoscale with an aspect ratio +of 1.

+

robot.plot_fellipse(vellipse) specifies a custon ellipse to plot.

+
+
+q
+

The joint configuration of the robot

+
+ +
+
+block
+

Block operation of the code and keep the figure open

+
+ +
+
+fellipse
+

the vellocity ellipsoid to plot

+
+ +
+
+limits
+

Custom view limits for the plot. If not supplied will +autoscale, [x1, x2, y1, y2, z1, z2]

+
+ +
+
+opt
+

‘trans’ or ‘rot’ will plot either the translational or +rotational force ellipsoid

+
+ +
+
+centre
+

The coordinates to plot the fellipse [x, y, z] or “ee” +to plot at the end-effector location

+
+ +
+
+jointaxes
+

(Plot Option) Plot an arrow indicating the axes in +which the joint revolves around(revolute joint) or translates +along (prosmatic joint)

+
+ +
+
+eeframe
+

(Plot Option) Plot the end-effector coordinate frame +at the location of the end-effector. Uses three arrows, red, +green and blue to indicate the x, y, and z-axes.

+
+ +
+
+shadow
+

(Plot Option) Plot a shadow of the robot in the x-y +plane

+
+ +
+
+name
+

(Plot Option) Plot the name of the robot near its base

+
+ +
+
Raises:
+

ValueError – q or fellipse must be supplied

+
+
Returns:
+

A reference to the PyPlot object which controls the +matplotlib figure

+
+
Return type:
+

env

+
+
+

Notes

+
    +
  • +
    By default the ellipsoid related to translational motion is

    drawn. Use opt='rot' to draw the rotational velocity +ellipsoid.

    +
    +
    +
  • +
  • +
    By default the ellipsoid is drawn at the origin. The option

    centre allows its origin to set to set to the specified +3-vector, or the string “ee” ensures it is drawn at the +end-effector position.

    +
    +
    +
  • +
+
+ +
+
+plot_vellipse(q, block=True, vellipse=None, limits=None, opt='trans', centre=[0, 0, 0], jointaxes=True, eeframe=True, shadow=True, name=True)
+

Plot the velocity ellipsoid for manipulator

+

robot.plot_vellipse(q) displays the velocity ellipsoid for the +robot at pose q. The plot will autoscale with an aspect ratio +of 1.

+

robot.plot_vellipse(vellipse) specifies a custon ellipse to plot.

+
+
block

Block operation of the code and keep the figure open

+
+
q

The joint configuration of the robot

+
+
vellipse

the vellocity ellipsoid to plot

+
+
limits

Custom view limits for the plot. If not supplied will +autoscale, [x1, x2, y1, y2, z1, z2]

+
+
opt

‘trans’ or ‘rot’ will plot either the translational or +rotational velocity ellipsoid

+
+
centre

The coordinates to plot the vellipse [x, y, z] or “ee” +to plot at the end-effector location

+
+
jointaxes

(Plot Option) Plot an arrow indicating the axes in +which the joint revolves around(revolute joint) or translates +along (prosmatic joint)

+
+
eeframe

(Plot Option) Plot the end-effector coordinate frame +at the location of the end-effector. Uses three arrows, red, +green and blue to indicate the x, y, and z-axes.

+
+
shadow

(Plot Option) Plot a shadow of the robot in the x-y +plane

+
+
name

(Plot Option) Plot the name of the robot near its base

+
+
+
+
Raises:
+

ValueError – q or fellipse must be supplied

+
+
Returns:
+

A reference to the PyPlot object which controls the +matplotlib figure

+
+
Return type:
+

env

+
+
+

Notes

+
    +
  • +
    By default the ellipsoid related to translational motion is

    drawn. Use opt='rot' to draw the rotational velocity +ellipsoid.

    +
    +
    +
  • +
  • +
    By default the ellipsoid is drawn at the origin. The option

    centre allows its origin to set to set to the specified +3-vector, or the string “ee” ensures it is drawn at the +end-effector position.

    +
    +
    +
  • +
+
+ +
+
+property prismaticjoints: List[bool]
+

Revolute joints as bool array

+
+
Returns:
+

array of joint type, True if prismatic

+
+
Return type:
+

prismaticjoints

+
+
+

Examples

+

+
+
+

Notes

+

Fixed joints, that maintain a constant link relative pose, +are not included.

+
+

See also

+

Link.isprismatic(), revolutejoints()

+
+
+ +
+
+property q: ndarray
+

Get/set robot joint configuration

+
    +
  • robot.q is the robot joint configuration

  • +
  • robot.q = ... checks and sets the joint configuration

  • +
+
+
Parameters:
+

q – the new robot joint configuration

+
+
Returns:
+

robot joint configuration

+
+
Return type:
+

q

+
+
+
+ +
+
+property qd: ndarray
+

Get/set robot joint velocity

+
    +
  • robot.qd is the robot joint velocity

  • +
  • robot.qd = ... checks and sets the joint velocity

  • +
+
+
Returns:
+

robot joint velocity

+
+
Return type:
+

qd

+
+
+
+ +
+
+property qdd: ndarray
+

Get/set robot joint acceleration

+
    +
  • robot.qdd is the robot joint acceleration

  • +
  • robot.qdd = ... checks and sets the robot joint acceleration

  • +
+
+
Returns:
+

robot joint acceleration

+
+
Return type:
+

qdd

+
+
+
+ +
+
+property qlim: ndarray
+

Joint limits

+

Limits are extracted from the link objects. If joints limits are +not set for:

+
    +
  • a revolute joint [-𝜋. 𝜋] is returned

  • +
  • a prismatic joint an exception is raised

  • +
+
+
+qlim
+

An array of joints limits (2, n)

+
+ +
+
Raises:
+

ValueError – unset limits for a prismatic joint

+
+
Returns:
+

Array of joint limit values

+
+
Return type:
+

qlim

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> robot = rtb.models.DH.Puma560()
+>>> robot.qlim
+array([[-2.7925, -1.9199, -2.3562, -4.6426, -1.7453, -4.6426],
+       [ 2.7925,  1.9199,  2.3562,  4.6426,  1.7453,  4.6426]])
+
+
+
+ +
+
+random_q()
+

Return a random joint configuration

+

The value for each joint is uniform randomly distributed between the +limits set for the robot.

+
+

Note

+

The joint limit for all joints must be set.

+
+
+
Returns:
+

Random joint configuration :rtype: ndarray(n)

+
+
Return type:
+

q

+
+
+
+

See also

+

Robot.qlim(), Link.qlim()

+
+
+ +
+
+property revolutejoints: List[bool]
+

Revolute joints as bool array

+
+
Returns:
+

array of joint type, True if revolute

+
+
Return type:
+

revolutejoints

+
+
+

Examples

+

+
+
+

Notes

+

Fixed joints, that maintain a constant link relative pose, +are not included.

+
+

See also

+

Link.isrevolute(), prismaticjoints()

+
+
+ +
+
+property scene_children: List[SceneNode]
+

Returns the child nodes of this object

+
+ +
+
+property scene_parent: Type[SceneNode]
+

Returns the parent node of this object

+
+ +
+
+segments()
+

Segments of branched robot

+

For a single-chain robot with structure:

+
L1 - L2 - L3
+
+
+

the return is [[None, L1, L2, L3]]

+

For a robot with structure:

+
L1 - L2 +-  L3 - L4
+        +- L5 - L6
+
+
+

the return is [[None, L1, L2], [L2, L3, L4], [L2, L5, L6]]

+
+
Returns:
+

Segment list

+
+
Return type:
+

segments

+
+
+

Notes

+
    +
  • the length of the list is the number of segments in the robot

  • +
  • +
    the first segment always starts with None which represents

    the base transform (since there is no base link)

    +
    +
    +
  • +
  • +
    the last link of one segment is also the first link of subsequent

    segments

    +
    +
    +
  • +
+
+ +
+
+showgraph(display_graph=True, **kwargs)
+

Display a link transform graph in browser

+

robot.showgraph() displays a graph of the robot’s link frames +and the ETS between them. It uses GraphViz dot.

+
+
The nodes are:
    +
  • Base is shown as a grey square. This is the world frame origin, +but can be changed using the base attribute of the robot.

  • +
  • Link frames are indicated by circles

  • +
  • ETS transforms are indicated by rounded boxes

  • +
+
+
The edges are:
    +
  • an arrow if jtype is False or the joint is fixed

  • +
  • an arrow with a round head if jtype is True and the joint is +revolute

  • +
  • an arrow with a box head if jtype is True and the joint is +prismatic

  • +
+
+
+

Edge labels or nodes in blue have a fixed transformation to the +preceding link.

+
+
Parameters:
+
    +
  • display_graph (bool) – Open the graph in a browser if True. Otherwise will return the +file path

  • +
  • etsbox – Put the link ETS in a box, otherwise an edge label

  • +
  • jtype – Arrowhead to node indicates revolute or prismatic type

  • +
  • static – Show static joints in blue and bold

  • +
+
+
Return type:
+

Optional[str]

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.URDF.Panda()
+>>> panda.showgraph()
+
+
+_images/panda-graph.svg +
+

See also

+

dotfile()

+
+
+ +
+
+property structure: str
+

Return the joint structure string

+

A string with one letter per joint: R for a revolute +joint, and P for a prismatic joint.

+
+
Returns:
+

joint configuration string

+
+
Return type:
+

structure

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> puma.structure
+'RRRRRR'
+>>> stanford = rtb.models.DH.Stanford()
+>>> stanford.structure
+'RRPRRR'
+
+
+

Notes

+

Fixed joints, that maintain a constant link relative pose, +are not included. +len(self.structure) == self.n.

+
+ +
+
+property symbolic: bool
+
+ +
+
+teach(q, block=True, limits=None, vellipse=False, fellipse=False, backend=None)
+

Graphical teach pendant

+

robot.teach(q) creates a matplotlib plot which allows the user to +“drive” a graphical robot using a graphical slider panel. The robot’s +inital joint configuration is q. The plot will autoscale with an +aspect ratio of 1.

+

robot.teach() as above except the robot’s stored value of q +is used.

+
+
q

The joint configuration of the robot (Optional, +if not supplied will use the stored q values).

+
+
block

Block operation of the code and keep the figure open

+
+
limits

Custom view limits for the plot. If not supplied will +autoscale, [x1, x2, y1, y2, z1, z2]

+
+
vellipse

(Plot Option) Plot the velocity ellipse at the +end-effector (this option is for ‘pyplot’ only)

+
+
fellipse

(Plot Option) Plot the force ellipse at the +end-effector (this option is for ‘pyplot’ only)

+
+
+
+
Returns:
+

A reference to the PyPlot object which controls the +matplotlib figure

+
+
Return type:
+

env

+
+
+

Notes

+
    +
  • +
    Program execution is blocked until the teach window is

    dismissed. If block=False the method is non-blocking but +you need to poll the window manager to ensure that the window +remains responsive.

    +
    +
    +
  • +
  • +
    The slider limits are derived from the joint limit properties.

    If not set then: +- For revolute joints they are assumed to be [-pi, +pi] +- For prismatic joint they are assumed unknown and an error

    +
    +

    occurs.

    +
    +
    +
    +
  • +
+
+ +
+
+todegrees(q)
+

Convert joint angles to degrees

+
+
Parameters:
+

q – The joint configuration of the robot

+
+
Return type:
+

ndarray

+
+
Returns:
+

    +
  • q – a vector of joint coordinates in degrees and metres

  • +
  • robot.todegrees(q) converts joint coordinates q to degrees

  • +
  • taking into account whether elements of q correspond to revolute

  • +
  • or prismatic joints, ie. prismatic joint values are not converted.

  • +
  • If q is a matrix, with one column per joint, the conversion is

  • +
  • performed columnwise.

  • +
+

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> from math import pi
+>>> stanford = rtb.models.DH.Stanford()
+>>> stanford.todegrees([pi/4, pi/8, 2, -pi/4, pi/6, pi/3])
+array([ 45. ,  22.5,   2. , -45. ,  30. ,  60. ])
+
+
+
+ +
+
+property tool: SE3
+

Get/set robot tool transform

+
    +
  • robot.tool is the robot tool transform as an SE3 object

  • +
  • robot._tool is the robot tool transform as a numpy array

  • +
  • robot.tool = ... checks and sets the robot tool transform

  • +
+
+
Parameters:
+

tool – the new robot tool transform (as an SE(3))

+
+
Returns:
+

robot tool transform

+
+
Return type:
+

tool

+
+
+
+ +
+
+toradians(q)
+

Convert joint angles to radians

+

robot.toradians(q) converts joint coordinates q to radians +taking into account whether elements of q correspond to revolute +or prismatic joints, ie. prismatic joint values are not converted.

+

If q is a matrix, with one column per joint, the conversion is +performed columnwise.

+
+
Parameters:
+

q – The joint configuration of the robot

+
+
Returns:
+

a vector of joint coordinates in radians and metres

+
+
Return type:
+

q

+
+
+

Examples

+
>>> import roboticstoolbox as rtb
+>>> stanford = rtb.models.DH.Stanford()
+>>> stanford.toradians([10, 20, 2, 30, 40, 50])
+array([0.1745, 0.3491, 2.    , 0.5236, 0.6981, 0.8727])
+
+
+
+ +
+
+property urdf_filepath: str
+
+ +
+
+property urdf_string: str
+
+ +
+
+vellipse(q, opt='trans', unit='rad', centre=[0, 0, 0], scale=0.1)
+

Create a velocity ellipsoid object for plotting with PyPlot

+

robot.vellipse(q) creates a force ellipsoid for the robot at +pose q. The ellipsoid is centered at the origin.

+
+
+q
+

The joint configuration of the robot.

+
+ +
+
+opt
+

‘trans’ or ‘rot’ will plot either the translational or +rotational force ellipsoid

+
+ +
+
+unit
+

‘rad’ or ‘deg’ will plot the ellipsoid in radians or +degrees

+
+ +
+
+centre
+

The centre of the ellipsoid, either ‘ee’ for the end-effector +or a 3-vector [x, y, z] in the world frame

+
+ +
+
+scale
+

The scale factor for the ellipsoid

+
+ +
+
Returns:
+

An EllipsePlot object

+
+
Return type:
+

env

+
+
+

Notes

+
    +
  • +
    By default the ellipsoid related to translational motion is

    drawn. Use opt='rot' to draw the rotational velocity +ellipsoid.

    +
    +
    +
  • +
  • +
    By default the ellipsoid is drawn at the origin. The option

    centre allows its origin to set to set to the specified +3-vector, or the string “ee” ensures it is drawn at the +end-effector position.

    +
    +
    +
  • +
+
+ +
+
+property base: SE2
+

Get/set robot base transform (Robot superclass)

+

robot.base is the robot base transform

+
+
Returns:
+

    +
  • base – robot tool transform

  • +
    • +
    • robot.base = ... checks and sets the robot base transform

    • +
    +
  • +
+

+
+
+

Notes

+
    +
  • +
    The private attribute _base will be None in the case of

    no base transform, but this property will return SE3() which +is an identity matrix.

    +
    +
    +
  • +
+
+ +
+
+jacob0(q, start=None, end=None)[source]
+
+ +
+
+jacobe(q, start=None, end=None)[source]
+
+ +
+
+fkine(q, end=None, start=None)[source]
+
+ +
+
+property reach: float
+

Reach of the robot

+

A conservative estimate of the reach of the robot. It is computed as +the sum of the translational ETs that define the link transform.

+
+

Note

+

Computed on the first access. If kinematic parameters +subsequently change this will not be reflected.

+
+
+
Returns:
+

Maximum reach of the robot

+
+
Return type:
+

reach

+
+
+

Notes

+
    +
  • Probably an overestimate of reach

  • +
  • Used by numerical inverse kinematics to scale translational +error.

  • +
  • For a prismatic joint, uses qlim if it is set

  • +
+
+ +
+
+fkine_all(q)[source]
+

Compute the pose of every link frame

+

T = robot.fkine_all(q) is an SE3 instance with robot.nlinks + +1 values:

+
    +
  • T[0] is the base transform

  • +
  • T[i] is the pose of link whose number is i

  • +
+
+
Parameters:
+

q (Union[ndarray, List[float], Tuple[float], Set[float]]) – The joint configuration

+
+
Returns:
+

Pose of all links

+
+
Return type:
+

fkine_all

+
+
+

References

+
    +
  • J. Haviland, and P. Corke. “Manipulator Differential Kinematics Part I: +Kinematics, Velocity, and Applications.” arXiv preprint arXiv:2207.01796 (2022).

  • +
+
+ +
+ +
+
+

Kinematic cache

+
+ +
+ + +
+
+
+ +
+ +
+

© Copyright 2023, Jesse Haviland and Peter Corke. + Last updated on 11-May-2024. +

+
+ + + +
+
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/arm_trajectory-1.hires.png b/arm_trajectory-1.hires.png new file mode 100644 index 00000000..8b6f51ee Binary files /dev/null and b/arm_trajectory-1.hires.png differ diff --git a/arm_trajectory-1.pdf b/arm_trajectory-1.pdf new file mode 100644 index 00000000..dc1289ec Binary files /dev/null and b/arm_trajectory-1.pdf differ diff --git a/arm_trajectory-1.png b/arm_trajectory-1.png new file mode 100644 index 00000000..70ef478e Binary files /dev/null and b/arm_trajectory-1.png differ diff --git a/arm_trajectory-1.py b/arm_trajectory-1.py new file mode 100644 index 00000000..1ea3d9b8 --- /dev/null +++ b/arm_trajectory-1.py @@ -0,0 +1,3 @@ +from roboticstoolbox import quintic +tg = quintic(1, 2, 10) +tg.plot() \ No newline at end of file diff --git a/arm_trajectory-2.hires.png b/arm_trajectory-2.hires.png new file mode 100644 index 00000000..ae92a92a Binary files /dev/null and b/arm_trajectory-2.hires.png differ diff --git a/arm_trajectory-2.pdf b/arm_trajectory-2.pdf new file mode 100644 index 00000000..347011af Binary files /dev/null and b/arm_trajectory-2.pdf differ diff --git a/arm_trajectory-2.png b/arm_trajectory-2.png new file mode 100644 index 00000000..efe9c890 Binary files /dev/null and b/arm_trajectory-2.png differ diff --git a/arm_trajectory-2.py b/arm_trajectory-2.py new file mode 100644 index 00000000..0f51443a --- /dev/null +++ b/arm_trajectory-2.py @@ -0,0 +1,3 @@ +from roboticstoolbox import trapezoidal +tg = trapezoidal(1, 2, 10) +tg.plot() \ No newline at end of file diff --git a/arm_trajectory.html b/arm_trajectory.html new file mode 100644 index 00000000..984d1d6b --- /dev/null +++ b/arm_trajectory.html @@ -0,0 +1,868 @@ + + + + + + + + + + + Trajectories — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

Trajectories

+
+
+class roboticstoolbox.tools.trajectory.Trajectory(name, t, s, sd=None, sdd=None, istime=False)[source]
+

Bases: object

+

A container class for trajectory data.

+
+
+__init__(name, t, s, sd=None, sdd=None, istime=False)[source]
+

Construct a new trajectory instance

+
+
Parameters:
+
    +
  • name (str) – name of the function that created the trajectory

  • +
  • t (ndarray(m)) – independent variable, eg. time or step

  • +
  • s (ndarray(m) or ndarray(m,n)) – position

  • +
  • sd (ndarray(m) or ndarray(m,n)) – velocity

  • +
  • sdd (ndarray(m) or ndarray(m,n)) – acceleration

  • +
  • istime (float) – t is time, otherwise step number

  • +
  • tblend – blend duration (trapezoidal only)

  • +
+
+
+

The object has attributes:

+
    +
  • t the independent variable

  • +
  • s the position

  • +
  • sd the velocity

  • +
  • sdd the acceleration

  • +
+

If t is time, ie. istime is True, then the units of sd and +sdd are \(s^{-1}\) and \(s^{-2}\) respectively, otherwise +with respect to t.

+
+

Note

+

Data is stored with timesteps as rows and axes as columns.

+
+
+
References:
+
    +
  • Robotics, Vision & Control in Python, 3e, P. Corke, Springer 2023, Chap 3.

  • +
+
+
+
+ +
+
+__str__()[source]
+

Return str(self).

+
+ +
+
+__repr__()[source]
+

Return repr(self).

+
+ +
+
+__len__()[source]
+

Length of trajectory

+
+
Returns:
+

number of steps in the trajectory

+
+
Return type:
+

int

+
+
+
+ +
+
+property q
+

Position trajectory

+
+
Returns:
+

trajectory with one row per timestep, one column per axis

+
+
Return type:
+

ndarray(n,m)

+
+
+
+

Note

+

This is a synonym for .s, for compatibility with other +applications.

+
+
+ +
+
+property qd
+

Velocity trajectory

+
+
Returns:
+

trajectory velocity with one row per timestep, one column per axis

+
+
Return type:
+

ndarray(n,m)

+
+
+
+

Note

+

This is a synonym for .sd, for compatibility with other +applications.

+
+
+ +
+
+property qdd
+

Acceleration trajectory

+
+
Returns:
+

trajectory acceleration with one row per timestep, one column per axis

+
+
Return type:
+

ndarray(n,m)

+
+
+
+

Note

+

This is a synonym for .sdd, for compatibility with other +applications.

+
+
+ +
+
+property naxes
+

Number of axes in the trajectory

+
+
Returns:
+

number of axes or dimensions

+
+
Return type:
+

int

+
+
+
+ +
+
+plot(block=False, plotargs=None, textargs=None)[source]
+

Plot trajectory

+
+
Parameters:
+

block (bool) – wait till plot is dismissed

+
+
+

Plot the position, velocity and acceleration data. The format of the +plot depends on the function that created it.

+
    +
  • quintic and trapezoidal show the individual points with markers

  • +
  • trapezoidal color code the different motion phases

  • +
  • jtraj general m-axis trajectory, show legend

  • +
+
+
Seealso:
+

quintic(), trapezoidal()

+
+
+
+ +
+
+qplot(**kwargs)[source]
+

Plot multi-axis trajectory

+
+
Parameters:
+

**kwargs

optional arguments passed to qplot

+

+
+
+

Plots a multi-axis trajectory, held within the object, as position against time.

+
+
Seealso:
+

qplot()

+
+
+
+ +
+ +
+
+roboticstoolbox.tools.trajectory.quintic(q0, qf, t, qd0=0, qdf=0)[source]
+

Generate scalar polynomial trajectory

+
+
Parameters:
+
    +
  • q0 (float) – initial value

  • +
  • qf (float) – final value

  • +
  • t (int or array_like(m)) – time

  • +
  • qd0 – initial velocity, optional

  • +
  • qdf – final velocity, optional

  • +
+
+
Returns:
+

trajectory

+
+
Return type:
+

Trajectory instance

+
+
+
    +
  • tg = quintic(q0, q1, m) is a scalar trajectory (Mx1) that varies +smoothly from q0 to qf using a quintic polynomial. The initial +and final velocity and acceleration are zero. m is an integer scalar, +indicating the total number of timesteps and

    +
    +
      +
    • Velocity is in units of distance per trajectory step, not per +second.

    • +
    • Acceleration is in units of distance per trajectory step squared, +not per second squared.

    • +
    +
    +
  • +
  • tg = quintic(q0, q1, t) as above but t is a uniformly-spaced time +vector

    +
    +
      +
    • Velocity is in units of distance per second.

    • +
    • Acceleration is in units of distance per second squared.

    • +
    +
    +
  • +
+

The return value is an object that contains position, velocity and +acceleration data.

+

Example:

+
>>> from roboticstoolbox import quintic
+>>> tg = quintic(1, 2, 10)
+>>> tg
+Trajectory created by quintic: 10 time steps x 1 axes
+>>> len(tg)
+10
+>>> tg.q
+array([1.    , 1.0115, 1.0764, 1.2099, 1.3967, 1.6033, 1.7901, 1.9236,
+       1.9885, 2.    ])
+>>> tg.plot()
+
+
+

(Source code, png, hires.png, pdf)

+
+_images/arm_trajectory-1.png +
+
+

Note

+

The time vector T is assumed to be monotonically increasing, and +time scaling is based on the first and last element.

+
+
+
References:
+
    +
  • Robotics, Vision & Control in Python, 3e, P. Corke, Springer 2023, Chap 3.

  • +
+
+
Seealso:
+

trapezoidal(), mtraj().

+
+
+
+ +
+
+roboticstoolbox.tools.trajectory.quintic_func(q0, qf, T, qd0=0, qdf=0)[source]
+

Quintic scalar polynomial as a function

+
+
Parameters:
+
    +
  • q0 (float, optional) – initial value

  • +
  • qf (float) – final value

  • +
  • T (float) – trajectory time

  • +
  • qd0 – initial velocity, defaults to 0

  • +
  • qdf – final velocity, defaults to 0

  • +
+
+
Returns:
+

polynomial function \(f: t \mapsto (q(t), \dot{q}(t), \ddot{q}(t))\)

+
+
Return type:
+

callable

+
+
+

Returns a function which computes the specific quintic polynomial, and its +derivatives, as described by the parameters.

+

Example:

+
>>> from roboticstoolbox import quintic_func
+>>> f = quintic_func(1, 2, 5)
+>>> f(0)
+(1.000000000000001, 2.782740400955608e-16, -8.565197162635485e-18)
+>>> f(5)
+(2.000000000000005, -3.233998897671231e-15, -7.225014857226153e-15)
+>>> f(2.5)
+(1.5000000000000029, 0.37500000000000167, 8.241020713062319e-16)
+
+
+
+
Seealso:
+

quintic() trapezoidal_func()

+
+
+
+ +
+
+roboticstoolbox.tools.trajectory.lspb(*args, **kwargs)[source]
+
+

Warning

+

Deprecated, use trapezoidal instead.

+
+
+ +
+
+roboticstoolbox.tools.trajectory.trapezoidal(q0, qf, t, V=None)[source]
+

Scalar trapezoidal trajectory

+
+
Parameters:
+
    +
  • q0 (float) – initial value

  • +
  • qf (float) – final value

  • +
  • t (float or array_like) – time

  • +
  • V (float) – velocity of linear segment, optional

  • +
+
+
Returns:
+

trajectory

+
+
Return type:
+

Trajectory instance

+
+
+

Computes a trapezoidal trajectory, which has a linear motion segment with +parabolic blends.

+
    +
  • tg = trapezoidal(q0, qf, t) is a scalar trajectory (Mx1) that varies +smoothly from q0 to qf in M steps using a constant velocity +segment and parabolic blends. Time t can be either:

    +
    +
      +
    • an integer scalar, indicating the total number of timesteps

      +
      +
        +
      • Velocity is in units of distance per trajectory step, not per +second.

      • +
      • Acceleration is in units of distance per trajectory step squared, +not per second squared.

      • +
      +
      +
    • +
    • an array_like, containing the time steps.

      +
      +
        +
      • Results are scaled to units of time.

      • +
      +
      +
    • +
    +
    +
  • +
  • tg = trapezoidal(q0, q1, t, V) as above but specifies the velocity of the +linear segment which is normally computed automatically.

  • +
+

The return value is an object that contains position, velocity and +acceleration data.

+

Example:

+
>>> from roboticstoolbox import trapezoidal
+>>> tg = trapezoidal(1, 2, 10)
+>>> tg
+Trajectory created by trapezoidal: 10 time steps x 1 axes
+>>> len(tg)
+10
+>>> tg.q
+array([1.    , 1.0278, 1.1111, 1.25  , 1.4167, 1.5833, 1.75  , 1.8889,
+       1.9722, 2.    ])
+
+
+

(Source code, png, hires.png, pdf)

+
+_images/arm_trajectory-2.png +
+
+

Note

+
    +
  • For some values of V no solution is possible and an error is flagged.

  • +
  • The time vector, if given, is assumed to be monotonically increasing, +and time scaling is based on the first and last element.

  • +
  • tg has an extra attribute xblend which is the blend duration.

  • +
+
+
+
References:
+
    +
  • Robotics, Vision & Control in Python, 3e, P. Corke, Springer 2023, Chap 3.

  • +
+
+
Seealso:
+

quintic(), trapezoidal_func() mtraj().

+
+
+
+ +
+
+roboticstoolbox.tools.trajectory.trapezoidal_func(q0, qf, T, V=None)[source]
+

Trapezoidal scalar profile as a function

+
+
Parameters:
+
    +
  • q0 (float) – initial value

  • +
  • qf (float) – final value

  • +
  • T (float) – maximum time

  • +
  • V (float, optional) – velocity of linear segment

  • +
+
+
Returns:
+

trapezoidal profile function \(f: t \mapsto (q(t), \dot{q}(t), \ddot{q}(t))\)

+
+
Return type:
+

callable

+
+
+

Returns a function which computes the specific trapezoidal profile, and its +derivatives, as described by the parameters.

+

Example:

+
>>> from roboticstoolbox import trapezoidal_func
+>>> f = trapezoidal_func(1, 2, 5)
+>>> f(0)
+(array([1.]), array([0.]), array([0.18]))
+>>> f(5)
+(array([2.]), array([0.]), array([-0.18]))
+>>> f(2.5)
+(array([1.5]), array([0.3]), array([0]))
+
+
+
+ +
+
+roboticstoolbox.tools.trajectory.mtraj(tfunc, q0, qf, t)[source]
+

Multi-axis trajectory

+
+
Parameters:
+
    +
  • tfunc (callable) – a 1D trajectory function, eg. quintic() or trapezoidal()

  • +
  • q0 (ndarray(m)) – initial configuration

  • +
  • qf (ndarray(m)) – final configuration

  • +
  • t (array_like or int) – time vector or number of steps

  • +
+
+
Raises:
+
+
+
Returns:
+

trajectory

+
+
Return type:
+

Trajectory instance

+
+
+
    +
  • tg = mtraj(func, q0, qf, n) is a multi-axis trajectory varying +from configuration q0 (M) to qf (M) according to the scalar trajectory +function tfunc in n steps.

  • +
  • tg = mtraj(func, q0, qf, t) as above but t is a uniformly-spaced time +vector

  • +
+

The scalar trajectory function is applied to each axis:

+
tg = tfunc(s0, sF, n)
+
+
+

and possible values of tfunc include trapezoidal for a trapezoidal trajectory, or +quintic for a polynomial trajectory.

+

The return value is an object that contains position, velocity and +acceleration data.

+
+

Note

+

The time vector, if given, is assumed to be monotonically increasing, and +time scaling is based on the first and last element.

+
+
+
References:
+
    +
  • Robotics, Vision & Control in Python, 3e, P. Corke, Springer 2023, Chap 3.

  • +
+
+
Seealso:
+

quintic(), trapezoidal()

+
+
+
+ +
+
+roboticstoolbox.tools.trajectory.jtraj(q0, qf, t, qd0=None, qd1=None)[source]
+

Compute a joint-space trajectory

+
+
Parameters:
+
    +
  • q0 (array_like(n)) – initial joint coordinate

  • +
  • qf (array_like(n)) – final joint coordinate

  • +
  • t (array_like or int) – time vector or number of steps

  • +
  • qd0 (array_like(n), optional) – initial velocity, defaults to zero

  • +
  • qd1 (array_like(n), optional) – final velocity, defaults to zero

  • +
+
+
Returns:
+

trajectory

+
+
Return type:
+

Trajectory instance

+
+
+
    +
  • tg = jtraj(q0, qf, N) is a joint space trajectory where the joint +coordinates vary from q0 (M) to qf (M). A quintic (5th order) +polynomial is used with default zero boundary conditions for velocity and +acceleration. Time is assumed to vary from 0 to 1 in N steps.

  • +
  • tg = jtraj(q0, qf, t) as above but t is a uniformly-spaced time +vector

  • +
+

The return value is an object that contains position, velocity and +acceleration data.

+
+

Note

+

The time vector, if given, scales the velocity and acceleration outputs +assuming that the time vector starts at zero and increases linearly.

+
+
+
References:
+
    +
  • Robotics, Vision & Control in Python, 3e, P. Corke, Springer 2023, Chap 3.

  • +
+
+
Seealso:
+

ctraj(), qplot(), jtraj()

+
+
+
+ +
+
+roboticstoolbox.tools.trajectory.ctraj(T0, T1, t=None, s=None)[source]
+

Cartesian trajectory between two poses

+
+
Parameters:
+
    +
  • T0 (SE3) – initial pose

  • +
  • T1 (SE3) – final pose

  • +
  • t (int or ndarray(n)) – number of samples or time vector

  • +
  • s (ndarray(s)) – array of distance along the path, in the interval [0, 1]

  • +
+
+
Return T0:
+

smooth path from T0 to T1

+
+
Return type:
+

SE3

+
+
+

ctraj(T0, T1, n) is a Cartesian trajectory from SE3 pose T0 to +T1 with n points that follow a trapezoidal velocity profile along +the path. The Cartesian trajectory is an SE3 instance containing n +values.

+

ctraj(T0, T1, t) as above but the trajectory is sampled at +the points in the array t.

+

ctraj(T0, T1, s=s) as above but the elements of s specify the +fractional distance along the path, and these values are in the +range [0 1]. The i’th point corresponds to a distance s[i] along +the path.

+

Examples:

+
>>> tg = ctraj(SE3.Rand(), SE3.Rand(), 20)
+>>> len(tg)
+20
+
+
+
+

Note

+
    +
  • In the second case s could be generated by a scalar trajectory

  • +
+

generator such as quintic or trapezoidal (default). +- Orientation interpolation is performed using unit-quaternion +interpolation.

+
+
+
References:
+
    +
  • Robotics, Vision & Control in Python, 3e, P. Corke, Springer 2023, Chap 3.

  • +
+
+
Seealso:
+

trapezoidal(), +interp()

+
+
+
+ +
+
+roboticstoolbox.tools.trajectory.cmstraj()[source]
+
+ +
+
+roboticstoolbox.tools.trajectory.mstraj(viapoints, dt, tacc, qdmax=None, tsegment=None, q0=None, qd0=None, qdf=None, verbose=False)[source]
+

Multi-segment multi-axis trajectory

+
+
Parameters:
+
    +
  • viapoints (ndarray(m,n)) – A set of viapoints, one per row

  • +
  • dt (float (seconds)) – time step

  • +
  • tacc (float) – acceleration time (seconds)

  • +
  • qdmax (array_like(n) or float, optional) – maximum speed, defaults to None

  • +
  • tsegment (array_like, optional) – maximum time of each motion segment (seconds), defaults +to None

  • +
  • q0 (array_like(n), optional) – initial coordinates, defaults to first row of viapoints

  • +
  • qd0 (array_like(n), optional) – inital velocity, defaults to zero

  • +
  • qdf (array_like(n), optional) – final velocity, defaults to zero

  • +
  • verbose (bool, optional) – print debug information, defaults to False

  • +
+
+
Returns:
+

trajectory

+
+
Return type:
+

Trajectory instance

+
+
+

Computes a trajectory for N axes moving smoothly through a set of +viapoints. +The motion comprises M segments:

+
    +
  • The initial coordinates are the first row of viapoints or q0 if +provided.

  • +
  • The final coordinates are the last row of viapoints

  • +
  • Each segment is linear motion and polynomial blends connect the +viapoints.

  • +
  • All joints arrive at each via point at the same time, ie. the motion is +coordinated across axes

  • +
+

The time of the segments can be specified in two different ways:

+
    +
  1. In terms of segment time where tsegment is an array of segment times +which is the number of via points minus one:

    +
    traj = mstraj(viapoints, dt, tacc, tsegment=TS)
    +
    +
    +
  2. +
  3. Governed by the speed of the slowest axis for the segment. The axis +speed is a scalar (all axes have the same speed) or an N-vector of speed +per axis:

    +
    traj = mstraj(viapoints, dt, tacc, qdmax=SPEED)
    +
    +
    +
  4. +
+

The return value is a namedtuple (named mstraj) with elements:

+
+
    +
  • t the time coordinate as a numpy ndarray, shape=(K,)

  • +
  • q the axis values as a numpy ndarray, shape=(K,N)

  • +
  • arrive a list of arrival times for each segment

  • +
  • info a list of named tuples, one per segment that describe the +slowest axis, segment time, and time stamp

  • +
  • via the passed set of via points

  • +
+
+

The trajectory proper is (traj.t, traj.q). The trajectory is a +matrix has one row per time step, and one column per axis.

+
+

Note

+
    +
  • Only one of qdmag or tsegment can be specified

  • +
  • If tacc is greater than zero then the path smoothly accelerates +between segments using a polynomial blend. This means that the the via +point is not actually reached.

  • +
  • The path length K is a function of the number of via +points and the time or velocity limits that apply.

  • +
  • Can be used to create joint space trajectories where each axis is a +joint coordinate.

  • +
  • Can be used to create Cartesian trajectories where the “axes” +correspond to translation and orientation in RPY or Euler angle form.

  • +
  • If qdmax is a scalar then all axes are assumed to have the same +maximum speed.

  • +
  • tg has extra attributes arrive, info and via

  • +
+
+
+
References:
+
    +
  • Robotics, Vision & Control in Python, 3e, P. Corke, Springer 2023, Chap 3.

  • +
+
+
Seealso:
+

trapezoidal(), ctraj(), mtraj()

+
+
+
+ +
+ + +
+
+
+ +
+ +
+

© Copyright 2023, Jesse Haviland and Peter Corke. + Last updated on 11-May-2024. +

+
+ + + +
+
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/blocks-arm.html b/blocks-arm.html new file mode 100644 index 00000000..74e88b2d --- /dev/null +++ b/blocks-arm.html @@ -0,0 +1,136 @@ + + + + + + + + + + + Robot manipulator blocks — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

Robot manipulator blocks

+https://raw.githubusercontent.com/petercorke/bdsim/master/figs/BDSimLogo_NoBackgnd@2x.png +
+ + +
+
+
+ +
+ +
+

© Copyright 2023, Jesse Haviland and Peter Corke. + Last updated on 11-May-2024. +

+
+ + + +
+
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/blocks-mobile.html b/blocks-mobile.html new file mode 100644 index 00000000..af67a5bb --- /dev/null +++ b/blocks-mobile.html @@ -0,0 +1,136 @@ + + + + + + + + + + + Mobile robot blocks — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

Mobile robot blocks

+https://raw.githubusercontent.com/petercorke/bdsim/master/figs/BDSimLogo_NoBackgnd@2x.png +
+ + +
+
+
+ +
+ +
+

© Copyright 2023, Jesse Haviland and Peter Corke. + Last updated on 11-May-2024. +

+
+ + + +
+
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/blocks-spatial.html b/blocks-spatial.html new file mode 100644 index 00000000..2b7a41e8 --- /dev/null +++ b/blocks-spatial.html @@ -0,0 +1,136 @@ + + + + + + + + + + + Spatial maths blocks — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

Spatial maths blocks

+https://raw.githubusercontent.com/petercorke/bdsim/master/figs/BDSimLogo_NoBackgnd@2x.png +
+ + +
+
+
+ +
+ +
+

© Copyright 2023, Jesse Haviland and Peter Corke. + Last updated on 11-May-2024. +

+
+ + + +
+
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/blocks-uav.html b/blocks-uav.html new file mode 100644 index 00000000..e4ea6f9a --- /dev/null +++ b/blocks-uav.html @@ -0,0 +1,135 @@ + + + + + + + + + + + UAV blocks — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

UAV blocks

+https://raw.githubusercontent.com/petercorke/bdsim/master/figs/BDSimLogo_NoBackgnd@2x.png +
+ + +
+
+
+ +
+ +
+

© Copyright 2023, Jesse Haviland and Peter Corke. + Last updated on 11-May-2024. +

+
+ + + +
+
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/blocks.html b/blocks.html new file mode 100644 index 00000000..dda98a20 --- /dev/null +++ b/blocks.html @@ -0,0 +1,145 @@ + + + + + + + + + + + bdsim blocks — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+
    +
  • + +
  • +
  • +
+
+
+
+
+ +
+

bdsim blocks

+https://raw.githubusercontent.com/petercorke/bdsim/master/figs/BDSimLogo_NoBackgnd@2x.png +

A set of block definitions that add robotic capability to the bdsim +block diagram simulation environment.

+ +
+ + +
+
+
+ +
+ +
+

© Copyright 2023, Jesse Haviland and Peter Corke. + Last updated on 11-May-2024. +

+
+ + + +
+
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/dgraph.html b/dgraph.html new file mode 100644 index 00000000..370a3960 --- /dev/null +++ b/dgraph.html @@ -0,0 +1,900 @@ + + + + + + + + + Directed graph — Simple graph functionality for Python documentation + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

Directed graph

+
+
+class PGraph.DGraph(arg=None, metric=None, heuristic=None, verbose=False)[source]
+

Bases: PGraph

+

Class for directed graphs

+
Inheritance diagram of DGraph
+ + + +
+
+classmethod Adjacency(A, coords=None, names=None)
+

Create graph from adjacency matrix

+
+
Parameters:
+
    +
  • A (ndarray(N,N)) – adjacency matrix

  • +
  • coords (ndarray(N,M), optional) – coordinates of vertices, defaults to None

  • +
  • names (list(N) of str, optional) – names of vertices, defaults to None

  • +
+
+
Returns:
+

[description]

+
+
Return type:
+

[type]

+
+
+

Create a directed or undirected graph where non-zero elements A[i,j] +correspond to edges from vertex i to vertex j.

+
+

Warning

+

For undirected graph A should be symmetric but this +is not checked. Only the upper triangular part is used.

+
+
+ +
+
+classmethod Dict(d, reverse=False)
+

Create graph from parent/child dictionary

+
+
Parameters:
+
    +
  • d (dict) – dictionary that maps from Vertex subclass to Vertex subclass

  • +
  • reverse (bool, optional) – reverse link direction, defaults to False

  • +
+
+
Returns:
+

graph

+
+
Return type:
+

UGraph or DGraph

+
+
+

Behaves like a constructor for a DGraph or UGraph from a +dictionary that maps vertices to parents. From this information it +can create a tree graph.

+

By default parent vertices are linked their children. If reverse is +True then children are linked to their parents.

+
+ +
+
+Laplacian()
+

Laplacian matrix for the graph

+
+
Returns:
+

Laplacian matrix

+
+
Return type:
+

NumPy ndarray

+
+
+

g.Laplacian() is the Laplacian matrix (NxN) of the graph where N +is the number of vertices.

+
+

Note

+
    +
  • Laplacian is always positive-semidefinite.

  • +
  • Laplacian has at least one zero eigenvalue.

  • +
  • +
    The number of zero-valued eigenvalues is the number of connected

    components in the graph.

    +
    +
    +
  • +
+
+
+
Seealso:
+

adjacency() incidence() degree()

+
+
+
+ +
+
+__init__(arg=None, metric=None, heuristic=None, verbose=False)
+
+ +
+
+add_edge(v1, v2, **kwargs)
+

Add an edge to the graph (superclass method)

+
+
Parameters:
+
    +
  • v1 (Vertex subclass) – first vertex (start if a directed graph)

  • +
  • v2 (Vertex subclass) – second vertex (end if a directed graph)

  • +
  • kwargs – optional arguments to pass to Vertex.connect

  • +
+
+
Returns:
+

edge

+
+
Return type:
+

Edge

+
+
+

Create an edge between a vertex pair and adds it to the graph.

+

This is a graph centric way of creating an edge. The +alternative is the connect method of a vertex.

+
+
Seealso:
+

Edge.connect() Vertex.connect()

+
+
+
+ +
+
+add_vertex(coord=None, name=None)[source]
+

Add vertex to directed graph

+
+
Parameters:
+
    +
  • coord (array-like, optional) – coordinate for an embedded graph, defaults to None

  • +
  • name (str, optional) – vertex name, defaults to “#i”

  • +
+
+
Returns:
+

new vertex

+
+
Return type:
+

DVertex

+
+
+
    +
  • g.add_vertex() creates a new vertex with optional coord and +name.

  • +
  • g.add_vertex(v) takes an instance or subclass of DVertex and adds +it to the graph

  • +
+
+ +
+
+adjacency()
+

Adjacency matrix of graph

+
+
Returns:
+

adjacency matrix

+
+
Return type:
+

ndarray(N,N)

+
+
+

The elements of the adjacency matrix [i,j] are 1 if vertex i is +connected to vertex j, else 0.

+
+

Note

+
    +
  • vertices are numbered in their order of creation. A vertex index +can be resolved to a vertex reference by graph[i].

  • +
  • for an undirected graph the matrix is symmetric

  • +
  • Eigenvalues of A are real and are known as the spectrum of the graph.

  • +
  • The element A[i,j] can be considered the number of walks of length one +edge from vertex i to vertex j (either zero or one).

  • +
  • If Ak = A ** k the element Ak[i,j] is the number of +walks of length k from vertex i to vertex j.

  • +
+
+
+
Seealso:
+

Laplacian() incidence() degree()

+
+
+
+ +
+
+average_degree()
+

Average degree of the graph

+
+
Returns:
+

average degree

+
+
Return type:
+

float

+
+
+

Average degree is \(2 E / N\) for an undirected graph and +\(E / N\) for a directed graph where \(E\) is the total number of +edges and \(N\) is the number of vertices.

+
+ +
+
+closest(coord)
+

Vertex closest to point

+
+
Parameters:
+

coord (ndarray(n)) – coordinates of a point

+
+
Returns:
+

closest vertex

+
+
Return type:
+

Vertex subclass

+
+
+

Returns the vertex closest to the given point. Distance is computed +according to the graph’s metric.

+
+
Seealso:
+

metric()

+
+
+
+ +
+
+component(c)
+

All vertices in specified graph component

+

graph.component(c) is a list of all vertices in graph component c.

+
+ +
+
+connectivity(vertices=None)
+

Graph connectivity

+
+
Returns:
+

a list with the number of edges per vertex

+
+
Return type:
+

list

+
+
+

The average vertex connectivity is:

+
mean(g.connectivity())
+
+
+

and the minimum vertex connectivity is:

+
min(g.connectivity())
+
+
+
+ +
+
+copy()
+

Deepcopy of graph

+
+
Parameters:
+

g (PGraph) – A graph

+
+
Returns:
+

deep copy

+
+
Return type:
+

PGraph

+
+
+
+ +
+
+degree()
+

Degree matrix of graph

+
+
Returns:
+

degree matrix

+
+
Return type:
+

ndarray(N,N)

+
+
+

This is a diagonal matrix where element [i,i] is the number +of edges connected to vertex id i.

+
+
Seealso:
+

adjacency() incidence() laplacian()

+
+
+
+ +
+
+distance()
+

Distance matrix of graph

+
+
Returns:
+

distance matrix

+
+
Return type:
+

ndarray(n,n)

+
+
+

The elements of the distance matrix D[i,j] is the edge cost of moving +from vertex i to vertex j. It is zero if the vertices are not +connected.

+
+ +
+
+dotfile(filename=None, direction=None)
+

Export graph as a GraphViz dot file

+
+
Parameters:
+

filename (str, optional) – filename to save graph to, defaults to None

+
+
+

g.dotfile() creates the specified file which contains the +GraphViz code to represent the embedded graph. By default output +is to the console.

+
+

Note

+
    +
  • The graph is undirected if it is a subclass of UGraph

  • +
  • The graph is directed if it is a subclass of DGraph

  • +
  • Use neato rather than dot to get the embedded layout

  • +
+
+
+

Note

+

If filename is a file object then the file will not +be closed after the GraphViz model is written.

+
+
+
Seealso:
+

showgraph()

+
+
+
+ +
+
+edges()
+

Get all edges in graph (superclass method)

+
+
Returns:
+

All edges in the graph

+
+
Return type:
+

list of Edge references

+
+
+

We can iterate over all edges in the graph by:

+
for e in g.edges():
+    print(e)
+
+
+
+

Note

+

The edges() of a Vertex is a list of all edges connected +to that vertex.

+
+
+
Seealso:
+

Vertex.edges()

+
+
+
+ +
+
+property heuristic
+

Get the heuristic distance metric for graph

+
+
Returns:
+

heuristic distance metric

+
+
Return type:
+

callable

+
+
+

This is a function of a vector and returns a scalar.

+
+ +
+
+highlight_edge(edge, scale=2, color='r', alpha=0.5)
+

Highlight an edge in the graph

+
+
Parameters:
+
    +
  • edge (Edge subclass) – The edge to highlight

  • +
  • scale (float, optional) – Overwrite with a line this much bigger than the original, +defaults to 1.5

  • +
  • color (str, optional) – Overwrite with a line in this color, defaults to ‘r’

  • +
+
+
+
+ +
+
+highlight_path(path, block=False, **kwargs)
+

Highlight a path through the graph

+
+
Parameters:
+
    +
  • path ([type]) – [description]

  • +
  • block (bool, optional) – [description], defaults to True

  • +
+
+
+

The vertices and edges along the path are overwritten with a different +size/width and color.

+
+
Seealso:
+

highlight_vertex() highlight_edge()

+
+
+
+ +
+
+highlight_vertex(vertex, scale=2, color='r', alpha=0.5)
+

Highlight a vertex in the graph

+
+
Parameters:
+
    +
  • edge (Vertex subclass) – The vertex to highlight

  • +
  • scale (float, optional) – Overwrite with a line this much bigger than the original, +defaults to 1.5

  • +
  • color (str, optional) – Overwrite with a line in this color, defaults to ‘r’

  • +
+
+
+
+ +
+
+incidence()
+

Incidence matrix of graph

+
+
Returns:
+

incidence matrix

+
+
Return type:
+

ndarray(n,ne)

+
+
+

The elements of the incidence matrix I[i,j] are 1 if vertex i is +connected to edge j, else 0.

+
+

Note

+
    +
  • vertices are numbered in their order of creation. A vertex index +can be resolved to a vertex reference by graph[i].

  • +
  • edges are numbered in the order they appear in graph.edges().

  • +
+
+
+
Seealso:
+

Laplacian() adjacency() degree()

+
+
+
+ +
+
+iscyclic()
+
+ +
+
+property metric
+

Get the distance metric for graph

+
+
Returns:
+

distance metric

+
+
Return type:
+

callable

+
+
+

This is a function of a vector and returns a scalar.

+
+ +
+
+property n
+

Number of vertices

+
+
Returns:
+

Number of vertices

+
+
Return type:
+

int

+
+
+
+ +
+
+property nc
+

Number of components

+
+
Returns:
+

Number of components

+
+
Return type:
+

int

+
+
+
+

Note

+
    +
  • Components are labeled from 0 to g.nc-1.

  • +
  • A graph coloring algorithm is run if the graph connectivity +has changed.

  • +
+
+
+

Note

+

A lazy approach is used, and if a connectivity changing +operation has been performed since the last call, the graph +coloring algorithm is run which is potentially expensive for +a large graph.

+
+
+ +
+
+property ne
+

Number of edges

+
+
Returns:
+

Number of vertices

+
+
Return type:
+

int

+
+
+
+ +
+
+path_Astar(S, G, verbose=False, summary=False)
+

A* search for path

+
+
Parameters:
+
    +
  • S (Vertex subclass) – start vertex

  • +
  • G (Vertex subclass) – goal vertex

  • +
+
+
Returns:
+

list of vertices from S to G inclusive, path length, tree

+
+
Return type:
+

list of Vertex subclass, float, dict

+
+
+

Returns a list of vertices that form a path from vertex S to +vertex G if possible, otherwise return None.

+

The search tree is returned as dict that maps a vertex to its parent.

+

The heuristic is the distance metric of the graph, which defaults to +Euclidean distance.

+
+
Seealso:
+

heuristic()

+
+
+
+ +
+
+path_BFS(S, G, verbose=False, summary=False)
+

Breadth-first search for path

+
+
Parameters:
+
    +
  • S (Vertex subclass) – start vertex

  • +
  • G (Vertex subclass) – goal vertex

  • +
+
+
Returns:
+

list of vertices from S to G inclusive, path length

+
+
Return type:
+

list of Vertex subclass, float

+
+
+

Returns a list of vertices that form a path from vertex S to +vertex G if possible, otherwise return None.

+
+ +
+
+path_UCS(S, G, verbose=False, summary=False)
+

Uniform cost search for path

+
+
Parameters:
+
    +
  • S (Vertex subclass) – start vertex

  • +
  • G (Vertex subclass) – goal vertex

  • +
+
+
Returns:
+

list of vertices from S to G inclusive, path length, tree

+
+
Return type:
+

list of Vertex subclass, float, dict

+
+
+

Returns a list of vertices that form a path from vertex S to +vertex G if possible, otherwise return None.

+

The search tree is returned as dict that maps a vertex to its parent.

+

The heuristic is the distance metric of the graph, which defaults to +Euclidean distance.

+
+ +
+
+plot(colorcomponents=True, force2d=False, vopt={}, eopt={}, text={}, block=False, grid=True, ax=None)
+

Plot the graph

+
+
Parameters:
+
    +
  • vopt (dict, optional) – vertex format, defaults to 12pt o-marker

  • +
  • eopt (dict, optional) – edge format, defaults to None

  • +
  • text (False or dict, optional) – text label format, defaults to None

  • +
  • colorcomponents – color vertices and edges by component, defaults to None

  • +
  • block (bool, optional) – block until figure is dismissed, defaults to True

  • +
+
+
+

The graph is plotted using matplotlib.

+

If colorcomponents is True then each component is assigned a unique +color. vertex and edge cannot include a color keyword.

+

If text is a dict it is used to format text labels for the vertices +which are the vertex names. If text is None default formatting is +used. If text is False no labels are added.

+
+ +
+
+remove(x)
+

Remove element from graph (superclass method)

+
+
Parameters:
+

x (Edge or Vertex subclass) – element to remove from graph

+
+
Raises:
+

TypeError – unknown type

+
+
+

The edge or vertex is removed, and all references and lists are +updated.

+
+

Warning

+

The connectivity of the network may be changed.

+
+
+ +
+
+samecomponent(v1, v2)
+

Test if vertices belong to same graph component

+
+
Parameters:
+
    +
  • v1 (Vertex subclass) – vertex

  • +
  • v2 (Vertex subclass) – vertex

  • +
+
+
Returns:
+

true if vertices belong to same graph component

+
+
Return type:
+

bool

+
+
+

Test whether vertices belong to the same component. For a:

+
    +
  • directed graph this implies a path between them

  • +
  • undirected graph there is not necessarily a path between them

  • +
+
+ +
+
+show()
+
+ +
+
+showgraph(**kwargs)
+

Display graph in a browser tab

+
+
Parameters:
+

kwargs – arguments passed to dotfile()

+
+
+

g.showgraph() renders and displays the graph in a browser tab. The +graph is exported in GraphViz format, rendered to +PDF using dot and then displayed in a browser tab.

+
+
Seealso:
+

dotfile()

+
+
+
+ +
+
+classmethod vertex_copy(vertex)[source]
+
+ +
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/directed.html b/directed.html new file mode 100644 index 00000000..47d82e13 --- /dev/null +++ b/directed.html @@ -0,0 +1,146 @@ + + + + + + + + + Directed graphs — Simple graph functionality for Python documentation + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

Directed graphs

+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/docs/Makefile b/docs/Makefile deleted file mode 100644 index d0c3cbf1..00000000 --- a/docs/Makefile +++ /dev/null @@ -1,20 +0,0 @@ -# Minimal makefile for Sphinx documentation -# - -# You can set these variables from the command line, and also -# from the environment for the first two. -SPHINXOPTS ?= -SPHINXBUILD ?= sphinx-build -SOURCEDIR = source -BUILDDIR = build - -# Put it first so that "make" without argument is like "make help". -help: - @$(SPHINXBUILD) -M help "$(SOURCEDIR)" "$(BUILDDIR)" $(SPHINXOPTS) $(O) - -.PHONY: help Makefile - -# Catch-all target: route all unknown targets to Sphinx using the new -# "make mode" option. $(O) is meant as a shortcut for $(SPHINXOPTS). -%: Makefile - @$(SPHINXBUILD) -M $@ "$(SOURCEDIR)" "$(BUILDDIR)" $(SPHINXOPTS) $(O) diff --git a/docs/make.bat b/docs/make.bat deleted file mode 100644 index 6247f7e2..00000000 --- a/docs/make.bat +++ /dev/null @@ -1,35 +0,0 @@ -@ECHO OFF - -pushd %~dp0 - -REM Command file for Sphinx documentation - -if "%SPHINXBUILD%" == "" ( - set SPHINXBUILD=sphinx-build -) -set SOURCEDIR=source -set BUILDDIR=build - -if "%1" == "" goto help - -%SPHINXBUILD% >NUL 2>NUL -if errorlevel 9009 ( - echo. - echo.The 'sphinx-build' command was not found. Make sure you have Sphinx - echo.installed, then set the SPHINXBUILD environment variable to point - echo.to the full path of the 'sphinx-build' executable. Alternatively you - echo.may add the Sphinx directory to PATH. - echo. - echo.If you don't have Sphinx installed, grab it from - echo.http://sphinx-doc.org/ - exit /b 1 -) - -%SPHINXBUILD% -M %1 %SOURCEDIR% %BUILDDIR% %SPHINXOPTS% %O% -goto end - -:help -%SPHINXBUILD% -M help %SOURCEDIR% %BUILDDIR% %SPHINXOPTS% %O% - -:end -popd diff --git a/docs/source/conf.py b/docs/source/conf.py deleted file mode 100644 index 23e13be9..00000000 --- a/docs/source/conf.py +++ /dev/null @@ -1,69 +0,0 @@ -# Configuration file for the Sphinx documentation builder. -# -# This file only contains a selection of the most common options. For a full -# list see the documentation: -# https://www.sphinx-doc.org/en/master/usage/configuration.html - -# -- Path setup -------------------------------------------------------------- - -# If extensions (or modules to document with autodoc) are in another directory, -# add these directories to sys.path here. If the directory is relative to the -# documentation root, use os.path.abspath to make it absolute, like shown here. -# - -import os -import sys -sys.path.insert(0, os.path.abspath('../../pgraph')) - -# -- Project information ----------------------------------------------------- - -project = 'Simple graph functionality for Python' -copyright = '2020, Peter Corke' -author = 'Peter Corke' - - -# -- General configuration --------------------------------------------------- - -# Add any Sphinx extension module names here, as strings. They can be -# extensions coming with Sphinx (named 'sphinx.ext.*') or your custom -# ones. -extensions = [ - 'sphinx.ext.autodoc', - 'sphinx.ext.todo', - 'sphinx.ext.viewcode', - 'sphinx.ext.mathjax', - 'sphinx.ext.coverage', - 'sphinx.ext.inheritance_diagram', -] - -html_theme_options = { - 'logo_only': False, - 'display_version': True, - 'prev_next_buttons_location': 'both', - 'analytics_id': 'G-11Q6WJM565', - - } - -# Add any paths that contain templates here, relative to this directory. -templates_path = ['_templates'] - -# List of patterns, relative to source directory, that match files and -# directories to ignore when looking for source files. -# This pattern also affects html_static_path and html_extra_path. -exclude_patterns = ['test_*'] - - -# -- Options for HTML output ------------------------------------------------- - -# The theme to use for HTML and HTML Help pages. 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18 - GridSpacingY - 18 - LinkGridToRulers - NO - PrintLineWidth - 0.5 - PrintsGrid - NO - ShowsGrid - YES - SnapsToGrid - NO - - SnapDrawing - NO - SnapEnds - NO - SnapGuidelines - NO - SnapRadiusGrid - 3 - SnapSound - None - SoftSnap - YES - - DKDHideExtension - YES - DKDLayersList - - - CloakLayerGuidelines - NO - CloakLayerVertices - NO - FullLayerScale - - ArchivePrecision - 12 - ScaleOriginX - 0 - ScaleOriginY - 0 - ScalePlusDown - YES - ScalePlusToRight - YES - ScaleScale - 1 - ScaleUnits - Centimeters - - GraphicsList - - - AttributedText - - String - Vertex subclass - TextAttribute - 0 - - Bounds - {{584.594428198, 92.5444270325}, {111.11092077, 21.4163415091}} - Class - DKDTextArea - DKDLockInfo - 14 - GraphicID - 9852BB83 - GraphicStyle - 10 - - - Bounds - {{122.115430323, 283.075572388}, {108.290742591, 108.290742591}} - Class - DKDCircle - DKDLockInfo - 10 - GraphicID - C0B95773 - GraphicStyle - 8 - - - Bounds - {{331.405592037, 134.116666667}, {108.290742591, 108.290742591}} - Class - DKDCircle - DKDLockInfo - 26 - GraphicID - 6F7E8773 - GraphicStyle - 8 - - - Bounds - {{156.726360719, 135.484214289}, {78.4881890125, 78.4881890125}} - Class - DKDSquare - DKDLockInfo - 14 - GraphicID - 17CBE773 - GraphicStyle - 8 - - - Class - DKDLine - DKDLockInfo - 14 - GraphicID - 73CA6873 - GraphicStyle - 24 - SVGPath - M215.290624951,317.021883591 L357.33324706,212.161226418 - - - Bounds - {{133.871950192, 329.397794982}, {87.782600853, 16.648764451}} - Class - DKDRectangle - DKDLockInfo - 14 - GraphicID - 614BBC73 - GraphicStyle - 8 - - - Class - DKDContinuousBezier - DKDLockInfo - 14 - DkBezArrow - 10 - GraphicID - C2545D73 - GraphicStyle - 40 - SVGPath - M156.299321488,337.93308586 C156.299321488,337.93308586 156.299321488,337.93308586 156.299321488,337.93308586 C158.092340586,301.632198305 114.961293747,276.145578952 109.075031946,246.591412895 C104.100700363,221.615931586 106.298408981,187.190683552 153.333481954,186.323614649 - - - AttributedText - - String - _edges - TextAttribute - 16 - - Bounds - {{152.127877297, 345.736705982}, {78.3773221901, 21.0231661209}} - Class - DKDTextArea - DKDLockInfo - 14 - GraphicID - 35D9CF73 - GraphicStyle - 10 - - - Class - DKDContinuousBezier - DKDLockInfo - 14 - DkBezArrow - 26 - GraphicID - 2BF2D183 - GraphicStyle - 56 - SVGPath - M216.910249349,159.454893916 C216.910249349,159.454893916 216.91024935,159.454893916 216.91024935,159.454893916 C238.194121828,159.412376723 268.628145605,170.476785973 274.440766176,190.385559057 C283.259505097,220.590570711 225.746142372,267.158044238 204.393312879,291.212913739 - - - Class - DKDContinuousBezier - DKDLockInfo - 14 - DkBezArrow - 26 - GraphicID - AC19D283 - GraphicStyle - 56 - SVGPath - M216.543099767,179.475592987 C216.543099767,179.475592987 216.543099767,179.475592987 216.543099767,179.475592987 C260.318439114,179.951153176 265.431057913,156.218627451 290.610971186,151.117492638 C306.284970846,147.942136771 322.063452074,148.951122287 338.970560146,158.82216796 - - - AttributedText - - String - v1 - TextAttribute - 16 - - Bounds - {{186.134058263, 150.294574897}, {31.0306494531, 21.0231661209}} - Class - DKDTextArea - DKDLockInfo - 14 - GraphicID - 38F37683 - GraphicStyle - 8 - - - AttributedText - - String - v2 - - TextAttribute - 16 - - Bounds - {{186.128126855, 171.134378779}, {30.9110329578, 20.8018422269}} - Class - DKDTextArea - DKDLockInfo - 14 - GraphicID - 19EDA683 - GraphicStyle - 8 - - - AttributedText - - String - Vertex subclass - TextAttributes - - {0, 6} - 32 - {6, 9} - 0 - - - Bounds - {{198.775981826, 381.234341154}, {111.11092077, 21.4163415091}} - Class - DKDTextArea - DKDLockInfo - 14 - GraphicID - 5B25DB83 - GraphicStyle - 10 - - - AttributedText - - String - Vertex subclass - TextAttributes - - {0, 6} - 32 - {6, 9} - 0 - - - Bounds - {{414.510790464, 228.736856939}, {111.11092077, 21.4163415091}} - Class - DKDTextArea - DKDLockInfo - 14 - GraphicID - 09BB7C83 - GraphicStyle - 10 - - - AttributedText - - String - Edge subclass - TextAttributes - - {0, 4} - 32 - {4, 9} - 0 - - - Bounds - {{157.882924378, 215.60312307}, {90.1938101669, 21.4163415091}} - Class - DKDTextArea - DKDLockInfo - 14 - GraphicID - 18ED7C83 - GraphicStyle - 10 - - - Bounds - {{140.115430322, 589.075572388}, {108.290742591, 108.290742591}} - Class - DKDCircle - DKDLockInfo - 26 - GraphicID - 5943FD83 - GraphicStyle - 8 - - - Bounds - {{349.405592037, 440.116666667}, {108.290742591, 108.290742591}} - Class - DKDCircle - DKDLockInfo - 26 - GraphicID - 6943FD83 - GraphicStyle - 8 - - - Bounds - {{174.726360719, 441.484214289}, {78.4881890125, 78.4881890125}} - Class - DKDSquare - DKDLockInfo - 14 - GraphicID - 7943FD83 - GraphicStyle - 8 - - - Class - DKDLine - DKDLockInfo - 14 - DkBezArrow - 42 - GraphicID - 8943FD83 - GraphicStyle - 24 - SVGPath - M233.290624951,623.021883591 L375.333247059,518.161226418 - - - Bounds - {{151.871950192, 635.397794983}, {87.782600853, 16.648764451}} - Class - DKDRectangle - DKDLockInfo - 14 - GraphicID - 9943FD83 - 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Class - DKDTextArea - DKDLockInfo - 14 - GraphicID - 0A43FD83 - GraphicStyle - 10 - - - AttributedText - - String - Vertex subclass - TextAttributes - - {0, 6} - 32 - {6, 9} - 0 - - - Bounds - {{432.510790464, 534.736856939}, {111.11092077, 21.4163415091}} - Class - DKDTextArea - DKDLockInfo - 14 - GraphicID - 1A43FD83 - GraphicStyle - 10 - - - AttributedText - - String - Edge subclass - TextAttributes - - {0, 4} - 32 - {4, 9} - 0 - - - Bounds - {{175.882924378, 521.60312307}, {90.1938101669, 21.4163415091}} - Class - DKDTextArea - DKDLockInfo - 14 - GraphicID - 2A43FD83 - GraphicStyle - 10 - - - Bounds - {{343.72361627, 179.302318817}, {87.782600853, 16.648764451}} - Class - DKDRectangle - DKDLockInfo - 14 - GraphicID - 43C31293 - GraphicStyle - 8 - - - AttributedText - - String - _edges - TextAttribute - 16 - - Bounds - {{361.979543375, 195.641229816}, {78.3773221901, 21.0231661209}} - Class - DKDTextArea - DKDLockInfo - 14 - GraphicID - 53C31293 - GraphicStyle - 10 - - - Bounds - {{361.864971875, 484.284040959}, {87.782600853, 16.648764451}} - Class - DKDRectangle - DKDLockInfo - 14 - GraphicID - 895D5293 - GraphicStyle - 8 - - - AttributedText - - String - _edges - TextAttribute - 16 - - Bounds - {{380.12089898, 500.622951958}, {78.3773221901, 21.0231661209}} - Class - DKDTextArea - DKDLockInfo - 14 - GraphicID - 995D5293 - GraphicStyle - 10 - - - Class - DKDBezier - DKDLockInfo - 14 - DkBezArrow - 2 - GraphicID - AA5AC793 - GraphicStyle - 56 - SVGPath - M392.38395542,187.385848569 C392.38395542,187.385848569 392.38395542,187.385848569 392.38395542,187.385848569 C388.874616953,82.0118025472 314.836279053,62.9552738792 252.242178667,64.2646682342 C74.660984366,73.469899094 94.9233564416,159.621417018 155.871923044,158.837670052 - - - AttributedText - - String - Undirected graph - TextAttribute - 48 - - Bounds - {{278.536338118, 269.434500953}, {202.348893933, 34.1385596992}} - Class - DKDTextArea - DKDLockInfo - 14 - GraphicID - 14DEC993 - GraphicStyle - 10 - - - AttributedText - - String - Directed graph - TextAttribute - 48 - 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TextAttribute - 64 - - Bounds - {{261.400575239, 323.584485991}, {85.897378884, 18.9174921865}} - Class - DKDTextArea - DKDLockInfo - 14 - GraphicID - 8BCA09A3 - GraphicStyle - 72 - - - Class - DKDLine - DKDLockInfo - 14 - GraphicID - 7462B9A3 - GraphicStyle - 72 - SVGPath - M221.084225911,339.165550309 L260.524401441,334.093988961 - - - HideDimensions - NO - LayerColorMod - - DKDOnColorMod - NO - DKDOpacityColorMod - 0.5 - DKDOutlineColorMod - NO - DKDTintColorColorMod - - ColorSpace - NSCalibratedWhiteColorSpace - Opacity - 1 - White - 0.5 - - DKDTintFractionColorMod - 0.5 - - LayerLock - NO - LayerName - Paper - LayerState - Active - OutlineLayer - NO - - - DKDPagesSpec - - BackgroundDisplay - Background - CanvasBorder - - Blue - 0.45904 - BluePlus - 0.533333 - ColorSpace - NSCalibratedRGBColorSpace - ColorSpacePlus - DKDsRgbColorSpace - Green - 0.458672 - GreenPlus - 0.533333 - Opacity - 1 - OpacityPlus - 1 - Red - 0.475001 - RedPlus - 0.54902 - - CanvasColor - - Catalog - System - Catalog-Color - windowBackgroundColor - - CanvasMargin - 0 - DetailsDrawerWidth - 260 - DisplayAttributesBar - YES - DisplayRulers - NO - FullScreen - NO - FullScreenCanvasMargin - 141.73228 - LayersDrawerWidth - 266 - NonFullScreenCanvasMargin - 0 - NumberAcrossFirst - YES - PagesAcross - 1 - PagesDown - 1 - PagesSpecBackgroundRGB - - ColorSpace - NSCalibratedWhiteColorSpace - Opacity - 1 - White - 1 - - PagesSpecPrintBackground - NO - ShowPageBreaks - NO - SizeChecker - 8 - aCheckerColor - - Blue - 0.926349 - BluePlus - 0.941176 - ColorSpace - NSCalibratedRGBColorSpace - ColorSpacePlus - DKDsRgbColorSpace - Green - 0.926333 - GreenPlus - 0.941176 - Opacity - 1 - OpacityPlus - 1 - Red - 0.926361 - RedPlus - 0.941176 - - bCheckerColor - - Blue - 0.737155 - BluePlus - 0.784314 - ColorSpace - NSCalibratedRGBColorSpace - ColorSpacePlus - DKDsRgbColorSpace - Green - 0.737142 - GreenPlus - 0.784314 - Opacity - 1 - OpacityPlus - 1 - Red - 0.737164 - RedPlus - 0.784314 - - - DKDPrintInfo - - BottomMargin - 18 - Copies - 1 - FallBackPaperSizeHeight - 841.889770508 - FallBackPaperSizeWidth - 595.27557373 - FirstPage - 1 - HorizontalPagination - 2 - HorizontallyCentered - YES - LastPage - -1 - LeftMargin - 18 - MustCollate - YES - Orientation - 0 - PaperName - A4 - PaperSizeHeight - 841.889770508 - PaperSizeWidth - 595.27557373 - PreviewPageNumber - 1 - PrintAllPages - YES - PrintJobDisposition - NSPrintSpoolJob - PrintSavePath - - PrintScalingFactor - 1 - PrinterName - Brother MFC-9340CDW - ReversePageOrder - NO - RightMargin - 18 - TopMargin - 18 - VerticalPagination - 0 - VerticallyCentered - YES - XPrintMirror - NO - YPrintMirror - NO - - DKDSaveTimeStamp - 2020-09-21 23:12:02 +0000 - DKDTablet - - BrushDynamic - NO - BrushFit - 6 - PenMax - 25 - PenMin - 0.5 - PenPressureFactor - 0.5 - PencilDynamic - NO - PencilFit - 7 - - DKDTimeFormat - - Field 0 Include - Weekday - Field 0 Type - Long - Field 1 Include - Month - Field 1 Type - Short - Field 2 Include - Day - Field 2 Type - Number - Field 3 Include - Year - Field 3 Type - Long - Include GMT - NO - Include Title - YES - IncludeDate - YES - IncludeTime - YES - Seperator 0 - - - Seperator 1 - . - Seperator 2 - , - Seperator 3 - - Time Seperator - : - TimeAfterDate - YES - Twelve Hour Clock - YES - Used Once - YES - - DKDToolbarSelectedButtonPairs - - ColorTextToolbarItemIdentifier_0 - - HSB_10FFFF - - ColorFillToolbarItemIdentifier_0 - - HSB_0000D0 - - ColorStrokeToolbarItemIdentifier_0 - - HSB_000020 - - GradientToolbarItemIdentifier_0 - - - EndGradientColor - - Blue - 0.999991 - BluePlus - 1 - ColorSpace - NSCalibratedRGBColorSpace - ColorSpacePlus - DKDsRgbColorSpace - Green - 0.999974 - GreenPlus - 1 - Opacity - 1 - OpacityPlus - 1 - Red - 0.999999 - RedPlus - 0.999996 - - GradientFillClass - DKDHorizontalGradientFill - StartGradientColor - - Blue - 0 - BluePlus - 0 - ColorSpace - NSCalibratedRGBColorSpace - ColorSpacePlus - DKDsRgbColorSpace - Green - 0 - GreenPlus - 0 - Opacity - 1 - OpacityPlus - 1 - Red - 0 - RedPlus - 0 - - - - PatternForToolbarItemIdentifier_0 - - Toolbar_02 - - TextureForToolbarItemIdentifier_0 - - Ancient - - - DKDWindowState - - CursorMode - Nothing - DocCenter - {279.637786865, 269.670318249} - DrawersOnMainView - YES - GDetailsLayersDrawerEdgePreference - Auto - GraphicDetailsOpen - NO - LayerActiveAbove - NO - LayerOpen - NO - LayerSelect - Active Only - LayersDrawerEdgePreference - Auto - OutlineDrawing - NO - WindowLocation - 852 319 909 912 0 0 2560 1417 - ZoomPercent - 160.180664 - - GroupEdit - Fixed - NumberColorsToListInContextMenu - 12 - NumberPairColorsToListInContextMenu - 6 - - diff --git a/docs/source/datastructures.png b/docs/source/datastructures.png deleted file mode 100644 index e7225cbb..00000000 Binary files a/docs/source/datastructures.png and /dev/null differ diff --git a/dvertex.html b/dvertex.html new file mode 100644 index 00000000..9c8f59d5 --- /dev/null +++ b/dvertex.html @@ -0,0 +1,428 @@ + + + + + + + + + Directed graph vertex — Simple graph functionality for Python documentation + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

Directed graph vertex

+
+
+class PGraph.DVertex(coord=None, name=None)[source]
+

Bases: Vertex

+

Vertex subclass for directed graphs

+

This class can be inherited to provide user objects with graph capability.

+
Inheritance diagram of DVertex
+ + + +
+
+__init__(coord=None, name=None)
+
+ +
+
+closest()
+
+ +
+
+connect(other, **kwargs)[source]
+

Connect two vertices with an edge

+
+
Parameters:
+
    +
  • dest (Vertex subclass) – The vertex to connect to

  • +
  • edge (Edge subclass, optional) – Use this as the edge object, otherwise a new Edge +object is created from the vertices being connected, +and the cost and edge parameters, defaults to None

  • +
  • cost (float, optional) – the cost to traverse this edge, defaults to None

  • +
  • data (Any, optional) – reference to arbitrary data associated with the edge, +defaults to None

  • +
+
+
Raises:
+

TypeError – vertex types are different subclasses

+
+
Returns:
+

the edge connecting the vertices

+
+
Return type:
+

Edge

+
+
+

v1.connect(v2) connects vertex v1 to vertex v2.

+
+

Note

+
    +
  • If the vertices subclass UVertex the edge is undirected, and if +they subclass DVertex the edge is directed.

  • +
  • Vertices must both be of the same Vertex subclass

  • +
+
+
+
Seealso:
+

Edge()

+
+
+
+ +
+
+copy(cls=None)
+
+ +
+
+property degree
+

Degree of vertex

+
+
Returns:
+

degree of the vertex

+
+
Return type:
+

int

+
+
+

Returns the number of edges connected to the vertex.

+
+

Note

+

For a DGraph only outgoing edges are considered.

+
+
+
Seealso:
+

edges()

+
+
+
+ +
+
+distance(coord)
+

Distance from vertex to point

+
+
Parameters:
+

coord (ndarray(n) or Vertex) – coordinates of the point

+
+
Returns:
+

distance

+
+
Return type:
+

float

+
+
+

Distance is computed according to the graph’s metric.

+
+
Seealso:
+

metric()

+
+
+
+ +
+
+edges()
+

All outgoing edges of vertex

+
+
Returns:
+

List of all edges leaving this vertex

+
+
Return type:
+

list of Edge

+
+
+
+

Note

+
    +
  • For a directed graph the edges are those leaving this vertex

  • +
  • +
    For a non-directed graph the edges are those leaving or entering

    this vertex

    +
    +
    +
  • +
+
+
+ +
+
+edgeto(dest)
+

Get edge connecting vertex to specific neighbour

+
+
Parameters:
+

dest (Vertex subclass) – a neigbouring vertex

+
+
Raises:
+

ValueErrordest is not a neighbour

+
+
Returns:
+

the edge from this vertex to dest

+
+
Return type:
+

Edge

+
+
+
+

Note

+
    +
  • For a directed graph dest must be at the arrow end of the edge

  • +
+
+
+ +
+
+heuristic_distance(v2)
+
+ +
+
+incidences()
+

Neighbours and edges of a vertex

+

v.incidences() is a generator that returns a list of incidences, +tuples of (vertex, edge) for all neighbours of the vertex v.

+
+

Note

+

For a directed graph the edges are those leaving this vertex

+
+
+ +
+
+isneighbour(vertex)
+

Test if vertex is a neigbour

+
+
Parameters:
+

vertex (Vertex subclass) – vertex reference

+
+
Returns:
+

true if a neighbour

+
+
Return type:
+

bool

+
+
+

For a directed graph this is true only if the edge is from self to +vertex.

+
+ +
+
+neighbors()
+

Neighbors of a vertex

+

v.neighbors() is a list of neighbors of this vertex.

+
+

Note

+

For a directed graph the neighbours are those on edges leaving this vertex

+
+
+ +
+
+neighbours()
+

Neighbours of a vertex

+

v.neighbours() is a list of neighbours of this vertex.

+
+

Note

+

For a directed graph the neighbours are those on edges leaving this vertex

+
+
+ +
+
+remove()[source]
+
+ +
+
+property x
+

The x-coordinate of an embedded vertex

+
+
Returns:
+

The x-coordinate

+
+
Return type:
+

float

+
+
+
+ +
+
+property y
+

The y-coordinate of an embedded vertex

+
+
Returns:
+

The y-coordinate

+
+
Return type:
+

float

+
+
+
+ +
+
+property z
+

The z-coordinate of an embedded vertex

+
+
Returns:
+

The z-coordinate

+
+
Return type:
+

float

+
+
+
+ +
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/edge.html b/edge.html new file mode 100644 index 00000000..206b173f --- /dev/null +++ b/edge.html @@ -0,0 +1,249 @@ + + + + + + + + + Edge — Simple graph functionality for Python documentation + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

Edge

+
+
+class PGraph.Edge(v1=None, v2=None, cost=None, data=None)[source]
+

Bases: object

+

Edge class

+

Is used to represent directed directed and undirected edges.

+

Each edge has: +- cost cost of traversing this edge, required for planning methods +- data reference to arbitrary data associated with the edge +- v1 first vertex, start vertex for a directed edge +- v2 second vertex, end vertex for a directed edge

+
+

Note

+
    +
  • An undirected graph is created by having a single edge object in the +edgelist of _each_ vertex.

  • +
  • This class can be inherited to provide user objects with graph capability.

  • +
  • Inheritance is an alternative to providing arbitrary user data.

  • +
+
+

An Edge points to a pair of vertices. At connect time the vertices +get references back to the Edge object.

+

graph.add_edge(v1, v2) calls v1.connect(v2)

+
+
+__init__(v1=None, v2=None, cost=None, data=None)[source]
+

Create an edge object

+
+
Parameters:
+
    +
  • v1 (Vertex subclass, optional) – start of the edge, defaults to None

  • +
  • v2 (Vertex subclass, optional) – end of the edge, defaults to None

  • +
  • cost (any, optional) – edge cost, defaults to None

  • +
  • data (any, optional) – edge data, defaults to None

  • +
+
+
+

Creates an edge but does not connect it to the vertices or add it to the +graph.

+

If vertices are given, and have associated coordinates, the edge cost +will be computed according to the distance measure associated with the +graph.

+

data is a way of associating any object with the edge, its value +can be found as the .data attribute of the edge. An alternative +approach is to subclass the Edge class.

+
+

Note

+

To compute edge cost from the vertices, the vertices must have +been added to the graph.

+
+
+
Seealso:
+

Edge.connect() Vertex.connect()

+
+
+
+ +
+
+connect(v1, v2)[source]
+

Add edge to the graph

+
+
Parameters:
+
    +
  • v1 (Vertex subclass) – start of the edge

  • +
  • v2 (Vertex subclass) – end of the edge

  • +
+
+
+

The edge is added to the graph and connects vertices v1 and v2.

+
+

Note

+

The vertices must already be added to the graph.

+
+
+ +
+
+property endpoints
+
+ +
+
+next(vertex)[source]
+

Return other end of an edge

+
+
Parameters:
+

vertex (Vertex subclass) – one vertex on the edge

+
+
Raises:
+

ValueErrorvertex is not on the edge

+
+
Returns:
+

the other vertex on the edge

+
+
Return type:
+

Vertex subclass

+
+
+

e.next(v1) is the vertex at the other end of edge e, ie. the +vertex that is not v1.

+
+ +
+
+vertices()[source]
+
+ +
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/examples/README.md b/examples/README.md deleted file mode 100644 index f3350512..00000000 --- a/examples/README.md +++ /dev/null @@ -1,8 +0,0 @@ -# Examples - -This folder contains two JSON files that describe towns and major roads in Queensland Australia. - -- `places.json` describes place names, UTM coordinates -- `routes.json` describes routes between towns as a set of tuples (start town, end town, distance, road type) where road type is 1 for a major road, and 2 for a minor road (worse quality, lower average speed) - -`roads.py` loads these files, creates an embedded graph, and find an \ No newline at end of file diff --git a/examples/places.json b/examples/places.json deleted file mode 100644 index ec41d22c..00000000 --- a/examples/places.json +++ /dev/null @@ -1,182 +0,0 @@ - { - "Brisbane": { - "x":2.6268952905813734, - "y":-0.41986028855831437, - "lat":-27.4697707, - "lon":153.0251235, - "dbne":0.090443497313396154, - "utmzone":"", - "utm":[1.0959145867144743E+6,6.9470436580185965E+6] - }, - "Camooweal": { - "x":2.6268952905813734, - "y":-0.41986028855831437, - "lat":-19.920667, - "lon":138.1213324, - "dbne":0.090443497313396154, - "utmzone":"", - "utm":[-432103.885525028,7.7725517093757875E+6] - }, - "Mount Isa": { - "x":2.6268952905813734, - "y":-0.41986028855831437, - "lat":-20.7255748, - "lon":139.4927085, - "dbne":0.090443497313396154, - "utmzone":"", - "utm":[-283332.59264152544,7.6899891627694024E+6] - }, - "Cloncurry": { - "x":2.6268952905813734, - "y":-0.41986028855831437, - "lat":-20.7063927, - "lon":140.5093258, - "dbne":0.090443497313396154, - "utmzone":"", - "utm":[-176971.789275437,7.6967476106785247E+6] - }, - "Hughenden": { - "x":2.6268952905813734, - "y":-0.41986028855831437, - "lat":-20.85, - "lon":144.2, - "dbne":0.090443497313396154, - "utmzone":"", - "utm":[208622.39381884906,7.6919176609326946E+6] - }, - "Charters Towers": { - "x":2.6268952905813734, - "y":-0.41986028855831437, - "lat":-20.0769637, - "lon":146.2601362, - "dbne":0.090443497313396154, - "utmzone":"", - "utm":[422642.40870479314,7.77983038891146E+6] - }, - "Townsville": { - "x":2.6268952905813734, - "y":-0.41986028855831437, - "lat":-19.2589635, - "lon":146.8169483, - "dbne":0.090443497313396154, - "utmzone":"", - "utm":[480763.802457357,7.8705082707399866E+6] - }, - "Boulia": { - "x":2.6268952905813734, - "y":-0.41986028855831437, - "lat":-22.911306, - "lon":139.910444, - "dbne":0.090443497313396154, - "utmzone":"", - "utm":[-228326.08073938196,7.4486950275674257E+6] - }, - "Winton": { - "x":2.6268952905813734, - "y":-0.41986028855831437, - "lat":-22.3913, - "lon":143.0381, - "dbne":0.090443497313396154, - "utmzone":"", - "utm":[91948.357084013987,7.51848052843709E+6] - }, - "Bedourie": { - "x":2.6268952905813734, - "y":-0.41986028855831437, - "lat":-24.3322136, - "lon":139.4619731, - "dbne":0.090443497313396154, - "utmzone":"", - "utm":[-266188.87148301676,7.2881442884511622E+6] - }, - "Birdsville": { - "x":2.6268952905813734, - "y":-0.41986028855831437, - "lat":-25.898889, - "lon":139.351667, - "dbne":0.090443497313396154, - "utmzone":"", - "utm":[-267499.55060334341,7.1130502189418562E+6] - }, - "Windorah": { - "x":2.6268952905813734, - "y":-0.41986028855831437, - "lat":-25.420556, - "lon":142.654722, - "dbne":0.090443497313396154, - "utmzone":"", - "utm":[62757.101391285716,7.1813570953489337E+6] - }, - "Longreach": { - "x":2.6268952905813734, - "y":-0.41986028855831437, - "lat":-23.4421917, - "lon":144.2555041, - "dbne":0.090443497313396154, - "utmzone":"", - "utm":[219590.58651533368,7.4048552480418114E+6] - }, - "Barcaldine": { - "x":2.6268952905813734, - "y":-0.41986028855831437, - "lat":-23.5563211, - "lon":145.293339, - "dbne":0.090443497313396154, - "utmzone":"", - "utm":[325806.09588169667,7.393856437148137E+6] - }, - "Blackall": { - "x":2.6268952905813734, - "y":-0.41986028855831437, - "lat":-24.416667, - "lon":145.466667, - "dbne":0.090443497313396154, - "utmzone":"", - "utm":[344536.99564438895,7.2987813273016019E+6] - }, - "Charleville": { - "x":2.6268952905813734, - "y":-0.41986028855831437, - "lat":-26.4020616, - "lon":146.2453597, - "dbne":0.090443497313396154, - "utmzone":"", - "utm":[424734.96995957423,7.0795687281981325E+6] - }, - "Roma": { - "x":2.6268952905813734, - "y":-0.41986028855831437, - "lat":-26.5694477, - "lon":148.7837619, - "dbne":0.090443497313396154, - "utmzone":"", - "utm":[677662.62520218431,7.0600136921939394E+6] - }, - "Emerald": { - "x":2.6268952905813734, - "y":-0.41986028855831437, - "lat":-23.527291, - "lon":148.164573, - "dbne":0.090443497313396154, - "utmzone":"", - "utm":[618884.226692918,7.3976249803609233E+6] - }, - "Mackay": { - "x":2.6268952905813734, - "y":-0.41986028855831437, - "lat":-21.1424956, - "lon":149.1821469, - "dbne":0.090443497313396154, - "utmzone":"", - "utm":[726612.80254785565,7.6605248314652806E+6] - }, - "Rockhampton": { - "x":2.6268952905813734, - "y":-0.41986028855831437, - "lat":-23.3790772, - "lon":150.510016, - "dbne":0.090443497313396154, - "utmzone":"", - "utm":[858854.31053935736,7.4101500345370444E+6] - } -} diff --git a/examples/queensland.json b/examples/queensland.json deleted file mode 100644 index 3dc05570..00000000 --- a/examples/queensland.json +++ /dev/null @@ -1 +0,0 @@ -{"places": {"Brisbane": {"lat": -27.4697707, "lon": 153.0251235, "utm": [1095.9145867144744, 6947.043658018596]}, "Camooweal": {"lat": -19.920667, "lon": 138.1213324, "utm": [-432.103885525028, 7772.551709375787]}, "Mount Isa": {"lat": -20.7255748, "lon": 139.4927085, "utm": [-283.3325926415254, 7689.989162769402]}, "Cloncurry": {"lat": -20.7063927, "lon": 140.5093258, "utm": [-176.971789275437, 7696.747610678524]}, "Hughenden": {"lat": -20.85, "lon": 144.2, "utm": [208.62239381884905, 7691.917660932694]}, "Charters Towers": {"lat": -20.0769637, "lon": 146.2601362, "utm": [422.6424087047931, 7779.8303889114595]}, "Townsville": {"lat": -19.2589635, "lon": 146.8169483, "utm": [480.763802457357, 7870.508270739987]}, "Boulia": {"lat": -22.911306, "lon": 139.910444, "utm": [-228.32608073938195, 7448.695027567425]}, "Winton": {"lat": -22.3913, "lon": 143.0381, "utm": [91.94835708401399, 7518.4805284370905]}, "Bedourie": {"lat": -24.3322136, "lon": 139.4619731, "utm": [-266.1888714830168, 7288.1442884511625]}, "Birdsville": {"lat": -25.898889, "lon": 139.351667, "utm": [-267.4995506033434, 7113.050218941856]}, "Windorah": {"lat": -25.420556, "lon": 142.654722, "utm": [62.757101391285715, 7181.357095348933]}, "Longreach": {"lat": -23.4421917, "lon": 144.2555041, "utm": [219.59058651533368, 7404.855248041811]}, "Barcaldine": {"lat": -23.5563211, "lon": 145.293339, "utm": [325.80609588169665, 7393.856437148137]}, "Blackall": {"lat": -24.416667, "lon": 145.466667, "utm": [344.536995644389, 7298.781327301602]}, "Charleville": {"lat": -26.4020616, "lon": 146.2453597, "utm": [424.73496995957424, 7079.568728198133]}, "Roma": {"lat": -26.5694477, "lon": 148.7837619, "utm": [677.6626252021844, 7060.013692193939]}, "Emerald": {"lat": -23.527291, "lon": 148.164573, "utm": [618.884226692918, 7397.6249803609235]}, "Mackay": {"lat": -21.1424956, "lon": 149.1821469, "utm": [726.6128025478556, 7660.524831465281]}, "Rockhampton": {"lat": -23.3790772, "lon": 150.510016, "utm": [858.8543105393574, 7410.150034537044]}}, "routes": [{"start": "Camooweal", "end": "Mount Isa", "distance": 188, "speed": 100}, {"start": "Mount Isa", "end": "Cloncurry", "distance": 118, "speed": 100}, {"start": "Cloncurry", "end": "Hughenden", "distance": 401, "speed": 100}, {"start": "Cloncurry", "end": "Winton", "distance": 350, "speed": 100}, {"start": "Hughenden", "end": "Charters Towers", "distance": 248, "speed": 100}, {"start": "Charters Towers", "end": "Townsville", "distance": 135, "speed": 100}, {"start": "Townsville", "end": "Mackay", "distance": 388, "speed": 100}, {"start": "Mackay", "end": "Rockhampton", "distance": 334, "speed": 100}, {"start": "Mackay", "end": "Emerald", "distance": 384, "speed": 100}, {"start": "Rockhampton", "end": "Brisbane", "distance": 682, "speed": 100}, {"start": "Rockhampton", "end": "Emerald", "distance": 270, "speed": 100}, {"start": "Brisbane", "end": "Roma", "distance": 482, "speed": 100}, {"start": "Roma", "end": "Charleville", "distance": 266, "speed": 100}, {"start": "Charleville", "end": "Blackall", "distance": 305, "speed": 100}, {"start": "Blackall", "end": "Barcaldine", "distance": 106, "speed": 100}, {"start": "Blackall", "end": "Windorah", "distance": 530, "speed": 50}, {"start": "Barcaldine", "end": "Emerald", "distance": 307, "speed": 100}, {"start": "Barcaldine", "end": "Hughenden", "distance": 500, "speed": 50}, {"start": "Barcaldine", "end": "Longreach", "distance": 106, "speed": 100}, {"start": "Longreach", "end": "Windorah", "distance": 311, "speed": 50}, {"start": "Longreach", "end": "Winton", "distance": 180, "speed": 100}, {"start": "Winton", "end": "Hughenden", "distance": 216, "speed": 100}, {"start": "Winton", "end": "Boulia", "distance": 363, "speed": 50}, {"start": "Winton", "end": "Windorah", "distance": 487, "speed": 50}, {"start": "Boulia", "end": "Mount Isa", "distance": 304, "speed": 50}, {"start": "Boulia", "end": "Bedourie", "distance": 194, "speed": 50}, {"start": "Bedourie", "end": "Birdsville", "distance": 193, "speed": 50}, {"start": "Birdsville", "end": "Windorah", "distance": 380, "speed": 50}, {"start": "Bedourie", "end": "Windorah", "distance": 411, "speed": 50}]} \ No newline at end of file diff --git a/examples/roads.png b/examples/roads.png deleted file mode 100644 index 4938b223..00000000 Binary files a/examples/roads.png and /dev/null differ diff --git a/examples/roads.py b/examples/roads.py deleted file mode 100644 index 6c373a61..00000000 --- a/examples/roads.py +++ /dev/null @@ -1,39 +0,0 @@ -import json -from pgraph import UGraph, DGraph - -# load the JSON file -with open('queensland.json', 'r') as f: - data = json.loads(f.read()) - -# create an undirected graph -g = UGraph() - -# create a vertex for every place, by providing the coordinate the -# resulting graph will be embedded -for name, info in data['places'].items(): - g.add_vertex(name=name, coord=info["utm"]) - -# create an edge for every route, and the cost is the driving distance -for route in data['routes']: - g.add_edge(route['start'], route['end'], cost=route['distance']) - -# print the graph in tabular form -print(g) -print(g.edges()) - -# solve for minimum distance path -p, length, parents = g.path_Astar('Hughenden', 'Brisbane', - verbose=True, summary=True) - -# compute the path using A*, the result is a list of UVertex objects -print(f"shortest path has length {length:.1f}:", - '->'.join([str(x.name) for x in p])) - -# plot the graph, and overlay the path -g.plot() -g.highlight_path(p, alpha=0.5, scale=2) - -# turn the vertex parent information into a tree, it's a dict that maps a -# vertex to its parent. -tree = DGraph.Dict(parents) -tree.showgraph() # display it via the browser diff --git a/examples/routes.json b/examples/routes.json deleted file mode 100644 index 63f585e8..00000000 --- a/examples/routes.json +++ /dev/null @@ -1,31 +0,0 @@ - [ - ["Camooweal","Mount Isa",188,1], - ["Mount Isa","Cloncurry",118,1], - ["Cloncurry","Hughenden",401,1], - ["Cloncurry","Winton",350,1], - ["Hughenden","Charters Towers",248,1], - ["Charters Towers","Townsville",135,1], - ["Townsville","Mackay",388,1], - ["Mackay","Rockhampton",334,1], - ["Mackay","Emerald",384,1], - ["Rockhampton","Brisbane",682,1], - ["Rockhampton","Emerald",270,1], - ["Brisbane","Roma",482,1], - ["Roma","Charleville",266,1], - ["Charleville","Blackall",305,1], - ["Blackall","Barcaldine",106,1], - ["Blackall","Windorah",530,2], - ["Barcaldine","Emerald",307,1,1], - ["Barcaldine","Hughenden",500,2], - ["Barcaldine","Longreach",106,1], - ["Longreach","Windorah",311,2], - ["Longreach","Winton",180,1], - ["Winton","Hughenden",216,1], - ["Winton","Boulia",363,2], - ["Winton","Windorah",487,2], - ["Boulia","Mount Isa",304,2], - ["Boulia","Bedourie",194,2], - ["Bedourie","Birdsville",193,2], - ["Birdsville","Windorah",380,2], - ["Bedourie","Windorah",411,2] - ] diff --git a/genindex.html b/genindex.html new file mode 100644 index 00000000..8db218e5 --- /dev/null +++ b/genindex.html @@ -0,0 +1,700 @@ + + + + + + + + Index — Simple graph functionality for Python documentation + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+
    +
  • + +
  • +
  • +
+
+
+
+
+ + +

Index

+ +
+ _ + | A + | C + | D + | E + | H + | I + | L + | M + | N + | P + | R + | S + | U + | V + | X + | Y + | Z + +
+

_

+ + +
+ +

A

+ + + +
+ +

C

+ + + +
+ +

D

+ + + +
+ +

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+ + + +
+ +

H

+ + + +
+ +

I

+ + + +
+ +

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+ + +
+ +

M

+ + + +
    +
  • + module + +
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+ +

N

+ + + +
+ +

P

+ + + +
+ +

R

+ + +
+ +

S

+ + + +
+ +

U

+ + + +
+ +

V

+ + + +
+ +

X

+ + +
+ +

Y

+ + +
+ +

Z

+ + +
+ + + +
+
+
+ +
+ +
+

© Copyright 2020, Peter Corke. + Last updated on 08-Jan-2025. +

+
+ + Built with Sphinx using a + theme + provided by Read the Docs. + + +
+
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/index.html b/index.html new file mode 100644 index 00000000..9b363e86 --- /dev/null +++ b/index.html @@ -0,0 +1,161 @@ + + + + + + + + + Graphs for Python — Simple graph functionality for Python documentation + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

Graphs for Python

+

This package provides a set of classes for manipulating simple directed and undirected graphs in Python.

+ +
+
+

Indices and tables

+ +
+ + +
+
+
+ +
+ +
+

© Copyright 2020, Peter Corke. + Last updated on 08-Jan-2025. +

+
+ + Built with Sphinx using a + theme + provided by Read the Docs. + + +
+
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/intro.html b/intro.html new file mode 100644 index 00000000..e77e8357 --- /dev/null +++ b/intro.html @@ -0,0 +1,1014 @@ + + + + + + + + + + + Introduction — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
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+ +
+

Introduction

+
+

Introduction

+

This is a modified version of a paper accepted to ICRA2021 [corke21a].

+

The Robotics Toolbox for MATLAB® (RTB-M) was created around 1991 to support +Peter Corke’s PhD research and was first published in 1995-6 [Corke95] +[Corke96]. It has evolved over 25 years to track changes and improvements to +the MATLAB language and ecosystem, such as the addition of structures, objects, +lists (cell arrays) and strings, myriad of other improvements to the language, +new graphics and new tools such as IDE, debugger, notebooks (LiveScripts), apps +and continuous integration. An adverse consequence is that many poor (in +retrospect) early design decisions hinder development.

+

Over time additional functionality was added, in particular for vision, and two +major refactorings led to the current state of three toolboxes: Robotics Toolbox +for MATLAB and Machine Vision Toolbox for MATLAB (1999) both of which are built +on the Spatial Math Toolbox for MATLAB (SMTB-M) in 2019.

+

The code was formally open sourced to support its use for the third edition of +John Craig’s book [Craig2005]. It was hosted on ftp sites, personal web +servers, Google code and currently GitHub and maintained under a succession of +version control tools including rcs, cvs, svn and git. A support forum on +Google Groups was established in 2008 and as of 2020 has over 1400 members.

+
+
+

A Python version

+

The imperative for a Python version has long existed and the first port was +started in 2008 but ultimately failed for lack of ongoing resources to complete +a sufficient subset of functionality. Subsequent attempts have all met the same +fate.

+

The design goals of this version can be summarised as new functionality:

+
    +
  • A superset of the MATLAB Toolbox functionality

  • +
  • Build on the Spatial Math Toolbox for Python [SMTB-P] which provides objects to +represent rotations as SO(2) and SE(3) matrices as well as unit-quaternions; +rigid-body motions as SE(2) and SE(3) matrices or twists in +se(2) and se(3); and Featherstone’s spatial vectors [Featherstone87].

  • +
  • Support models expressed using Denavit-Hartenberg notation (standard and +modified), elementary transform sequences [Corke07] [Haviland20], and URDF-style +rigid-body trees. Support branched, but not closed-loop or parallel, robots

  • +
  • Collision checking

  • +
+

and improved software engineering:

+
    +
  • Use Python 3 (3.6 and greater)

  • +
  • Utilize WebGL and Javascript graphics technologies

  • +
  • Documentation in ReStructured Text using Sphinx and delivered via GitHub pages.

  • +
  • Hosted on GitHub with continuous integration using GitHub actions

  • +
  • High code-quality metrics for test coverage and automated code review and security analysis

  • +
  • As few dependencies as possible, in particular being able to work with ROS but not be dependent on ROS. This sidesteps ROS constraints on operating system and Python versions.

  • +
  • Modular approach to interfacing to different graphics libraries, simulators and physical robots.

  • +
  • Support Python notebooks which allows publication of static notebooks (for example via GitHub) and interactive online notebooks (MyBinder.org).

  • +
  • Use of UniCode characters to make console output easier to read

  • +
+
+
+

Spatial math layer

+

Robotics and computer vision require us to describe position, orientation and +pose in 3D space. Mobile robotics has the same requirement, but generally for 2D +space. We therefore need tools to represent quantities such as rigid-body +transformations (matrices \(\in \SE{n}\) or twists \(\in \se{n}\)), +rotations (matrices \(\in \SO{n}\) or \(\so{n}\), Euler or roll-pitch-yaw +angles, or unit quaternions \(\in \mathrm{S}^3\)). Such capability is amongst the oldest in +RTB-M and the equivalent functionality exists in RTB-P which makes use of the +Spatial Maths Toolbox for Python (SMTB-P) [SMTB-P]. For example:

+
>>> from spatialmath.base import *
+>>> T = transl(0.5, 0.0, 0.0) @ rpy2tr(0.1, 0.2, 0.3, order='xyz') @ trotx(-90, 'deg')
+>>> print(T)
+[[ 0.9752 -0.1987 -0.0978  0.5   ]
+ [ 0.1538  0.2896  0.9447  0.    ]
+ [-0.1593 -0.9363  0.313   0.    ]
+ [ 0.      0.      0.      1.    ]]
+
+
+

There is strong similarity to the equivalent MATLAB case apart from the use of +the @ operator, the use of keyword arguments instead of keyword-value pairs, +and the format of the printed array. All the classic RTB-M functions are +provided in the spatialmath.base package as well as additional functions for +quaternions, vectors, twists and argument handling. There are also functions to +perform interpolation, plot and animate coordinate frames, and create movies, +using matplotlib. The underlying datatypes in all cases are 1D and 2D NumPy +arrays.

+
+

Warning

+

For a user transitioning from MATLAB the most significant difference is +the use of 1D arrays – all MATLAB arrays have two dimensions, even if one of +them is equal to one.

+
+

However some challenges arise when using arrays, whether native MATLAB matrices +or NumPy arrays as in this case. Firstly, arrays are not typed and for example a +\(3 \times 3\) array could be an element of \(\SE{2}\) or +\(\SO{3}\) or an arbitrary matrix.

+

Secondly, the operators we need for poses are a subset of those available for +matrices, and some operators may need to be redefined in a specific way. For +example, \(\SE{3} * \SE{3} \rightarrow \SE{3}\) but \(\SE{3} + \SE{3} \rightarrow \mathbb{R}^{4 \times 4}\), and equality testing for a +unit-quaternion has to respect the double mapping.

+

Thirdly, in robotics we often need to represent time sequences of poses. We +could add an extra dimension to the matrices representing rigid-body +transformations or unit-quaternions, or place them in a list. The first +approach is cumbersome and reduces code clarity, while the second cannot ensure +that all elements of the list have the same type.

+

We use classes and data encapsulation to address all these issues. SMTB-P +provides abstraction classes SE3, Twist3, SO3, UnitQuaternion, +SE2, Twist2 and SO2. For example, the previous example could be written +as:

+
 1>>> from spatialmath import *
+ 2>>> T = SE3(0.5, 0.0, 0.0) * SE3.RPY([0.1, 0.2, 0.3], order='xyz') * SE3.Rx(-90, unit='deg')
+ 3>>> print(T)
+ 4   0.9752   -0.1987   -0.09784   0.5       
+ 5   0.1538    0.2896    0.9447    0         
+ 6  -0.1593   -0.9363    0.313     0         
+ 7   0         0         0         1         
+ 8
+ 9>>> T.eul()
+10array([ 1.674 ,  1.2525, -1.4022])
+11>>> T.R
+12array([[ 0.9752, -0.1987, -0.0978],
+13       [ 0.1538,  0.2896,  0.9447],
+14       [-0.1593, -0.9363,  0.313 ]])
+15>>> T.t
+16array([0.5, 0. , 0. ])
+
+
+

where composition is denoted by the * operator and the matrix is printed more elegantly (and elements are color +coded at the console or in ipython). +SE3.RPY() is a class method that acts like a constructor, creating an SE3 instance from a set of roll-pitch-yaw angles, +and SE3.Rx() creates an SE3 instance from a pure rotation about the x-axis. +Attempts to compose with a non SE3 instance would result in a TypeError.

+

The orientation of the new coordinate frame may be expressed in terms of Euler angles (line 9) +and components can be extracted such as the rotation submatrix (line 11) and translation (line 15).

+

The pose T can also be displayed as a 3D coordinate frame:

+
>>> T.plot(color='red', label='2')
+
+
+

Rotation can also be represented by a unit quaternion

+
>>> from spatialmath import UnitQuaternion
+>>> print(UnitQuaternion.Rx(0.3))
+ 0.9888 <<  0.1494,  0.0000,  0.0000 >>
+>>> print(UnitQuaternion.AngVec(0.3, [1, 0, 0]))
+ 0.9888 <<  0.1494,  0.0000,  0.0000 >>
+
+
+

which again demonstrates several alternative constructors.

+
+

Multiple values

+

To support sequences of values each of these types inherits list properties from collections.UserList

+
+Any of the SMTB-P pose classes can contain a list of values +
+

Any of the SMTB-P pose classes can contain a list of values

+
+
+

We can index the values, iterate over the values, assign to values. +Some constructors take an array-like argument allowing creation of multi-valued pose objects, +for example:

+
>>> from spatialmath import SE3
+>>> import numpy as np
+>>> R = SE3.Rx(np.linspace(0, np.pi/2, num=100))
+>>> len(R)
+100
+
+
+

where the instance R contains a sequence of 100 rotation matrices. +Composition with a single-valued (scalar) pose instance broadcasts the scalar +across the sequence

+
+Overloaded operators support broadcasting +
+

Overloaded operators support broadcasting

+
+
+
+
+

Common constructors

+

The Toolboxes classes are somewhat polymorphic and share many “variant constructors” that allow object construction:

+
    +
  • with orientation expressed in terms of canonic axis rotations, Euler vectors, angle-vector pair, +Euler or roll-pitch-yaw angles or orientation- and approach-vectors.

  • +
  • from random values .Rand()

  • +
  • SE3, SE2, SO3 and SO2 also support a matrix exponential constructor where the argument is the +corresponding Lie algebra element.

  • +
  • empty, i.e. having no values or a length of 0 .Empty()

  • +
  • an array of N values initialized to the object’s identity value .Alloc(N)

  • +
+
+
+

Common methods and operators

+

The types all have an inverse method .inv() and support composition with the inverse using the / operator +and integer exponentiation (repeated composition) using the ** operator. +Other overloaded operators include *, *=, **, **=, /, /=, ==, !=, +, -.

+

All of this allows for concise and readable code. +The use of classes ensures type safety and that the matrices abstracted by the class are always valid members of +the group. +Operations such as addition, which are not group operations, yield a NumPy array rather than a class instance.

+
+
+

Performance

+

These benefits come at a price in terms of execution time due to the overhead of +constructors, methods which wrap base functions, and type checking. The +Toolbox supports SymPy which provides powerful symbolic support for Python and +it works well in conjunction with NumPy, ie. a NumPy array can contain symbolic +elements. Many the Toolbox methods and functions contain extra logic to ensure +that symbolic operations work as expected. While this adds to the overhead it +means that for the user, working with symbols is as easy as working with +numbers.

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Performance on a 3.6GHz Intel Core i9

Function/method

Execution time

base.rotx()

4.07 μs

base.trotx()

5.79 μs

SE3.Rx()

12.3 μs

SE3 * SE3

4.69 μs

4x4 @

0.986 μs

SE3.inv()

7.62 μs

base.trinv()

4.19 μs

np.linalg.inv()

4.49 μs

+
+
+
+

Robotics Toolbox

+
+

Robot models

+

The Toolbox ships with over 30 robot models, most of which are purely kinematic +but some have inertial and frictional parameters. Kinematic models can be +specified in a variety of ways: standard or modified Denavit-Hartenberg (DH, +MDH) notation, as an ETS string [Corke07], as a rigid-body tree, or from a URDF +file.

+
+
+

Denavit-Hartenberg parameters

+

To specify a kinematic model using DH notation, we create a new subclass of DHRobot and pass the superclass constructor +a list of link objects. For example, a Puma560 is simply:

+
>>> robot = DHRobot(
+    [
+        RevoluteDH(alpha=pi/2),
+        RevoluteDH(a=0.4318),
+        RevoluteDH(d=0.15005, a=0.0203, alpha=-pi/2),
+        RevoluteDH(d=0.4318, alpha=pi/2)
+        RevoluteDH(alpha=-pi/2),
+        RevoluteDH()
+    ], name="Puma560")
+
+
+

where only the non-zero parameters need to be specified. In this case we used +RevoluteDH objects for a revolute joint described using standard DH +conventions. Other classes available are PrismaticDH, RevoluteMDH and +PrismaticMDH. Other parameters such as mass, CoG, link inertia, motor +inertia, viscous friction, Coulomb friction, and joint limits can also be +specified using additional keyword arguments.

+

The toolbox provides such definitions wrapped as class definitions, for example:

+
class Puma560(DHRobot):
+
+    def __init__(self):
+        super().__init__(
+                [
+                    RevoluteDH(alpha=pi/2),
+                    RevoluteDH(a=0.4318),
+                    RevoluteDH(d=0.15005, a=0.0203, alpha=-pi/2),
+                    RevoluteDH(d=0.4318, alpha=pi/2)
+                    RevoluteDH(alpha=-pi/2),
+                    RevoluteDH()
+                ], name="Puma560"
+                        )
+
+
+

We can now easily perform standard kinematic operations

+
 1>>> import roboticstoolbox as rtb
+ 2>>> puma = rtb.models.DH.Puma560()                  # instantiate robot model
+ 3>>> print(puma)
+ 4DHRobot: Puma 560 (by Unimation), 6 joints (RRRRRR), dynamics, geometry, standard DH parameters
+ 5┌────┬────────┬────────┬────────┬─────────┬────────┐
+ 6│θⱼ  │   dⱼ   │   aⱼ   │   ⍺ⱼ   │   q⁻    │   q⁺   │
+ 7├────┼────────┼────────┼────────┼─────────┼────────┤
+ 8│ q1 │ 0.6718 │      0 │  90.0° │ -160.0° │ 160.0° │
+ 9│ q2 │      0 │ 0.4318 │   0.0° │ -110.0° │ 110.0° │
+10│ q3 │   0.15 │ 0.0203 │ -90.0° │ -135.0° │ 135.0° │
+11│ q4 │ 0.4318 │      0 │  90.0° │ -266.0° │ 266.0° │
+12│ q5 │      0 │      0 │ -90.0° │ -100.0° │ 100.0° │
+13│ q6 │      0 │      0 │   0.0° │ -266.0° │ 266.0° │
+14└────┴────────┴────────┴────────┴─────────┴────────┘
+15
+16┌─┬──┐
+17└─┴──┘
+18
+19┌─────┬─────┬──────┬───────┬─────┬──────┬─────┐
+20│name │ q0  │ q1   │ q2    │ q3  │ q4   │ q5  │
+21├─────┼─────┼──────┼───────┼─────┼──────┼─────┤
+22│  qr │  0° │  90° │ -90°  │  0° │  0°  │  0° │
+23│  qz │  0° │  0°  │  0°   │  0° │  0°  │  0° │
+24│  qn │  0° │  45° │  180° │  0° │  45° │  0° │
+25│  qs │  0° │  0°  │ -90°  │  0° │  0°  │  0° │
+26└─────┴─────┴──────┴───────┴─────┴──────┴─────┘
+27
+28>>> print(puma.qr)
+29[ 0.      1.5708 -1.5708  0.      0.      0.    ]
+30>>> T = puma.fkine([0.1, 0.2, 0.3, 0.4, 0.5, 0.6])  # forward kinematics
+31>>> print(T)
+32   0.1217   -0.6067   -0.7856    0.2478    
+33   0.8184    0.5092   -0.2665   -0.1259    
+34   0.5617   -0.6105    0.5584    1.146     
+35   0         0         0         1         
+36
+37>>> sol = puma.ikine_LM(T)                          # inverse kinematics
+38>>> print(sol)
+39IKSolution: q=[0.1, 2.025, 2.936, -2.894, -2.273, -2.025], success=True, iterations=14, searches=1, residual=4.91e-10
+
+
+

The Toolbox supports named joint configurations and these are shown in the table +at lines 16-22.

+

ikine_LM is a generalised iterative numerical solution based on +Levenberg-Marquadt minimization, and additional status results are also +returned as part of a named tuple.

+

The default plot method:

+
>>> puma.plot(q)
+
+
+

uses matplotlib to produce a “noodle robot” plot like

+
+Puma560, with a velocity ellipsoid, rendered using the default matplotlib visualizer +
+

Puma560, with a velocity ellipsoid, rendered using the default matplotlib visualizer.

+
+
+

and we can use the mouse to rotate and zoom the plot.

+
+

Note

+

The initial joint configuration for the inverse-kinematic solution may be specified, but +defaults to zero, and affects both the search time and the solution found, since in general +a manipulator may have several multiple joint configurations which result in the same end-effector pose. +For a redundant manipulator, a solution will be found but there is no +explicit control over the null-space. For a manipulator with \(n < 6\) DOF +an additional argument is required to indicate which of the +\(6-n\) Cartesian DOF are to be unconstrained in the solution.

+
+
+

Note

+

A solution is not possible if the specified transform describes +a point out of reach of the manipulator – in such a case the function will +return with an error.

+
+

The inverse kinematic procedure for most robots can +be derived symbolically +and an efficient closed-form solution obtained. +Some provided robot models have an analytical solution coded, for example:

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()       # instantiate robot model
+>>> T = puma.fkine([0.1, 0.2, 0.3, 0.4, 0.5, 0.6])
+>>> puma.ikine_a(T, config="lun")        # analytic inverse kinematics
+IKSolution(q=array([ 2.1012,  1.1163,  0.3   ,  0.9528, -1.6505, -0.986 ]), success=True, iterations=0, searches=0, residual=0.0, reason='')
+
+
+

where we have specified a left-handed, elbow up and wrist no-flip configuration.

+
+
+

ETS notation

+

A Puma robot can also be specified in ETS format [Corke07] as a sequence of simple rigid-body transformations – pure translation +or pure rotation – each with either a constant parameter or a free parameter which is a joint variable.

+
 1>>> from roboticstoolbox import ET
+ 2>>> import roboticstoolbox as rtb
+ 3>>> # Puma dimensions (m), see RVC2 Fig. 7.4 for details
+ 4>>> l1 = 0.672; l2 = -0.2337; l3 = 0.4318; l4 = 0.0203; l5 = 0.0837; l6 = 0.4318
+ 5>>> e = ET.tz(l1) * ET.Rz() * ET.ty(l2) * ET.Ry() * ET.tz(l3) * ET.tx(l4) * ET.ty(l5) * ET.Ry() * ET.tz(l6) * ET.Rz() * ET.Ry() * ET.Rz()
+ 6>>> print(e)
+ 7tz(0.672) ⊕ Rz(q0) ⊕ ty(-0.2337) ⊕ Ry(q1) ⊕ tz(0.4318) ⊕ tx(0.0203) ⊕ ty(0.0837) ⊕ Ry(q2) ⊕ tz(0.4318) ⊕ Rz(q3) ⊕ Ry(q4) ⊕ Rz(q5)
+ 8>>> robot = rtb.Robot(e)
+ 9>>> print(robot)
+10ERobot: , 6 joints (RRRRRR)
+11┌─────┬────────┬───────┬────────┬───────────────────────────────────────────────┐
+12│link │  link  │ joint │ parent │              ETS: parent to link              │
+13├─────┼────────┼───────┼────────┼───────────────────────────────────────────────┤
+14│   0 │ link0  │     0 │ BASE   │ tz(0.672) ⊕ Rz(q0)                            │
+15│   1 │ link1  │     1 │ link0  │ ty(-0.2337) ⊕ Ry(q1)                          │
+16│   2 │ link2  │     2 │ link1  │ tz(0.4318) ⊕ tx(0.0203) ⊕ ty(0.0837) ⊕ Ry(q2) │
+17│   3 │ link3  │     3 │ link2  │ tz(0.4318) ⊕ Rz(q3)                           │
+18│   4 │ link4  │     4 │ link3  │ Ry(q4)                                        │
+19│   5 │ @link5 │     5 │ link4  │ Rz(q5)                                        │
+20└─────┴────────┴───────┴────────┴───────────────────────────────────────────────┘
+21
+
+
+

Line 3 defines the unique lengths of the Puma robot, and line 4 defines the kinematic chain in +terms of elementary transforms. +In line 7 we pass this to the constructor for a Robot which partitions the +elementary transform sequence into a series of links and joints – link frames are declared +after each joint variable as well as the start and end of the sequence. +The Robot can represent single-branched robots with any combination of revolute and prismatic joints, but +can also represent more general branched mechanisms.

+
+
+

Robot: rigid-body tree and URDF import

+

The final approach to manipulator modeling is to an import a URDF file. The Toolbox includes a parser with built-in xacro processor +which makes many models from the ROS universe available.

+

Provided models, such as for Panda or Puma, are again encapsulated as classes:

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.DH.Panda()
+>>> print(panda)
+DHRobot: Panda (by Franka Emika), 7 joints (RRRRRRR), dynamics, geometry, modified DH parameters
+┌────────┬────────┬─────┬───────┬─────────┬────────┐
+│ aⱼ₋₁   │  ⍺ⱼ₋₁  │ θⱼ  │  dⱼ   │   q⁻    │   q⁺   │
+├────────┼────────┼─────┼───────┼─────────┼────────┤
+│    0.0 │   0.0° │  q1 │ 0.333 │ -166.0° │ 166.0° │
+│    0.0 │ -90.0° │  q2 │   0.0 │ -101.0° │ 101.0° │
+│    0.0 │  90.0° │  q3 │ 0.316 │ -166.0° │ 166.0° │
+│ 0.0825 │  90.0° │  q4 │   0.0 │ -176.0° │  -4.0° │
+│-0.0825 │ -90.0° │  q5 │ 0.384 │ -166.0° │ 166.0° │
+│    0.0 │  90.0° │  q6 │   0.0 │   -1.0° │ 215.0° │
+│  0.088 │  90.0° │  q7 │ 0.107 │ -166.0° │ 166.0° │
+└────────┴────────┴─────┴───────┴─────────┴────────┘
+
+┌─────┬───────────────────────────────────────┐
+│tool │ t = 0, 0, 0.1; rpy/xyz = -45°, 0°, 0° │
+└─────┴───────────────────────────────────────┘
+
+┌─────┬─────┬────────┬─────┬───────┬─────┬───────┬──────┐
+│name │ q0  │ q1     │ q2  │ q3    │ q4  │ q5    │ q6   │
+├─────┼─────┼────────┼─────┼───────┼─────┼───────┼──────┤
+│  qr │  0° │ -17.2° │  0° │ -126° │  0° │  115° │  45° │
+│  qz │  0° │  0°    │  0° │  0°   │  0° │  0°   │  0°  │
+└─────┴─────┴────────┴─────┴───────┴─────┴───────┴──────┘
+
+>>> T = panda.fkine(panda.qz)
+>>> print(T)
+   0.7071    0.7071    0         0.088     
+   0.7071   -0.7071    0         0         
+   0         0        -1         0.823     
+   0         0         0         1         
+
+
+
+

and kinematic operations are performed using methods with the same name +as discussed above. +For branched robots, with multiple end-effectors, the name of the frame of interest must be provided.

+

Some URDF models have multiple end-effectors, in which case the particular +end-effector must be specified.

+
>>> import roboticstoolbox as rtb
+>>> panda = rtb.models.URDF.Panda()
+>>> print(panda)
+ERobot: panda (by Franka Emika), 7 joints (RRRRRRR), 1 gripper, geometry, collision
+┌─────┬──────────────┬───────┬─────────────┬────────────────────────────────────────────────┐
+│link │     link     │ joint │   parent    │              ETS: parent to link               │
+├─────┼──────────────┼───────┼─────────────┼────────────────────────────────────────────────┤
+│   0 │ panda_link0  │       │ BASE        │ SE3()                                          │
+│   1 │ panda_link1  │     0 │ panda_link0 │ SE3(0, 0, 0.333) ⊕ Rz(q0)                      │
+│   2 │ panda_link2  │     1 │ panda_link1 │ SE3(-90°, -0°, 0°) ⊕ Rz(q1)                    │
+│   3 │ panda_link3  │     2 │ panda_link2 │ SE3(0, -0.316, 0; 90°, -0°, 0°) ⊕ Rz(q2)       │
+│   4 │ panda_link4  │     3 │ panda_link3 │ SE3(0.0825, 0, 0; 90°, -0°, 0°) ⊕ Rz(q3)       │
+│   5 │ panda_link5  │     4 │ panda_link4 │ SE3(-0.0825, 0.384, 0; -90°, -0°, 0°) ⊕ Rz(q4) │
+│   6 │ panda_link6  │     5 │ panda_link5 │ SE3(90°, -0°, 0°) ⊕ Rz(q5)                     │
+│   7 │ panda_link7  │     6 │ panda_link6 │ SE3(0.088, 0, 0; 90°, -0°, 0°) ⊕ Rz(q6)        │
+│   8 │ @panda_link8 │       │ panda_link7 │ SE3(0, 0, 0.107)                               │
+└─────┴──────────────┴───────┴─────────────┴────────────────────────────────────────────────┘
+
+┌─────┬─────┬────────┬─────┬───────┬─────┬───────┬──────┐
+│name │ q0  │ q1     │ q2  │ q3    │ q4  │ q5    │ q6   │
+├─────┼─────┼────────┼─────┼───────┼─────┼───────┼──────┤
+│  qr │  0° │ -17.2° │  0° │ -126° │  0° │  115° │  45° │
+│  qz │  0° │  0°    │  0° │  0°   │  0° │  0°   │  0°  │
+└─────┴─────┴────────┴─────┴───────┴─────┴───────┴──────┘
+
+>>> T = panda.fkine(panda.qz, end='panda_hand')
+>>> print(T)
+   0.7071    0.7071    0         0.088     
+   0.7071   -0.7071    0         0         
+   0         0        -1         0.926     
+   0         0         0         1         
+
+
+
+

In the table above we see the end-effectors indicated by @ (determined automatically +from the URDF file), so we specify one of these. We can also specify any +other link in order to determine the pose of that link’s coordinate frame.

+

This URDF model comes with meshes provided as Collada file which provide +detailed geometry and color. This can be visualized using the Swift simulator:

+
>>> panda.plot(qz, backend="swift")
+
+
+

which produces the 3-D plot

+
+Panda robot rendered using the Toolbox’s Swift visualizer +
+

Panda robot rendered using the Toolbox’s Swift visualizer.

+
+
+

Swift is a web-based visualizer using three.js to provide high-quality 3D animations. +It can produce vivid 3D effects using anaglyphs viewed with colored glasses. +Animations can be recorded as MP4 files or animated GIF files which are useful for inclusion in GitHub markdown documents.

+
+
+
+

Trajectories

+

A joint-space trajectory for the Puma robot from its zero angle +pose to the upright (or READY) pose in 100 steps is

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> traj = rtb.jtraj(puma.qz, puma.qr, 100)
+>>> traj.q.shape
+(100, 6)
+
+
+

where puma.qr is an example of a named joint configuration. +traj is named tuple with elements q = \(\vec{q}_k\), qd = \(\dvec{q}_k\) and qdd = \(\ddvec{q}_k\). +Each element is an array with one row per time step, and each row is a joint coordinate vector. +The trajectory is a fifth order polynomial which has continuous jerk. +By default, the initial and final velocities are zero, but these may be specified by additional +arguments.

+

We could plot the joint coordinates as a function of time using the convenience +function:

+
>>> rtb.qplot(traj.q)
+
+
+

Straight line (Cartesian) paths can be generated in a similar way between +two points specified by a pair of poses in \(\SE{3}\)

+

At line 9 we see that the resulting trajectory, Ts, is an SE3 instance with 200 values.

+

At line 11 we compute the inverse kinematics of each pose in the trajectory +using a single call to the ikine_LM method. +The result is a list of named tuples, which gives the IK success status for +each time step. +At line 12 we convert this into an array, with one row per time step, and each +row is a joint coordinate. +The starting +joint coordinates for each inverse kinematic solution +is taken as the result of the solution at the previous time step.

+
+
+

Symbolic manipulation

+

As mentioned earlier, the Toolbox supports symbolic manipulation using SymPy. For example:

+
>>> import spatialmath.base as base
+>>> phi, theta, psi = base.sym.symbol('φ, ϴ, ψ')
+>>> base.rpy2r(phi, theta, psi)
+array([[cos(ψ)*cos(ϴ), sin(φ)*sin(ϴ)*cos(ψ) - sin(ψ)*cos(φ),
+        sin(φ)*sin(ψ) + sin(ϴ)*cos(φ)*cos(ψ)],
+       [sin(ψ)*cos(ϴ), sin(φ)*sin(ψ)*sin(ϴ) + cos(φ)*cos(ψ),
+        -sin(φ)*cos(ψ) + sin(ψ)*sin(ϴ)*cos(φ)],
+       [-sin(ϴ), sin(φ)*cos(ϴ), cos(φ)*cos(ϴ)]], dtype=object)
+
+
+

The capability extends to forward kinematics

+
1>>> import roboticstoolbox as rtb
+2>>> import spatialmath.base as base
+3>>> puma = rtb.models.DH.Puma560(symbolic=True)
+4>>> q = base.sym.symbol("q_:6") # q = (q_1, q_2, ... q_5)
+5>>> T = puma.fkine(q)
+6>>> T.t[0]
+70.15005*sin(q_0) - 0.0203*sin(q_1)*sin(q_2)*cos(q_0) - 0.4318*sin(q_1)*cos(q_0)*cos(q_2) - 0.4318*sin(q_2)*cos(q_0)*cos(q_1) + 0.0203*cos(q_0)*cos(q_1)*cos(q_2) + 0.4318*cos(q_0)*cos(q_1)
+
+
+

If we display the value of puma we see that the \(\alpha_j\) values are +now displayed in red to indicate that they are symbolic constants. The +x-coordinate of the end-effector is given by line 7.

+

SymPy allows any expression to be converted to LaTeX or a variety of languages +including C, Python and Octave/MATLAB.

+
+
+

Differential kinematics

+

The Toolbox computes Jacobians:

+
>>> J = puma.jacob0(q)
+>>> J = puma.jacobe(q)
+
+
+

in the base or end-effector frames respectively, as NumPy arrays. +At a singular configuration

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> J = puma.jacob0(puma.qr)
+>>> np.linalg.matrix_rank(J)
+5
+>>> rtb.jsingu(J)
+column 5 =2.6e-15 column_0 + 1 column_3
+
+
+

Jacobians can also be computed for symbolic joint variables as for forward kinematics above.

+

For Robot instances we can also compute the Hessians:

+
>>> H = puma.hessian0(q)
+>>> H = puma.hessiane(q)
+
+
+

in the base or end-effector frames respectively, as 3D NumPy arrays in \(\mathbb{R}^{6 \times n \times n}\).

+

For all robot classes we can compute manipulability

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> m = puma.manipulability(puma.qn)
+>>> print("Yoshikawa manipulability is", m)
+Yoshikawa manipulability is 0.07861716534599998
+>>> m = puma.manipulability(puma.qn, method="asada")
+>>> print("Asada manipulability is", m)
+Asada manipulability is 0.0043746137281665005
+
+
+

for the Yoshikawa and Asada measures respectively, and

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> m = puma.manipulability(puma.qn, axes="trans")
+>>> print("Yoshikawa manipulability is", m)
+Yoshikawa manipulability is 0.11118146146764128
+
+
+

is the Yoshikawa measure computed for just the task space translational degrees +of freedom. +For Robot instances we can also compute the manipulability +Jacobian:

+
>>> Jm = puma.manipm(q, J, H)
+
+
+

such that \(\dot{m} = \mat{J}_m(\vec{q}) \dvec{q}\).

+
+

Dynamics

+

The Python Toolbox supports several approaches to computing dynamics. +For models defined using standard or modified DH notation we use a classical version of the recursive Newton-Euler +algorithm implemented in Python or C.

+
+

Note

+

The same C code as used by RTB-M is called directly from Python, and does not use NumPy.

+
+

For example, the inverse dynamics

+
>>> import roboticstoolbox as rtb
+>>> puma = rtb.models.DH.Puma560()
+>>> tau = puma.rne(puma.qn, np.zeros((6,)), np.zeros((6,)))
+>>> print(tau)
+[-0.     31.6399  6.0351  0.      0.0283  0.    ]
+
+
+

is the gravity torque for the robot in the configuration qn.

+

Inertia, Coriolis/centripetal and gravity terms are computed by:

+
>>> puma.inertia(q)
+>>> puma.coriolis(q, qd)
+>>> puma.gravload(q)
+
+
+

respectively, using the method of Orin and Walker from the inverse dynamics. These values include the effect of motor inertia and friction.

+

Forward dynamics are given by:

+
>>> qdd = puma.accel(q, tau, qd)
+
+
+

We can integrate this over time by:

+
>>> q = puma.fdyn(5, q0, mycontrol, ...)
+
+
+

which uses an RK45 numerical integration from the SciPy package to solve for the joint trajectory q given the +optional control function called as:

+
tau = mycontrol(robot, t, q, qd, **args)
+
+
+

The fast C implementation is not capable of symbolic operation so a Python +version of RNE rne_python has been implemented as well. For a 6- or 7-DoF +manipulator the torque expressions have thousands of terms yet are computed in +less than a second. However, subsequent expression manipulation is slow.

+

For the Puma560 robot the C version of inverse dynamics takes 23μs while the +Python version takes 1.5ms (\(65\times\) slower). With symbolic operands it +takes 170ms (\(113\times\) slower) to produce the unsimplified torque +expressions.

+

For Robot subclasses there is also an implementation of Featherstone’s spatial vector +method, rne(), and SMTB-P provides a set of classes for spatial +velocity, acceleration, momentum, force and inertia.

+
+
+
+

New capability

+

There are several areas of innovation compared to the MATLAB version of the Toolbox.

+
+

Branched mechanisms

+

The RTB-M SerialLink class had no option to express branching. In RTB-P the +equivalent class is DHRobot is similarly limited, but a new class ERobot +is more general and allows for branching (but not closed kinematic loops). The +robot is described by a set of ELink objects, each of which points to its +parent link. The ERobot has references to the root and leaf ELink objects. This +structure closely mirrors the URDF representation, allowing for easy import of +URDF models.

+
+
+

Collision checking

+

RTB-M had a simple, contributed but unsupported, collision checking capability. +This is dramatically improved in the Python version using [PyBullet] +which supports primitive shapes such as Cylinders, Spheres and Boxes as well as +mesh objects. Every robot link can have a collision shape in addition to the shape +used for rendering. +We can conveniently perform collision checks between links as well as between +whole robots, discrete links, and objects in the world. For example a \(1 +\times 1 \times 1\) box centered at \((1,0,0)\) can be tested against all, or +just one link, of the robot by:

+
>>> panda = rtb.models.Panda()
+>>> obstacle = rtb.Box([1, 1, 1], SE3(1, 0, 0))
+>>> iscollision = panda.collided(obstacle) # boolean
+>>> iscollision = panda.links[0].collided(obstacle)
+
+
+

Additionally, we can compute the minimum Euclidean distance between whole +robots, discrete links, or objects. Each distance is the length of a line +segment defined by two points in the world frame

+
>>> d, p1, p2 = panda.closest_point(obstacle)
+>>> d, p1, p2 = panda.links[0].closest_point(obstacle)
+
+
+
+
+

Interfaces

+

RTB-M could only animate a robot in a figure, and there was limited but +not-well-supported ability to interface to V-REP and a physical robot. The +Python version supports a simple, but universal API to a robot inspired by the +simplicity and expressiveness of the OpenAI Gym API which was designed as a +toolkit for developing and comparing reinforcement learning algorithms. Whether +simulating a robot or controlling a real physical robot, the API operates in the +same manner, providing users with a common interface which is not found among +other robotics packages.

+

By default the Toolbox behaves like the MATLAB version with a plot method:

+
>>> puma.plot(q)
+
+
+

which will plot the robot at the specified joint configurmation, or animate it if q is an \(m \times 6\) matrix, using +the default PyPlot backend which draws a “noodle robot” using the PyPlot backend.

+

The more general solution, and what is implemented inside plot in the example above, is:

+
>>> pyplot = roboticstoolbox.backends.PyPlot()
+>>> pyplot.launch()
+>>> pyplot.add(puma)
+>>> puma.q = q
+>>> puma.step()
+
+
+

This makes it possible to animate multiple robots in the one graphical window, or the one robot in various environments either graphical +or real.

+
+Puma560 rendered using the web-based VPython visualizer. +
+

Puma560 rendered using the web-based VPython visualizer.

+
+
+

The VPython backend provides browser-based 3D graphics based on WebGL. This is advantageous for displaying on mobile +devices. Still frames and animations can be recorded.

+
+
+

Code engineering

+

The code is implemented in Python \(\ge 3.6\) and all code is hosted on GitHub and +unit-testing is performed using GitHub-actions. Test coverage is uploaded to +codecov.io for visualization and trending, and we use lgtm.com to perform +automated code review. The code is documented with ReStructured Text format +docstrings which provides powerful markup including cross-referencing, +equations, class inheritance diagrams and figures – all of which is converted +to HTML documentation whenever a change is pushed, and this is accessible via +GitHub pages. Issues can be reported via GitHub issues or patches submitted as +pull requests.

+

RTB-P, and its dependencies, can be installed simply by either of:

+
$ pip install roboticstoolbox-python
+
+$ conda install -c conda-forge roboticstoolbox-python
+
+
+

which includes basic visualization using matplotlib. +Options such as vpython can be used to specify additional dependencies to be installed. +The Toolbox adopts a “when needed” approach to many dependencies and will only attempt +to import them if the user attempts to exploit a functionality that requires it.

+

If a dependency is not installed, a warning provides instructions on how to install it using pip. +More details are given on the project home page. +This applies to the visualizers Vpython and Swift, as well as pybullet and ROS. +The Toolbox provides capability to import URDF-xacro files without ROS. +The backend architecture allows a user to connect to a ROS environment if required, and only then does ROS have to be +installed.

+
+
+
+

Conclusion

+

This article has introduced and demonstrated in tutorial form the principle +features of the Robotics Toolbox for Python which runs on Mac, Windows and Linux +using Python 3.6 or better. The code is free and open, and released under the +MIT licence. It provides many of the essential tools necessary for robotic +manipulator modelling, simulation and control which is essential for robotics +education and research. It is familiar yet new, and we hope it will serve the +community well for the next 25 years.

+

A high-performance reactive motion controller, NEO, is based on this toolbox +[neo]. Currently under development are backend interfaces for CoppeliaSim, +Dynamixel servo chains, and ROS; symbolic dynamics, simplification and code +generation; mobile robotics motion models, planners, EKF localization, map +making and SLAM; and a minimalist block-diagram simulation tool [bdsim].

+
+
+

References

+ +
+
+ + +
+
+
+ +
+ +
+

© Copyright 2023, Jesse Haviland and Peter Corke. + Last updated on 11-May-2024. +

+
+ + + +
+
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/matrices.html b/matrices.html new file mode 100644 index 00000000..f809143b --- /dev/null +++ b/matrices.html @@ -0,0 +1,276 @@ + + + + + + + + + Convert graph to matrix form — Simple graph functionality for Python documentation + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

Convert graph to matrix form

+

These methods convert a graph to a matrix representation .

+
+
+class PGraph.PGraph(arg=None, metric=None, heuristic=None, verbose=False)[source]
+
+
+Laplacian()[source]
+

Laplacian matrix for the graph

+
+
Returns:
+

Laplacian matrix

+
+
Return type:
+

NumPy ndarray

+
+
+

g.Laplacian() is the Laplacian matrix (NxN) of the graph where N +is the number of vertices.

+
+

Note

+
    +
  • Laplacian is always positive-semidefinite.

  • +
  • Laplacian has at least one zero eigenvalue.

  • +
  • +
    The number of zero-valued eigenvalues is the number of connected

    components in the graph.

    +
    +
    +
  • +
+
+
+
Seealso:
+

adjacency() incidence() degree()

+
+
+
+ +
+
+adjacency()[source]
+

Adjacency matrix of graph

+
+
Returns:
+

adjacency matrix

+
+
Return type:
+

ndarray(N,N)

+
+
+

The elements of the adjacency matrix [i,j] are 1 if vertex i is +connected to vertex j, else 0.

+
+

Note

+
    +
  • vertices are numbered in their order of creation. A vertex index +can be resolved to a vertex reference by graph[i].

  • +
  • for an undirected graph the matrix is symmetric

  • +
  • Eigenvalues of A are real and are known as the spectrum of the graph.

  • +
  • The element A[i,j] can be considered the number of walks of length one +edge from vertex i to vertex j (either zero or one).

  • +
  • If Ak = A ** k the element Ak[i,j] is the number of +walks of length k from vertex i to vertex j.

  • +
+
+
+
Seealso:
+

Laplacian() incidence() degree()

+
+
+
+ +
+
+degree()[source]
+

Degree matrix of graph

+
+
Returns:
+

degree matrix

+
+
Return type:
+

ndarray(N,N)

+
+
+

This is a diagonal matrix where element [i,i] is the number +of edges connected to vertex id i.

+
+
Seealso:
+

adjacency() incidence() laplacian()

+
+
+
+ +
+
+distance()[source]
+

Distance matrix of graph

+
+
Returns:
+

distance matrix

+
+
Return type:
+

ndarray(n,n)

+
+
+

The elements of the distance matrix D[i,j] is the edge cost of moving +from vertex i to vertex j. It is zero if the vertices are not +connected.

+
+ +
+
+incidence()[source]
+

Incidence matrix of graph

+
+
Returns:
+

incidence matrix

+
+
Return type:
+

ndarray(n,ne)

+
+
+

The elements of the incidence matrix I[i,j] are 1 if vertex i is +connected to edge j, else 0.

+
+

Note

+
    +
  • vertices are numbered in their order of creation. A vertex index +can be resolved to a vertex reference by graph[i].

  • +
  • edges are numbered in the order they appear in graph.edges().

  • +
+
+
+
Seealso:
+

Laplacian() adjacency() degree()

+
+
+
+ +
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/mobile-SLAM.html b/mobile-SLAM.html new file mode 100644 index 00000000..b67ec151 --- /dev/null +++ b/mobile-SLAM.html @@ -0,0 +1,2160 @@ + + + + + + + + + + + Localization and Mapping — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

Localization and Mapping

+

These classes support simulation of vehicle and map estimation in a simple +planar world with point landmarks.

+
+

State estimation

+

Two state estimators are included.

+
+

Extended Kalman filter

+

The EKF is capable of vehicle localization, map estimation or SLAM.

+
+
+class roboticstoolbox.mobile.EKF(robot, sensor=None, map=None, P0=None, x_est=None, joseph=True, animate=True, x0=[0, 0, 0], verbose=False, history=True, workspace=None)[source]
+

Bases: object

+
+
+__init__(robot, sensor=None, map=None, P0=None, x_est=None, joseph=True, animate=True, x0=[0, 0, 0], verbose=False, history=True, workspace=None)[source]
+

Extended Kalman filter

+
+
Parameters:
+
    +
  • robot (2-tuple) – robot motion model

  • +
  • sensor (2-tuple, optional) – vehicle mounted sensor model, defaults to None

  • +
  • map (LandmarkMap, optional) – landmark map, defaults to None

  • +
  • P0 (ndarray(n,n), optional) – initial covariance matrix, defaults to None

  • +
  • x_est (array_like(n), optional) – initial state estimate, defaults to None

  • +
  • joseph (bool, optional) – use Joseph update of covariance, defaults to True

  • +
  • animate (bool, optional) – show animation of vehicle motion, defaults to True

  • +
  • x0 (array_like(n), optional) – initial EKF state, defaults to [0, 0, 0]

  • +
  • verbose (bool, optional) – display extra debug information, defaults to False

  • +
  • history (bool, optional) – retain step-by-step history, defaults to True

  • +
  • workspace (scalar, array_like(2), array_like(4)) – dimension of workspace, see expand_dims()

  • +
+
+
+

This class solves several classical robotic estimation problems, which are +selected according to the arguments:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

Problem

len(x)

robot

sensor

map

P0

Dead reckoning

3

(veh,V)

None

None

P0

Map-based localization

3

(veh,V)

(smodel,W)

yes

P0

Map creation

2N

(veh,None)

(smodel,W)

None

None

SLAM

3+2N

(veh,V)

(smodel,W)

None

P0

+

where:

+
    +
  • veh models the robotic vehicle kinematics and odometry and is a VehicleBase subclass

  • +
  • V is the estimated odometry (process) noise covariance as an ndarray(3,3)

  • +
  • smodel models the robot mounted sensor and is a SensorBase subclass

  • +
  • W is the estimated sensor (measurement) noise covariance as an ndarray(2,2)

  • +
+

The state vector has different lengths depending on the particular +estimation problem, see below.

+

At each iteration of the EKF:

+
    +
  • invoke the step method of the robot

  • +
+
+
    +
  • obtains the next control input from the driver agent, and apply it +as the vehicle control input

  • +
  • the vehicle returns a noisy odometry estimate

  • +
+
+
    +
  • the state prediction is computed

  • +
  • the true pose is used to determine a noisy sensor observation

  • +
  • the state is corrected, new landmarks are added to the map

  • +
+

The working area of the robot is defined by workspace or inherited +from the landmark map attached to the sensor (see +expand_dims()):

+ + + + + + + + + + + + + + + + + + + + + +

workspace

x-range

y-range

A (scalar)

-A:A

-A:A

[A, B]

A:B

A:B

[A, B, C, D]

A:B

C:D

+

Dead-reckoning localization

+

The state \(\vec{x} = (x, y, \theta)\) is the estimated vehicle +configuration.

+

Create a vehicle with odometry covariance V, add a driver to it, +run the Kalman filter with estimated covariances V and initial +vehicle state covariance P0:

+
V = np.diag([0.02, np.radians(0.5)]) ** 2;
+robot = Bicycle(covar=V)
+robot.control = RandomPath(workspace=10)
+
+x_sdev = [0.05, 0.05, np.radians(0.5)]
+P0 = np.diag(x_sdev) ** 2
+ekf = EKF(robot=(robot, V), P0=P0)
+
+ekf.run(T=20)  # run the simulation for 20 seconds
+
+robot.plot_xy(color="b")  # plot the true vehicle path
+ekf.plot_xy(color="r")    # overlay the estimated path
+ekf.plot_ellipse(filled=True, facecolor="g", alpha=0.3)  # overlay uncertainty ellipses
+
+# plot the covariance against time
+t = ekf.get_t();
+pn = ekf.get_Pnorm()
+plt.plot(t, pn);
+
+
+

Map-based vehicle localization

+

The state \(\vec{x} = (x, y, \theta)\) is the estimated vehicle +configuration.

+

Create a vehicle with odometry covariance V, add a driver to it, +create a map with 20 point landmarks, create a sensor that uses the map +and vehicle state to estimate landmark range and bearing with covariance +W, the Kalman filter with estimated covariances V and W and +initial vehicle state covariance P0:

+
V = np.diag([0.02, np.radians(0.5)]) ** 2;
+robot = Bicycle(covar=V)
+robot.control = RandomPath(workspace=10)
+
+map = LandmarkMap(nlandmarks=20, workspace=10)
+
+W = np.diag([0.1, np.radians(1)]) ** 2
+sensor = RangeBearingSensor(robot=robot, map=map, covar=W, angle=[-np.pi/2, np.pi/2], range=4, animate=True)
+
+x_sdev = [0.05, 0.05, np.radians(0.5)]
+P0 = np.diag(x_sdev) ** 2
+ekf = EKF(robot=(robot, V), P0=P0, map=map, sensor=(sensor, W))
+
+ekf.run(T=20)  # run the simulation for 20 seconds
+
+map.plot()  #  plot the map
+robot.plot_xy(color="b")  # plot the true vehicle path
+ekf.plot_xy(color="r")    # overlay the estimated path
+ekf.plot_ellipse()  # overlay uncertainty ellipses
+
+# plot the covariance against time
+t = ekf.get_t();
+pn = ekf.get_Pnorm()
+plt.plot(t, pn);
+
+
+

Vehicle-based map making

+

The state \(\vec{x} = (x_0, y_0, \dots, x_{N-1}, y_{N-1})\) is the +estimated landmark positions where \(N\) is the number of landmarks. +The state vector is initially empty, and is extended by 2 elements every +time a new landmark is observed.

+

Create a vehicle with perfect odometry (no covariance), add a driver to it, +create a sensor that uses the map and vehicle state to estimate landmark range +and bearing with covariance W, the Kalman filter with estimated sensor +covariance W, then run the filter for N time steps:

+
robot = Bicycle()
+robot.add_driver(RandomPath(20, 2))
+
+map = LandmarkMap(nlandmarks=20, workspace=10, seed=0)
+
+W = np.diag([0.1, np.radians(1)]) ** 2
+sensor = RangeBearingSensor(robot, map, W)
+
+ekf = EKF(robot=(robot, None), sensor=(sensor, W))
+
+ekf.run(T=20)  # run the simulation for 20 seconds
+
+map.plot()  #  plot the map
+robot.plot_xy(color="b")  # plot the true vehicle path
+
+
+

Simultaneous localization and mapping (SLAM)

+

The state \(\vec{x} = (x, y, \theta, x_0, y_0, \dots, x_{N-1}, +y_{N-1})\) is the estimated vehicle configuration followed by the +estimated landmark positions where \(N\) is the number of landmarks. +The state vector is initially of length 3, and is extended by 2 elements +every time a new landmark is observed.

+

Create a vehicle with odometry covariance V, add a driver to it, +create a map with 20 point landmarks, create a sensor that uses the map +and vehicle state to estimate landmark range and bearing with covariance +W, the Kalman filter with estimated covariances V and W and +initial state covariance P0, then run the filter to estimate the +vehicle state at each time step and the map:

+
V = np.diag([0.02, np.radians(0.5)]) ** 2;
+robot = Bicycle(covar=V)
+robot.control = RandomPath(workspace=10)
+
+map = LandmarkMap(nlandmarks=20, workspace=10)
+
+W = np.diag([0.1, np.radians(1)]) ** 2
+sensor = RangeBearingSensor(robot=robot, map=map, covar=W, angle=[-np.pi/2, np.pi/2], range=4, animate=True)
+
+ekf = EKF(robot=(robot, V), P0=P0, sensor=(sensor, W))
+
+ekf.run(T=20)  # run the simulation for 20 seconds
+
+map.plot(); # plot true map
+ekf.plot_map(); # plot estimated landmark position
+
+robot.plot_xy(); # plot true path
+ekf.plot_xy(); # plot estimated robot path
+ekf.plot_ellipse(); # plot estimated covariance
+
+# plot the covariance against time
+t = ekf.get_t();
+pn = ekf.get_Pnorm()
+plt.plot(t, pn);
+
+
+
+
Seealso:
+

run()

+
+
+
+ +
+
+property x_est
+

Get EKF state

+
+
Returns:
+

state vector

+
+
Return type:
+

ndarray(n)

+
+
+

Returns the value of the estimated state vector at the end of +simulation. The dimensions depend on the problem being solved.

+
+ +
+
+property P_est
+

Get EKF covariance

+
+
Returns:
+

covariance matrix

+
+
Return type:
+

ndarray(n,n)

+
+
+

Returns the value of the estimated covariance matrix at the end of +simulation. The dimensions depend on the problem being solved.

+
+ +
+
+property P0
+

Get initial EKF covariance

+
+
Returns:
+

covariance matrix

+
+
Return type:
+

ndarray(n,n)

+
+
+

Returns the value of the covariance matrix passed to the constructor.

+
+ +
+
+property V_est
+

Get estimated odometry covariance

+
+
Returns:
+

odometry covariance

+
+
Return type:
+

ndarray(2,2)

+
+
+

Returns the value of the estimated odometry covariance matrix passed to +the constructor

+
+ +
+
+property W_est
+

Get estimated sensor covariance

+
+
Returns:
+

sensor covariance

+
+
Return type:
+

ndarray(2,2)

+
+
+

Returns the value of the estimated sensor covariance matrix passed to +the constructor

+
+ +
+
+property robot
+

Get robot object

+
+
Returns:
+

robot used in simulation

+
+
Return type:
+

VehicleBase subclass

+
+
+
+ +
+
+property sensor
+

Get sensor object

+
+
Returns:
+

sensor used in simulation

+
+
Return type:
+

SensorBase subclass

+
+
+
+ +
+
+property map
+

Get map object

+
+
Returns:
+

map used in simulation

+
+
Return type:
+

LandmarkMap subclass

+
+
+
+ +
+
+property verbose
+

Get verbosity state

+
+
Returns:
+

verbosity

+
+
Return type:
+

bool

+
+
+
+ +
+
+property history
+

Get EKF simulation history

+
+
Returns:
+

simulation history

+
+
Return type:
+

list of namedtuples

+
+
+

At each simulation timestep a namedtuple of is appended to the history +list. It contains, for that time step, estimated state and covariance, +and sensor observation.

+
+
Seealso:
+

get_t() get_xyt() get_map() get_P() +get_Pnorm()

+
+
+
+ +
+
+property workspace
+

Size of robot workspace

+
+
Returns:
+

workspace bounds [xmin, xmax, ymin, ymax]

+
+
Return type:
+

ndarray(4)

+
+
+

Returns the bounds of the workspace as specified by the constructor +option workspace

+
+ +
+
+property landmarks
+

Get landmark information

+
+
Returns:
+

landmark information

+
+
Return type:
+

dict

+
+
+

The dictionary is indexed by the landmark id and gives a 3-tuple:

+
    +
  • order in which landmark was seen

  • +
  • number of times seen

  • +
+

The order in which the landmark was first seen. The first observed +landmark has order 0 and so on.

+
+
Seealso:
+

landmark()

+
+
+
+ +
+
+landmark(id)[source]
+

Landmark information

+
+
Parameters:
+

id (int) – landmark index

+
+
Returns:
+

order in which it was first seen, number of times seen

+
+
Return type:
+

int, int

+
+
+

The first observed landmark has order 0 and so on.

+
+
Seealso:
+

landmarks() landmark_index() landmark_mindex()

+
+
+
+ +
+
+landmark_index(id)[source]
+

Landmark index in complete state vector

+
+
Parameters:
+

id (int) – landmark index

+
+
Returns:
+

index in the state vector

+
+
Return type:
+

int

+
+
+

The return value j is the index of the x-coordinate of the landmark +in the EKF state vector, and j+1 is the index of the y-coordinate.

+
+
Seealso:
+

landmark()

+
+
+
+ +
+
+landmark_mindex(id)[source]
+

Landmark index in map state vector

+
+
Parameters:
+

id (int) – landmark index

+
+
Returns:
+

index in the state vector

+
+
Return type:
+

int

+
+
+

The return value j is the index of the x-coordinate of the landmark +in the map vector, and j+1 is the index of the y-coordinate.

+
+
Seealso:
+

landmark()

+
+
+
+ +
+
+landmark_x(id)[source]
+

Landmark position

+
+
Parameters:
+

id (int) – landmark index

+
+
Returns:
+

landmark position \((x,y)\)

+
+
Return type:
+

ndarray(2)

+
+
+

Returns the landmark position from the current state vector.

+
+ +
+
+init()[source]
+
+ +
+
+run(T, animate=False)[source]
+

Run the EKF simulation

+
+
Parameters:
+
    +
  • T (float) – maximum simulation time in seconds

  • +
  • animate (bool, optional) – animate motion of vehicle, defaults to False

  • +
+
+
+

Simulates the motion of a vehicle (under the control of a driving agent) +and the EKF estimator. The steps are:

+
    +
  • initialize the filter, vehicle and vehicle driver agent, sensor

  • +
  • for each time step:

    +
    +
      +
    • step the vehicle and its driver agent, obtain odometry

    • +
    • take a sensor reading

    • +
    • execute the EKF

    • +
    • save information as a namedtuple to the history list for later display

    • +
    +
    +
  • +
+
+
Seealso:
+

history() landmark() landmarks() +get_xyt() get_t() get_map() get_P() get_Pnorm() +plot_xy() plot_ellipse() plot_error() plot_map() +run_animation()

+
+
+
+ +
+
+run_animation(T=10, x0=None, control=None, format=None, file=None)[source]
+

Run the EKF simulation

+
+
Parameters:
+
    +
  • T (float) – maximum simulation time in seconds

  • +
  • format (str, optional) – Output format

  • +
  • file (str, optional) – File name

  • +
+
+
Returns:
+

Matplotlib animation object

+
+
Return type:
+

matplotlib.animation.FuncAnimation()

+
+
+

Simulates the motion of a vehicle (under the control of a driving agent) +and the EKF estimator for T seconds and returns an animation +in various formats:

+
``format``    ``file``   description
+============  =========  ============================
+``"html"``    str, None  return HTML5 video
+``"jshtml"``  str, None  return JS+HTML video
+``"gif"``     str        return animated GIF
+``"mp4"``     str        return MP4/H264 video
+``None``                 return a ``FuncAnimation`` object
+
+
+

If file can be None then return the video as a string, otherwise it +must be a filename.

+

The simulation steps are:

+
    +
  • initialize the filter, vehicle and vehicle driver agent, sensor

  • +
  • for each time step:

    +
    +
      +
    • step the vehicle and its driver agent, obtain odometry

    • +
    • take a sensor reading

    • +
    • execute the EKF

    • +
    • save information as a namedtuple to the history list for later display

    • +
    +
    +
  • +
+
+
Seealso:
+

history() landmark() landmarks() +get_xyt() get_t() get_map() get_P() get_Pnorm() +plot_xy() plot_ellipse() plot_error() plot_map() +run_animation()

+
+
+
+ +
+
+step(pause=None)[source]
+

Execute one timestep of the simulation

+
+ +
+
+get_t()[source]
+

Get time from simulation

+
+
Returns:
+

simulation time vector

+
+
Return type:
+

ndarray(n)

+
+
+

Return simulation time vector, starts at zero. The timestep is an +attribute of the robot object.

+
+
Seealso:
+

run() history()

+
+
+
+ +
+
+get_xyt()[source]
+

Get estimated vehicle trajectory

+
+
Returns:
+

vehicle trajectory where each row is configuration \((x, y, \theta)\)

+
+
Return type:
+

ndarray(n,3)

+
+
Seealso:
+

plot_xy() run() history()

+
+
+
+ +
+
+plot_xy(*args, block=None, **kwargs)[source]
+

Plot estimated vehicle position

+
+
Parameters:
+
    +
  • args – position arguments passed to plot()

  • +
  • kwargs – keywords arguments passed to plot()

  • +
  • block (bool, optional) – hold plot until figure is closed, defaults to None

  • +
+
+
+

Plot the estimated vehicle path in the xy-plane.

+
+
Seealso:
+

get_xyt() plot_error() plot_ellipse() plot_P() +run() history()

+
+
+
+ +
+
+plot_ellipse(confidence=0.95, N=10, block=None, **kwargs)[source]
+

Plot uncertainty ellipses

+
+
Parameters:
+
    +
  • confidence (float, optional) – ellipse confidence interval, defaults to 0.95

  • +
  • N (int, optional) – number of ellipses to plot, defaults to 10

  • +
  • block (bool, optional) – hold plot until figure is closed, defaults to None

  • +
  • kwargs – arguments passed to spatialmath.base.graphics.plot_ellipse()

  • +
+
+
+

Plot N uncertainty ellipses spaced evenly along the trajectory.

+
+
Seealso:
+

get_P() run() history()

+
+
+
+ +
+
+plot_error(bgcolor='r', confidence=0.95, ax=None, block=None, **kwargs)[source]
+

Plot error with uncertainty bounds

+
+
Parameters:
+
    +
  • bgcolor (str, optional) – background color, defaults to ‘r’

  • +
  • confidence (float, optional) – confidence interval, defaults to 0.95

  • +
  • block (bool, optional) – hold plot until figure is closed, defaults to None

  • +
+
+
+

Plot the error between actual and estimated vehicle +path \((x, y, \theta)`\) versus time as three stacked plots. +Heading error is wrapped into the range \([-\pi,\pi)\)

+

Behind each line draw a shaded polygon bgcolor showing the specified +confidence bounds based on the covariance at each time step. +Ideally the lines should be within the shaded polygon confidence +of the time.

+
+

Note

+

Observations will decrease the uncertainty while periods of dead-reckoning increase it.

+
+
+
Seealso:
+

get_P() run() history()

+
+
+
+ +
+
+get_map()[source]
+

Get estimated map

+
+
Returns:
+

landmark coordinates \((x, y)\)

+
+
Return type:
+

ndarray(n,2)

+
+
+

Landmarks are returned in the order they were first observed.

+
+
Seealso:
+

landmarks() run() history()

+
+
+
+ +
+
+plot_map(marker=None, ellipse=None, confidence=0.95, block=None)[source]
+

Plot estimated landmarks

+
+
Parameters:
+
    +
  • marker (dict, optional) – plot marker for landmark, arguments passed to plot(), defaults to “r+”

  • +
  • ellipse (dict, optional) – arguments passed to plot_ellipse(), defaults to None

  • +
  • confidence (float, optional) – ellipse confidence interval, defaults to 0.95

  • +
  • block (bool, optional) – hold plot until figure is closed, defaults to None

  • +
+
+
+

Plot a marker and covariance ellipses for each estimated landmark.

+
+
Seealso:
+

get_map() run() history()

+
+
+
+ +
+
+get_P(k=None)[source]
+

Get covariance matrices from simulation

+
+
Parameters:
+

k (int, optional) – timestep, defaults to None

+
+
Returns:
+

covariance matrix

+
+
Return type:
+

ndarray(n,n) or list of ndarray(n,n)

+
+
+

If k is given return covariance from simulation timestep k, else +return a list of all covariance matrices.

+
+
Seealso:
+

get_Pnorm() run() history()

+
+
+
+ +
+
+get_Pnorm(k=None)[source]
+

Get covariance norm from simulation

+
+
Parameters:
+

k (int, optional) – timestep, defaults to None

+
+
Returns:
+

covariance matrix norm

+
+
Return type:
+

float or ndarray(n)

+
+
+

If k is given return covariance norm from simulation timestep k, else +return all covariance norms as a 1D NumPy array.

+
+
Seealso:
+

get_P() run() history()

+
+
+
+ +
+
+disp_P(P, colorbar=False)[source]
+

Display covariance matrix

+
+
Parameters:
+
    +
  • P (ndarray(n,n)) – covariance matrix

  • +
  • colorbar – add a colorbar

  • +
+
+
Type:
+

bool or dict

+
+
+

Plot the elements of the covariance matrix as an image. If colorbar +is True add a color bar, if colorbar is a dict add a color bar with +these options passed to colorbar.

+
+

Note

+

A log scale is used.

+
+
+
Seealso:
+

imshow() matplotlib.pyplot.colorbar()

+
+
+
+ +
+
+get_transform(map)[source]
+

Transformation from estimated map to true map frame

+
+
Parameters:
+

map (LandmarkMap) – known landmark positions

+
+
Returns:
+

transform from map to estimated map frame

+
+
Return type:
+

SE2 instance

+
+
+

Uses a least squares technique to find the transform between the +landmark is world frame and the estimated landmarks in the SLAM +reference frame.

+
+
Seealso:
+

points2tr2()

+
+
+
+ +
+ +
+
+

Particle filter

+

The particle filter is capable of map-based vehicle localization.

+
+
+class roboticstoolbox.mobile.ParticleFilter(robot, sensor, R, L, nparticles=500, seed=0, x0=None, verbose=False, animate=False, history=True, workspace=None)[source]
+

Bases: object

+
+
+__init__(robot, sensor, R, L, nparticles=500, seed=0, x0=None, verbose=False, animate=False, history=True, workspace=None)[source]
+

Particle filter

+
+
Parameters:
+
    +
  • robot (VehicleBase subclass,) – robot motion model

  • +
  • sensor (SensorBase subclass) – vehicle mounted sensor model

  • +
  • R (ndarray(3,3)) – covariance of the zero-mean Gaussian noise added to the particles at each step (diffusion)

  • +
  • L (ndarray(2,2)) – covariance used in the sensor likelihood model

  • +
  • nparticles (int, optional) – number of particles, defaults to 500

  • +
  • seed (int, optional) – random number seed, defaults to 0

  • +
  • x0 (array_like(3), optional) – initial state, defaults to [0, 0, 0]

  • +
  • verbose (bool, optional) – display extra debug information, defaults to False

  • +
  • history (bool, optional) – retain step-by-step history, defaults to True

  • +
  • workspace (scalar, array_like(2), array_like(4)) – dimension of workspace, see expand_dims()

  • +
+
+
+

This class implements a Monte-Carlo estimator or particle filter for +vehicle state, based on odometry, a landmark map, and landmark +observations. The state of each particle is a possible vehicle +configuration \((x,y, heta)\). Bootstrap particle resampling is +used.

+

The working area is defined by workspace or inherited from the +landmark map attached to the sensor (see +expand_dims()):

+ + + + + + + + + + + + + + + + + + + + + +

workspace

x-range

y-range

A (scalar)

-A:A

-A:A

[A, B]

A:B

A:B

[A, B, C, D]

A:B

C:D

+

Particles are initially distributed uniform randomly over this area.

+

Example:

+
V = np.diag([0.02, np.radians(0.5)]) ** 2
+robot = Bicycle(covar=V, animation="car", workspace=10)
+robot.control = RandomPath(workspace=robot)
+
+map = LandmarkMap(nlandmarks=20, workspace=robot.workspace)
+
+W = np.diag([0.1, np.radians(1)]) ** 2
+sensor = RangeBearingSensor(robot, map, covar=W, plot=True)
+
+R = np.diag([0.1, 0.1, np.radians(1)]) ** 2
+L = np.diag([0.1, 0.1])
+pf = ParticleFilter(robot, sensor, R, L, nparticles=1000)
+
+pf.run(T=10)
+
+map.plot()
+robot.plot_xy()
+pf.plot_xy()
+
+plt.plot(pf.get_std()[:100,:])
+
+
+
+

Note

+

Set seed=0 to get different behaviour from run to run.

+
+
+
Seealso:
+

run()

+
+
+
+ +
+
+property robot
+

Get robot object

+
+
Returns:
+

robot used in simulation

+
+
Return type:
+

VehicleBase subclass

+
+
+
+ +
+
+property sensor
+

Get sensor object

+
+
Returns:
+

sensor used in simulation

+
+
Return type:
+

SensorBase subclass

+
+
+
+ +
+
+property map
+

Get map object

+
+
Returns:
+

map used in simulation

+
+
Return type:
+

LandmarkMap subclass

+
+
+
+ +
+
+property verbose
+

Get verbosity state

+
+
Returns:
+

verbosity

+
+
Return type:
+

bool

+
+
+
+ +
+
+property history
+

Get EKF simulation history

+
+
Returns:
+

simulation history

+
+
Return type:
+

list of namedtuples

+
+
+

At each simulation timestep a namedtuple of is appended to the history +list. It contains, for that time step, estimated state and covariance, +and sensor observation.

+
+
Seealso:
+

get_t() get_xy() get_std() +get_Pnorm()

+
+
+
+ +
+
+property workspace
+

Size of robot workspace

+
+
Returns:
+

workspace bounds [xmin, xmax, ymin, ymax]

+
+
Return type:
+

ndarray(4)

+
+
+

Returns the bounds of the workspace as specified by constructor +option workspace

+
+ +
+
+property random
+

Get private random number generator

+
+
Returns:
+

NumPy random number generator

+
+
Return type:
+

numpy.random.Generator

+
+
+

Has methods including:

+
    +
  • integers(low, high, size, endpoint)

  • +
  • random(size)

  • +
  • uniform

  • +
  • normal(mean, std, size)

  • +
  • multivariate_normal(mean, covar, size)

  • +
+

The generator is initialized with the seed provided at constructor +time every time init is called.

+
+ +
+
+run(T=10, x0=None)[source]
+

Run the particle filter simulation

+
+
Parameters:
+
    +
  • T (float) – maximum simulation time in seconds

  • +
  • x0 (array_like(3) or array_like(2)) – Initial state, defaults to value given to Vehicle constructor

  • +
+
+
+

Simulates the motion of a vehicle (under the control of a driving agent) +and the particle-filter estimator. The steps are:

+
    +
  • initialize the filter, vehicle and vehicle driver agent, sensor

  • +
  • for each time step:

    +
    +
      +
    • step the vehicle and its driver agent, obtain odometry

    • +
    • take a sensor reading

    • +
    • execute the EKF

    • +
    • save information as a namedtuple to the history list for later display

    • +
    +
    +
  • +
+
+
Seealso:
+

history() landmark() landmarks() +get_xy() get_t() get_std() +plot_xy()

+
+
+
+ +
+
+run_animation(T=10, x0=None, format=None, file=None)[source]
+

Run the particle filter simulation

+
+
Parameters:
+
    +
  • T (float) – maximum simulation time in seconds

  • +
  • x0 (array_like(3) or array_like(2)) – Initial state, defaults to value given to Vehicle constructor

  • +
  • format (str, optional) – Output format

  • +
  • file (str, optional) – File name

  • +
+
+
Returns:
+

Matplotlib animation object

+
+
Return type:
+

matplotlib.animation.FuncAnimation()

+
+
+

Simulates the motion of a vehicle (under the control of a driving agent) +and the particle-filter estimator and returns an animation +in various formats:

+
``format``    ``file``   description
+============  =========  ============================
+``"html"``    str, None  return HTML5 video
+``"jshtml"``  str, None  return JS+HTML video
+``"gif"``     str        return animated GIF
+``"mp4"``     str        return MP4/H264 video
+``None``                 return a ``FuncAnimation`` object
+
+
+

The allowables types for file are given in the second column. A str +value is the file name. If None is an option then return the video as a string.

+

For the last case, a reference to the animation object must be held if used for +animation in a Jupyter cell:

+
anim = robot.run_animation(T=20)
+
+
+

The steps are:

+
    +
  • initialize the filter, vehicle and vehicle driver agent, sensor

  • +
  • for each time step:

    +
    +
      +
    • step the vehicle and its driver agent, obtain odometry

    • +
    • take a sensor reading

    • +
    • execute the EKF

    • +
    • save information as a namedtuple to the history list for later display

    • +
    +
    +
  • +
+
+
Seealso:
+

history() landmark() landmarks() +get_xy() get_t() get_std() +plot_xy()

+
+
+
+ +
+
+plot_pdf()[source]
+

Plot particle PDF

+

Displays a discrete PDF of vehicle position. Creates a 3D plot where +the x- and y-axes are the estimated vehicle position and the z-axis is +the particle weight. Each particle is represented by a a vertical line +segment of height equal to particle weight.

+
+ +
+
+get_t()[source]
+

Get time from simulation

+
+
Returns:
+

simulation time vector

+
+
Return type:
+

ndarray(n)

+
+
+

Return simulation time vector, starts at zero. The timestep is an +attribute of the robot object.

+
+ +
+
+get_xyt()[source]
+

Get estimated vehicle trajectory

+
+
Returns:
+

vehicle trajectory where each row is configuration \((x, y, \theta)\)

+
+
Return type:
+

ndarray(n,3)

+
+
Seealso:
+

plot_xy() run() history()

+
+
+
+ +
+
+get_std()[source]
+

Get standard deviation of particles

+
+
Returns:
+

standard deviation of vehicle position estimate

+
+
Return type:
+

ndarray(n,2)

+
+
+

Return the standard deviation \((\sigma_x, \sigma_y)\) of the +particle cloud at each time step.

+
+
Seealso:
+

get_xyt()

+
+
+
+ +
+
+plot_xy(block=None, **kwargs)[source]
+

Plot estimated vehicle position

+
+
Parameters:
+
    +
  • args – position arguments passed to plot()

  • +
  • kwargs – keywords arguments passed to plot()

  • +
  • block (bool, optional) – hold plot until figure is closed, defaults to None

  • +
+
+
+

Plot the estimated vehicle path in the xy-plane.

+
+
Seealso:
+

get_xy()

+
+
+
+ +
+ +
+
+
+

Sensor models

+
+
+class roboticstoolbox.mobile.RangeBearingSensor(robot, map, line_style=None, poly_style=None, covar=None, range=None, angle=None, plot=False, seed=0, **kwargs)[source]
+

Bases: SensorBase

+
+
+__init__(robot, map, line_style=None, poly_style=None, covar=None, range=None, angle=None, plot=False, seed=0, **kwargs)[source]
+

Range and bearing angle sensor

+
+
Parameters:
+
    +
  • robot (VehicleBase subclass) – model of robot carrying the sensor

  • +
  • map (LandmarkMap instance) – map of landmarks

  • +
  • polygon (dict, optional) – polygon style for sensing region, see plot_polygon, defaults to None

  • +
  • covar (ndarray(2,2), optional) – covariance matrix for sensor readings, defaults to None

  • +
  • range (float or array_like(2), optional) – maximum range \(r_{max}\) or range span \([r_{min}, r_{max}]\), defaults to None

  • +
  • angle (float, optional) – angular field of view, from \([-\theta, \theta]\) defaults to None

  • +
  • plot (bool, optional) – animate the sensor beams, defaults to False

  • +
  • seed (int, optional) – random number seed, defaults to 0

  • +
  • kwargs – arguments passed to SensorBase

  • +
+
+
+

Sensor object that returns the range and bearing angle \((r, +\beta)\) to a point landmark from a robot-mounted sensor. The sensor +measurements are corrupted with zero-mean Gaussian noise with covariance +covar.

+

The sensor can have a maximum range, or a minimum and maximum range. The +sensor can also have a restricted angular field of view.

+

The sensing region can be displayed by setting the polygon parameter +which can show an outline or a filled polygon. This is updated every +time reading() is called, based on the current configuration of +the robot.

+
>>> from roboticstoolbox import Bicycle, LandmarkMap, RangeBearingSensor
+>>> from math import pi
+>>> robot = Bicycle()
+>>> map = LandmarkMap(20)
+>>> sensor = RangeBearingSensor(robot, map, range=(0.5, 20), angle=pi/4)
+>>> print(sensor)
+RangeBearingSensor sensor class
+  LandmarkMap object with 20 landmarks, workspace=(-10.0: 10.0, -10.0: 10.0)
+  W = [ 0, 0 | 0, 0 ]
+  sampled every 1 samples
+  range: (0.5: 20.0)
+  angle: (-0.785: 0.785)
+
+
+
+
Seealso:
+

LandmarkMap EKF

+
+
+
+ +
+
+init()[source]
+

Initialize sensor

+
    +
  • reseed the random number generator

  • +
  • reset the counter for handling the every and fail options

  • +
  • reset the landmark log

  • +
  • initalize plots

  • +
+
+
Seealso:
+

SensorBase.init()

+
+
+
+ +
+
+property W
+

Get sensor covariance

+
+
Returns:
+

sensor covariance

+
+
Return type:
+

ndarray(2,2)

+
+
+

Returns the value of the sensor covariance matrix passed to +the constructor.

+
+ +
+
+reading()[source]
+

Choose landmark and return observation

+
+
Returns:
+

range and bearing angle to a landmark, and landmark id

+
+
Return type:
+

ndarray(2), int

+
+
+

Returns an observation of a random visible landmark (range, bearing) and +the id of that landmark. The landmark is chosen randomly from the +set of all visible landmarks, those within the angular field of view and +range limit.

+

If constructor argument every is set then only return a valid +reading on every every calls.

+

If constructor argument fail is set then do not return a reading +during that specified time interval.

+

If no valid reading is available then return (None, None)

+
>>> from roboticstoolbox import Bicycle, LandmarkMap, RangeBearingSensor
+>>> from math import pi
+>>> robot = Bicycle()
+>>> map = LandmarkMap(20)
+>>> sensor = RangeBearingSensor(robot, map, range=(0.5, 20), angle=pi/4)
+>>> print(sensor.reading())
+(array([ 7.2691, -0.039 ]), 16)
+
+
+
+

Note

+
    +
  • Noise with covariance W (set by constructor) is added to the +reading

  • +
  • If animate option is set then show a line from the vehicle to +the landmark

    +
    +
      +
    • If animate option set and the angular and distance limits +are set then display the sensor field of view as a polygon.

    • +
    +
    +
  • +
+
+
+
Seealso:
+

h()

+
+
+
+ +
+
+visible()[source]
+

List of all visible landmarks

+
+
Returns:
+

list of visible landmarks

+
+
Return type:
+

list of int

+
+
+

Return a list of the id of all landmarks that are visible, that is, it +lies with the sensing range and field of view of the sensor at the +robot’s current configuration.

+
+
Seealso:
+

isvisible() h()

+
+
+
+ +
+
+isvisible(id)[source]
+

Test if landmark is visible

+
+
Parameters:
+

id (int) – landmark id

+
+
Returns:
+

visibility

+
+
Return type:
+

bool

+
+
+

The landmark id is visible if it lies with the sensing range and +field of view of the sensor at the robot’s current configuration.

+
+
Seealso:
+

visible() h()

+
+
+
+ +
+
+h(x, landmark=None)[source]
+

Landmark observation function

+
+
Parameters:
+
    +
  • x (array_like(3), array_like(N,3)) – vehicle state \((x, y, \theta)\)

  • +
  • landmark (int or array_like(2), optional) – landmark id or position, defaults to None

  • +
+
+
Returns:
+

range and bearing angle to landmark math:(r,beta)

+
+
Return type:
+

ndarray(2) or ndarray(N,2)

+
+
+

Return the range and bearing to a landmark:

+
    +
  • .h(x) is range and bearing to all landmarks, one row per landmark

  • +
  • .h(x, id) is range and bearing to landmark id

  • +
  • .h(x, p) is range and bearing to landmark with coordinates p

  • +
+
>>> from roboticstoolbox import Bicycle, LandmarkMap, RangeBearingSensor
+>>> from math import pi
+>>> robot = Bicycle()
+>>> map = LandmarkMap(20)
+>>> sensor = RangeBearingSensor(robot, map, range=(0.5, 20), angle=pi/4)
+>>> z = sensor.h((1, 2, pi/2), 3)
+>>> print(z)
+[10.7111  1.4826]
+
+
+
+

Note

+
    +
  • Noise with covariance (property W) is added to each row of z.

  • +
  • Performs fast vectorized operation where x is an ndarray(n,3).

  • +
  • The landmark is assumed to be visible, field of view and range limits are not +applied.

  • +
+
+
+
Seealso:
+

reading() Hx() Hw() Hp()

+
+
+
+ +
+
+Hx(x, landmark)[source]
+

Jacobian dh/dx

+
+
Parameters:
+
    +
  • x (array_like(3)) – vehicle state \((x, y, \theta)\)

  • +
  • arg (int or array_like(2)) – landmark id or coordinate

  • +
+
+
Returns:
+

Jacobian matrix

+
+
Return type:
+

ndarray(2,3)

+
+
+

Compute the Jacobian of the observation function with respect to vehicle +configuration \(\partial h/\partial x\)

+
    +
  • sensor.Hx(q, id) is Jacobian for landmark id

  • +
  • sensor.h(q, p) is Jacobian for landmark with coordinates p

  • +
+
+
Seealso:
+

h() Hp() Hw()

+
+
+
+ +
+
+Hp(x, landmark)[source]
+

Jacobian dh/dp

+
+
Parameters:
+
    +
  • x (array_like(3)) – vehicle state \((x, y, \theta)\)

  • +
  • arg (int or array_like(2)) – landmark id or coordinate

  • +
+
+
Returns:
+

Jacobian matrix

+
+
Return type:
+

ndarray(2,2)

+
+
+

Compute the Jacobian of the observation function with respect +to landmark position \(\partial h/\partial p\)

+
    +
  • sensor.Hp(x, id) is Jacobian for landmark id

  • +
  • sensor.Hp(x, p) is Jacobian for landmark with coordinates p

  • +
+
+
Seealso:
+

h() Hx() Hw()

+
+
+
+ +
+
+Hw(x, landmark)[source]
+

Jacobian dh/dw

+
+
Parameters:
+
    +
  • x (array_like(3)) – vehicle state \((x, y, \theta)\)

  • +
  • arg (int or array_like(2)) – landmark id or coordinate

  • +
+
+
Returns:
+

Jacobian matrix

+
+
Return type:
+

ndarray(2,2)

+
+
+

Compute the Jacobian of the observation function with respect +to sensor noise \(\partial h/\partial w\)

+
    +
  • sensor.Hw(x, id) is Jacobian for landmark id

  • +
  • sensor.Hw(x, p) is Jacobian for landmark with coordinates p

  • +
+
+

Note

+

x and landmark are not used to compute this.

+
+
+
Seealso:
+

h() Hx() Hp()

+
+
+
+ +
+
+g(x, z)[source]
+

Landmark position from sensor observation

+
+
Parameters:
+
    +
  • x (array_like(3)) – vehicle state \((x, y, \theta)\)

  • +
  • z (array_like(2)) – landmark observation \((r, \beta)\)

  • +
+
+
Returns:
+

landmark position \((x, y)\)

+
+
Return type:
+

ndarray(2)

+
+
+

Compute the world coordinate of a landmark given +the observation z from a vehicle state with x.

+
+
Seealso:
+

h() Gx() Gz()

+
+
+
+ +
+
+Gx(x, z)[source]
+

Jacobian dg/dx

+
+
Parameters:
+
    +
  • x (array_like(3)) – vehicle state \((x, y, \theta)\)

  • +
  • z (array_like(2)) – landmark observation \((r, \beta)\)

  • +
+
+
Returns:
+

Jacobian matrix

+
+
Return type:
+

ndarray(2,3)

+
+
+

Compute the Jacobian of the landmark position function with respect +to landmark position \(\partial g/\partial x\)

+
+
Seealso:
+

g()

+
+
+
+ +
+
+Gz(x, z)[source]
+

Jacobian dg/dz

+
+
Parameters:
+
    +
  • x (array_like(3)) – vehicle state \((x, y, \theta)\)

  • +
  • z (array_like(2)) – landmark observation \((r, \beta)\)

  • +
+
+
Returns:
+

Jacobian matrix

+
+
Return type:
+

ndarray(2,2)

+
+
+

Compute the Jacobian of the landmark position function with respect +to sensor observation \(\partial g/\partial z\)

+
+
Seealso:
+

g()

+
+
+
+ +
+
+property map
+

Landmark map associated with sensor (superclass)

+
+
Returns:
+

robot

+
+
Return type:
+

LandmarkMap

+
+
+
+ +
+
+plot(id)
+

Plot sensor observation

+
+
Parameters:
+

id (int) – landmark id

+
+
+

Draws a line from the robot to landmark id.

+
+

Note

+
    +
  • The line is drawn using the line_style given at constructor time

  • +
+
+
+ +
+
+property random
+

Get private random number generator (superclass)

+
+
Returns:
+

NumPy random number generator

+
+
Return type:
+

numpy.random.Generator

+
+
+

Has methods including:

+
+
+

The generator is initialized with the seed provided at constructor +time every time init() is called.

+
+
Seealso:
+

init()

+
+
+
+ +
+
+property robot
+

Robot associated with sensor (superclass)

+
+
Returns:
+

robot

+
+
Return type:
+

VehicleBase subclass

+
+
+
+ +
+
+property verbose
+

Get verbosity state

+
+
Returns:
+

verbosity

+
+
Return type:
+

bool

+
+
+
+ +
+ +
+
+

Map models

+
+
+class roboticstoolbox.mobile.landmarkmap.LandmarkMap(map, workspace=10, verbose=True, seed=0)[source]
+

Bases: object

+

Map of planar point landmarks

+
+
Parameters:
+
    +
  • map (ndarray(2, N) or int) – map or number of landmarks

  • +
  • workspace (scalar, array_like(2), array_like(4), optional) – workspace or map bounds, defaults to 10

  • +
  • verbose (bool, optional) – display debug information, defaults to True

  • +
  • seed (int, optional) – random number seed, defaults to 0

  • +
+
+
Returns:
+

a landmark map object

+
+
Return type:
+

LandmarkMap

+
+
+

A LandmarkMap object represents a rectangular 2D environment with a number +of point landmarks.

+

The landmarks can be specified explicitly or be uniform randomly positioned +inside a region defined by the workspace. The workspace can be numeric:

+ + + + + + + + + + + + + + + + + + + + + +

workspace

x-range

y-range

A (scalar)

-A:A

-A:A

[A, B]

A:B

A:B

[A, B, C, D]

A:B

C:D

+

or any object that has a workspace attribute.

+

Example:

+
>>> from roboticstoolbox import LandmarkMap
+>>> map = LandmarkMap(20)
+>>> print(map)
+LandmarkMap object with 20 landmarks, workspace=(-10.0: 10.0, -10.0: 10.0)
+>>> print(map[3])  # coordinates of landmark 3
+[-9.6694  2.9438]
+
+
+

The object is an iterator that returns consecutive landmark coordinates.

+
+
Reference:
+

Robotics, Vision & Control, Chap 6, +Peter Corke, +Springer 2011

+
+
+

See also RangeBearingSensor EKF

+
+
+__init__(map, workspace=10, verbose=True, seed=0)[source]
+
+ +
+
+__len__()[source]
+

Number of landmarks in map

+
+
Returns:
+

number of landmarks in the map

+
+
Return type:
+

int

+
+
+
+ +
+
+property landmarks
+

xy-coordinates of all landmarks

+
+
Returns:
+

xy-coordinates for landmark points

+
+
Return type:
+

ndarray(2, n)

+
+
+
+ +
+
+property workspace
+

Size of map workspace

+
+
Returns:
+

workspace bounds [xmin, xmax, ymin, ymax]

+
+
Return type:
+

ndarray(4)

+
+
+

Returns the bounds of the workspace as specified by constructor +option workspace.

+
+ +
+
+__getitem__(k)[source]
+

Get landmark coordinates from map

+
+
Parameters:
+

k (int) – landmark index

+
+
Returns:
+

coordinate \((x,y)\) of k’th landmark

+
+
Return type:
+

ndarray(2)

+
+
+
+ +
+
+plot(labels=False, block=None, **kwargs)[source]
+

Plot landmark map

+
+
Parameters:
+
    +
  • labels (bool, optional) – number the points on the plot, defaults to False

  • +
  • block (bool, optional) – block until figure is closed, defaults to False

  • +
  • kwargsplot() options

  • +
+
+
+

Plot landmark points using Matplotlib options. Default style is black +crosses.

+
+ +
+ +
+
+ + +
+
+
+ +
+ +
+

© Copyright 2023, Jesse Haviland and Peter Corke. + Last updated on 11-May-2024. +

+
+ + + +
+
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/mobile-drivers.html b/mobile-drivers.html new file mode 100644 index 00000000..9845ed4f --- /dev/null +++ b/mobile-drivers.html @@ -0,0 +1,375 @@ + + + + + + + + + + + Drivers — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

Drivers

+

These classes define agents thta can drive a mobile robot around a workplace, and are +useful for testing localization algorithms.

+
+

Random Path

+
+
+
+class roboticstoolbox.mobile.drivers.RandomPath(workspace, speed=1, dthresh=0.05, seed=0, headinggain=0.3, goalmarkerstyle=None)[source]
+

Bases: VehicleDriverBase

+
+
+__init__(workspace, speed=1, dthresh=0.05, seed=0, headinggain=0.3, goalmarkerstyle=None)[source]
+

Driving agent for random path

+
+
Parameters:
+
    +
  • workspace (scalar, array_like(2), array_like(4)) – dimension of workspace, see spatialmath.base.exand_dims()

  • +
  • speed (float, optional) – forward speed, defaults to 1

  • +
  • dthresh (float, optional) – distance threshold, defaults to 0.05

  • +
+
+
Raises:
+

ValueError – bad workspace specified

+
+
+

Returns a driver object that drives the attached vehicle to a +sequence of random waypoints.

+

The driver is connected to the vehicle by:

+
Vehicle(control=driver)
+
+
+

or:

+
veh = Vehicle()
+veh.control = driver
+
+
+

The waypoints are positioned inside a rectangular region defined by +the vehicle that is specified by (see plotvol2):

+ + + + + + + + + + + + + + + + + + + + + +

workspace

x-range

y-range

A (scalar)

-A:A

-A:A

[A, B]

A:B

A:B

[A, B, C, D]

A:B

C:D

+
+

Note

+
    +
  • It is possible in some cases for the vehicle to move outside the desired +region, for instance if moving to a waypoint near the edge, the limited +turning circle may cause the vehicle to temporarily move outside.

  • +
  • The vehicle chooses a new waypoint when it is closer than dthresh +to the current waypoint.

  • +
  • Uses its own random number generator so as to not influence the performance +of other randomized algorithms such as path planning. Set seed=None to have it randomly initialized from the +operating system.

  • +
+
+
+
Seealso:
+

Bicycle Unicycle plotvol2()

+
+
+
+ +
+
+__str__()[source]
+

%RandomPath.char Convert to string +% +% s = R.char() is a string showing driver parameters and state in in +% a compact human readable format.

+
+ +
+
+property workspace
+

Size of robot driving workspace

+
+
Returns:
+

workspace bounds [xmin, xmax, ymin, ymax]

+
+
Return type:
+

ndarray(4)

+
+
+

Returns the bounds of the workspace as specified by constructor +option workspace

+
+ +
+
+init(ax=None)[source]
+

Initialize random path driving agent

+
+
Parameters:
+

ax (Axes, optional) – axes in which to draw via points, defaults to None

+
+
+

Called at the start of a simulation run. Ensures that the +random number generated is reseeded to ensure that +the sequence of random waypoints is repeatable.

+
+ +
+
+demand()[source]
+
+
Compute speed and heading for random waypoint

% +% [SPEED,STEER] = R.demand() is the speed and steer angle to +% drive the vehicle toward the next waypoint. When the vehicle is +% within R.dtresh a new waypoint is chosen. +% +% See also Vehicle.

+
+
+
+ +
+
+property vehicle
+

Set/get the vehicle under control

+
+
Getter:
+

return VehicleBase instance

+
+
Setter:
+

set VehicleBase instance

+
+
+
+

Note

+

The setter is invoked by vehicle.control = driver

+
+
+ +
+ +
+
+
+

Superclass

+
+
+
+class roboticstoolbox.mobile.drivers.VehicleDriverBase[source]
+

Bases: ABC

+

Abstract Vehicle driver class

+

Abtract class that can drive a mobile robot.

+
+
Seealso:
+

RandomPath

+
+
+
+
+abstract demand()[source]
+

Compute speed and heading

+
+
Returns:
+

speed and steering for VehicleBase

+
+
+

When an instance of a VehicleDriverBase class is attached as +the control for an instance of a VehicleBase class, this method +is called at each time step to provide the control input.

+

Has access to the vehicle and its state through the vehicle() +property.

+
+ +
+
+abstract init()[source]
+

Initialize driving agent

+

Called at the start of a simulation run. Used to initialize state +including random number generator state.

+
+ +
+
+property vehicle
+

Set/get the vehicle under control

+
+
Getter:
+

return VehicleBase instance

+
+
Setter:
+

set VehicleBase instance

+
+
+
+

Note

+

The setter is invoked by vehicle.control = driver

+
+
+ +
+ +
+
+
+ + +
+
+
+ +
+ +
+

© Copyright 2023, Jesse Haviland and Peter Corke. + Last updated on 11-May-2024. +

+
+ + + +
+
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/mobile-planner-base.html b/mobile-planner-base.html new file mode 100644 index 00000000..a11560aa --- /dev/null +++ b/mobile-planner-base.html @@ -0,0 +1,953 @@ + + + + + + + + + + + Supporting classes — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

Supporting classes

+
+

Planner superclass

+
+
+class roboticstoolbox.mobile.PlannerBase(occgrid=None, inflate=0, ndims=None, start=None, goal=None, verbose=False, msgcolor='yellow', progress=True, marker=None, seed=None, **unused)[source]
+

Bases: ABC

+

Mobile robot motion planner (superclass)

+
+
Parameters:
+
    +
  • occgrid (OccGrid instance of ndarray(N,M), optional) – occupancy grid, defaults to None

  • +
  • start (array_like(2) or array_like(3), optional) – start position \((x, y)\) or configuration \((x, y, \theta)\), defaults to None

  • +
  • goal (array_like(2) or array_like(3), optional) – goal position \((x, y)\) or configuration \((x, y, \theta)\), defaults to None

  • +
  • inflate (float, optional) – obstacle inflation, defaults to 0

  • +
  • ndims – dimensionality of the planning, either 2 for \(\mathbb{R}^2\) or +3 for \(\SE{2}\)

  • +
  • ndims – int, optional

  • +
  • verbose (bool, optional) – verbosity, defaults to False

  • +
  • msgcolor (str, defaults to yellow) – color for message channel printing

  • +
  • seed (int, optional) – seed provided to private random number generator, defaults to None

  • +
+
+
+

Superclass for all mobile robot motion planners. Key functionality +includes:

+
    +
  • encapsulates an occupancy grid and optionally inflates it

  • +
  • validates start and goal if given

  • +
  • encapsulates a private random number generator with specifiable seed

  • +
  • encapsulates state such as start, goal, and the plan

  • +
  • provides a message channel for diagnostic output

  • +
+

The start and goal can be specifed in various ways:

+
    +
  • at constructor time by the arguments start or goal

  • +
  • by assigning the attributes start or goal

  • +
  • at planning time by specifying goal to plan()

  • +
  • at query time by specifying start to query()

  • +
+
+
Seealso:
+

OccGrid

+
+
+
+
+__str__()[source]
+

Compact representation of the planner

+
+
Returns:
+

pretty printed representation

+
+
Return type:
+

str

+
+
+
+ +
+
+property start
+

Set/get start point or configuration (superclass)

+
+
Getter:
+

Return start point or configuration

+
+
Return type:
+

ndarray(2) or ndarray(3)

+
+
Setter:
+

Set start point or configuration

+
+
Param:
+

array_like(2) or array_like(3)

+
+
+

The start is either a point \((x, y)\) or a configuration \((x, y, \theta)\).

+
+
Seealso:
+

goal()

+
+
+
+ +
+
+property goal
+

Set/get goal point or configuration (superclass)

+
+
Getter:
+

Return goal pointor configuration

+
+
Return type:
+

ndarray(2) or ndarray(3)

+
+
Setter:
+

Set goal point or configuration

+
+
Param:
+

array_like(2) or array_like(3)

+
+
+

The goal is either a point \((x, y)\) or a configuration \((x, y, \theta)\).

+
+
Seealso:
+

goal()

+
+
+
+ +
+
+property verbose
+

Get verbosity

+
+
Returns:
+

verbosity

+
+
Return type:
+

bool

+
+
+

If verbosity print more diagnostic messages to the planner’s +message channel.

+
+ +
+
+property random
+

Get private random number generator

+
+
Returns:
+

NumPy random number generator

+
+
Return type:
+

numpy.random.Generator

+
+
+

Has methods including:

+ +

The generator is initialized with the seed provided at constructor +time.

+
+
Seealso:
+

numpy.random.default_rng()

+
+
+
+ +
+
+random_init(seed=None)[source]
+

Initialize private random number generator

+
+
Parameters:
+

seed (int) – random number seed, defaults to value given to constructor

+
+
+

The private random number generator is initialized. The seed is seed +or the value given to the constructor. If None, the generator will +be randomly seeded using a seed from the operating system.

+
+ +
+
+plan()[source]
+

Plan path (superclass)

+
+
Parameters:
+
    +
  • start (array_like(2) or array_like(3), optional) – start position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value passed to constructor

  • +
  • goal (array_like(2) or array_like(3), optional) – goal position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value passed to constructor

  • +
+
+
+

The implementation depends on the particular planner. Some may have +no planning phase. The plan may also depend on just the start or goal.

+
+ +
+
+property occgrid
+

Occupancy grid

+
+
Returns:
+

occupancy grid used for planning

+
+
Return type:
+

OccGrid or subclass or None

+
+
+

Returns the occupancy grid that was optionally inflated at constructor time.

+
+
Seealso:
+

validate_endpoint() isoccupied()

+
+
+
+ +
+
+isoccupied(p)[source]
+

Test if point is occupied (superclass)

+
+
Parameters:
+

p (array_like(2)) – world coordinate (x, y)

+
+
Returns:
+

occupancy status of corresponding grid cell

+
+
Return type:
+

bool

+
+
+

The world coordinate is transformed and the status of the occupancy +grid cell is returned. If the point lies outside the bounds of +the occupancy grid return True (obstacle)

+

If there is no occupancy grid this function always returns False (free).

+
+
Seealso:
+

occgrid() validate_endpoint() BinaryOccGrid.isoccupied()

+
+
+
+ +
+
+validate_endpoint(p, dtype=None)[source]
+

Validate start or goal point

+
+
Parameters:
+
    +
  • p (array_like(2)) – the point

  • +
  • dtype (str, optional) – data type for point coordinates, defaults to None

  • +
+
+
Raises:
+

ValueError – point is inside obstacle

+
+
Returns:
+

point as a NumPy array of specified dtype

+
+
Return type:
+

ndarray(2)

+
+
+

The coordinate is tested to be a free cell in the occupancy grid.

+
+
Seealso:
+

isoccupied() occgrid()

+
+
+
+ +
+
+progress_start(n)[source]
+

Initialize a progress bar (superclass)

+
+
Parameters:
+

n (int) – Number of iterations in the operation

+
+
+

Create a progress bar for an operation that has n steps, for +example:

+
planner.progress_start(100)
+for i in range(100):
+    # ...
+    planner.progress_next()
+planner.progress_end()
+
+
+
+
Seealso:
+

progress_next() progress_end()

+
+
+
+ +
+
+progress_next()[source]
+

Increment a progress bar (superclass)

+

Create a progress bar for an operation that has n steps, for +example:

+
planner.progress_start(100)
+for i in range(100):
+    # ...
+    planner.progress_next()
+planner.progress_end()
+
+
+
+
Seealso:
+

progress_start() progress_end()

+
+
+
+ +
+
+progress_end()[source]
+

Finalize a progress bar (superclass)

+

Remove/cleanip a progress bar, for +example:

+
planner.progress_start(100)
+for i in range(100):
+    # ...
+    planner.progress_next()
+planner.progress_end()
+
+
+
+
Seealso:
+

progress_start() progress_next()

+
+
+
+ +
+
+query(start=None, goal=None, dtype=None, next=True, animate=False, movie=None)[source]
+

Find a path from start to goal using planner (superclass)

+
+
Parameters:
+
    +
  • start (array_like(), optional) – start position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value specified to constructor

  • +
  • goal (array_like(), optional) – goal position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value specified to constructor

  • +
  • dtype (str, optional) – data type for point coordinates, defaults to None

  • +
  • next (bool, optional) – invoke next() method of class, defaults to True

  • +
  • animate (bool, optional) – show the vehicle path, defaults to False

  • +
+
+
Returns:
+

path from start to goal, one point \((x, y)\) or configuration \((x, y, \theta)\) per row

+
+
Return type:
+

ndarray(N,2) or ndarray(N,3)

+
+
+

Find a path from start to goal using a previously computed plan.

+

This is a generic method that works for any planner +(\(\mathbb{R}^2\) or \(\SE{2}\)) that can incrementally +determine the next point on the path. The method performs the following +steps:

+
    +
  • Validate start and goal

  • +
  • If animate, visualize the environment using plot()

  • +
  • Iterate on the class’s next() method until the goal is +achieved, and if animate plot points.

  • +
+
+
Seealso:
+

next() plan()

+
+
+
+ +
+
+plot(path=None, line=None, line_r=None, configspace=False, unwrap=True, direction=None, background=True, path_marker=None, path_marker_reverse=None, start_marker=None, goal_marker=None, start_vehicle=None, goal_vehicle=None, start=None, goal=None, ax=None, block=None, bgargs={}, **unused)[source]
+

Plot vehicle path (superclass)

+
+
Parameters:
+
    +
  • path ((N, 2) or ndarray(N, 3)) – path, defaults to None

  • +
  • direction (array_like(N), optional) – travel direction associated with each point on path, is either >0 or <0, defaults to None

  • +
  • line (sequence of dict of arguments for plot) – line style for forward motion, default is striped yellow on black

  • +
  • line_r (sequence of dict of arguments for plot) – line style for reverse motion, default is striped red on black

  • +
  • configspace (bool, optional) – plot the path in 3D configuration space, input must be 3xN. +Start and goal style will be given by qstart_marker and qgoal_marker, defaults to False

  • +
  • unwrap (bool, optional) – for configuration space plot unwrap \(\theta\) so +there are no discontinuities at \(\pm \pi\), defaults to True

  • +
  • background (bool, optional) – plot occupancy grid if present, default True

  • +
  • start_marker (dict, optional) – style for marking start point

  • +
  • goal_marker (dict, optional) – style for marking goal point

  • +
  • start_vehicle (dict) – style for vehicle animation object at start configuration

  • +
  • goal_vehicle (dict) – style for vehicle animation object at goal configuration

  • +
  • start (array_like(2) or array_like(3), optional) – start position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value previously set

  • +
  • goal (array_like(2) or array_like(3), optional) – goal position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value previously set

  • +
  • bgargs (dict, optional) – arguments passed to plot_bg(), defaults to None

  • +
  • ax (matplotlib axes) – axes to plot into

  • +
  • block (bool, optional) – block after displaying the plot

  • +
+
+
+

Plots the start and goal location/pose if they are specified by +start or goal or were set by the object constructor or +plan or query method.

+

If the start and goal have length 2, planning in +\(\mathbb{R}^2\), then markers are drawn using styles specified by +start_marker and end_marker which are dicts using Matplotlib +keywords, for example:

+
planner.plot(path, start=dict(marker='s', color='b'))
+
+
+

If the start and goal have length 3, planning in \(\SE{2}\), +and configspace is False, then direction-indicating markers are used +to display start and goal configuration. These are also given as dicts +but have two items: 'shape' which is the shape of the polygonal +marker and is either 'triangle' or 'car'. The second item +'args' is passed to base.plot_poly() and Matplotlib and could +have values such as filled=True or color.

+

If configspace is False then a 3D plot is created and the start and +goal are indicated by Matplotlib markers specified by the dicts +start_marker and end_marker

+

Default values are provided for all markers:

+
+
    +
  • the start point is a circle

  • +
  • the goal point is a star

  • +
  • the start vehicle style is a VehiclePolygon(shape='car') as +an unfilled outline

  • +
  • the goal vehicle style is a VehiclePolygon(shape='car') as +a transparent filled shape

  • +
+
+

If background is True then the background of the plot is either or +both of:

+
    +
  • the occupancy grid

  • +
  • the distance field of the planner

  • +
+

Additional arguments bgargs can be passed through to plot_bg()

+

If path is specified it has one column per point and either 2 or 3 rows:

+
    +
  • 2 rows describes motion in the \(xy\)-plane and a 2D plot is created

  • +
  • 3 rows describes motion in the \(xy\theta\)-configuration space. By +default only the \(xy\)-plane is plotted unless configspace +is True in which case motion in \(xy\theta\)-configuration space +is shown as a 3D plot.

  • +
+

If the planner supports bi-directional motion then the direction +option gives the direction for each point on the path.

+

Forward motion segments are drawn using style information from line +while reverse motion segments are drawn using style information from +line_r. These are each a sequence of dicts of Matplotlib plot +options which can draw an arbitrary number of lines on top of each +other. The default:

+
line = (
+        {color:'black', linewidth:4},
+        {color:'yellow', linewidth:3, dashes:(5,5)}
+    )
+
+
+

will draw a blackline of width 4 with a dashed yellow line of width 3 +plotted on top, giving a line of alternating black and yellow dashes.

+
+
Seealso:
+

plot_bg() base.plot_poly()

+
+
+
+ +
+
+plot_bg(distance=None, cmap='gray', ax=None, inflated=True, colorbar=True, block=None, **unused)[source]
+

Plot background (superclass)

+
+
Parameters:
+
    +
  • distance (ndarray(N,M), optional) – override distance field, defaults to None

  • +
  • cmap (str or Colormap, optional) – Specify a colormap for the distance field, defaults to ‘gray’

  • +
+
+
+

Displays the background which is either the occupancy grid or a distance +field. The distance field encodes the distance of a point from the goal, small +distance is dark, a large distance is bright.

+
+
If the planner has an occupancy grid then that will be displayed with:
    +
  • free cells in white

  • +
  • occupied cells in red

  • +
  • inflated occupied cells in pink

  • +
+
+
+

If distance is provided, or the planner has a distancemap attribute +the the distance field will be used as the background and obstacle cells +(actual or inflated) will be shown in red. A colorbar is added.

+
+ +
+
+message(s, color=None)[source]
+

Print message to message channel

+
+
Parameters:
+
    +
  • s (str) – message to print

  • +
  • color (str, optional) – color to print it, defaults to color specified at +constructor time.

  • +
+
+
+
+ +
+ +
+
+

Occupancy grid base classes

+
+
+class roboticstoolbox.mobile.OccGrid.BaseMap(workspace=None, name=None, **unused)[source]
+

Bases: ABC

+
+ +
+
+class roboticstoolbox.mobile.OccGrid.BaseOccupancyGrid(grid=None, origin=(0, 0), value=0, cellsize=1, **kwargs)[source]
+

Bases: BaseMap

+
+
+copy()[source]
+

Copy an occupancy grid (superclass)

+
+
Returns:
+

copy of the ocupancy grid

+
+
Return type:
+

OccGrid

+
+
+
+ +
+
+__str__()[source]
+

Compact string description of occupancy grid (superclass)

+
+
Returns:
+

summary of occupancy grid characteristics

+
+
Return type:
+

str

+
+
+
+ +
+
+property grid
+

Occupancy grid as a NumPy array (superclass)

+
+
Returns:
+

binary occupancy grid

+
+
Return type:
+

ndarray(N,M) of bool

+
+
+

If inflate() has been called, this will return the inflated +occupancy grid.

+
+ +
+
+property xmin
+

Minimum x-coordinate of this grid (superclass)

+
+
Returns:
+

minimum world x-coordinate

+
+
Return type:
+

float

+
+
+
+ +
+
+property xmax
+

Maximum x-coordinate of this grid (superclass)

+
+
Returns:
+

maximum world x-coordinate

+
+
Return type:
+

float

+
+
+
+ +
+
+property ymin
+

Minimum y-coordinate of this grid (superclass)

+
+
Returns:
+

minimum world y-coordinate

+
+
Return type:
+

float

+
+
+
+ +
+
+property ymax
+

Maximum y-coordinate of this grid (superclass)

+
+
Returns:
+

maximum world y-coordinate

+
+
Return type:
+

float

+
+
+
+ +
+
+property shape
+

Shape of the occupancy grid array (superclass)

+
+
Returns:
+

shape of the occupancy grid array

+
+
Return type:
+

2-tuple

+
+
+

This is the shape of the NumPy array that holds the occupancy grid.

+
+ +
+
+property maxdim
+

Maximum dimension of grid in world coordinates (superclass)

+
+
Returns:
+

maximum side length of the occupancy grid

+
+
Return type:
+

float

+
+
+
+ +
+
+property workspace
+

Bounds of the occupancy grid in world coordinates (superclass)

+
+
Returns:
+

workspace bounds [xmin, xmax, ymin, ymax]

+
+
Return type:
+

ndarray(4)

+
+
+

Returns the bounds of the occupancy grid in world coordinates.

+
+ +
+
+property name
+

Occupancy grid name (superclass)

+
+
Returns:
+

name of the occupancy grid

+
+
Return type:
+

str

+
+
+
+ +
+
+set(region, value)[source]
+

Set region of map (superclass)

+
+
Parameters:
+
    +
  • region (array_like(4)) – The region [xmin, ymin, xmax, ymax]

  • +
  • value (int, bool, float) – value to set cells to

  • +
+
+
+
+ +
+
+g2w(p)[source]
+

Convert grid coordinate to world coordinate (superclass)

+
+
Parameters:
+

p (array_like(2)) – grid coordinate (column, row)

+
+
Returns:
+

world coordinate (x, y)

+
+
Return type:
+

ndarray(2)

+
+
+

The grid cell size and offset are used to convert occupancy grid +coordinate p to a world coordinate.

+
+ +
+
+w2g(p)[source]
+

Convert world coordinate to grid coordinate (superclass)

+
+
Parameters:
+

p (array_like(2)) – world coordinate (x, y)

+
+
Returns:
+

grid coordinate (column, row)

+
+
Return type:
+

ndarray(2)

+
+
+

The grid cell size and offset are used to convert p to an occupancy +grid coordinate. The grid coordinate is rounded and cast to integer +value. No check is made on the validity of the coordinate.

+
+ +
+
+plot(map=None, ax=None, block=None, **kwargs)[source]
+

Plot the occupancy grid (superclass)

+
+
Parameters:
+
    +
  • map (ndarray(N,M), optional) – array which is plotted instead of the grid, must be same +size as the occupancy grid,defaults to None

  • +
  • ax (Axes2D, optional) – matplotlib axes to plot into, defaults to None

  • +
  • block (bool, optional) – block until plot is dismissed, defaults to None

  • +
  • kwargs – arguments passed to imshow

  • +
+
+
+

The grid is plotted as an image but with axes in world coordinates.

+

The grid is a NumPy boolean array which has values 0 (false=unoccupied) +and 1 (true=occupied). Passing a cmap option to imshow can be used +to control the displayed color of free space and obstacles.

+
+ +
+
+line_w(p1, p2)[source]
+

Get index of cells along a line segment (superclass)

+
+
Parameters:
+
    +
  • p1 (array_like(2)) – start

  • +
  • p2 (array_like(2)) – end

  • +
+
+
Returns:
+

index into grid

+
+
Return type:
+

ndarray(N)

+
+
+

Get the indices of cells along a line segment defined by the end +points given in world coordinates.

+

The returned indices can be applied to a raveled view of the grid.

+
+
Seealso:
+

ravel() w2g()

+
+
+
+ +
+
+property ravel
+

Ravel the grid (superclass)

+
+
Returns:
+

1D view of the occupancy grid

+
+
Return type:
+

ndarray(N)

+
+
+
+ +
+ +
+
+ + +
+
+
+ +
+ +
+

© Copyright 2023, Jesse Haviland and Peter Corke. + Last updated on 11-May-2024. +

+
+ + + +
+
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/mobile-planner-continuous.html b/mobile-planner-continuous.html new file mode 100644 index 00000000..03799099 --- /dev/null +++ b/mobile-planner-continuous.html @@ -0,0 +1,2346 @@ + + + + + + + + + + + Continuous configuration-space planners — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

Continuous configuration-space planners

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

Planner

Plans in

Discrete/Continuous

Obstacle avoidance

DubinsPlanner

\(\SE{2}\)

continuous

no

ReedsSheppPlanner

\(\SE{2}\)

continuous

no

CurvaturePolyPlanner

\(\SE{2}\)

continuous

no

QuinticPolyPlanner

\(\SE{2}\)

continuous

no

RRTPlanner

\(\SE{2}\)

continuous

yes

+

These planners do not support planning around obstacles, but allow for the +start and goal configuration \((x, y, \theta)\) to be specified.

+
+

PRM planner

+
+
+class roboticstoolbox.mobile.PRMPlanner(occgrid=None, npoints=100, dist_thresh=None, **kwargs)[source]
+

Bases: PlannerBase

+

Distance transform path planner

+
+
Parameters:
+
    +
  • occgrid (BinaryOccGrid or ndarray(h,w)) – occupancy grid

  • +
  • npoints (int, optional) – number of random points, defaults to 100

  • +
  • dist_thresh (float, optional) – distance threshold, a new point is only added to the +roadmap if it is closer than this distance to an existing vertex, +defaults to None

  • +
  • Planner – probabilistic roadmap path planner

  • +
  • kwargs – common planner options, see PlannerBase

  • +
+
+
+ + + + + + + + + + + + + + + + + + + + +

Feature

Capability

Plan

Cartesian space

Obstacle avoidance

Yes, occupancy grid

Curvature

Discontinuous

Motion

Omnidirectional

+

Creates a planner that finds the path between two points in the +plane using omnidirectional motion. The path comprises a set of way points.

+

Example:

+
>>> from roboticstoolbox import PRMPlanner
+>>> import numpy as np
+>>> simplegrid = np.zeros((6, 6));
+>>> simplegrid[2:5, 3:5] = 1
+>>> prm = PRMPlanner(simplegrid);
+>>> prm.plan()
+>>> path = prm.query(start=(5, 4), goal=(1,1))
+>>> print(path.T)
+[[5.     4.9748 4.6733 4.5757 3.1849 2.1734 1.0059 1.    ]
+ [4.     4.0343 3.0184 2.0005 1.351  1.2122 1.0357 1.    ]]
+
+
+
+
Author:
+

Peter Corke

+
+
Seealso:
+

PlannerBase

+
+
+
+
+property npoints
+

Number of points in the roadmap

+
+
Returns:
+

Number of points

+
+
Return type:
+

int

+
+
+
+ +
+
+property dist_thresh
+

Distance threshold

+
+
Returns:
+

distance threshold

+
+
Return type:
+

float

+
+
+

Edges are created between points if the distance between them is less +than this value.

+
+ +
+
+property graph
+

Roadmap graph

+
+
Returns:
+

roadmap as an undirected graph

+
+
Return type:
+

pgraph.UGraph instance

+
+
+
+ +
+
+plan(npoints=None, dist_thresh=None, animate=None)[source]
+

Plan PRM path

+
+
Parameters:
+
    +
  • npoints (int, optional) – number of random points, defaults to npoints given +to constructor

  • +
  • dist_thresh (float, optional) – distance threshold, defaults to dist_thresh given +to constructor

  • +
  • animate (bool, optional) – animate the planning algorithm iterations, defaults to False

  • +
+
+
+

Create a probablistic roadmap. This is a graph connecting points +randomly selected from the free space of the occupancy grid. Edges are +created between points if the distance between them is less than +dist_thresh.

+

The roadmap is a pgraph UGraph +UGraph +UGraph

+
+
Seealso:
+

query() graph()

+
+
+
+ +
+
+query(start, goal, **kwargs)[source]
+

Find a path from start to goal using planner

+
+
Parameters:
+
    +
  • start (array_like(), optional) – start position \((x, y)\), defaults to previously set value

  • +
  • goal (array_like(), optional) – goal position \((x, y)\), defaults to previously set value

  • +
  • kwargs – options passed to PlannerBase.query()

  • +
+
+
Returns:
+

path from start to goal, one point \((x, y)\) per row

+
+
Return type:
+

ndarray(N,2)

+
+
+

The path is a sparse sequence of waypoints, with variable distance +between them.

+
+

Warning

+

Waypoints 1 to N-2 are part of the roadmap, while waypoints +0 and N-1 are the start and goal respectively. The first and last +motion segment is not guaranteed to be obstacle free.

+
+
+ +
+
+plot(*args, vertex={}, edge={}, **kwargs)[source]
+

Plot PRM path

+
+
Parameters:
+
    +
  • vertex (dict, optional) – vertex style, defaults to {}

  • +
  • edge (dict, optional) – edge style, defaults to {}

  • +
+
+
+

Displays:

+
    +
  • the planner background (obstacles)

  • +
  • the roadmap graph

  • +
  • the path

  • +
+
+
Seealso:
+

UGraph.plot()

+
+
+
+ +
+
+property goal
+

Set/get goal point or configuration (superclass)

+
+
Getter:
+

Return goal pointor configuration

+
+
Return type:
+

ndarray(2) or ndarray(3)

+
+
Setter:
+

Set goal point or configuration

+
+
Param:
+

array_like(2) or array_like(3)

+
+
+

The goal is either a point \((x, y)\) or a configuration \((x, y, \theta)\).

+
+
Seealso:
+

goal()

+
+
+
+ +
+
+property occgrid
+

Occupancy grid

+
+
Returns:
+

occupancy grid used for planning

+
+
Return type:
+

OccGrid or subclass or None

+
+
+

Returns the occupancy grid that was optionally inflated at constructor time.

+
+
Seealso:
+

validate_endpoint() isoccupied()

+
+
+
+ +
+
+property start
+

Set/get start point or configuration (superclass)

+
+
Getter:
+

Return start point or configuration

+
+
Return type:
+

ndarray(2) or ndarray(3)

+
+
Setter:
+

Set start point or configuration

+
+
Param:
+

array_like(2) or array_like(3)

+
+
+

The start is either a point \((x, y)\) or a configuration \((x, y, \theta)\).

+
+
Seealso:
+

goal()

+
+
+
+ +
+
+validate_endpoint(p, dtype=None)
+

Validate start or goal point

+
+
Parameters:
+
    +
  • p (array_like(2)) – the point

  • +
  • dtype (str, optional) – data type for point coordinates, defaults to None

  • +
+
+
Raises:
+

ValueError – point is inside obstacle

+
+
Returns:
+

point as a NumPy array of specified dtype

+
+
Return type:
+

ndarray(2)

+
+
+

The coordinate is tested to be a free cell in the occupancy grid.

+
+
Seealso:
+

isoccupied() occgrid()

+
+
+
+ +
+
+property verbose
+

Get verbosity

+
+
Returns:
+

verbosity

+
+
Return type:
+

bool

+
+
+

If verbosity print more diagnostic messages to the planner’s +message channel.

+
+ +
+ +
+
+

Dubins path planner

+
+
+class roboticstoolbox.mobile.DubinsPlanner(curvature=1, stepsize=0.1, **kwargs)[source]
+

Bases: PlannerBase

+

Dubins path planner

+
+
Parameters:
+
    +
  • curvature (float, optional) – maximum path curvature, defaults to 1.0

  • +
  • stepsize (float, optional) – spacing between points on the path, defaults to 0.1

  • +
+
+
Returns:
+

Dubins path planner

+
+
Return type:
+

DubinsPlanner instance

+
+
+ + + + + + + + + + + + + + + + + + + + +

Feature

Capability

Plan

\(\SE{2}\)

Obstacle avoidance

No

Curvature

Discontinuous

Motion

Forwards only

+

Creates a planner that finds the path between two configurations in the +plane using forward motion only. The path comprises upto 3 segments that are +straight lines, or arcs with curvature of \(\pm\) curvature.

+

Example:

+
>>> from roboticstoolbox import DubinsPlanner
+>>> from math import pi
+>>> dubins = DubinsPlanner(curvature=1.0)
+>>> path, status = dubins.query(start=(0, 0, pi/2), goal=(1, 0, pi/2))
+>>> print(path[:5,:])
+[[ 0.      0.      1.5708]
+ [-0.005   0.0998  1.6708]
+ [-0.0199  0.1987  1.7708]
+ [-0.0447  0.2955  1.8708]
+ [-0.0789  0.3894  1.9708]]
+>>> print(status)
+DubinsStatus(segments=['L', 'S', 'L'], length=7.283185307179586, seglengths=[4.71238898038469, 1.0, 1.5707963267948966])
+
+
+
+
Reference:
+

On Curves of Minimal Length with a Constraint on Average +Curvature, and with Prescribed Initial and Terminal Positions and +Tangents, Dubins, L.E. (July 1957), American Journal of Mathematics. +79(3): 497–516.

+
+
Thanks:
+

based on Dubins path planning from Python Robotics

+
+
Seealso:
+

ReedsSheppPlanner PlannerBase

+
+
+
+
+property curvature
+
+ +
+
+property stepsize
+
+ +
+
+query(start, goal, **kwargs)[source]
+

Find a path between two configurations

+
+
Parameters:
+
    +
  • start (array_like(3), optional) – start configuration \((x, y, \theta)\)

  • +
  • goal (array_like(3), optional) – goal configuration \((x, y, \theta)\)

  • +
+
+
Returns:
+

path and status

+
+
Return type:
+

ndarray(N,3), namedtuple

+
+
+

The path comprises points equally spaced at a distance of stepsize.

+

The returned status value has elements:

+ + + + + + + + + + + + + + + +

Element

Description

segments

a list containing the type of each path segment as +a single letter code: either “L”, “R” or “S” for +left turn, right turn or straight line respectively.

length

total path length

lengths

the length of each path segment. The sign of the +length indicates the direction of travel.

+
+ +
+
+property goal
+

Set/get goal point or configuration (superclass)

+
+
Getter:
+

Return goal pointor configuration

+
+
Return type:
+

ndarray(2) or ndarray(3)

+
+
Setter:
+

Set goal point or configuration

+
+
Param:
+

array_like(2) or array_like(3)

+
+
+

The goal is either a point \((x, y)\) or a configuration \((x, y, \theta)\).

+
+
Seealso:
+

goal()

+
+
+
+ +
+
+property occgrid
+

Occupancy grid

+
+
Returns:
+

occupancy grid used for planning

+
+
Return type:
+

OccGrid or subclass or None

+
+
+

Returns the occupancy grid that was optionally inflated at constructor time.

+
+
Seealso:
+

validate_endpoint() isoccupied()

+
+
+
+ +
+
+plan()
+

Plan path (superclass)

+
+
Parameters:
+
    +
  • start (array_like(2) or array_like(3), optional) – start position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value passed to constructor

  • +
  • goal (array_like(2) or array_like(3), optional) – goal position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value passed to constructor

  • +
+
+
+

The implementation depends on the particular planner. Some may have +no planning phase. The plan may also depend on just the start or goal.

+
+ +
+
+plot(path=None, line=None, line_r=None, configspace=False, unwrap=True, direction=None, background=True, path_marker=None, path_marker_reverse=None, start_marker=None, goal_marker=None, start_vehicle=None, goal_vehicle=None, start=None, goal=None, ax=None, block=None, bgargs={}, **unused)
+

Plot vehicle path (superclass)

+
+
Parameters:
+
    +
  • path ((N, 2) or ndarray(N, 3)) – path, defaults to None

  • +
  • direction (array_like(N), optional) – travel direction associated with each point on path, is either >0 or <0, defaults to None

  • +
  • line (sequence of dict of arguments for plot) – line style for forward motion, default is striped yellow on black

  • +
  • line_r (sequence of dict of arguments for plot) – line style for reverse motion, default is striped red on black

  • +
  • configspace (bool, optional) – plot the path in 3D configuration space, input must be 3xN. +Start and goal style will be given by qstart_marker and qgoal_marker, defaults to False

  • +
  • unwrap (bool, optional) – for configuration space plot unwrap \(\theta\) so +there are no discontinuities at \(\pm \pi\), defaults to True

  • +
  • background (bool, optional) – plot occupancy grid if present, default True

  • +
  • start_marker (dict, optional) – style for marking start point

  • +
  • goal_marker (dict, optional) – style for marking goal point

  • +
  • start_vehicle (dict) – style for vehicle animation object at start configuration

  • +
  • goal_vehicle (dict) – style for vehicle animation object at goal configuration

  • +
  • start (array_like(2) or array_like(3), optional) – start position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value previously set

  • +
  • goal (array_like(2) or array_like(3), optional) – goal position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value previously set

  • +
  • bgargs (dict, optional) – arguments passed to plot_bg(), defaults to None

  • +
  • ax (matplotlib axes) – axes to plot into

  • +
  • block (bool, optional) – block after displaying the plot

  • +
+
+
+

Plots the start and goal location/pose if they are specified by +start or goal or were set by the object constructor or +plan or query method.

+

If the start and goal have length 2, planning in +\(\mathbb{R}^2\), then markers are drawn using styles specified by +start_marker and end_marker which are dicts using Matplotlib +keywords, for example:

+
planner.plot(path, start=dict(marker='s', color='b'))
+
+
+

If the start and goal have length 3, planning in \(\SE{2}\), +and configspace is False, then direction-indicating markers are used +to display start and goal configuration. These are also given as dicts +but have two items: 'shape' which is the shape of the polygonal +marker and is either 'triangle' or 'car'. The second item +'args' is passed to base.plot_poly() and Matplotlib and could +have values such as filled=True or color.

+

If configspace is False then a 3D plot is created and the start and +goal are indicated by Matplotlib markers specified by the dicts +start_marker and end_marker

+

Default values are provided for all markers:

+
+
    +
  • the start point is a circle

  • +
  • the goal point is a star

  • +
  • the start vehicle style is a VehiclePolygon(shape='car') as +an unfilled outline

  • +
  • the goal vehicle style is a VehiclePolygon(shape='car') as +a transparent filled shape

  • +
+
+

If background is True then the background of the plot is either or +both of:

+
    +
  • the occupancy grid

  • +
  • the distance field of the planner

  • +
+

Additional arguments bgargs can be passed through to plot_bg()

+

If path is specified it has one column per point and either 2 or 3 rows:

+
    +
  • 2 rows describes motion in the \(xy\)-plane and a 2D plot is created

  • +
  • 3 rows describes motion in the \(xy\theta\)-configuration space. By +default only the \(xy\)-plane is plotted unless configspace +is True in which case motion in \(xy\theta\)-configuration space +is shown as a 3D plot.

  • +
+

If the planner supports bi-directional motion then the direction +option gives the direction for each point on the path.

+

Forward motion segments are drawn using style information from line +while reverse motion segments are drawn using style information from +line_r. These are each a sequence of dicts of Matplotlib plot +options which can draw an arbitrary number of lines on top of each +other. The default:

+
line = (
+        {color:'black', linewidth:4},
+        {color:'yellow', linewidth:3, dashes:(5,5)}
+    )
+
+
+

will draw a blackline of width 4 with a dashed yellow line of width 3 +plotted on top, giving a line of alternating black and yellow dashes.

+
+
Seealso:
+

plot_bg() base.plot_poly()

+
+
+
+ +
+
+property start
+

Set/get start point or configuration (superclass)

+
+
Getter:
+

Return start point or configuration

+
+
Return type:
+

ndarray(2) or ndarray(3)

+
+
Setter:
+

Set start point or configuration

+
+
Param:
+

array_like(2) or array_like(3)

+
+
+

The start is either a point \((x, y)\) or a configuration \((x, y, \theta)\).

+
+
Seealso:
+

goal()

+
+
+
+ +
+
+validate_endpoint(p, dtype=None)
+

Validate start or goal point

+
+
Parameters:
+
    +
  • p (array_like(2)) – the point

  • +
  • dtype (str, optional) – data type for point coordinates, defaults to None

  • +
+
+
Raises:
+

ValueError – point is inside obstacle

+
+
Returns:
+

point as a NumPy array of specified dtype

+
+
Return type:
+

ndarray(2)

+
+
+

The coordinate is tested to be a free cell in the occupancy grid.

+
+
Seealso:
+

isoccupied() occgrid()

+
+
+
+ +
+
+property verbose
+

Get verbosity

+
+
Returns:
+

verbosity

+
+
Return type:
+

bool

+
+
+

If verbosity print more diagnostic messages to the planner’s +message channel.

+
+ +
+ +
+
+

Reeds-Shepp path planner

+
+
+class roboticstoolbox.mobile.ReedsSheppPlanner(curvature=1, stepsize=0.1, **kwargs)[source]
+

Bases: PlannerBase

+

Reeds-Shepp path planner

+
+
Parameters:
+
    +
  • curvature (float, optional) – maximum path curvature, defaults to 1.0

  • +
  • stepsize (float, optional) – spacing between points on the path, defaults to 0.1

  • +
  • Planner (ReedsSheppPlanner instance) – Reeds-Shepp path planner

  • +
+
+
+ + + + + + + + + + + + + + + + + + + + +

Feature

Capability

Plan

\(\SE{2}\)

Obstacle avoidance

No

Curvature

Discontinuous

Motion

Bidirectional

+

Creates a planner that finds the path between two configurations in the +plane using forward and backward motion. The path comprises upto 3 segments +that are straight lines, or arcs with curvature of \(\pm\) +curvature.

+

Example:

+
>>> from roboticstoolbox import ReedsSheppPlanner
+>>> from math import pi
+>>> reedshepp = ReedsSheppPlanner(curvature=1.0)
+>>> path, status = reedshepp.query(start=(0, 0, pi/2), goal=(1, 0, pi/2))
+>>> print(path[:5,:])
+[[0.     0.     1.5708]
+ [0.005  0.0998 1.4708]
+ [0.0199 0.1987 1.3708]
+ [0.0447 0.2955 1.2708]
+ [0.0789 0.3894 1.1708]]
+>>> print(status)
+ReedsSheppStatus(segments=['R', 'L', 'R'], length=6.283185307179586, seglengths=[4.459708725242611, -0.5053605102841573, 1.3181160716528177], direction=[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1])
+
+
+
+
Reference:
+

Optimal paths for a car that goes both forwards and backwards, +Reeds, J.A. and L.A. Shepp, Pacific J. Math., 145 (1990), +pp. 367–393.

+
+
Thanks:
+

based on Reeds-Shepp path planning from Python Robotics

+
+
Seealso:
+

DubinsPlanner PlannerBase

+
+
+
+
+query(start, goal, **kwargs)[source]
+

Find a path between two configurations

+
+
Parameters:
+
    +
  • start (array_like(3), optional) – start configuration \((x, y, \theta)\)

  • +
  • goal (array_like(3), optional) – goal configuration \((x, y, \theta)\)

  • +
+
+
Returns:
+

path and status

+
+
Return type:
+

ndarray(N,3), namedtuple

+
+
+

The path comprises points equally spaced at a distance of stepsize.

+

The returned status value has elements:

+ + + + + + + + + + + + + + + + + + +

Element

Description

segments

a list containing the type of each path segment as +a single letter code: either “L”, “R” or “S” for +left turn, right turn or straight line respectively.

length

total path length

lengths

the length of each path segment. The sign of the +length indicates the direction of travel.

direction

the direction of motion at each point on the path

+
+

Note

+

The direction of turning is reversed when travelling +backwards.

+
+
+ +
+
+property goal
+

Set/get goal point or configuration (superclass)

+
+
Getter:
+

Return goal pointor configuration

+
+
Return type:
+

ndarray(2) or ndarray(3)

+
+
Setter:
+

Set goal point or configuration

+
+
Param:
+

array_like(2) or array_like(3)

+
+
+

The goal is either a point \((x, y)\) or a configuration \((x, y, \theta)\).

+
+
Seealso:
+

goal()

+
+
+
+ +
+
+property occgrid
+

Occupancy grid

+
+
Returns:
+

occupancy grid used for planning

+
+
Return type:
+

OccGrid or subclass or None

+
+
+

Returns the occupancy grid that was optionally inflated at constructor time.

+
+
Seealso:
+

validate_endpoint() isoccupied()

+
+
+
+ +
+
+plan()
+

Plan path (superclass)

+
+
Parameters:
+
    +
  • start (array_like(2) or array_like(3), optional) – start position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value passed to constructor

  • +
  • goal (array_like(2) or array_like(3), optional) – goal position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value passed to constructor

  • +
+
+
+

The implementation depends on the particular planner. Some may have +no planning phase. The plan may also depend on just the start or goal.

+
+ +
+
+plot(path=None, line=None, line_r=None, configspace=False, unwrap=True, direction=None, background=True, path_marker=None, path_marker_reverse=None, start_marker=None, goal_marker=None, start_vehicle=None, goal_vehicle=None, start=None, goal=None, ax=None, block=None, bgargs={}, **unused)
+

Plot vehicle path (superclass)

+
+
Parameters:
+
    +
  • path ((N, 2) or ndarray(N, 3)) – path, defaults to None

  • +
  • direction (array_like(N), optional) – travel direction associated with each point on path, is either >0 or <0, defaults to None

  • +
  • line (sequence of dict of arguments for plot) – line style for forward motion, default is striped yellow on black

  • +
  • line_r (sequence of dict of arguments for plot) – line style for reverse motion, default is striped red on black

  • +
  • configspace (bool, optional) – plot the path in 3D configuration space, input must be 3xN. +Start and goal style will be given by qstart_marker and qgoal_marker, defaults to False

  • +
  • unwrap (bool, optional) – for configuration space plot unwrap \(\theta\) so +there are no discontinuities at \(\pm \pi\), defaults to True

  • +
  • background (bool, optional) – plot occupancy grid if present, default True

  • +
  • start_marker (dict, optional) – style for marking start point

  • +
  • goal_marker (dict, optional) – style for marking goal point

  • +
  • start_vehicle (dict) – style for vehicle animation object at start configuration

  • +
  • goal_vehicle (dict) – style for vehicle animation object at goal configuration

  • +
  • start (array_like(2) or array_like(3), optional) – start position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value previously set

  • +
  • goal (array_like(2) or array_like(3), optional) – goal position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value previously set

  • +
  • bgargs (dict, optional) – arguments passed to plot_bg(), defaults to None

  • +
  • ax (matplotlib axes) – axes to plot into

  • +
  • block (bool, optional) – block after displaying the plot

  • +
+
+
+

Plots the start and goal location/pose if they are specified by +start or goal or were set by the object constructor or +plan or query method.

+

If the start and goal have length 2, planning in +\(\mathbb{R}^2\), then markers are drawn using styles specified by +start_marker and end_marker which are dicts using Matplotlib +keywords, for example:

+
planner.plot(path, start=dict(marker='s', color='b'))
+
+
+

If the start and goal have length 3, planning in \(\SE{2}\), +and configspace is False, then direction-indicating markers are used +to display start and goal configuration. These are also given as dicts +but have two items: 'shape' which is the shape of the polygonal +marker and is either 'triangle' or 'car'. The second item +'args' is passed to base.plot_poly() and Matplotlib and could +have values such as filled=True or color.

+

If configspace is False then a 3D plot is created and the start and +goal are indicated by Matplotlib markers specified by the dicts +start_marker and end_marker

+

Default values are provided for all markers:

+
+
    +
  • the start point is a circle

  • +
  • the goal point is a star

  • +
  • the start vehicle style is a VehiclePolygon(shape='car') as +an unfilled outline

  • +
  • the goal vehicle style is a VehiclePolygon(shape='car') as +a transparent filled shape

  • +
+
+

If background is True then the background of the plot is either or +both of:

+
    +
  • the occupancy grid

  • +
  • the distance field of the planner

  • +
+

Additional arguments bgargs can be passed through to plot_bg()

+

If path is specified it has one column per point and either 2 or 3 rows:

+
    +
  • 2 rows describes motion in the \(xy\)-plane and a 2D plot is created

  • +
  • 3 rows describes motion in the \(xy\theta\)-configuration space. By +default only the \(xy\)-plane is plotted unless configspace +is True in which case motion in \(xy\theta\)-configuration space +is shown as a 3D plot.

  • +
+

If the planner supports bi-directional motion then the direction +option gives the direction for each point on the path.

+

Forward motion segments are drawn using style information from line +while reverse motion segments are drawn using style information from +line_r. These are each a sequence of dicts of Matplotlib plot +options which can draw an arbitrary number of lines on top of each +other. The default:

+
line = (
+        {color:'black', linewidth:4},
+        {color:'yellow', linewidth:3, dashes:(5,5)}
+    )
+
+
+

will draw a blackline of width 4 with a dashed yellow line of width 3 +plotted on top, giving a line of alternating black and yellow dashes.

+
+
Seealso:
+

plot_bg() base.plot_poly()

+
+
+
+ +
+
+property start
+

Set/get start point or configuration (superclass)

+
+
Getter:
+

Return start point or configuration

+
+
Return type:
+

ndarray(2) or ndarray(3)

+
+
Setter:
+

Set start point or configuration

+
+
Param:
+

array_like(2) or array_like(3)

+
+
+

The start is either a point \((x, y)\) or a configuration \((x, y, \theta)\).

+
+
Seealso:
+

goal()

+
+
+
+ +
+
+validate_endpoint(p, dtype=None)
+

Validate start or goal point

+
+
Parameters:
+
    +
  • p (array_like(2)) – the point

  • +
  • dtype (str, optional) – data type for point coordinates, defaults to None

  • +
+
+
Raises:
+

ValueError – point is inside obstacle

+
+
Returns:
+

point as a NumPy array of specified dtype

+
+
Return type:
+

ndarray(2)

+
+
+

The coordinate is tested to be a free cell in the occupancy grid.

+
+
Seealso:
+

isoccupied() occgrid()

+
+
+
+ +
+
+property verbose
+

Get verbosity

+
+
Returns:
+

verbosity

+
+
Return type:
+

bool

+
+
+

If verbosity print more diagnostic messages to the planner’s +message channel.

+
+ +
+ +
+
+

Curvature-polynomial planner

+
+
+class roboticstoolbox.mobile.CurvaturePolyPlanner(curvature=None)[source]
+

Bases: PlannerBase

+
+
+query(start, goal)[source]
+

Find a path betwee two configurations

+
+
Parameters:
+
    +
  • start (array_like(3), optional) – start configuration \((x, y, \theta)\)

  • +
  • goal (array_like(3), optional) – goal configuration \((x, y, \theta)\)

  • +
+
+
Returns:
+

path and status

+
+
Return type:
+

ndarray(N,3), namedtuple

+
+
+

The path comprises points equally spaced at a distance of stepsize.

+

The returned status value has elements:

+ + + + + + + + + + + + + + + +

Element

Description

length

total path length

maxcurvature

maximum curvature on path

poly

curvature polynomial coefficients

+
+ +
+
+__str__()
+

Compact representation of the planner

+
+
Returns:
+

pretty printed representation

+
+
Return type:
+

str

+
+
+
+ +
+
+property goal
+

Set/get goal point or configuration (superclass)

+
+
Getter:
+

Return goal pointor configuration

+
+
Return type:
+

ndarray(2) or ndarray(3)

+
+
Setter:
+

Set goal point or configuration

+
+
Param:
+

array_like(2) or array_like(3)

+
+
+

The goal is either a point \((x, y)\) or a configuration \((x, y, \theta)\).

+
+
Seealso:
+

goal()

+
+
+
+ +
+
+property occgrid
+

Occupancy grid

+
+
Returns:
+

occupancy grid used for planning

+
+
Return type:
+

OccGrid or subclass or None

+
+
+

Returns the occupancy grid that was optionally inflated at constructor time.

+
+
Seealso:
+

validate_endpoint() isoccupied()

+
+
+
+ +
+
+plan()
+

Plan path (superclass)

+
+
Parameters:
+
    +
  • start (array_like(2) or array_like(3), optional) – start position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value passed to constructor

  • +
  • goal (array_like(2) or array_like(3), optional) – goal position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value passed to constructor

  • +
+
+
+

The implementation depends on the particular planner. Some may have +no planning phase. The plan may also depend on just the start or goal.

+
+ +
+
+plot(path=None, line=None, line_r=None, configspace=False, unwrap=True, direction=None, background=True, path_marker=None, path_marker_reverse=None, start_marker=None, goal_marker=None, start_vehicle=None, goal_vehicle=None, start=None, goal=None, ax=None, block=None, bgargs={}, **unused)
+

Plot vehicle path (superclass)

+
+
Parameters:
+
    +
  • path ((N, 2) or ndarray(N, 3)) – path, defaults to None

  • +
  • direction (array_like(N), optional) – travel direction associated with each point on path, is either >0 or <0, defaults to None

  • +
  • line (sequence of dict of arguments for plot) – line style for forward motion, default is striped yellow on black

  • +
  • line_r (sequence of dict of arguments for plot) – line style for reverse motion, default is striped red on black

  • +
  • configspace (bool, optional) – plot the path in 3D configuration space, input must be 3xN. +Start and goal style will be given by qstart_marker and qgoal_marker, defaults to False

  • +
  • unwrap (bool, optional) – for configuration space plot unwrap \(\theta\) so +there are no discontinuities at \(\pm \pi\), defaults to True

  • +
  • background (bool, optional) – plot occupancy grid if present, default True

  • +
  • start_marker (dict, optional) – style for marking start point

  • +
  • goal_marker (dict, optional) – style for marking goal point

  • +
  • start_vehicle (dict) – style for vehicle animation object at start configuration

  • +
  • goal_vehicle (dict) – style for vehicle animation object at goal configuration

  • +
  • start (array_like(2) or array_like(3), optional) – start position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value previously set

  • +
  • goal (array_like(2) or array_like(3), optional) – goal position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value previously set

  • +
  • bgargs (dict, optional) – arguments passed to plot_bg(), defaults to None

  • +
  • ax (matplotlib axes) – axes to plot into

  • +
  • block (bool, optional) – block after displaying the plot

  • +
+
+
+

Plots the start and goal location/pose if they are specified by +start or goal or were set by the object constructor or +plan or query method.

+

If the start and goal have length 2, planning in +\(\mathbb{R}^2\), then markers are drawn using styles specified by +start_marker and end_marker which are dicts using Matplotlib +keywords, for example:

+
planner.plot(path, start=dict(marker='s', color='b'))
+
+
+

If the start and goal have length 3, planning in \(\SE{2}\), +and configspace is False, then direction-indicating markers are used +to display start and goal configuration. These are also given as dicts +but have two items: 'shape' which is the shape of the polygonal +marker and is either 'triangle' or 'car'. The second item +'args' is passed to base.plot_poly() and Matplotlib and could +have values such as filled=True or color.

+

If configspace is False then a 3D plot is created and the start and +goal are indicated by Matplotlib markers specified by the dicts +start_marker and end_marker

+

Default values are provided for all markers:

+
+
    +
  • the start point is a circle

  • +
  • the goal point is a star

  • +
  • the start vehicle style is a VehiclePolygon(shape='car') as +an unfilled outline

  • +
  • the goal vehicle style is a VehiclePolygon(shape='car') as +a transparent filled shape

  • +
+
+

If background is True then the background of the plot is either or +both of:

+
    +
  • the occupancy grid

  • +
  • the distance field of the planner

  • +
+

Additional arguments bgargs can be passed through to plot_bg()

+

If path is specified it has one column per point and either 2 or 3 rows:

+
    +
  • 2 rows describes motion in the \(xy\)-plane and a 2D plot is created

  • +
  • 3 rows describes motion in the \(xy\theta\)-configuration space. By +default only the \(xy\)-plane is plotted unless configspace +is True in which case motion in \(xy\theta\)-configuration space +is shown as a 3D plot.

  • +
+

If the planner supports bi-directional motion then the direction +option gives the direction for each point on the path.

+

Forward motion segments are drawn using style information from line +while reverse motion segments are drawn using style information from +line_r. These are each a sequence of dicts of Matplotlib plot +options which can draw an arbitrary number of lines on top of each +other. The default:

+
line = (
+        {color:'black', linewidth:4},
+        {color:'yellow', linewidth:3, dashes:(5,5)}
+    )
+
+
+

will draw a blackline of width 4 with a dashed yellow line of width 3 +plotted on top, giving a line of alternating black and yellow dashes.

+
+
Seealso:
+

plot_bg() base.plot_poly()

+
+
+
+ +
+
+property start
+

Set/get start point or configuration (superclass)

+
+
Getter:
+

Return start point or configuration

+
+
Return type:
+

ndarray(2) or ndarray(3)

+
+
Setter:
+

Set start point or configuration

+
+
Param:
+

array_like(2) or array_like(3)

+
+
+

The start is either a point \((x, y)\) or a configuration \((x, y, \theta)\).

+
+
Seealso:
+

goal()

+
+
+
+ +
+
+validate_endpoint(p, dtype=None)
+

Validate start or goal point

+
+
Parameters:
+
    +
  • p (array_like(2)) – the point

  • +
  • dtype (str, optional) – data type for point coordinates, defaults to None

  • +
+
+
Raises:
+

ValueError – point is inside obstacle

+
+
Returns:
+

point as a NumPy array of specified dtype

+
+
Return type:
+

ndarray(2)

+
+
+

The coordinate is tested to be a free cell in the occupancy grid.

+
+
Seealso:
+

isoccupied() occgrid()

+
+
+
+ +
+
+property verbose
+

Get verbosity

+
+
Returns:
+

verbosity

+
+
Return type:
+

bool

+
+
+

If verbosity print more diagnostic messages to the planner’s +message channel.

+
+ +
+ +
+
+

Quintic-polynomial planner

+
+
+class roboticstoolbox.mobile.QuinticPolyPlanner(dt=0.1, start_vel=0, start_acc=0, goal_vel=0, goal_acc=0, max_acc=1, max_jerk=0.5, min_t=5, max_t=100)[source]
+

Bases: PlannerBase

+

Quintic polynomial path planner

+
+
Parameters:
+
    +
  • dt (float, optional) – time step, defaults to 0.1

  • +
  • start_vel (float, optional) – initial velocity, defaults to 0

  • +
  • start_acc (float, optional) – initial acceleration, defaults to 0

  • +
  • goal_vel (float, optional) – goal velocity, defaults to 0

  • +
  • goal_acc (float, optional) – goal acceleration, defaults to 0

  • +
  • max_acc (int, optional) – maximum acceleration, defaults to 1

  • +
  • max_jerk (float, optional) – maximum jerk, defaults to 0.5

  • +
  • min_t (float, optional) – minimum path time, defaults to 5

  • +
  • max_t (float, optional) – maximum path time, defaults to 100

  • +
+
+
Returns:
+

Quintic polynomial path planner

+
+
Return type:
+

QuinticPolyPlanner instance

+
+
+ + + + + + + + + + + + + + + + + + + + +

Feature

Capability

Plan

\(\SE{2}\)

Obstacle avoidance

No

Curvature

Continuous

Motion

Forwards only

+

Creates a planner that finds the path between two configurations in the +plane using forward motion only. The path is a continuous quintic polynomial +for x and y

+
+\[\begin{split}x(t) &= a_0 + a_1 t + a_2 t^2 + a_3 t^3 + a_4 t^4 + a_5 t^5 \\ +y(t) &= b_0 + b_1 t + b_2 t^2 + b_3 t^3 + b_4 t^4 + b_5 t^5\end{split}\]
+

that meets the given constraints. Trajectory time is given as a range.

+
+
Reference:
+

“Local Path Planning And Motion Control For AGV In +Positioning”, Takahashi, T. Hongo, Y. Ninomiya and G. +Sugimoto; Proceedings. IEEE/RSJ International Workshop on +Intelligent Robots and Systems (IROS ‘89) doi: 10.1109/IROS.1989.637936

+
+
+
+

Note

+

The path time is searched in the interval [min_t, max_t] in steps +of min_t.

+
+

Example:

+
>>> from roboticstoolbox import QuinticPolyPlanner
+>>> import numpy as np
+>>> start = (10, 10, np.deg2rad(10.0))
+>>> goal = (30, -10, np.deg2rad(20.0))
+>>> quintic = QuinticPolyPlanner(start_vel=1)
+>>> path, status = quintic.query(start, goal)
+find path!!
+>>> print(path[:5,:])
+[[10.     10.      0.1745]
+ [10.0985 10.0173  0.1724]
+ [10.1971 10.0342  0.1662]
+ [10.2959 10.0503  0.156 ]
+ [10.3949 10.0652  0.142 ]]
+
+
+
+
Thanks:
+

based on quintic polynomial planning from Python Robotics

+
+
Seealso:
+

Planner

+
+
+
+
+query(start, goal)[source]
+

Find a quintic polynomial path

+
+
Parameters:
+
    +
  • start (array_like(3), optional) – start configuration \((x, y, \theta)\)

  • +
  • goal (array_like(3), optional) – goal configuration \((x, y, \theta)\)

  • +
+
+
Returns:
+

path and status

+
+
Return type:
+

ndarray(N,3), namedtuple

+
+
+

The returned status value has elements:

+ + + + + + + + + + + + + + + + + + + + +

Element

Description

t

time to execute the path

vel

velocity profile along the path

accel

acceleration profile along the path

jerk

jerk profile along the path

+
+
Seealso:
+

Planner.query()

+
+
+
+ +
+
+__str__()
+

Compact representation of the planner

+
+
Returns:
+

pretty printed representation

+
+
Return type:
+

str

+
+
+
+ +
+
+property goal
+

Set/get goal point or configuration (superclass)

+
+
Getter:
+

Return goal pointor configuration

+
+
Return type:
+

ndarray(2) or ndarray(3)

+
+
Setter:
+

Set goal point or configuration

+
+
Param:
+

array_like(2) or array_like(3)

+
+
+

The goal is either a point \((x, y)\) or a configuration \((x, y, \theta)\).

+
+
Seealso:
+

goal()

+
+
+
+ +
+
+property occgrid
+

Occupancy grid

+
+
Returns:
+

occupancy grid used for planning

+
+
Return type:
+

OccGrid or subclass or None

+
+
+

Returns the occupancy grid that was optionally inflated at constructor time.

+
+
Seealso:
+

validate_endpoint() isoccupied()

+
+
+
+ +
+
+plan()
+

Plan path (superclass)

+
+
Parameters:
+
    +
  • start (array_like(2) or array_like(3), optional) – start position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value passed to constructor

  • +
  • goal (array_like(2) or array_like(3), optional) – goal position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value passed to constructor

  • +
+
+
+

The implementation depends on the particular planner. Some may have +no planning phase. The plan may also depend on just the start or goal.

+
+ +
+
+plot(path=None, line=None, line_r=None, configspace=False, unwrap=True, direction=None, background=True, path_marker=None, path_marker_reverse=None, start_marker=None, goal_marker=None, start_vehicle=None, goal_vehicle=None, start=None, goal=None, ax=None, block=None, bgargs={}, **unused)
+

Plot vehicle path (superclass)

+
+
Parameters:
+
    +
  • path ((N, 2) or ndarray(N, 3)) – path, defaults to None

  • +
  • direction (array_like(N), optional) – travel direction associated with each point on path, is either >0 or <0, defaults to None

  • +
  • line (sequence of dict of arguments for plot) – line style for forward motion, default is striped yellow on black

  • +
  • line_r (sequence of dict of arguments for plot) – line style for reverse motion, default is striped red on black

  • +
  • configspace (bool, optional) – plot the path in 3D configuration space, input must be 3xN. +Start and goal style will be given by qstart_marker and qgoal_marker, defaults to False

  • +
  • unwrap (bool, optional) – for configuration space plot unwrap \(\theta\) so +there are no discontinuities at \(\pm \pi\), defaults to True

  • +
  • background (bool, optional) – plot occupancy grid if present, default True

  • +
  • start_marker (dict, optional) – style for marking start point

  • +
  • goal_marker (dict, optional) – style for marking goal point

  • +
  • start_vehicle (dict) – style for vehicle animation object at start configuration

  • +
  • goal_vehicle (dict) – style for vehicle animation object at goal configuration

  • +
  • start (array_like(2) or array_like(3), optional) – start position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value previously set

  • +
  • goal (array_like(2) or array_like(3), optional) – goal position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value previously set

  • +
  • bgargs (dict, optional) – arguments passed to plot_bg(), defaults to None

  • +
  • ax (matplotlib axes) – axes to plot into

  • +
  • block (bool, optional) – block after displaying the plot

  • +
+
+
+

Plots the start and goal location/pose if they are specified by +start or goal or were set by the object constructor or +plan or query method.

+

If the start and goal have length 2, planning in +\(\mathbb{R}^2\), then markers are drawn using styles specified by +start_marker and end_marker which are dicts using Matplotlib +keywords, for example:

+
planner.plot(path, start=dict(marker='s', color='b'))
+
+
+

If the start and goal have length 3, planning in \(\SE{2}\), +and configspace is False, then direction-indicating markers are used +to display start and goal configuration. These are also given as dicts +but have two items: 'shape' which is the shape of the polygonal +marker and is either 'triangle' or 'car'. The second item +'args' is passed to base.plot_poly() and Matplotlib and could +have values such as filled=True or color.

+

If configspace is False then a 3D plot is created and the start and +goal are indicated by Matplotlib markers specified by the dicts +start_marker and end_marker

+

Default values are provided for all markers:

+
+
    +
  • the start point is a circle

  • +
  • the goal point is a star

  • +
  • the start vehicle style is a VehiclePolygon(shape='car') as +an unfilled outline

  • +
  • the goal vehicle style is a VehiclePolygon(shape='car') as +a transparent filled shape

  • +
+
+

If background is True then the background of the plot is either or +both of:

+
    +
  • the occupancy grid

  • +
  • the distance field of the planner

  • +
+

Additional arguments bgargs can be passed through to plot_bg()

+

If path is specified it has one column per point and either 2 or 3 rows:

+
    +
  • 2 rows describes motion in the \(xy\)-plane and a 2D plot is created

  • +
  • 3 rows describes motion in the \(xy\theta\)-configuration space. By +default only the \(xy\)-plane is plotted unless configspace +is True in which case motion in \(xy\theta\)-configuration space +is shown as a 3D plot.

  • +
+

If the planner supports bi-directional motion then the direction +option gives the direction for each point on the path.

+

Forward motion segments are drawn using style information from line +while reverse motion segments are drawn using style information from +line_r. These are each a sequence of dicts of Matplotlib plot +options which can draw an arbitrary number of lines on top of each +other. The default:

+
line = (
+        {color:'black', linewidth:4},
+        {color:'yellow', linewidth:3, dashes:(5,5)}
+    )
+
+
+

will draw a blackline of width 4 with a dashed yellow line of width 3 +plotted on top, giving a line of alternating black and yellow dashes.

+
+
Seealso:
+

plot_bg() base.plot_poly()

+
+
+
+ +
+
+property start
+

Set/get start point or configuration (superclass)

+
+
Getter:
+

Return start point or configuration

+
+
Return type:
+

ndarray(2) or ndarray(3)

+
+
Setter:
+

Set start point or configuration

+
+
Param:
+

array_like(2) or array_like(3)

+
+
+

The start is either a point \((x, y)\) or a configuration \((x, y, \theta)\).

+
+
Seealso:
+

goal()

+
+
+
+ +
+
+validate_endpoint(p, dtype=None)
+

Validate start or goal point

+
+
Parameters:
+
    +
  • p (array_like(2)) – the point

  • +
  • dtype (str, optional) – data type for point coordinates, defaults to None

  • +
+
+
Raises:
+

ValueError – point is inside obstacle

+
+
Returns:
+

point as a NumPy array of specified dtype

+
+
Return type:
+

ndarray(2)

+
+
+

The coordinate is tested to be a free cell in the occupancy grid.

+
+
Seealso:
+

isoccupied() occgrid()

+
+
+
+ +
+
+property verbose
+

Get verbosity

+
+
Returns:
+

verbosity

+
+
Return type:
+

bool

+
+
+

If verbosity print more diagnostic messages to the planner’s +message channel.

+
+ +
+ +
+
+

RRT planner

+
+
+class roboticstoolbox.mobile.RRTPlanner(map, vehicle, curvature=1.0, stepsize=0.2, npoints=50, **kwargs)[source]
+

Bases: PlannerBase

+

Rapidly exploring tree planner

+
+
Parameters:
+
    +
  • map (PolygonMap) – occupancy grid

  • +
  • vehicle (VehicleBase subclass) – vehicle kinematic model

  • +
  • curvature (float, optional) – maximum path curvature, defaults to 1.0

  • +
  • stepsize (float, optional) – spacing between points on the path, defaults to 0.2

  • +
  • npoints (int, optional) – number of vertices in random tree, defaults to 50

  • +
+
+
+ + + + + + + + + + + + + + + + + + + + +

Feature

Capability

Plan

\(\SE{2}\)

Obstacle avoidance

Yes, polygons

Curvature

Discontinuous

Motion

Bidirectional

+

Creates a planner that finds the obstacle-free path between two +configurations in the plane using forward and backward motion. The path +comprises multiple Dubins curves comprising straight lines, or arcs with +curvature of \(\pm\) curvature. Motion along the segments may be in +the forward or backward direction.

+

Polygons are used for obstacle avoidance:

+
    +
  • the environment is defined by a set of polygons represented by a PolygonMap

  • +
  • the vehicle is defined by a single polygon specified by the polygon +argument to its constructor

  • +
+

Example:

+
from roboticstoolbox import RRTPlanner
+from spatialmath import Polygon2
+from math import pi
+
+# create polygonal obstacles
+map = PolygonMap(workspace=[0, 10])
+map.add([(5, 50), (5, 6), (6, 6), (6, 50)])
+map.add([(5, 4), (5, -50), (6, -50), (6, 4)])
+
+# create outline polygon for vehicle
+l, w = 3, 1.5
+vpolygon = Polygon2([(-l/2, w/2), (-l/2, -w/2), (l/2, -w/2), (l/2, w/2)])
+
+# create vehicle model
+vehicle = Bicycle(steer_max=1, L=2, polygon=vpolygon)
+
+# create planner
+rrt = RRTPlanner(map=map, vehicle=vehicle, npoints=50, seed=0)
+# start and goal configuration
+qs = (2, 8, -pi/2)
+qg = (8, 2, -pi/2)
+
+# plan path
+rrt.plan(goal=qg)
+path, status = rrt.query(start=qs)
+print(path[:5,:])
+print(status)
+
+
+
+
Seealso:
+

DubinsPlanner Vehicle PlannerBase

+
+
+
+
+plan(goal, showsamples=True, showvalid=True, animate=False)[source]
+

Plan paths to goal using RRT

+
+
Parameters:
+
    +
  • goal (array_like(3), optional) – goal pose \((x, y, \theta)\), defaults to previously set value

  • +
  • showsamples (bool, optional) – display position part of configurations overlaid on the map, defaults to True

  • +
  • showvalid (bool, optional) – display valid configurations as vehicle polygons overlaid on the map, defaults to False

  • +
  • animate (bool, optional) – update the display as configurations are tested, defaults to False

  • +
+
+
+

Compute a rapidly exploring random tree with its root at the goal. +The tree will have npoints vertices spread uniformly randomly over +the workspace which is an attribute of the map.

+

For every new point added, a Dubins path is computed to the nearest +vertex already in the graph. Each configuration on that path, with +spacing of stepsize, is tested for obstacle intersection.

+

The configurations tested are displayed (translation only) if showsamples is +True. The valid configurations are displayed as vehicle polygones if showvalid +is True. If animate is True these points are displayed during the search +process, otherwise a single figure is displayed at the end.

+
+
Seealso:
+

query()

+
+
+
+ +
+
+query(start)[source]
+

Find a path from start configuration

+
+
Parameters:
+

start (array_like(3), optional) – start configuration \((x, y, \theta)\)

+
+
Returns:
+

path and status

+
+
Return type:
+

ndarray(N,3), namedtuple

+
+
+

The path comprises points equally spaced at a distance of stepsize.

+

The returned status value has elements:

+ + + + + + + + + + + + + +

Element

Description

length | total path length

initial_d | distance from start to first vertex in graph

vertices

sequence of vertices in the graph

+
+ +
+
+qrandom()[source]
+

Random configuration

+
+
Returns:
+

random configuration \((x, y, \theta)\)

+
+
Return type:
+

ndarray(3)

+
+
+

Returns a random configuration where position \((x, y)\) +lies within the bounds of the map associated with this planner.

+
+
Seealso:
+

qrandom_free()

+
+
+
+ +
+
+qrandom_free()[source]
+

Random obstacle free configuration

+
+
Returns:
+

random configuration \((x, y, \theta)\)

+
+
Return type:
+

ndarray(3)

+
+
+

Returns a random obstacle free configuration where position \((x, +y)\) lies within the bounds of the map associated with this planner. +Iterates on qrandom()

+
+
Seealso:
+

qrandom() iscollision()

+
+
+
+ +
+
+iscollision(q)[source]
+

Test if configuration is collision

+
+
Parameters:
+

q (array_like(3)) – vehicle configuration \((x, y, \theta)\)

+
+
Returns:
+

collision status

+
+
Return type:
+

bool

+
+
+

Transforms the vehicle polygon and tests for intersection against +the polygonal obstacle map.

+
+ +
+
+__str__()
+

Compact representation of the planner

+
+
Returns:
+

pretty printed representation

+
+
Return type:
+

str

+
+
+
+ +
+
+property goal
+

Set/get goal point or configuration (superclass)

+
+
Getter:
+

Return goal pointor configuration

+
+
Return type:
+

ndarray(2) or ndarray(3)

+
+
Setter:
+

Set goal point or configuration

+
+
Param:
+

array_like(2) or array_like(3)

+
+
+

The goal is either a point \((x, y)\) or a configuration \((x, y, \theta)\).

+
+
Seealso:
+

goal()

+
+
+
+ +
+
+property occgrid
+

Occupancy grid

+
+
Returns:
+

occupancy grid used for planning

+
+
Return type:
+

OccGrid or subclass or None

+
+
+

Returns the occupancy grid that was optionally inflated at constructor time.

+
+
Seealso:
+

validate_endpoint() isoccupied()

+
+
+
+ +
+
+plot(path=None, line=None, line_r=None, configspace=False, unwrap=True, direction=None, background=True, path_marker=None, path_marker_reverse=None, start_marker=None, goal_marker=None, start_vehicle=None, goal_vehicle=None, start=None, goal=None, ax=None, block=None, bgargs={}, **unused)
+

Plot vehicle path (superclass)

+
+
Parameters:
+
    +
  • path ((N, 2) or ndarray(N, 3)) – path, defaults to None

  • +
  • direction (array_like(N), optional) – travel direction associated with each point on path, is either >0 or <0, defaults to None

  • +
  • line (sequence of dict of arguments for plot) – line style for forward motion, default is striped yellow on black

  • +
  • line_r (sequence of dict of arguments for plot) – line style for reverse motion, default is striped red on black

  • +
  • configspace (bool, optional) – plot the path in 3D configuration space, input must be 3xN. +Start and goal style will be given by qstart_marker and qgoal_marker, defaults to False

  • +
  • unwrap (bool, optional) – for configuration space plot unwrap \(\theta\) so +there are no discontinuities at \(\pm \pi\), defaults to True

  • +
  • background (bool, optional) – plot occupancy grid if present, default True

  • +
  • start_marker (dict, optional) – style for marking start point

  • +
  • goal_marker (dict, optional) – style for marking goal point

  • +
  • start_vehicle (dict) – style for vehicle animation object at start configuration

  • +
  • goal_vehicle (dict) – style for vehicle animation object at goal configuration

  • +
  • start (array_like(2) or array_like(3), optional) – start position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value previously set

  • +
  • goal (array_like(2) or array_like(3), optional) – goal position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value previously set

  • +
  • bgargs (dict, optional) – arguments passed to plot_bg(), defaults to None

  • +
  • ax (matplotlib axes) – axes to plot into

  • +
  • block (bool, optional) – block after displaying the plot

  • +
+
+
+

Plots the start and goal location/pose if they are specified by +start or goal or were set by the object constructor or +plan or query method.

+

If the start and goal have length 2, planning in +\(\mathbb{R}^2\), then markers are drawn using styles specified by +start_marker and end_marker which are dicts using Matplotlib +keywords, for example:

+
planner.plot(path, start=dict(marker='s', color='b'))
+
+
+

If the start and goal have length 3, planning in \(\SE{2}\), +and configspace is False, then direction-indicating markers are used +to display start and goal configuration. These are also given as dicts +but have two items: 'shape' which is the shape of the polygonal +marker and is either 'triangle' or 'car'. The second item +'args' is passed to base.plot_poly() and Matplotlib and could +have values such as filled=True or color.

+

If configspace is False then a 3D plot is created and the start and +goal are indicated by Matplotlib markers specified by the dicts +start_marker and end_marker

+

Default values are provided for all markers:

+
+
    +
  • the start point is a circle

  • +
  • the goal point is a star

  • +
  • the start vehicle style is a VehiclePolygon(shape='car') as +an unfilled outline

  • +
  • the goal vehicle style is a VehiclePolygon(shape='car') as +a transparent filled shape

  • +
+
+

If background is True then the background of the plot is either or +both of:

+
    +
  • the occupancy grid

  • +
  • the distance field of the planner

  • +
+

Additional arguments bgargs can be passed through to plot_bg()

+

If path is specified it has one column per point and either 2 or 3 rows:

+
    +
  • 2 rows describes motion in the \(xy\)-plane and a 2D plot is created

  • +
  • 3 rows describes motion in the \(xy\theta\)-configuration space. By +default only the \(xy\)-plane is plotted unless configspace +is True in which case motion in \(xy\theta\)-configuration space +is shown as a 3D plot.

  • +
+

If the planner supports bi-directional motion then the direction +option gives the direction for each point on the path.

+

Forward motion segments are drawn using style information from line +while reverse motion segments are drawn using style information from +line_r. These are each a sequence of dicts of Matplotlib plot +options which can draw an arbitrary number of lines on top of each +other. The default:

+
line = (
+        {color:'black', linewidth:4},
+        {color:'yellow', linewidth:3, dashes:(5,5)}
+    )
+
+
+

will draw a blackline of width 4 with a dashed yellow line of width 3 +plotted on top, giving a line of alternating black and yellow dashes.

+
+
Seealso:
+

plot_bg() base.plot_poly()

+
+
+
+ +
+
+property start
+

Set/get start point or configuration (superclass)

+
+
Getter:
+

Return start point or configuration

+
+
Return type:
+

ndarray(2) or ndarray(3)

+
+
Setter:
+

Set start point or configuration

+
+
Param:
+

array_like(2) or array_like(3)

+
+
+

The start is either a point \((x, y)\) or a configuration \((x, y, \theta)\).

+
+
Seealso:
+

goal()

+
+
+
+ +
+
+validate_endpoint(p, dtype=None)
+

Validate start or goal point

+
+
Parameters:
+
    +
  • p (array_like(2)) – the point

  • +
  • dtype (str, optional) – data type for point coordinates, defaults to None

  • +
+
+
Raises:
+

ValueError – point is inside obstacle

+
+
Returns:
+

point as a NumPy array of specified dtype

+
+
Return type:
+

ndarray(2)

+
+
+

The coordinate is tested to be a free cell in the occupancy grid.

+
+
Seealso:
+

isoccupied() occgrid()

+
+
+
+ +
+
+property verbose
+

Get verbosity

+
+
Returns:
+

verbosity

+
+
Return type:
+

bool

+
+
+

If verbosity print more diagnostic messages to the planner’s +message channel.

+
+ +
+ +
+
+ + +
+
+
+ +
+ +
+

© Copyright 2023, Jesse Haviland and Peter Corke. + Last updated on 11-May-2024. +

+
+ + + +
+
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/mobile-planner-discrete.html b/mobile-planner-discrete.html new file mode 100644 index 00000000..bded8a41 --- /dev/null +++ b/mobile-planner-discrete.html @@ -0,0 +1,3287 @@ + + + + + + + + + + + Discrete (Grid-based) planners — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

Discrete (Grid-based) planners

+ + + + + + + + + + + + + + + + + + + + + + + + + +

Planner

Plans in

Discrete/Continuous

Obstacle avoidance

Bug2

\(\mathbb{R}^2\)

discrete

yes

DistanceTransformPlanner

\(\mathbb{R}^2\)

discrete

yes

DstarPlanner

\(\mathbb{R}^2\)

discrete

yes

+
+

Distance transform planner

+
+
+class roboticstoolbox.mobile.DistanceTransformPlanner(occgrid=None, metric='euclidean', **kwargs)[source]
+

Bases: PlannerBase

+

Distance transform path planner

+
+
Parameters:
+
    +
  • occgrid (BinaryOccGrid or ndarray(h,w)) – occupancy grid

  • +
  • metric (str optional) – distane metric, one of: “euclidean” [default], “manhattan”

  • +
  • kwargs – common planner options, see PlannerBase

  • +
+
+
+ + + + + + + + + + + + + + + + + + + + +

Feature

Capability

Plan

\(\mathbb{R}^2\), discrete

Obstacle avoidance

Yes, occupancy grid

Curvature

Discontinuous

Motion

Omnidirectional

+

Creates a planner that finds the path between two points in the 2D grid +using omnidirectional motion and avoiding occupied cells. The path +comprises a set of 4- or -way connected points in adjacent cells. Also +known as the wavefront, grassfire or brushfire planning algorithm.

+

The map is described by a 2D occupancy occgrid whose elements are zero +if traversable of nonzero if untraversable, ie. an obstacle.

+

The cells are assumed to be unit squares. Crossing the cell horizontally or +vertically is a travel distance of 1, and for the Euclidean distance +measure, the crossing the cell diagonally is a distance of \(\sqrt{2}\).

+

Example:

+
>>> from roboticstoolbox import DistanceTransformPlanner
+>>> import numpy as np
+>>> simplegrid = np.zeros((6, 6));
+>>> simplegrid[2:5, 3:5] = 1
+>>> dx = DistanceTransformPlanner(simplegrid, goal=(1, 1), distance="manhattan");
+>>> dx.plan()
+>>> path = dx.query(start=(5, 4))
+>>> print(path.T)
+[[5 5 5 4 3 2 1]
+ [4 3 2 1 1 1 1]]
+
+
+
+

Warning

+

The distance planner is iterative and implemented in Python, will +be slow for very large occupancy grids.

+
+
+
Author:
+

Peter Corke

+
+
Seealso:
+

plan() query() PlannerBase

+
+
+
+
+property metric
+

Get the distance metric

+
+
Returns:
+

Get the metric, either “euclidean” or “manhattan”

+
+
Return type:
+

str

+
+
+
+ +
+
+property distancemap
+

Get the distance map

+
+
Returns:
+

distance map

+
+
Return type:
+

ndarray(h,w)

+
+
+

The 2D array, the same size as the passed occupancy grid, has elements +equal to nan if they contain an obstacle, otherwise the minimum +obstacle-free distance to the goal using the particular distance metric.

+
+ +
+
+plan(goal=None, animate=False, summary=False)[source]
+

Plan path using distance transform

+
+
Parameters:
+

goal (array_like(2), optional) – goal position \((x, y)\), defaults to previously set value

+
+
+

Compute the distance transform for all non-obstacle cells, which is the +minimum obstacle-free distance to the goal using the particular distance +metric.

+
+
Seealso:
+

query()

+
+
+
+ +
+
+next(position)[source]
+

Find next point on the path

+
+
Parameters:
+

position (array_like(2)) – current robot position

+
+
Raises:
+

RuntimeError – no plan has been computed

+
+
Returns:
+

next robot position

+
+
Return type:
+

ndarray(2)

+
+
+

Return the robot position that is one step closer to the goal. Called +by query() to find a path from start to goal.

+
+
Seealso:
+

plan() query()

+
+
+
+ +
+
+plot_3d(path=None, ls=None)[source]
+

Plot path on 3D cost surface

+
+
Parameters:
+
    +
  • path (ndarray(N,2), optional) – robot path, defaults to None

  • +
  • ls (dict, optional) – dictionary of Matplotlib linestyle options, defaults to None

  • +
+
+
Returns:
+

Matplotlib 3D axes

+
+
Return type:
+

Axes

+
+
+

Creates a 3D plot showing distance from the goal as a cost surface. +Overlays the path if given.

+
+

Warning

+

The visualization is poor because of Matplotlib’s poor hidden +line/surface handling.

+
+
+ +
+
+property goal
+

Set/get goal point or configuration (superclass)

+
+
Getter:
+

Return goal pointor configuration

+
+
Return type:
+

ndarray(2) or ndarray(3)

+
+
Setter:
+

Set goal point or configuration

+
+
Param:
+

array_like(2) or array_like(3)

+
+
+

The goal is either a point \((x, y)\) or a configuration \((x, y, \theta)\).

+
+
Seealso:
+

goal()

+
+
+
+ +
+
+isoccupied(p)
+

Test if point is occupied (superclass)

+
+
Parameters:
+

p (array_like(2)) – world coordinate (x, y)

+
+
Returns:
+

occupancy status of corresponding grid cell

+
+
Return type:
+

bool

+
+
+

The world coordinate is transformed and the status of the occupancy +grid cell is returned. If the point lies outside the bounds of +the occupancy grid return True (obstacle)

+

If there is no occupancy grid this function always returns False (free).

+
+
Seealso:
+

occgrid() validate_endpoint() BinaryOccGrid.isoccupied()

+
+
+
+ +
+
+property occgrid
+

Occupancy grid

+
+
Returns:
+

occupancy grid used for planning

+
+
Return type:
+

OccGrid or subclass or None

+
+
+

Returns the occupancy grid that was optionally inflated at constructor time.

+
+
Seealso:
+

validate_endpoint() isoccupied()

+
+
+
+ +
+
+plot(path=None, line=None, line_r=None, configspace=False, unwrap=True, direction=None, background=True, path_marker=None, path_marker_reverse=None, start_marker=None, goal_marker=None, start_vehicle=None, goal_vehicle=None, start=None, goal=None, ax=None, block=None, bgargs={}, **unused)
+

Plot vehicle path (superclass)

+
+
Parameters:
+
    +
  • path ((N, 2) or ndarray(N, 3)) – path, defaults to None

  • +
  • direction (array_like(N), optional) – travel direction associated with each point on path, is either >0 or <0, defaults to None

  • +
  • line (sequence of dict of arguments for plot) – line style for forward motion, default is striped yellow on black

  • +
  • line_r (sequence of dict of arguments for plot) – line style for reverse motion, default is striped red on black

  • +
  • configspace (bool, optional) – plot the path in 3D configuration space, input must be 3xN. +Start and goal style will be given by qstart_marker and qgoal_marker, defaults to False

  • +
  • unwrap (bool, optional) – for configuration space plot unwrap \(\theta\) so +there are no discontinuities at \(\pm \pi\), defaults to True

  • +
  • background (bool, optional) – plot occupancy grid if present, default True

  • +
  • start_marker (dict, optional) – style for marking start point

  • +
  • goal_marker (dict, optional) – style for marking goal point

  • +
  • start_vehicle (dict) – style for vehicle animation object at start configuration

  • +
  • goal_vehicle (dict) – style for vehicle animation object at goal configuration

  • +
  • start (array_like(2) or array_like(3), optional) – start position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value previously set

  • +
  • goal (array_like(2) or array_like(3), optional) – goal position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value previously set

  • +
  • bgargs (dict, optional) – arguments passed to plot_bg(), defaults to None

  • +
  • ax (matplotlib axes) – axes to plot into

  • +
  • block (bool, optional) – block after displaying the plot

  • +
+
+
+

Plots the start and goal location/pose if they are specified by +start or goal or were set by the object constructor or +plan or query method.

+

If the start and goal have length 2, planning in +\(\mathbb{R}^2\), then markers are drawn using styles specified by +start_marker and end_marker which are dicts using Matplotlib +keywords, for example:

+
planner.plot(path, start=dict(marker='s', color='b'))
+
+
+

If the start and goal have length 3, planning in \(\SE{2}\), +and configspace is False, then direction-indicating markers are used +to display start and goal configuration. These are also given as dicts +but have two items: 'shape' which is the shape of the polygonal +marker and is either 'triangle' or 'car'. The second item +'args' is passed to base.plot_poly() and Matplotlib and could +have values such as filled=True or color.

+

If configspace is False then a 3D plot is created and the start and +goal are indicated by Matplotlib markers specified by the dicts +start_marker and end_marker

+

Default values are provided for all markers:

+
+
    +
  • the start point is a circle

  • +
  • the goal point is a star

  • +
  • the start vehicle style is a VehiclePolygon(shape='car') as +an unfilled outline

  • +
  • the goal vehicle style is a VehiclePolygon(shape='car') as +a transparent filled shape

  • +
+
+

If background is True then the background of the plot is either or +both of:

+
    +
  • the occupancy grid

  • +
  • the distance field of the planner

  • +
+

Additional arguments bgargs can be passed through to plot_bg()

+

If path is specified it has one column per point and either 2 or 3 rows:

+
    +
  • 2 rows describes motion in the \(xy\)-plane and a 2D plot is created

  • +
  • 3 rows describes motion in the \(xy\theta\)-configuration space. By +default only the \(xy\)-plane is plotted unless configspace +is True in which case motion in \(xy\theta\)-configuration space +is shown as a 3D plot.

  • +
+

If the planner supports bi-directional motion then the direction +option gives the direction for each point on the path.

+

Forward motion segments are drawn using style information from line +while reverse motion segments are drawn using style information from +line_r. These are each a sequence of dicts of Matplotlib plot +options which can draw an arbitrary number of lines on top of each +other. The default:

+
line = (
+        {color:'black', linewidth:4},
+        {color:'yellow', linewidth:3, dashes:(5,5)}
+    )
+
+
+

will draw a blackline of width 4 with a dashed yellow line of width 3 +plotted on top, giving a line of alternating black and yellow dashes.

+
+
Seealso:
+

plot_bg() base.plot_poly()

+
+
+
+ +
+
+plot_bg(distance=None, cmap='gray', ax=None, inflated=True, colorbar=True, block=None, **unused)
+

Plot background (superclass)

+
+
Parameters:
+
    +
  • distance (ndarray(N,M), optional) – override distance field, defaults to None

  • +
  • cmap (str or Colormap, optional) – Specify a colormap for the distance field, defaults to ‘gray’

  • +
+
+
+

Displays the background which is either the occupancy grid or a distance +field. The distance field encodes the distance of a point from the goal, small +distance is dark, a large distance is bright.

+
+
If the planner has an occupancy grid then that will be displayed with:
    +
  • free cells in white

  • +
  • occupied cells in red

  • +
  • inflated occupied cells in pink

  • +
+
+
+

If distance is provided, or the planner has a distancemap attribute +the the distance field will be used as the background and obstacle cells +(actual or inflated) will be shown in red. A colorbar is added.

+
+ +
+
+query(start=None, goal=None, dtype=None, next=True, animate=False, movie=None)
+

Find a path from start to goal using planner (superclass)

+
+
Parameters:
+
    +
  • start (array_like(), optional) – start position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value specified to constructor

  • +
  • goal (array_like(), optional) – goal position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value specified to constructor

  • +
  • dtype (str, optional) – data type for point coordinates, defaults to None

  • +
  • next (bool, optional) – invoke next() method of class, defaults to True

  • +
  • animate (bool, optional) – show the vehicle path, defaults to False

  • +
+
+
Returns:
+

path from start to goal, one point \((x, y)\) or configuration \((x, y, \theta)\) per row

+
+
Return type:
+

ndarray(N,2) or ndarray(N,3)

+
+
+

Find a path from start to goal using a previously computed plan.

+

This is a generic method that works for any planner +(\(\mathbb{R}^2\) or \(\SE{2}\)) that can incrementally +determine the next point on the path. The method performs the following +steps:

+
    +
  • Validate start and goal

  • +
  • If animate, visualize the environment using plot()

  • +
  • Iterate on the class’s next() method until the goal is +achieved, and if animate plot points.

  • +
+
+
Seealso:
+

next() plan()

+
+
+
+ +
+
+property start
+

Set/get start point or configuration (superclass)

+
+
Getter:
+

Return start point or configuration

+
+
Return type:
+

ndarray(2) or ndarray(3)

+
+
Setter:
+

Set start point or configuration

+
+
Param:
+

array_like(2) or array_like(3)

+
+
+

The start is either a point \((x, y)\) or a configuration \((x, y, \theta)\).

+
+
Seealso:
+

goal()

+
+
+
+ +
+
+validate_endpoint(p, dtype=None)
+

Validate start or goal point

+
+
Parameters:
+
    +
  • p (array_like(2)) – the point

  • +
  • dtype (str, optional) – data type for point coordinates, defaults to None

  • +
+
+
Raises:
+

ValueError – point is inside obstacle

+
+
Returns:
+

point as a NumPy array of specified dtype

+
+
Return type:
+

ndarray(2)

+
+
+

The coordinate is tested to be a free cell in the occupancy grid.

+
+
Seealso:
+

isoccupied() occgrid()

+
+
+
+ +
+
+property verbose
+

Get verbosity

+
+
Returns:
+

verbosity

+
+
Return type:
+

bool

+
+
+

If verbosity print more diagnostic messages to the planner’s +message channel.

+
+ +
+ +
+
+

D* planner

+
+
+class roboticstoolbox.mobile.DstarPlanner(costmap=None, **kwargs)[source]
+

Bases: PlannerBase

+

D* path planner

+
+
Parameters:
+
    +
  • costmap (OccGrid or ndarray(w,h)) – traversability costmap

  • +
  • kwargs – common planner options, see PlannerBase

  • +
+
+
+ + + + + + + + + + + + + + + + + + + + +

Feature

Capability

Plan

\(\mathbb{R}^2\), discrete

Obstacle avoidance

Yes, occupancy grid

Curvature

Discontinuous

Motion

Omnidirectional

+

Creates a planner that finds the minimum-cost path between two points in the +plane using omnidirectional motion. The path comprises a set of 8-way +connected points in adjacent cells.

+

The map is described by a 2D costmap whose elements indicate the cost +of traversing that cell. The cost of diagonal traverse is \(\sqrt{2}\) +the value of the cell. An infinite cost indicates an untraversable cell +or obstacle.

+

Example:

+
>>> from roboticstoolbox import DstarPlanner
+>>> import numpy as np
+>>> costmap = np.ones((6, 6));
+>>> costmap[2:5, 3:5] = 10
+>>> ds = DstarPlanner(costmap, goal=(1, 1));
+>>> ds.plan()
+>>> path, status = ds.query(start=(5, 4))
+>>> print(path.T)
+[[5 5 5 4 3 2 1]
+ [4 3 2 1 1 1 1]]
+>>> print(status)
+_DstarStatus(cost=6.414213562373095)
+
+
+
+
Thanks:
+

based on D* grid planning included from Python Robotics

+
+
Seealso:
+

PlannerBase

+
+
+
+
+plan(goal=None, animate=False, progress=True, summary=False)[source]
+

Plan D* path

+
+
Parameters:
+
    +
  • goal (array_like(2), optional) – goal position \((x, y)\), defaults to previously set value

  • +
  • animate (bool, optional) – animate the planning algorithm iterations, defaults to False

  • +
  • progress (bool, optional) – show progress bar, defaults to True

  • +
+
+
+

Compute the minimum-cost obstacle-free distance to the goal from all +points in the grid.

+
+ +
+
+property nexpand
+

Number of node expansions

+
+
Returns:
+

number of expansions

+
+
Return type:
+

int

+
+
+

This number will increase during initial planning, and also if +replanning is invoked during the query().

+
+ +
+
+query(start, sensor=None, animate=False, verbose=False)[source]
+

Find path with replanning

+
+
Parameters:
+
    +
  • start (array_like(2)) – start position \((x,y)\)

  • +
  • sensor (callable, optional) – sensor function, defaults to None

  • +
  • animate (bool, optional) – animate the motion of the robot, defaults to False

  • +
  • verbose (bool, optional) – display detailed diagnostic information about D* operations, defaults to False

  • +
+
+
Returns:
+

path from start to goal, one point \((x, y)\) per row

+
+
Return type:
+

ndarray(N,2)

+
+
+

If sensor is None then the plan determined by the plan phase +is used unaltered.

+

If sensor is not None it must be callable, and is called at each +step of the path with the current robot coordintes:

+
+

sensor((x, y))

+
+

and mimics the behaviour of a simple sensor onboard the robot which can +dynamically change the costmap. The function return a list (0 or more) +of 3-tuples (x, y, newcost) which are the coordinates of cells and their +cost. If the cost has changed this will trigger D* incremental +replanning. In this case the value returned by nexpand() will +increase, according to the severity of the replanning.

+
+
Seealso:
+

plan()

+
+
+
+ +
+
+__str__()
+

Compact representation of the planner

+
+
Returns:
+

pretty printed representation

+
+
Return type:
+

str

+
+
+
+ +
+
+property goal
+

Set/get goal point or configuration (superclass)

+
+
Getter:
+

Return goal pointor configuration

+
+
Return type:
+

ndarray(2) or ndarray(3)

+
+
Setter:
+

Set goal point or configuration

+
+
Param:
+

array_like(2) or array_like(3)

+
+
+

The goal is either a point \((x, y)\) or a configuration \((x, y, \theta)\).

+
+
Seealso:
+

goal()

+
+
+
+ +
+
+property occgrid
+

Occupancy grid

+
+
Returns:
+

occupancy grid used for planning

+
+
Return type:
+

OccGrid or subclass or None

+
+
+

Returns the occupancy grid that was optionally inflated at constructor time.

+
+
Seealso:
+

validate_endpoint() isoccupied()

+
+
+
+ +
+
+plot(path=None, line=None, line_r=None, configspace=False, unwrap=True, direction=None, background=True, path_marker=None, path_marker_reverse=None, start_marker=None, goal_marker=None, start_vehicle=None, goal_vehicle=None, start=None, goal=None, ax=None, block=None, bgargs={}, **unused)
+

Plot vehicle path (superclass)

+
+
Parameters:
+
    +
  • path ((N, 2) or ndarray(N, 3)) – path, defaults to None

  • +
  • direction (array_like(N), optional) – travel direction associated with each point on path, is either >0 or <0, defaults to None

  • +
  • line (sequence of dict of arguments for plot) – line style for forward motion, default is striped yellow on black

  • +
  • line_r (sequence of dict of arguments for plot) – line style for reverse motion, default is striped red on black

  • +
  • configspace (bool, optional) – plot the path in 3D configuration space, input must be 3xN. +Start and goal style will be given by qstart_marker and qgoal_marker, defaults to False

  • +
  • unwrap (bool, optional) – for configuration space plot unwrap \(\theta\) so +there are no discontinuities at \(\pm \pi\), defaults to True

  • +
  • background (bool, optional) – plot occupancy grid if present, default True

  • +
  • start_marker (dict, optional) – style for marking start point

  • +
  • goal_marker (dict, optional) – style for marking goal point

  • +
  • start_vehicle (dict) – style for vehicle animation object at start configuration

  • +
  • goal_vehicle (dict) – style for vehicle animation object at goal configuration

  • +
  • start (array_like(2) or array_like(3), optional) – start position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value previously set

  • +
  • goal (array_like(2) or array_like(3), optional) – goal position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value previously set

  • +
  • bgargs (dict, optional) – arguments passed to plot_bg(), defaults to None

  • +
  • ax (matplotlib axes) – axes to plot into

  • +
  • block (bool, optional) – block after displaying the plot

  • +
+
+
+

Plots the start and goal location/pose if they are specified by +start or goal or were set by the object constructor or +plan or query method.

+

If the start and goal have length 2, planning in +\(\mathbb{R}^2\), then markers are drawn using styles specified by +start_marker and end_marker which are dicts using Matplotlib +keywords, for example:

+
planner.plot(path, start=dict(marker='s', color='b'))
+
+
+

If the start and goal have length 3, planning in \(\SE{2}\), +and configspace is False, then direction-indicating markers are used +to display start and goal configuration. These are also given as dicts +but have two items: 'shape' which is the shape of the polygonal +marker and is either 'triangle' or 'car'. The second item +'args' is passed to base.plot_poly() and Matplotlib and could +have values such as filled=True or color.

+

If configspace is False then a 3D plot is created and the start and +goal are indicated by Matplotlib markers specified by the dicts +start_marker and end_marker

+

Default values are provided for all markers:

+
+
    +
  • the start point is a circle

  • +
  • the goal point is a star

  • +
  • the start vehicle style is a VehiclePolygon(shape='car') as +an unfilled outline

  • +
  • the goal vehicle style is a VehiclePolygon(shape='car') as +a transparent filled shape

  • +
+
+

If background is True then the background of the plot is either or +both of:

+
    +
  • the occupancy grid

  • +
  • the distance field of the planner

  • +
+

Additional arguments bgargs can be passed through to plot_bg()

+

If path is specified it has one column per point and either 2 or 3 rows:

+
    +
  • 2 rows describes motion in the \(xy\)-plane and a 2D plot is created

  • +
  • 3 rows describes motion in the \(xy\theta\)-configuration space. By +default only the \(xy\)-plane is plotted unless configspace +is True in which case motion in \(xy\theta\)-configuration space +is shown as a 3D plot.

  • +
+

If the planner supports bi-directional motion then the direction +option gives the direction for each point on the path.

+

Forward motion segments are drawn using style information from line +while reverse motion segments are drawn using style information from +line_r. These are each a sequence of dicts of Matplotlib plot +options which can draw an arbitrary number of lines on top of each +other. The default:

+
line = (
+        {color:'black', linewidth:4},
+        {color:'yellow', linewidth:3, dashes:(5,5)}
+    )
+
+
+

will draw a blackline of width 4 with a dashed yellow line of width 3 +plotted on top, giving a line of alternating black and yellow dashes.

+
+
Seealso:
+

plot_bg() base.plot_poly()

+
+
+
+ +
+
+plot_bg(distance=None, cmap='gray', ax=None, inflated=True, colorbar=True, block=None, **unused)
+

Plot background (superclass)

+
+
Parameters:
+
    +
  • distance (ndarray(N,M), optional) – override distance field, defaults to None

  • +
  • cmap (str or Colormap, optional) – Specify a colormap for the distance field, defaults to ‘gray’

  • +
+
+
+

Displays the background which is either the occupancy grid or a distance +field. The distance field encodes the distance of a point from the goal, small +distance is dark, a large distance is bright.

+
+
If the planner has an occupancy grid then that will be displayed with:
    +
  • free cells in white

  • +
  • occupied cells in red

  • +
  • inflated occupied cells in pink

  • +
+
+
+

If distance is provided, or the planner has a distancemap attribute +the the distance field will be used as the background and obstacle cells +(actual or inflated) will be shown in red. A colorbar is added.

+
+ +
+
+property start
+

Set/get start point or configuration (superclass)

+
+
Getter:
+

Return start point or configuration

+
+
Return type:
+

ndarray(2) or ndarray(3)

+
+
Setter:
+

Set start point or configuration

+
+
Param:
+

array_like(2) or array_like(3)

+
+
+

The start is either a point \((x, y)\) or a configuration \((x, y, \theta)\).

+
+
Seealso:
+

goal()

+
+
+
+ +
+
+validate_endpoint(p, dtype=None)
+

Validate start or goal point

+
+
Parameters:
+
    +
  • p (array_like(2)) – the point

  • +
  • dtype (str, optional) – data type for point coordinates, defaults to None

  • +
+
+
Raises:
+

ValueError – point is inside obstacle

+
+
Returns:
+

point as a NumPy array of specified dtype

+
+
Return type:
+

ndarray(2)

+
+
+

The coordinate is tested to be a free cell in the occupancy grid.

+
+
Seealso:
+

isoccupied() occgrid()

+
+
+
+ +
+
+property verbose
+

Get verbosity

+
+
Returns:
+

verbosity

+
+
Return type:
+

bool

+
+
+

If verbosity print more diagnostic messages to the planner’s +message channel.

+
+ +
+ +
+
+

Lattice planner

+
+
+class roboticstoolbox.mobile.LatticePlanner(costs=None, root=(0, 0, 0), **kwargs)[source]
+

Bases: PlannerBase

+

Lattice planner

+
+
Parameters:
+
    +
  • costs (array_like(3), optional) – cost for straight, left-turn, right-turn, defaults to \((1, \pi/2, \pi/2)\)

  • +
  • root (array_like(3), optional) – configuration of root node, defaults to (0,0,0)

  • +
  • kwargs – arguments passed to PlannerBase constructor

  • +
+
+
+ + + + + + + + + + + + + + + + + + + + +

Feature

Capability

Plan

\(\SE{2}\)

Obstacle avoidance

Yes, occupancy grid

Curvature

Discontinuous

Motion

Forwards only

+

The lattice planner incrementally builds a graph from the root vertex, at +each iteration adding three edges to the graph:

+ + + + + + + + + + + + + + + + + +

code

direction

‘S’

straight ahead

‘L’

turn left

‘R’

turn right

+

If the configuration is already in the graph, the edge connects to that +existing vertex. The vertex is named after the sequence of moves required +to reach it from the root. This means, that any configuration, ie. +\((x, y, \theta)\) can be reached by multiple paths and potentially have +multiple names. The first name assigned to a vertex is permanent and is not +overriden.

+

If an occupancy grid exists and the configuration is an obstacle, then +the vertex is not added.

+

The path through the lattice is found using A* graph search, and costs +changes the weighting for path costs at query time.

+

Example:

+
>>> from roboticstoolbox import LatticePlanner
+>>> import numpy as np
+>>> lattice = LatticePlanner();
+>>> lattice.plan(iterations=6)
+>>> path, status = lattice.query(start=(0, 0, 0), goal=(1, 2, np.pi/2))
+>>> print(path.T)
+[[0.     1.     1.    ]
+ [0.     1.     2.    ]
+ [0.     1.5708 1.5708]]
+>>> print(status)
+LatticeStatus(cost=2.5707963267948966, segments=['L', 'S'], edges=[Edge{[0] -- [0L], cost=1.571}, Edge{[0L] -- [0LS], cost=1}])
+
+
+
+
Seealso:
+

plan() query() PlannerBase

+
+
+
+
+plan(iterations=None, verbose=False, summary=False)[source]
+

Create a lattice plan

+
+
Parameters:
+
    +
  • iterations (int, optional) – number of iterations, defaults to None

  • +
  • verbose (bool, optional) – show frontier and added vertices/edges at each iteration, defaults to False

  • +
+
+
+

If an occupancy grid exists the if iterations is None the area of the +grid will be completely filled.

+
+
Seealso:
+

query()

+
+
+
+ +
+
+query(start, goal)[source]
+

Find a path through the lattice

+
+
Parameters:
+
    +
  • start (array_like(3), optional) – start configuration \((x, y, \theta)\)

  • +
  • goal (array_like(3), optional) – goal configuration \((x, y, \theta)\)

  • +
+
+
Returns:
+

path and status

+
+
Return type:
+

ndarray(N,3), namedtuple

+
+
+

The returned status value has elements:

+ + + + + + + + + + + + + + + +

Element

Description

cost

path cost

segments

a list containing the type of each path segment as +a single letter code: either “L”, “R” or “S” for +left turn, right turn or straight line respectively.

edges

successive edges of the graph LatticeEdge type

+
+
Seealso:
+

plan()

+
+
+
+ +
+
+plot(path=None, **kwargs)[source]
+

Plot vehicle path (superclass)

+
+
Parameters:
+
    +
  • path ((N, 2) or ndarray(N, 3)) – path, defaults to None

  • +
  • direction (array_like(N), optional) – travel direction associated with each point on path, is either >0 or <0, defaults to None

  • +
  • line (sequence of dict of arguments for plot) – line style for forward motion, default is striped yellow on black

  • +
  • line_r (sequence of dict of arguments for plot) – line style for reverse motion, default is striped red on black

  • +
  • configspace (bool, optional) – plot the path in 3D configuration space, input must be 3xN. +Start and goal style will be given by qstart_marker and qgoal_marker, defaults to False

  • +
  • unwrap (bool, optional) – for configuration space plot unwrap \(\theta\) so +there are no discontinuities at \(\pm \pi\), defaults to True

  • +
  • background (bool, optional) – plot occupancy grid if present, default True

  • +
  • start_marker (dict, optional) – style for marking start point

  • +
  • goal_marker (dict, optional) – style for marking goal point

  • +
  • start_vehicle (dict) – style for vehicle animation object at start configuration

  • +
  • goal_vehicle (dict) – style for vehicle animation object at goal configuration

  • +
  • start (array_like(2) or array_like(3), optional) – start position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value previously set

  • +
  • goal (array_like(2) or array_like(3), optional) – goal position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value previously set

  • +
  • bgargs (dict, optional) – arguments passed to plot_bg(), defaults to None

  • +
  • ax (matplotlib axes) – axes to plot into

  • +
  • block (bool, optional) – block after displaying the plot

  • +
+
+
+

Plots the start and goal location/pose if they are specified by +start or goal or were set by the object constructor or +plan or query method.

+

If the start and goal have length 2, planning in +\(\mathbb{R}^2\), then markers are drawn using styles specified by +start_marker and end_marker which are dicts using Matplotlib +keywords, for example:

+
planner.plot(path, start=dict(marker='s', color='b'))
+
+
+

If the start and goal have length 3, planning in \(\SE{2}\), +and configspace is False, then direction-indicating markers are used +to display start and goal configuration. These are also given as dicts +but have two items: 'shape' which is the shape of the polygonal +marker and is either 'triangle' or 'car'. The second item +'args' is passed to base.plot_poly() and Matplotlib and could +have values such as filled=True or color.

+

If configspace is False then a 3D plot is created and the start and +goal are indicated by Matplotlib markers specified by the dicts +start_marker and end_marker

+

Default values are provided for all markers:

+
+
    +
  • the start point is a circle

  • +
  • the goal point is a star

  • +
  • the start vehicle style is a VehiclePolygon(shape='car') as +an unfilled outline

  • +
  • the goal vehicle style is a VehiclePolygon(shape='car') as +a transparent filled shape

  • +
+
+

If background is True then the background of the plot is either or +both of:

+
    +
  • the occupancy grid

  • +
  • the distance field of the planner

  • +
+

Additional arguments bgargs can be passed through to plot_bg()

+

If path is specified it has one column per point and either 2 or 3 rows:

+
    +
  • 2 rows describes motion in the \(xy\)-plane and a 2D plot is created

  • +
  • 3 rows describes motion in the \(xy\theta\)-configuration space. By +default only the \(xy\)-plane is plotted unless configspace +is True in which case motion in \(xy\theta\)-configuration space +is shown as a 3D plot.

  • +
+

If the planner supports bi-directional motion then the direction +option gives the direction for each point on the path.

+

Forward motion segments are drawn using style information from line +while reverse motion segments are drawn using style information from +line_r. These are each a sequence of dicts of Matplotlib plot +options which can draw an arbitrary number of lines on top of each +other. The default:

+
line = (
+        {color:'black', linewidth:4},
+        {color:'yellow', linewidth:3, dashes:(5,5)}
+    )
+
+
+

will draw a blackline of width 4 with a dashed yellow line of width 3 +plotted on top, giving a line of alternating black and yellow dashes.

+
+
Seealso:
+

plot_bg() base.plot_poly()

+
+
+
+ +
+
+property goal
+

Set/get goal point or configuration (superclass)

+
+
Getter:
+

Return goal pointor configuration

+
+
Return type:
+

ndarray(2) or ndarray(3)

+
+
Setter:
+

Set goal point or configuration

+
+
Param:
+

array_like(2) or array_like(3)

+
+
+

The goal is either a point \((x, y)\) or a configuration \((x, y, \theta)\).

+
+
Seealso:
+

goal()

+
+
+
+ +
+
+property occgrid
+

Occupancy grid

+
+
Returns:
+

occupancy grid used for planning

+
+
Return type:
+

OccGrid or subclass or None

+
+
+

Returns the occupancy grid that was optionally inflated at constructor time.

+
+
Seealso:
+

validate_endpoint() isoccupied()

+
+
+
+ +
+
+property start
+

Set/get start point or configuration (superclass)

+
+
Getter:
+

Return start point or configuration

+
+
Return type:
+

ndarray(2) or ndarray(3)

+
+
Setter:
+

Set start point or configuration

+
+
Param:
+

array_like(2) or array_like(3)

+
+
+

The start is either a point \((x, y)\) or a configuration \((x, y, \theta)\).

+
+
Seealso:
+

goal()

+
+
+
+ +
+
+validate_endpoint(p, dtype=None)
+

Validate start or goal point

+
+
Parameters:
+
    +
  • p (array_like(2)) – the point

  • +
  • dtype (str, optional) – data type for point coordinates, defaults to None

  • +
+
+
Raises:
+

ValueError – point is inside obstacle

+
+
Returns:
+

point as a NumPy array of specified dtype

+
+
Return type:
+

ndarray(2)

+
+
+

The coordinate is tested to be a free cell in the occupancy grid.

+
+
Seealso:
+

isoccupied() occgrid()

+
+
+
+ +
+
+property verbose
+

Get verbosity

+
+
Returns:
+

verbosity

+
+
Return type:
+

bool

+
+
+

If verbosity print more diagnostic messages to the planner’s +message channel.

+
+ +
+ +
+
+
+

Continuous configuration-space planners

+

These planners do not support planning around obstacles, but allow for the +start and goal configuration \((x, y, \theta)\) to be specified.

+
+

Dubins path planner

+
+
+class roboticstoolbox.mobile.DubinsPlanner(curvature=1, stepsize=0.1, **kwargs)[source]
+

Bases: PlannerBase

+

Dubins path planner

+
+
Parameters:
+
    +
  • curvature (float, optional) – maximum path curvature, defaults to 1.0

  • +
  • stepsize (float, optional) – spacing between points on the path, defaults to 0.1

  • +
+
+
Returns:
+

Dubins path planner

+
+
Return type:
+

DubinsPlanner instance

+
+
+ + + + + + + + + + + + + + + + + + + + +

Feature

Capability

Plan

\(\SE{2}\)

Obstacle avoidance

No

Curvature

Discontinuous

Motion

Forwards only

+

Creates a planner that finds the path between two configurations in the +plane using forward motion only. The path comprises upto 3 segments that are +straight lines, or arcs with curvature of \(\pm\) curvature.

+

Example:

+
>>> from roboticstoolbox import DubinsPlanner
+>>> from math import pi
+>>> dubins = DubinsPlanner(curvature=1.0)
+>>> path, status = dubins.query(start=(0, 0, pi/2), goal=(1, 0, pi/2))
+>>> print(path[:5,:])
+[[ 0.      0.      1.5708]
+ [-0.005   0.0998  1.6708]
+ [-0.0199  0.1987  1.7708]
+ [-0.0447  0.2955  1.8708]
+ [-0.0789  0.3894  1.9708]]
+>>> print(status)
+DubinsStatus(segments=['L', 'S', 'L'], length=7.283185307179586, seglengths=[4.71238898038469, 1.0, 1.5707963267948966])
+
+
+
+
Reference:
+

On Curves of Minimal Length with a Constraint on Average +Curvature, and with Prescribed Initial and Terminal Positions and +Tangents, Dubins, L.E. (July 1957), American Journal of Mathematics. +79(3): 497–516.

+
+
Thanks:
+

based on Dubins path planning from Python Robotics

+
+
Seealso:
+

ReedsSheppPlanner PlannerBase

+
+
+
+
+property curvature
+
+ +
+
+property stepsize
+
+ +
+
+query(start, goal, **kwargs)[source]
+

Find a path between two configurations

+
+
Parameters:
+
    +
  • start (array_like(3), optional) – start configuration \((x, y, \theta)\)

  • +
  • goal (array_like(3), optional) – goal configuration \((x, y, \theta)\)

  • +
+
+
Returns:
+

path and status

+
+
Return type:
+

ndarray(N,3), namedtuple

+
+
+

The path comprises points equally spaced at a distance of stepsize.

+

The returned status value has elements:

+ + + + + + + + + + + + + + + +

Element

Description

segments

a list containing the type of each path segment as +a single letter code: either “L”, “R” or “S” for +left turn, right turn or straight line respectively.

length

total path length

lengths

the length of each path segment. The sign of the +length indicates the direction of travel.

+
+ +
+
+property goal
+

Set/get goal point or configuration (superclass)

+
+
Getter:
+

Return goal pointor configuration

+
+
Return type:
+

ndarray(2) or ndarray(3)

+
+
Setter:
+

Set goal point or configuration

+
+
Param:
+

array_like(2) or array_like(3)

+
+
+

The goal is either a point \((x, y)\) or a configuration \((x, y, \theta)\).

+
+
Seealso:
+

goal()

+
+
+
+ +
+
+property occgrid
+

Occupancy grid

+
+
Returns:
+

occupancy grid used for planning

+
+
Return type:
+

OccGrid or subclass or None

+
+
+

Returns the occupancy grid that was optionally inflated at constructor time.

+
+
Seealso:
+

validate_endpoint() isoccupied()

+
+
+
+ +
+
+plan()
+

Plan path (superclass)

+
+
Parameters:
+
    +
  • start (array_like(2) or array_like(3), optional) – start position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value passed to constructor

  • +
  • goal (array_like(2) or array_like(3), optional) – goal position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value passed to constructor

  • +
+
+
+

The implementation depends on the particular planner. Some may have +no planning phase. The plan may also depend on just the start or goal.

+
+ +
+
+plot(path=None, line=None, line_r=None, configspace=False, unwrap=True, direction=None, background=True, path_marker=None, path_marker_reverse=None, start_marker=None, goal_marker=None, start_vehicle=None, goal_vehicle=None, start=None, goal=None, ax=None, block=None, bgargs={}, **unused)
+

Plot vehicle path (superclass)

+
+
Parameters:
+
    +
  • path ((N, 2) or ndarray(N, 3)) – path, defaults to None

  • +
  • direction (array_like(N), optional) – travel direction associated with each point on path, is either >0 or <0, defaults to None

  • +
  • line (sequence of dict of arguments for plot) – line style for forward motion, default is striped yellow on black

  • +
  • line_r (sequence of dict of arguments for plot) – line style for reverse motion, default is striped red on black

  • +
  • configspace (bool, optional) – plot the path in 3D configuration space, input must be 3xN. +Start and goal style will be given by qstart_marker and qgoal_marker, defaults to False

  • +
  • unwrap (bool, optional) – for configuration space plot unwrap \(\theta\) so +there are no discontinuities at \(\pm \pi\), defaults to True

  • +
  • background (bool, optional) – plot occupancy grid if present, default True

  • +
  • start_marker (dict, optional) – style for marking start point

  • +
  • goal_marker (dict, optional) – style for marking goal point

  • +
  • start_vehicle (dict) – style for vehicle animation object at start configuration

  • +
  • goal_vehicle (dict) – style for vehicle animation object at goal configuration

  • +
  • start (array_like(2) or array_like(3), optional) – start position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value previously set

  • +
  • goal (array_like(2) or array_like(3), optional) – goal position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value previously set

  • +
  • bgargs (dict, optional) – arguments passed to plot_bg(), defaults to None

  • +
  • ax (matplotlib axes) – axes to plot into

  • +
  • block (bool, optional) – block after displaying the plot

  • +
+
+
+

Plots the start and goal location/pose if they are specified by +start or goal or were set by the object constructor or +plan or query method.

+

If the start and goal have length 2, planning in +\(\mathbb{R}^2\), then markers are drawn using styles specified by +start_marker and end_marker which are dicts using Matplotlib +keywords, for example:

+
planner.plot(path, start=dict(marker='s', color='b'))
+
+
+

If the start and goal have length 3, planning in \(\SE{2}\), +and configspace is False, then direction-indicating markers are used +to display start and goal configuration. These are also given as dicts +but have two items: 'shape' which is the shape of the polygonal +marker and is either 'triangle' or 'car'. The second item +'args' is passed to base.plot_poly() and Matplotlib and could +have values such as filled=True or color.

+

If configspace is False then a 3D plot is created and the start and +goal are indicated by Matplotlib markers specified by the dicts +start_marker and end_marker

+

Default values are provided for all markers:

+
+
    +
  • the start point is a circle

  • +
  • the goal point is a star

  • +
  • the start vehicle style is a VehiclePolygon(shape='car') as +an unfilled outline

  • +
  • the goal vehicle style is a VehiclePolygon(shape='car') as +a transparent filled shape

  • +
+
+

If background is True then the background of the plot is either or +both of:

+
    +
  • the occupancy grid

  • +
  • the distance field of the planner

  • +
+

Additional arguments bgargs can be passed through to plot_bg()

+

If path is specified it has one column per point and either 2 or 3 rows:

+
    +
  • 2 rows describes motion in the \(xy\)-plane and a 2D plot is created

  • +
  • 3 rows describes motion in the \(xy\theta\)-configuration space. By +default only the \(xy\)-plane is plotted unless configspace +is True in which case motion in \(xy\theta\)-configuration space +is shown as a 3D plot.

  • +
+

If the planner supports bi-directional motion then the direction +option gives the direction for each point on the path.

+

Forward motion segments are drawn using style information from line +while reverse motion segments are drawn using style information from +line_r. These are each a sequence of dicts of Matplotlib plot +options which can draw an arbitrary number of lines on top of each +other. The default:

+
line = (
+        {color:'black', linewidth:4},
+        {color:'yellow', linewidth:3, dashes:(5,5)}
+    )
+
+
+

will draw a blackline of width 4 with a dashed yellow line of width 3 +plotted on top, giving a line of alternating black and yellow dashes.

+
+
Seealso:
+

plot_bg() base.plot_poly()

+
+
+
+ +
+
+property start
+

Set/get start point or configuration (superclass)

+
+
Getter:
+

Return start point or configuration

+
+
Return type:
+

ndarray(2) or ndarray(3)

+
+
Setter:
+

Set start point or configuration

+
+
Param:
+

array_like(2) or array_like(3)

+
+
+

The start is either a point \((x, y)\) or a configuration \((x, y, \theta)\).

+
+
Seealso:
+

goal()

+
+
+
+ +
+
+validate_endpoint(p, dtype=None)
+

Validate start or goal point

+
+
Parameters:
+
    +
  • p (array_like(2)) – the point

  • +
  • dtype (str, optional) – data type for point coordinates, defaults to None

  • +
+
+
Raises:
+

ValueError – point is inside obstacle

+
+
Returns:
+

point as a NumPy array of specified dtype

+
+
Return type:
+

ndarray(2)

+
+
+

The coordinate is tested to be a free cell in the occupancy grid.

+
+
Seealso:
+

isoccupied() occgrid()

+
+
+
+ +
+
+property verbose
+

Get verbosity

+
+
Returns:
+

verbosity

+
+
Return type:
+

bool

+
+
+

If verbosity print more diagnostic messages to the planner’s +message channel.

+
+ +
+ +
+
+

Reeds-Shepp path planner

+
+
+class roboticstoolbox.mobile.ReedsSheppPlanner(curvature=1, stepsize=0.1, **kwargs)[source]
+

Bases: PlannerBase

+

Reeds-Shepp path planner

+
+
Parameters:
+
    +
  • curvature (float, optional) – maximum path curvature, defaults to 1.0

  • +
  • stepsize (float, optional) – spacing between points on the path, defaults to 0.1

  • +
  • Planner (ReedsSheppPlanner instance) – Reeds-Shepp path planner

  • +
+
+
+ + + + + + + + + + + + + + + + + + + + +

Feature

Capability

Plan

\(\SE{2}\)

Obstacle avoidance

No

Curvature

Discontinuous

Motion

Bidirectional

+

Creates a planner that finds the path between two configurations in the +plane using forward and backward motion. The path comprises upto 3 segments +that are straight lines, or arcs with curvature of \(\pm\) +curvature.

+

Example:

+
>>> from roboticstoolbox import ReedsSheppPlanner
+>>> from math import pi
+>>> reedshepp = ReedsSheppPlanner(curvature=1.0)
+>>> path, status = reedshepp.query(start=(0, 0, pi/2), goal=(1, 0, pi/2))
+>>> print(path[:5,:])
+[[0.     0.     1.5708]
+ [0.005  0.0998 1.4708]
+ [0.0199 0.1987 1.3708]
+ [0.0447 0.2955 1.2708]
+ [0.0789 0.3894 1.1708]]
+>>> print(status)
+ReedsSheppStatus(segments=['R', 'L', 'R'], length=6.283185307179586, seglengths=[4.459708725242611, -0.5053605102841573, 1.3181160716528177], direction=[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1])
+
+
+
+
Reference:
+

Optimal paths for a car that goes both forwards and backwards, +Reeds, J.A. and L.A. Shepp, Pacific J. Math., 145 (1990), +pp. 367–393.

+
+
Thanks:
+

based on Reeds-Shepp path planning from Python Robotics

+
+
Seealso:
+

DubinsPlanner PlannerBase

+
+
+
+
+query(start, goal, **kwargs)[source]
+

Find a path between two configurations

+
+
Parameters:
+
    +
  • start (array_like(3), optional) – start configuration \((x, y, \theta)\)

  • +
  • goal (array_like(3), optional) – goal configuration \((x, y, \theta)\)

  • +
+
+
Returns:
+

path and status

+
+
Return type:
+

ndarray(N,3), namedtuple

+
+
+

The path comprises points equally spaced at a distance of stepsize.

+

The returned status value has elements:

+ + + + + + + + + + + + + + + + + + +

Element

Description

segments

a list containing the type of each path segment as +a single letter code: either “L”, “R” or “S” for +left turn, right turn or straight line respectively.

length

total path length

lengths

the length of each path segment. The sign of the +length indicates the direction of travel.

direction

the direction of motion at each point on the path

+
+

Note

+

The direction of turning is reversed when travelling +backwards.

+
+
+ +
+
+property goal
+

Set/get goal point or configuration (superclass)

+
+
Getter:
+

Return goal pointor configuration

+
+
Return type:
+

ndarray(2) or ndarray(3)

+
+
Setter:
+

Set goal point or configuration

+
+
Param:
+

array_like(2) or array_like(3)

+
+
+

The goal is either a point \((x, y)\) or a configuration \((x, y, \theta)\).

+
+
Seealso:
+

goal()

+
+
+
+ +
+
+property occgrid
+

Occupancy grid

+
+
Returns:
+

occupancy grid used for planning

+
+
Return type:
+

OccGrid or subclass or None

+
+
+

Returns the occupancy grid that was optionally inflated at constructor time.

+
+
Seealso:
+

validate_endpoint() isoccupied()

+
+
+
+ +
+
+plan()
+

Plan path (superclass)

+
+
Parameters:
+
    +
  • start (array_like(2) or array_like(3), optional) – start position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value passed to constructor

  • +
  • goal (array_like(2) or array_like(3), optional) – goal position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value passed to constructor

  • +
+
+
+

The implementation depends on the particular planner. Some may have +no planning phase. The plan may also depend on just the start or goal.

+
+ +
+
+plot(path=None, line=None, line_r=None, configspace=False, unwrap=True, direction=None, background=True, path_marker=None, path_marker_reverse=None, start_marker=None, goal_marker=None, start_vehicle=None, goal_vehicle=None, start=None, goal=None, ax=None, block=None, bgargs={}, **unused)
+

Plot vehicle path (superclass)

+
+
Parameters:
+
    +
  • path ((N, 2) or ndarray(N, 3)) – path, defaults to None

  • +
  • direction (array_like(N), optional) – travel direction associated with each point on path, is either >0 or <0, defaults to None

  • +
  • line (sequence of dict of arguments for plot) – line style for forward motion, default is striped yellow on black

  • +
  • line_r (sequence of dict of arguments for plot) – line style for reverse motion, default is striped red on black

  • +
  • configspace (bool, optional) – plot the path in 3D configuration space, input must be 3xN. +Start and goal style will be given by qstart_marker and qgoal_marker, defaults to False

  • +
  • unwrap (bool, optional) – for configuration space plot unwrap \(\theta\) so +there are no discontinuities at \(\pm \pi\), defaults to True

  • +
  • background (bool, optional) – plot occupancy grid if present, default True

  • +
  • start_marker (dict, optional) – style for marking start point

  • +
  • goal_marker (dict, optional) – style for marking goal point

  • +
  • start_vehicle (dict) – style for vehicle animation object at start configuration

  • +
  • goal_vehicle (dict) – style for vehicle animation object at goal configuration

  • +
  • start (array_like(2) or array_like(3), optional) – start position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value previously set

  • +
  • goal (array_like(2) or array_like(3), optional) – goal position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value previously set

  • +
  • bgargs (dict, optional) – arguments passed to plot_bg(), defaults to None

  • +
  • ax (matplotlib axes) – axes to plot into

  • +
  • block (bool, optional) – block after displaying the plot

  • +
+
+
+

Plots the start and goal location/pose if they are specified by +start or goal or were set by the object constructor or +plan or query method.

+

If the start and goal have length 2, planning in +\(\mathbb{R}^2\), then markers are drawn using styles specified by +start_marker and end_marker which are dicts using Matplotlib +keywords, for example:

+
planner.plot(path, start=dict(marker='s', color='b'))
+
+
+

If the start and goal have length 3, planning in \(\SE{2}\), +and configspace is False, then direction-indicating markers are used +to display start and goal configuration. These are also given as dicts +but have two items: 'shape' which is the shape of the polygonal +marker and is either 'triangle' or 'car'. The second item +'args' is passed to base.plot_poly() and Matplotlib and could +have values such as filled=True or color.

+

If configspace is False then a 3D plot is created and the start and +goal are indicated by Matplotlib markers specified by the dicts +start_marker and end_marker

+

Default values are provided for all markers:

+
+
    +
  • the start point is a circle

  • +
  • the goal point is a star

  • +
  • the start vehicle style is a VehiclePolygon(shape='car') as +an unfilled outline

  • +
  • the goal vehicle style is a VehiclePolygon(shape='car') as +a transparent filled shape

  • +
+
+

If background is True then the background of the plot is either or +both of:

+
    +
  • the occupancy grid

  • +
  • the distance field of the planner

  • +
+

Additional arguments bgargs can be passed through to plot_bg()

+

If path is specified it has one column per point and either 2 or 3 rows:

+
    +
  • 2 rows describes motion in the \(xy\)-plane and a 2D plot is created

  • +
  • 3 rows describes motion in the \(xy\theta\)-configuration space. By +default only the \(xy\)-plane is plotted unless configspace +is True in which case motion in \(xy\theta\)-configuration space +is shown as a 3D plot.

  • +
+

If the planner supports bi-directional motion then the direction +option gives the direction for each point on the path.

+

Forward motion segments are drawn using style information from line +while reverse motion segments are drawn using style information from +line_r. These are each a sequence of dicts of Matplotlib plot +options which can draw an arbitrary number of lines on top of each +other. The default:

+
line = (
+        {color:'black', linewidth:4},
+        {color:'yellow', linewidth:3, dashes:(5,5)}
+    )
+
+
+

will draw a blackline of width 4 with a dashed yellow line of width 3 +plotted on top, giving a line of alternating black and yellow dashes.

+
+
Seealso:
+

plot_bg() base.plot_poly()

+
+
+
+ +
+
+property start
+

Set/get start point or configuration (superclass)

+
+
Getter:
+

Return start point or configuration

+
+
Return type:
+

ndarray(2) or ndarray(3)

+
+
Setter:
+

Set start point or configuration

+
+
Param:
+

array_like(2) or array_like(3)

+
+
+

The start is either a point \((x, y)\) or a configuration \((x, y, \theta)\).

+
+
Seealso:
+

goal()

+
+
+
+ +
+
+validate_endpoint(p, dtype=None)
+

Validate start or goal point

+
+
Parameters:
+
    +
  • p (array_like(2)) – the point

  • +
  • dtype (str, optional) – data type for point coordinates, defaults to None

  • +
+
+
Raises:
+

ValueError – point is inside obstacle

+
+
Returns:
+

point as a NumPy array of specified dtype

+
+
Return type:
+

ndarray(2)

+
+
+

The coordinate is tested to be a free cell in the occupancy grid.

+
+
Seealso:
+

isoccupied() occgrid()

+
+
+
+ +
+
+property verbose
+

Get verbosity

+
+
Returns:
+

verbosity

+
+
Return type:
+

bool

+
+
+

If verbosity print more diagnostic messages to the planner’s +message channel.

+
+ +
+ +
+
+

Curvature-polynomial planner

+
+
+class roboticstoolbox.mobile.CurvaturePolyPlanner(curvature=None)[source]
+

Bases: PlannerBase

+
+
+query(start, goal)[source]
+

Find a path betwee two configurations

+
+
Parameters:
+
    +
  • start (array_like(3), optional) – start configuration \((x, y, \theta)\)

  • +
  • goal (array_like(3), optional) – goal configuration \((x, y, \theta)\)

  • +
+
+
Returns:
+

path and status

+
+
Return type:
+

ndarray(N,3), namedtuple

+
+
+

The path comprises points equally spaced at a distance of stepsize.

+

The returned status value has elements:

+ + + + + + + + + + + + + + + +

Element

Description

length

total path length

maxcurvature

maximum curvature on path

poly

curvature polynomial coefficients

+
+ +
+
+__str__()
+

Compact representation of the planner

+
+
Returns:
+

pretty printed representation

+
+
Return type:
+

str

+
+
+
+ +
+
+property goal
+

Set/get goal point or configuration (superclass)

+
+
Getter:
+

Return goal pointor configuration

+
+
Return type:
+

ndarray(2) or ndarray(3)

+
+
Setter:
+

Set goal point or configuration

+
+
Param:
+

array_like(2) or array_like(3)

+
+
+

The goal is either a point \((x, y)\) or a configuration \((x, y, \theta)\).

+
+
Seealso:
+

goal()

+
+
+
+ +
+
+property occgrid
+

Occupancy grid

+
+
Returns:
+

occupancy grid used for planning

+
+
Return type:
+

OccGrid or subclass or None

+
+
+

Returns the occupancy grid that was optionally inflated at constructor time.

+
+
Seealso:
+

validate_endpoint() isoccupied()

+
+
+
+ +
+
+plan()
+

Plan path (superclass)

+
+
Parameters:
+
    +
  • start (array_like(2) or array_like(3), optional) – start position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value passed to constructor

  • +
  • goal (array_like(2) or array_like(3), optional) – goal position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value passed to constructor

  • +
+
+
+

The implementation depends on the particular planner. Some may have +no planning phase. The plan may also depend on just the start or goal.

+
+ +
+
+plot(path=None, line=None, line_r=None, configspace=False, unwrap=True, direction=None, background=True, path_marker=None, path_marker_reverse=None, start_marker=None, goal_marker=None, start_vehicle=None, goal_vehicle=None, start=None, goal=None, ax=None, block=None, bgargs={}, **unused)
+

Plot vehicle path (superclass)

+
+
Parameters:
+
    +
  • path ((N, 2) or ndarray(N, 3)) – path, defaults to None

  • +
  • direction (array_like(N), optional) – travel direction associated with each point on path, is either >0 or <0, defaults to None

  • +
  • line (sequence of dict of arguments for plot) – line style for forward motion, default is striped yellow on black

  • +
  • line_r (sequence of dict of arguments for plot) – line style for reverse motion, default is striped red on black

  • +
  • configspace (bool, optional) – plot the path in 3D configuration space, input must be 3xN. +Start and goal style will be given by qstart_marker and qgoal_marker, defaults to False

  • +
  • unwrap (bool, optional) – for configuration space plot unwrap \(\theta\) so +there are no discontinuities at \(\pm \pi\), defaults to True

  • +
  • background (bool, optional) – plot occupancy grid if present, default True

  • +
  • start_marker (dict, optional) – style for marking start point

  • +
  • goal_marker (dict, optional) – style for marking goal point

  • +
  • start_vehicle (dict) – style for vehicle animation object at start configuration

  • +
  • goal_vehicle (dict) – style for vehicle animation object at goal configuration

  • +
  • start (array_like(2) or array_like(3), optional) – start position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value previously set

  • +
  • goal (array_like(2) or array_like(3), optional) – goal position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value previously set

  • +
  • bgargs (dict, optional) – arguments passed to plot_bg(), defaults to None

  • +
  • ax (matplotlib axes) – axes to plot into

  • +
  • block (bool, optional) – block after displaying the plot

  • +
+
+
+

Plots the start and goal location/pose if they are specified by +start or goal or were set by the object constructor or +plan or query method.

+

If the start and goal have length 2, planning in +\(\mathbb{R}^2\), then markers are drawn using styles specified by +start_marker and end_marker which are dicts using Matplotlib +keywords, for example:

+
planner.plot(path, start=dict(marker='s', color='b'))
+
+
+

If the start and goal have length 3, planning in \(\SE{2}\), +and configspace is False, then direction-indicating markers are used +to display start and goal configuration. These are also given as dicts +but have two items: 'shape' which is the shape of the polygonal +marker and is either 'triangle' or 'car'. The second item +'args' is passed to base.plot_poly() and Matplotlib and could +have values such as filled=True or color.

+

If configspace is False then a 3D plot is created and the start and +goal are indicated by Matplotlib markers specified by the dicts +start_marker and end_marker

+

Default values are provided for all markers:

+
+
    +
  • the start point is a circle

  • +
  • the goal point is a star

  • +
  • the start vehicle style is a VehiclePolygon(shape='car') as +an unfilled outline

  • +
  • the goal vehicle style is a VehiclePolygon(shape='car') as +a transparent filled shape

  • +
+
+

If background is True then the background of the plot is either or +both of:

+
    +
  • the occupancy grid

  • +
  • the distance field of the planner

  • +
+

Additional arguments bgargs can be passed through to plot_bg()

+

If path is specified it has one column per point and either 2 or 3 rows:

+
    +
  • 2 rows describes motion in the \(xy\)-plane and a 2D plot is created

  • +
  • 3 rows describes motion in the \(xy\theta\)-configuration space. By +default only the \(xy\)-plane is plotted unless configspace +is True in which case motion in \(xy\theta\)-configuration space +is shown as a 3D plot.

  • +
+

If the planner supports bi-directional motion then the direction +option gives the direction for each point on the path.

+

Forward motion segments are drawn using style information from line +while reverse motion segments are drawn using style information from +line_r. These are each a sequence of dicts of Matplotlib plot +options which can draw an arbitrary number of lines on top of each +other. The default:

+
line = (
+        {color:'black', linewidth:4},
+        {color:'yellow', linewidth:3, dashes:(5,5)}
+    )
+
+
+

will draw a blackline of width 4 with a dashed yellow line of width 3 +plotted on top, giving a line of alternating black and yellow dashes.

+
+
Seealso:
+

plot_bg() base.plot_poly()

+
+
+
+ +
+
+property start
+

Set/get start point or configuration (superclass)

+
+
Getter:
+

Return start point or configuration

+
+
Return type:
+

ndarray(2) or ndarray(3)

+
+
Setter:
+

Set start point or configuration

+
+
Param:
+

array_like(2) or array_like(3)

+
+
+

The start is either a point \((x, y)\) or a configuration \((x, y, \theta)\).

+
+
Seealso:
+

goal()

+
+
+
+ +
+
+validate_endpoint(p, dtype=None)
+

Validate start or goal point

+
+
Parameters:
+
    +
  • p (array_like(2)) – the point

  • +
  • dtype (str, optional) – data type for point coordinates, defaults to None

  • +
+
+
Raises:
+

ValueError – point is inside obstacle

+
+
Returns:
+

point as a NumPy array of specified dtype

+
+
Return type:
+

ndarray(2)

+
+
+

The coordinate is tested to be a free cell in the occupancy grid.

+
+
Seealso:
+

isoccupied() occgrid()

+
+
+
+ +
+
+property verbose
+

Get verbosity

+
+
Returns:
+

verbosity

+
+
Return type:
+

bool

+
+
+

If verbosity print more diagnostic messages to the planner’s +message channel.

+
+ +
+ +
+
+

Quintic-polynomial planner

+
+
+class roboticstoolbox.mobile.QuinticPolyPlanner(dt=0.1, start_vel=0, start_acc=0, goal_vel=0, goal_acc=0, max_acc=1, max_jerk=0.5, min_t=5, max_t=100)[source]
+

Bases: PlannerBase

+

Quintic polynomial path planner

+
+
Parameters:
+
    +
  • dt (float, optional) – time step, defaults to 0.1

  • +
  • start_vel (float, optional) – initial velocity, defaults to 0

  • +
  • start_acc (float, optional) – initial acceleration, defaults to 0

  • +
  • goal_vel (float, optional) – goal velocity, defaults to 0

  • +
  • goal_acc (float, optional) – goal acceleration, defaults to 0

  • +
  • max_acc (int, optional) – maximum acceleration, defaults to 1

  • +
  • max_jerk (float, optional) – maximum jerk, defaults to 0.5

  • +
  • min_t (float, optional) – minimum path time, defaults to 5

  • +
  • max_t (float, optional) – maximum path time, defaults to 100

  • +
+
+
Returns:
+

Quintic polynomial path planner

+
+
Return type:
+

QuinticPolyPlanner instance

+
+
+ + + + + + + + + + + + + + + + + + + + +

Feature

Capability

Plan

\(\SE{2}\)

Obstacle avoidance

No

Curvature

Continuous

Motion

Forwards only

+

Creates a planner that finds the path between two configurations in the +plane using forward motion only. The path is a continuous quintic polynomial +for x and y

+
+\[\begin{split}x(t) &= a_0 + a_1 t + a_2 t^2 + a_3 t^3 + a_4 t^4 + a_5 t^5 \\ +y(t) &= b_0 + b_1 t + b_2 t^2 + b_3 t^3 + b_4 t^4 + b_5 t^5\end{split}\]
+

that meets the given constraints. Trajectory time is given as a range.

+
+
Reference:
+

“Local Path Planning And Motion Control For AGV In +Positioning”, Takahashi, T. Hongo, Y. Ninomiya and G. +Sugimoto; Proceedings. IEEE/RSJ International Workshop on +Intelligent Robots and Systems (IROS ‘89) doi: 10.1109/IROS.1989.637936

+
+
+
+

Note

+

The path time is searched in the interval [min_t, max_t] in steps +of min_t.

+
+

Example:

+
>>> from roboticstoolbox import QuinticPolyPlanner
+>>> import numpy as np
+>>> start = (10, 10, np.deg2rad(10.0))
+>>> goal = (30, -10, np.deg2rad(20.0))
+>>> quintic = QuinticPolyPlanner(start_vel=1)
+>>> path, status = quintic.query(start, goal)
+find path!!
+>>> print(path[:5,:])
+[[10.     10.      0.1745]
+ [10.0985 10.0173  0.1724]
+ [10.1971 10.0342  0.1662]
+ [10.2959 10.0503  0.156 ]
+ [10.3949 10.0652  0.142 ]]
+
+
+
+
Thanks:
+

based on quintic polynomial planning from Python Robotics

+
+
Seealso:
+

Planner

+
+
+
+
+query(start, goal)[source]
+

Find a quintic polynomial path

+
+
Parameters:
+
    +
  • start (array_like(3), optional) – start configuration \((x, y, \theta)\)

  • +
  • goal (array_like(3), optional) – goal configuration \((x, y, \theta)\)

  • +
+
+
Returns:
+

path and status

+
+
Return type:
+

ndarray(N,3), namedtuple

+
+
+

The returned status value has elements:

+ + + + + + + + + + + + + + + + + + + + +

Element

Description

t

time to execute the path

vel

velocity profile along the path

accel

acceleration profile along the path

jerk

jerk profile along the path

+
+
Seealso:
+

Planner.query()

+
+
+
+ +
+
+__str__()
+

Compact representation of the planner

+
+
Returns:
+

pretty printed representation

+
+
Return type:
+

str

+
+
+
+ +
+
+property goal
+

Set/get goal point or configuration (superclass)

+
+
Getter:
+

Return goal pointor configuration

+
+
Return type:
+

ndarray(2) or ndarray(3)

+
+
Setter:
+

Set goal point or configuration

+
+
Param:
+

array_like(2) or array_like(3)

+
+
+

The goal is either a point \((x, y)\) or a configuration \((x, y, \theta)\).

+
+
Seealso:
+

goal()

+
+
+
+ +
+
+property occgrid
+

Occupancy grid

+
+
Returns:
+

occupancy grid used for planning

+
+
Return type:
+

OccGrid or subclass or None

+
+
+

Returns the occupancy grid that was optionally inflated at constructor time.

+
+
Seealso:
+

validate_endpoint() isoccupied()

+
+
+
+ +
+
+plan()
+

Plan path (superclass)

+
+
Parameters:
+
    +
  • start (array_like(2) or array_like(3), optional) – start position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value passed to constructor

  • +
  • goal (array_like(2) or array_like(3), optional) – goal position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value passed to constructor

  • +
+
+
+

The implementation depends on the particular planner. Some may have +no planning phase. The plan may also depend on just the start or goal.

+
+ +
+
+plot(path=None, line=None, line_r=None, configspace=False, unwrap=True, direction=None, background=True, path_marker=None, path_marker_reverse=None, start_marker=None, goal_marker=None, start_vehicle=None, goal_vehicle=None, start=None, goal=None, ax=None, block=None, bgargs={}, **unused)
+

Plot vehicle path (superclass)

+
+
Parameters:
+
    +
  • path ((N, 2) or ndarray(N, 3)) – path, defaults to None

  • +
  • direction (array_like(N), optional) – travel direction associated with each point on path, is either >0 or <0, defaults to None

  • +
  • line (sequence of dict of arguments for plot) – line style for forward motion, default is striped yellow on black

  • +
  • line_r (sequence of dict of arguments for plot) – line style for reverse motion, default is striped red on black

  • +
  • configspace (bool, optional) – plot the path in 3D configuration space, input must be 3xN. +Start and goal style will be given by qstart_marker and qgoal_marker, defaults to False

  • +
  • unwrap (bool, optional) – for configuration space plot unwrap \(\theta\) so +there are no discontinuities at \(\pm \pi\), defaults to True

  • +
  • background (bool, optional) – plot occupancy grid if present, default True

  • +
  • start_marker (dict, optional) – style for marking start point

  • +
  • goal_marker (dict, optional) – style for marking goal point

  • +
  • start_vehicle (dict) – style for vehicle animation object at start configuration

  • +
  • goal_vehicle (dict) – style for vehicle animation object at goal configuration

  • +
  • start (array_like(2) or array_like(3), optional) – start position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value previously set

  • +
  • goal (array_like(2) or array_like(3), optional) – goal position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value previously set

  • +
  • bgargs (dict, optional) – arguments passed to plot_bg(), defaults to None

  • +
  • ax (matplotlib axes) – axes to plot into

  • +
  • block (bool, optional) – block after displaying the plot

  • +
+
+
+

Plots the start and goal location/pose if they are specified by +start or goal or were set by the object constructor or +plan or query method.

+

If the start and goal have length 2, planning in +\(\mathbb{R}^2\), then markers are drawn using styles specified by +start_marker and end_marker which are dicts using Matplotlib +keywords, for example:

+
planner.plot(path, start=dict(marker='s', color='b'))
+
+
+

If the start and goal have length 3, planning in \(\SE{2}\), +and configspace is False, then direction-indicating markers are used +to display start and goal configuration. These are also given as dicts +but have two items: 'shape' which is the shape of the polygonal +marker and is either 'triangle' or 'car'. The second item +'args' is passed to base.plot_poly() and Matplotlib and could +have values such as filled=True or color.

+

If configspace is False then a 3D plot is created and the start and +goal are indicated by Matplotlib markers specified by the dicts +start_marker and end_marker

+

Default values are provided for all markers:

+
+
    +
  • the start point is a circle

  • +
  • the goal point is a star

  • +
  • the start vehicle style is a VehiclePolygon(shape='car') as +an unfilled outline

  • +
  • the goal vehicle style is a VehiclePolygon(shape='car') as +a transparent filled shape

  • +
+
+

If background is True then the background of the plot is either or +both of:

+
    +
  • the occupancy grid

  • +
  • the distance field of the planner

  • +
+

Additional arguments bgargs can be passed through to plot_bg()

+

If path is specified it has one column per point and either 2 or 3 rows:

+
    +
  • 2 rows describes motion in the \(xy\)-plane and a 2D plot is created

  • +
  • 3 rows describes motion in the \(xy\theta\)-configuration space. By +default only the \(xy\)-plane is plotted unless configspace +is True in which case motion in \(xy\theta\)-configuration space +is shown as a 3D plot.

  • +
+

If the planner supports bi-directional motion then the direction +option gives the direction for each point on the path.

+

Forward motion segments are drawn using style information from line +while reverse motion segments are drawn using style information from +line_r. These are each a sequence of dicts of Matplotlib plot +options which can draw an arbitrary number of lines on top of each +other. The default:

+
line = (
+        {color:'black', linewidth:4},
+        {color:'yellow', linewidth:3, dashes:(5,5)}
+    )
+
+
+

will draw a blackline of width 4 with a dashed yellow line of width 3 +plotted on top, giving a line of alternating black and yellow dashes.

+
+
Seealso:
+

plot_bg() base.plot_poly()

+
+
+
+ +
+
+property start
+

Set/get start point or configuration (superclass)

+
+
Getter:
+

Return start point or configuration

+
+
Return type:
+

ndarray(2) or ndarray(3)

+
+
Setter:
+

Set start point or configuration

+
+
Param:
+

array_like(2) or array_like(3)

+
+
+

The start is either a point \((x, y)\) or a configuration \((x, y, \theta)\).

+
+
Seealso:
+

goal()

+
+
+
+ +
+
+validate_endpoint(p, dtype=None)
+

Validate start or goal point

+
+
Parameters:
+
    +
  • p (array_like(2)) – the point

  • +
  • dtype (str, optional) – data type for point coordinates, defaults to None

  • +
+
+
Raises:
+

ValueError – point is inside obstacle

+
+
Returns:
+

point as a NumPy array of specified dtype

+
+
Return type:
+

ndarray(2)

+
+
+

The coordinate is tested to be a free cell in the occupancy grid.

+
+
Seealso:
+

isoccupied() occgrid()

+
+
+
+ +
+
+property verbose
+

Get verbosity

+
+
Returns:
+

verbosity

+
+
Return type:
+

bool

+
+
+

If verbosity print more diagnostic messages to the planner’s +message channel.

+
+ +
+ +
+
+

RRT planner

+
+
+class roboticstoolbox.mobile.RRTPlanner(map, vehicle, curvature=1.0, stepsize=0.2, npoints=50, **kwargs)[source]
+

Bases: PlannerBase

+

Rapidly exploring tree planner

+
+
Parameters:
+
    +
  • map (PolygonMap) – occupancy grid

  • +
  • vehicle (VehicleBase subclass) – vehicle kinematic model

  • +
  • curvature (float, optional) – maximum path curvature, defaults to 1.0

  • +
  • stepsize (float, optional) – spacing between points on the path, defaults to 0.2

  • +
  • npoints (int, optional) – number of vertices in random tree, defaults to 50

  • +
+
+
+ + + + + + + + + + + + + + + + + + + + +

Feature

Capability

Plan

\(\SE{2}\)

Obstacle avoidance

Yes, polygons

Curvature

Discontinuous

Motion

Bidirectional

+

Creates a planner that finds the obstacle-free path between two +configurations in the plane using forward and backward motion. The path +comprises multiple Dubins curves comprising straight lines, or arcs with +curvature of \(\pm\) curvature. Motion along the segments may be in +the forward or backward direction.

+

Polygons are used for obstacle avoidance:

+
    +
  • the environment is defined by a set of polygons represented by a PolygonMap

  • +
  • the vehicle is defined by a single polygon specified by the polygon +argument to its constructor

  • +
+

Example:

+
from roboticstoolbox import RRTPlanner
+from spatialmath import Polygon2
+from math import pi
+
+# create polygonal obstacles
+map = PolygonMap(workspace=[0, 10])
+map.add([(5, 50), (5, 6), (6, 6), (6, 50)])
+map.add([(5, 4), (5, -50), (6, -50), (6, 4)])
+
+# create outline polygon for vehicle
+l, w = 3, 1.5
+vpolygon = Polygon2([(-l/2, w/2), (-l/2, -w/2), (l/2, -w/2), (l/2, w/2)])
+
+# create vehicle model
+vehicle = Bicycle(steer_max=1, L=2, polygon=vpolygon)
+
+# create planner
+rrt = RRTPlanner(map=map, vehicle=vehicle, npoints=50, seed=0)
+# start and goal configuration
+qs = (2, 8, -pi/2)
+qg = (8, 2, -pi/2)
+
+# plan path
+rrt.plan(goal=qg)
+path, status = rrt.query(start=qs)
+print(path[:5,:])
+print(status)
+
+
+
+
Seealso:
+

DubinsPlanner Vehicle PlannerBase

+
+
+
+
+plan(goal, showsamples=True, showvalid=True, animate=False)[source]
+

Plan paths to goal using RRT

+
+
Parameters:
+
    +
  • goal (array_like(3), optional) – goal pose \((x, y, \theta)\), defaults to previously set value

  • +
  • showsamples (bool, optional) – display position part of configurations overlaid on the map, defaults to True

  • +
  • showvalid (bool, optional) – display valid configurations as vehicle polygons overlaid on the map, defaults to False

  • +
  • animate (bool, optional) – update the display as configurations are tested, defaults to False

  • +
+
+
+

Compute a rapidly exploring random tree with its root at the goal. +The tree will have npoints vertices spread uniformly randomly over +the workspace which is an attribute of the map.

+

For every new point added, a Dubins path is computed to the nearest +vertex already in the graph. Each configuration on that path, with +spacing of stepsize, is tested for obstacle intersection.

+

The configurations tested are displayed (translation only) if showsamples is +True. The valid configurations are displayed as vehicle polygones if showvalid +is True. If animate is True these points are displayed during the search +process, otherwise a single figure is displayed at the end.

+
+
Seealso:
+

query()

+
+
+
+ +
+
+query(start)[source]
+

Find a path from start configuration

+
+
Parameters:
+

start (array_like(3), optional) – start configuration \((x, y, \theta)\)

+
+
Returns:
+

path and status

+
+
Return type:
+

ndarray(N,3), namedtuple

+
+
+

The path comprises points equally spaced at a distance of stepsize.

+

The returned status value has elements:

+ + + + + + + + + + + + + +

Element

Description

length | total path length

initial_d | distance from start to first vertex in graph

vertices

sequence of vertices in the graph

+
+ +
+
+qrandom()[source]
+

Random configuration

+
+
Returns:
+

random configuration \((x, y, \theta)\)

+
+
Return type:
+

ndarray(3)

+
+
+

Returns a random configuration where position \((x, y)\) +lies within the bounds of the map associated with this planner.

+
+
Seealso:
+

qrandom_free()

+
+
+
+ +
+
+qrandom_free()[source]
+

Random obstacle free configuration

+
+
Returns:
+

random configuration \((x, y, \theta)\)

+
+
Return type:
+

ndarray(3)

+
+
+

Returns a random obstacle free configuration where position \((x, +y)\) lies within the bounds of the map associated with this planner. +Iterates on qrandom()

+
+
Seealso:
+

qrandom() iscollision()

+
+
+
+ +
+
+iscollision(q)[source]
+

Test if configuration is collision

+
+
Parameters:
+

q (array_like(3)) – vehicle configuration \((x, y, \theta)\)

+
+
Returns:
+

collision status

+
+
Return type:
+

bool

+
+
+

Transforms the vehicle polygon and tests for intersection against +the polygonal obstacle map.

+
+ +
+
+__str__()
+

Compact representation of the planner

+
+
Returns:
+

pretty printed representation

+
+
Return type:
+

str

+
+
+
+ +
+
+property goal
+

Set/get goal point or configuration (superclass)

+
+
Getter:
+

Return goal pointor configuration

+
+
Return type:
+

ndarray(2) or ndarray(3)

+
+
Setter:
+

Set goal point or configuration

+
+
Param:
+

array_like(2) or array_like(3)

+
+
+

The goal is either a point \((x, y)\) or a configuration \((x, y, \theta)\).

+
+
Seealso:
+

goal()

+
+
+
+ +
+
+property occgrid
+

Occupancy grid

+
+
Returns:
+

occupancy grid used for planning

+
+
Return type:
+

OccGrid or subclass or None

+
+
+

Returns the occupancy grid that was optionally inflated at constructor time.

+
+
Seealso:
+

validate_endpoint() isoccupied()

+
+
+
+ +
+
+plot(path=None, line=None, line_r=None, configspace=False, unwrap=True, direction=None, background=True, path_marker=None, path_marker_reverse=None, start_marker=None, goal_marker=None, start_vehicle=None, goal_vehicle=None, start=None, goal=None, ax=None, block=None, bgargs={}, **unused)
+

Plot vehicle path (superclass)

+
+
Parameters:
+
    +
  • path ((N, 2) or ndarray(N, 3)) – path, defaults to None

  • +
  • direction (array_like(N), optional) – travel direction associated with each point on path, is either >0 or <0, defaults to None

  • +
  • line (sequence of dict of arguments for plot) – line style for forward motion, default is striped yellow on black

  • +
  • line_r (sequence of dict of arguments for plot) – line style for reverse motion, default is striped red on black

  • +
  • configspace (bool, optional) – plot the path in 3D configuration space, input must be 3xN. +Start and goal style will be given by qstart_marker and qgoal_marker, defaults to False

  • +
  • unwrap (bool, optional) – for configuration space plot unwrap \(\theta\) so +there are no discontinuities at \(\pm \pi\), defaults to True

  • +
  • background (bool, optional) – plot occupancy grid if present, default True

  • +
  • start_marker (dict, optional) – style for marking start point

  • +
  • goal_marker (dict, optional) – style for marking goal point

  • +
  • start_vehicle (dict) – style for vehicle animation object at start configuration

  • +
  • goal_vehicle (dict) – style for vehicle animation object at goal configuration

  • +
  • start (array_like(2) or array_like(3), optional) – start position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value previously set

  • +
  • goal (array_like(2) or array_like(3), optional) – goal position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value previously set

  • +
  • bgargs (dict, optional) – arguments passed to plot_bg(), defaults to None

  • +
  • ax (matplotlib axes) – axes to plot into

  • +
  • block (bool, optional) – block after displaying the plot

  • +
+
+
+

Plots the start and goal location/pose if they are specified by +start or goal or were set by the object constructor or +plan or query method.

+

If the start and goal have length 2, planning in +\(\mathbb{R}^2\), then markers are drawn using styles specified by +start_marker and end_marker which are dicts using Matplotlib +keywords, for example:

+
planner.plot(path, start=dict(marker='s', color='b'))
+
+
+

If the start and goal have length 3, planning in \(\SE{2}\), +and configspace is False, then direction-indicating markers are used +to display start and goal configuration. These are also given as dicts +but have two items: 'shape' which is the shape of the polygonal +marker and is either 'triangle' or 'car'. The second item +'args' is passed to base.plot_poly() and Matplotlib and could +have values such as filled=True or color.

+

If configspace is False then a 3D plot is created and the start and +goal are indicated by Matplotlib markers specified by the dicts +start_marker and end_marker

+

Default values are provided for all markers:

+
+
    +
  • the start point is a circle

  • +
  • the goal point is a star

  • +
  • the start vehicle style is a VehiclePolygon(shape='car') as +an unfilled outline

  • +
  • the goal vehicle style is a VehiclePolygon(shape='car') as +a transparent filled shape

  • +
+
+

If background is True then the background of the plot is either or +both of:

+
    +
  • the occupancy grid

  • +
  • the distance field of the planner

  • +
+

Additional arguments bgargs can be passed through to plot_bg()

+

If path is specified it has one column per point and either 2 or 3 rows:

+
    +
  • 2 rows describes motion in the \(xy\)-plane and a 2D plot is created

  • +
  • 3 rows describes motion in the \(xy\theta\)-configuration space. By +default only the \(xy\)-plane is plotted unless configspace +is True in which case motion in \(xy\theta\)-configuration space +is shown as a 3D plot.

  • +
+

If the planner supports bi-directional motion then the direction +option gives the direction for each point on the path.

+

Forward motion segments are drawn using style information from line +while reverse motion segments are drawn using style information from +line_r. These are each a sequence of dicts of Matplotlib plot +options which can draw an arbitrary number of lines on top of each +other. The default:

+
line = (
+        {color:'black', linewidth:4},
+        {color:'yellow', linewidth:3, dashes:(5,5)}
+    )
+
+
+

will draw a blackline of width 4 with a dashed yellow line of width 3 +plotted on top, giving a line of alternating black and yellow dashes.

+
+
Seealso:
+

plot_bg() base.plot_poly()

+
+
+
+ +
+
+property start
+

Set/get start point or configuration (superclass)

+
+
Getter:
+

Return start point or configuration

+
+
Return type:
+

ndarray(2) or ndarray(3)

+
+
Setter:
+

Set start point or configuration

+
+
Param:
+

array_like(2) or array_like(3)

+
+
+

The start is either a point \((x, y)\) or a configuration \((x, y, \theta)\).

+
+
Seealso:
+

goal()

+
+
+
+ +
+
+validate_endpoint(p, dtype=None)
+

Validate start or goal point

+
+
Parameters:
+
    +
  • p (array_like(2)) – the point

  • +
  • dtype (str, optional) – data type for point coordinates, defaults to None

  • +
+
+
Raises:
+

ValueError – point is inside obstacle

+
+
Returns:
+

point as a NumPy array of specified dtype

+
+
Return type:
+

ndarray(2)

+
+
+

The coordinate is tested to be a free cell in the occupancy grid.

+
+
Seealso:
+

isoccupied() occgrid()

+
+
+
+ +
+
+property verbose
+

Get verbosity

+
+
Returns:
+

verbosity

+
+
Return type:
+

bool

+
+
+

If verbosity print more diagnostic messages to the planner’s +message channel.

+
+ +
+ +
+
+ + +
+
+
+ +
+ +
+

© Copyright 2023, Jesse Haviland and Peter Corke. + Last updated on 11-May-2024. +

+
+ + + +
+
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/mobile-planner-map.html b/mobile-planner-map.html new file mode 100644 index 00000000..43d38ecc --- /dev/null +++ b/mobile-planner-map.html @@ -0,0 +1,947 @@ + + + + + + + + + + + Map classes — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

Map classes

+
+

Occupancy grid classes

+
+

Binary Occupancy grid

+
+
+class roboticstoolbox.mobile.BinaryOccupancyGrid(grid=None, **kwargs)[source]
+

Bases: BaseOccupancyGrid

+
+
+__init__(grid=None, **kwargs)[source]
+

Create a binary occupancy grid instance

+
+
Parameters:
+
    +
  • grid (ndarray(N,M)) – occupancy grid as a NumPy array

  • +
  • size (float, optional) – cell size, defaults to 1

  • +
  • origin (array_like(2), optional) – world coordinates of the grid element [0,0], defaults to (0, 0)

  • +
  • kwargs – options passed to BaseMap

  • +
+
+
+

The array is kept internally as a bool array. Cells are set to True +(occupied) corresponding to input values > 0.

+

This object supports a user-defined coordinate system and grid size. +World coordinates are converted to grid coordinates to lookup the +occupancy status.

+

Example:

+
>>> from roboticstoolbox import BinaryOccupancyGrid
+>>> import numpy as np
+>>> og = BinaryOccupancyGrid(np.zeros((5,5)))
+>>> print(og)
+BinaryOccupancyGrid: 5 x 5, cell size=1, x = [0.0, 4.0], y = [0.0, 4.0], 0.0% occupied
+>>> og = BinaryOccupancyGrid(workspace=[-5,5], cellsize=0.1, value=0)
+>>> print(og)
+BinaryOccupancyGrid: 101 x 101, cell size=0.1, x = [-5.0, 5.0], y = [-5.0, 5.0], 0.0% occupied
+
+
+
+
Seealso:
+

OccupancyGrid

+
+
+
+ +
+
+isoccupied(p)[source]
+

Test if coordinate is occupied

+
+
Parameters:
+

p (array_like(2)) – world coordinate (x, y)

+
+
Returns:
+

occupancy status of corresponding grid cell

+
+
Return type:
+

bool

+
+
+

The grid cell size and offset are used to convert p to an occupancy +grid coordinate. The grid coordinate is rounded and cast to integer +value. If the coordinate is outside the bounds of the occupancy grid +it is considered to be occupied.

+
+
Seealso:
+

w2g()

+
+
+
+ +
+
+inflate(radius)[source]
+

Inflate obstales

+
+
Parameters:
+

radius (float) – radius of circular structuring element in world units

+
+
+

A circular structuring element is created and used to dilate the +stored occupancy grid.

+

Successive calls to inflate will compound the inflation.

+
+
Seealso:
+

scipy.ndimage.binary_dilation()

+
+
+
+ +
+
+copy()
+

Copy an occupancy grid (superclass)

+
+
Returns:
+

copy of the ocupancy grid

+
+
Return type:
+

OccGrid

+
+
+
+ +
+
+g2w(p)
+

Convert grid coordinate to world coordinate (superclass)

+
+
Parameters:
+

p (array_like(2)) – grid coordinate (column, row)

+
+
Returns:
+

world coordinate (x, y)

+
+
Return type:
+

ndarray(2)

+
+
+

The grid cell size and offset are used to convert occupancy grid +coordinate p to a world coordinate.

+
+ +
+
+property grid
+

Occupancy grid as a NumPy array (superclass)

+
+
Returns:
+

binary occupancy grid

+
+
Return type:
+

ndarray(N,M) of bool

+
+
+

If inflate() has been called, this will return the inflated +occupancy grid.

+
+ +
+
+line_w(p1, p2)
+

Get index of cells along a line segment (superclass)

+
+
Parameters:
+
    +
  • p1 (array_like(2)) – start

  • +
  • p2 (array_like(2)) – end

  • +
+
+
Returns:
+

index into grid

+
+
Return type:
+

ndarray(N)

+
+
+

Get the indices of cells along a line segment defined by the end +points given in world coordinates.

+

The returned indices can be applied to a raveled view of the grid.

+
+
Seealso:
+

ravel() w2g()

+
+
+
+ +
+
+property maxdim
+

Maximum dimension of grid in world coordinates (superclass)

+
+
Returns:
+

maximum side length of the occupancy grid

+
+
Return type:
+

float

+
+
+
+ +
+
+property name
+

Occupancy grid name (superclass)

+
+
Returns:
+

name of the occupancy grid

+
+
Return type:
+

str

+
+
+
+ +
+
+plot(map=None, ax=None, block=None, **kwargs)
+

Plot the occupancy grid (superclass)

+
+
Parameters:
+
    +
  • map (ndarray(N,M), optional) – array which is plotted instead of the grid, must be same +size as the occupancy grid,defaults to None

  • +
  • ax (Axes2D, optional) – matplotlib axes to plot into, defaults to None

  • +
  • block (bool, optional) – block until plot is dismissed, defaults to None

  • +
  • kwargs – arguments passed to imshow

  • +
+
+
+

The grid is plotted as an image but with axes in world coordinates.

+

The grid is a NumPy boolean array which has values 0 (false=unoccupied) +and 1 (true=occupied). Passing a cmap option to imshow can be used +to control the displayed color of free space and obstacles.

+
+ +
+
+property ravel
+

Ravel the grid (superclass)

+
+
Returns:
+

1D view of the occupancy grid

+
+
Return type:
+

ndarray(N)

+
+
+
+ +
+
+set(region, value)
+

Set region of map (superclass)

+
+
Parameters:
+
    +
  • region (array_like(4)) – The region [xmin, ymin, xmax, ymax]

  • +
  • value (int, bool, float) – value to set cells to

  • +
+
+
+
+ +
+
+property shape
+

Shape of the occupancy grid array (superclass)

+
+
Returns:
+

shape of the occupancy grid array

+
+
Return type:
+

2-tuple

+
+
+

This is the shape of the NumPy array that holds the occupancy grid.

+
+ +
+
+w2g(p)
+

Convert world coordinate to grid coordinate (superclass)

+
+
Parameters:
+

p (array_like(2)) – world coordinate (x, y)

+
+
Returns:
+

grid coordinate (column, row)

+
+
Return type:
+

ndarray(2)

+
+
+

The grid cell size and offset are used to convert p to an occupancy +grid coordinate. The grid coordinate is rounded and cast to integer +value. No check is made on the validity of the coordinate.

+
+ +
+
+property workspace
+

Bounds of the occupancy grid in world coordinates (superclass)

+
+
Returns:
+

workspace bounds [xmin, xmax, ymin, ymax]

+
+
Return type:
+

ndarray(4)

+
+
+

Returns the bounds of the occupancy grid in world coordinates.

+
+ +
+
+property xmax
+

Maximum x-coordinate of this grid (superclass)

+
+
Returns:
+

maximum world x-coordinate

+
+
Return type:
+

float

+
+
+
+ +
+
+property xmin
+

Minimum x-coordinate of this grid (superclass)

+
+
Returns:
+

minimum world x-coordinate

+
+
Return type:
+

float

+
+
+
+ +
+
+property ymax
+

Maximum y-coordinate of this grid (superclass)

+
+
Returns:
+

maximum world y-coordinate

+
+
Return type:
+

float

+
+
+
+ +
+
+property ymin
+

Minimum y-coordinate of this grid (superclass)

+
+
Returns:
+

minimum world y-coordinate

+
+
Return type:
+

float

+
+
+
+ +
+ +
+
+

Occupancy grid

+
+
+class roboticstoolbox.mobile.OccupancyGrid(grid=None, origin=(0, 0), value=0, cellsize=1, **kwargs)[source]
+

Bases: BaseOccupancyGrid

+

General occupancy grid

+

The elements of the array are floats and can represent occupancy +probability or traversal cost.

+

Example:

+
>>> from roboticstoolbox import OccupancyGrid
+>>> import numpy as np
+>>> og = OccupancyGrid(np.zeros((5,5)))
+>>> print(og)
+OccupancyGrid: 5 x 5, cell size=1, x = [0.0, 4.0], y = [0.0, 4.0], dtype float64, min 0.0, max 0.0, mean 0.0
+>>> og = OccupancyGrid(workspace=[-5,5], cellsize=0.1, value=0.5)
+>>> print(og)
+OccupancyGrid: 101 x 101, cell size=0.1, x = [-5.0, 5.0], y = [-5.0, 5.0], dtype float64, min 0.5, max 0.5, mean 0.5
+
+
+
+
Seealso:
+

BinaryOccupancyGrid

+
+
+
+
+__init__(grid=None, origin=(0, 0), value=0, cellsize=1, **kwargs)
+

Occupancy grid (superclass)

+
+
Parameters:
+
    +
  • grid (ndarray(N,M)) – occupancy grid as a NumPy array

  • +
  • value (any, optional) – initial value of cells

  • +
  • origin (array_like(2), optional) – world coordinates of the grid element [0,0], defaults to (0, 0)

  • +
  • cellsize (float, optional) – cell size, defaults to 1

  • +
  • kwargs – options passed to BaseMap

  • +
+
+
+

This object supports a user-defined coordinate system and grid size. +World coordinates are converted to grid coordinates to lookup the +occupancy status.

+

The grid can be initialized by:

+
    +
  • a 2D NumPy array

  • +
  • specifying workspace and value arguments

  • +
+
+ +
+
+copy()
+

Copy an occupancy grid (superclass)

+
+
Returns:
+

copy of the ocupancy grid

+
+
Return type:
+

OccGrid

+
+
+
+ +
+
+g2w(p)
+

Convert grid coordinate to world coordinate (superclass)

+
+
Parameters:
+

p (array_like(2)) – grid coordinate (column, row)

+
+
Returns:
+

world coordinate (x, y)

+
+
Return type:
+

ndarray(2)

+
+
+

The grid cell size and offset are used to convert occupancy grid +coordinate p to a world coordinate.

+
+ +
+
+property grid
+

Occupancy grid as a NumPy array (superclass)

+
+
Returns:
+

binary occupancy grid

+
+
Return type:
+

ndarray(N,M) of bool

+
+
+

If inflate() has been called, this will return the inflated +occupancy grid.

+
+ +
+
+line_w(p1, p2)
+

Get index of cells along a line segment (superclass)

+
+
Parameters:
+
    +
  • p1 (array_like(2)) – start

  • +
  • p2 (array_like(2)) – end

  • +
+
+
Returns:
+

index into grid

+
+
Return type:
+

ndarray(N)

+
+
+

Get the indices of cells along a line segment defined by the end +points given in world coordinates.

+

The returned indices can be applied to a raveled view of the grid.

+
+
Seealso:
+

ravel() w2g()

+
+
+
+ +
+
+property maxdim
+

Maximum dimension of grid in world coordinates (superclass)

+
+
Returns:
+

maximum side length of the occupancy grid

+
+
Return type:
+

float

+
+
+
+ +
+
+property name
+

Occupancy grid name (superclass)

+
+
Returns:
+

name of the occupancy grid

+
+
Return type:
+

str

+
+
+
+ +
+
+plot(map=None, ax=None, block=None, **kwargs)
+

Plot the occupancy grid (superclass)

+
+
Parameters:
+
    +
  • map (ndarray(N,M), optional) – array which is plotted instead of the grid, must be same +size as the occupancy grid,defaults to None

  • +
  • ax (Axes2D, optional) – matplotlib axes to plot into, defaults to None

  • +
  • block (bool, optional) – block until plot is dismissed, defaults to None

  • +
  • kwargs – arguments passed to imshow

  • +
+
+
+

The grid is plotted as an image but with axes in world coordinates.

+

The grid is a NumPy boolean array which has values 0 (false=unoccupied) +and 1 (true=occupied). Passing a cmap option to imshow can be used +to control the displayed color of free space and obstacles.

+
+ +
+
+property ravel
+

Ravel the grid (superclass)

+
+
Returns:
+

1D view of the occupancy grid

+
+
Return type:
+

ndarray(N)

+
+
+
+ +
+
+set(region, value)
+

Set region of map (superclass)

+
+
Parameters:
+
    +
  • region (array_like(4)) – The region [xmin, ymin, xmax, ymax]

  • +
  • value (int, bool, float) – value to set cells to

  • +
+
+
+
+ +
+
+property shape
+

Shape of the occupancy grid array (superclass)

+
+
Returns:
+

shape of the occupancy grid array

+
+
Return type:
+

2-tuple

+
+
+

This is the shape of the NumPy array that holds the occupancy grid.

+
+ +
+
+w2g(p)
+

Convert world coordinate to grid coordinate (superclass)

+
+
Parameters:
+

p (array_like(2)) – world coordinate (x, y)

+
+
Returns:
+

grid coordinate (column, row)

+
+
Return type:
+

ndarray(2)

+
+
+

The grid cell size and offset are used to convert p to an occupancy +grid coordinate. The grid coordinate is rounded and cast to integer +value. No check is made on the validity of the coordinate.

+
+ +
+
+property workspace
+

Bounds of the occupancy grid in world coordinates (superclass)

+
+
Returns:
+

workspace bounds [xmin, xmax, ymin, ymax]

+
+
Return type:
+

ndarray(4)

+
+
+

Returns the bounds of the occupancy grid in world coordinates.

+
+ +
+
+property xmax
+

Maximum x-coordinate of this grid (superclass)

+
+
Returns:
+

maximum world x-coordinate

+
+
Return type:
+

float

+
+
+
+ +
+
+property xmin
+

Minimum x-coordinate of this grid (superclass)

+
+
Returns:
+

minimum world x-coordinate

+
+
Return type:
+

float

+
+
+
+ +
+
+property ymax
+

Maximum y-coordinate of this grid (superclass)

+
+
Returns:
+

maximum world y-coordinate

+
+
Return type:
+

float

+
+
+
+ +
+
+property ymin
+

Minimum y-coordinate of this grid (superclass)

+
+
Returns:
+

minimum world y-coordinate

+
+
Return type:
+

float

+
+
+
+ +
+ +
+
+
+

Polygon map

+
+
+class roboticstoolbox.mobile.PolygonMap(workspace=None, polygons=[])[source]
+

Bases: BaseMap

+
+
+__init__(workspace=None, polygons=[])[source]
+

Polygonal obstacle map

+
+
Parameters:
+
    +
  • workspace (float, array_like(2), array_like(4)) – dimensions of 2D plot area, defaults to (-10:10) x (-10:10), +see plotvol2()

  • +
  • polygons (list, optional) – obstacle polygons, defaults to []

  • +
+
+
+

The obstacle polygons are specified as instances of Polygon2 +or ndarray(2,N).

+

The workspace can be specified in several ways:

+ + + + + + + + + + + + + + + + + + + + + +

workspace

x-range

y-range

A (scalar)

-A:A

-A:A

[A, B]

A:B

A:B

[A, B, C, D]

A:B

C:D

+

Workspace is used only to set plot bounds.

+
+ +
+
+add(polygon)[source]
+

Add a polygon to map

+
+
Parameters:
+

polygon (Polygon2 or ndarray(2,N)) – a polygon

+
+
+
+ +
+
+iscollision(polygon)[source]
+

Test for collision

+
+
Parameters:
+

polygon (Polygon2 or ndarray(2,N)) – a polygon

+
+
Returns:
+

collision

+
+
Return type:
+

bool

+
+
+

The polygon is tested against polygons in the map, and returns True +on the first collision.

+
+
Seealso:
+

add() Polygon2

+
+
+
+ +
+
+plot(block=None)[source]
+
+ +
+
+isoccupied(p)[source]
+

Test if point lies inside an obstacle

+
+
Parameters:
+

p (array_like(2)) – a 2D point

+
+
Returns:
+

enclosure

+
+
Return type:
+

bool

+
+
+

The point is tested for enclosure by polygons in the map, and returns True +on the first enclosure.

+
+ +
+
+property workspace
+

Bounds of the occupancy grid

+
+
Returns:
+

workspace bounds [xmin, xmax, ymin, ymax]

+
+
Return type:
+

ndarray(4)

+
+
+

Returns the bounds of the occupancy grid.

+
+ +
+ +
+
+ + +
+
+
+ +
+ +
+

© Copyright 2023, Jesse Haviland and Peter Corke. + Last updated on 11-May-2024. +

+
+ + + +
+
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/mobile-planner.html b/mobile-planner.html new file mode 100644 index 00000000..8c816603 --- /dev/null +++ b/mobile-planner.html @@ -0,0 +1,259 @@ + + + + + + + + + + + Path planning — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

Path planning

+

A set of path planners for robots operating in planar environments with +configuration \(\vec{q} \in \mathbb{R}^2\) or \(\vec{q} \in \mathbb{R}^2 \times S^1 \sim \SE{2}\). +All inherit from PlannerBase.

+

Some planners are based on code from the PathPlanning category of +PythonRobotics by Atsushi Sakai.

+
Inheritance diagram of roboticstoolbox.mobile.DistanceTransformPlanner, roboticstoolbox.mobile.DstarPlanner, roboticstoolbox.mobile.DubinsPlanner, roboticstoolbox.mobile.ReedsSheppPlanner, roboticstoolbox.mobile.QuinticPolyPlanner, roboticstoolbox.mobile.CurvaturePolyPlanner, roboticstoolbox.mobile.RRTPlanner
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

Planner

Plans in

Discrete/Continuous

Obstacle avoidance

Bug2

\(\mathbb{R}^2\)

discrete

yes

DistanceTransformPlanner

\(\mathbb{R}^2\)

discrete

yes

DstarPlanner

\(\mathbb{R}^2\)

discrete

yes

PRMPlanner

\(\mathbb{R}^2\)

continuous

yes

LatticePlanner

\(\SE{2}\)

discrete

yes

DubinsPlanner

\(\SE{2}\)

continuous

no

ReedsSheppPlanner

\(\SE{2}\)

continuous

no

CurvaturePolyPlanner

\(\SE{2}\)

continuous

no

QuinticPolyPlanner

\(\SE{2}\)

continuous

no

RRTPlanner

\(\SE{2}\)

continuous

yes

+ +
+ + +
+
+
+ +
+ +
+

© Copyright 2023, Jesse Haviland and Peter Corke. + Last updated on 11-May-2024. +

+
+ + + +
+
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/mobile-reactive.html b/mobile-reactive.html new file mode 100644 index 00000000..1593adcf --- /dev/null +++ b/mobile-reactive.html @@ -0,0 +1,675 @@ + + + + + + + + + + + Reactive navigation — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

Reactive navigation

+

These algorithms work with an occupancy grid representation of the world. Start +and goal are specified by 2D \((x, y)\) coordinates in the plane.

+
+
+class roboticstoolbox.mobile.Bug2(**kwargs)[source]
+

Bases: PlannerBase

+

Construct a Bug2 reactive navigator

+
+
Parameters:
+
    +
  • occgrid (OccGrid instance or ndarray(N,M)) – occupancy grid

  • +
  • kwargs – common arguments for PlannerBase superclass

  • +
+
+
Returns:
+

Bug2 reactive navigator

+
+
Return type:
+

Bug2 instance

+
+
+

Creates an object which simulates an automaton, capable of omnidirectional +motion, finding a path across an occupancy grid using only a bump sensor.

+
+
Reference:
+

“Path-Planning Strategies for a Point Mobile Automaton Moving +Amidst Unknown Obstacles of Arbitrary Shape”, Lumelsky and Stepanov, +Algorithmica (1987)2, pp.403-430

+
+
+
+

Note

+

This class is not a planner, even though it subclasses +PlannerBase. It can produce very inefficient paths.

+
+
+
Author:
+

Kristian Gibson and Peter Corke, based on MATLAB version by Peter Corke

+
+
Seealso:
+

PlannerBase

+
+
+
+
+property m_line
+

Get m-line

+
+
Returns:
+

m-line in homogeneous form

+
+
Return type:
+

ndarray(3)

+
+
+

This is the m-line computed for the last run().

+
+ +
+
+run(start=None, goal=None, animate=False, pause=0.001, trail=True, movie=None, **kwargs)[source]
+

Find a path using Bug2 reactive navigation algorithm

+
+
Parameters:
+
    +
  • start (array_like(2)) – starting position

  • +
  • goal (array_like(2)) – goal position

  • +
  • animate (bool, optional) – show animation of robot moving over the map, +defaults to False

  • +
  • movie (str or tuple(str, float), optional) – save animation as a movie, defaults to None. Is either +name of movie or a tuple (filename, frame interval)

  • +
  • trail – show the path followed by the robot, defaults to True

  • +
+
+
Returns:
+

path from start to goal

+
+
Return type:
+

ndarray(2,N)

+
+
+

Compute the path from start to goal assuming the robot is capable +of 8-way motion from its current cell to the next.

+
+

Note

+

start and goal are given as (x,y) coordinates in the +occupancy grid map, not as matrix row and column coordinates.

+
+
+
Seealso:
+

Bug2.plot()

+
+
+
+ +
+
+plot_m_line(ls=None)[source]
+

Plot m-line

+
+
Parameters:
+

ls (str, optional) – linestyle, defaults to "k--"

+
+
+

Plots the m-line on the current axes.

+
+ +
+
+__str__()
+

Compact representation of the planner

+
+
Returns:
+

pretty printed representation

+
+
Return type:
+

str

+
+
+
+ +
+
+property goal
+

Set/get goal point or configuration (superclass)

+
+
Getter:
+

Return goal pointor configuration

+
+
Return type:
+

ndarray(2) or ndarray(3)

+
+
Setter:
+

Set goal point or configuration

+
+
Param:
+

array_like(2) or array_like(3)

+
+
+

The goal is either a point \((x, y)\) or a configuration \((x, y, \theta)\).

+
+
Seealso:
+

goal()

+
+
+
+ +
+
+isoccupied(p)
+

Test if point is occupied (superclass)

+
+
Parameters:
+

p (array_like(2)) – world coordinate (x, y)

+
+
Returns:
+

occupancy status of corresponding grid cell

+
+
Return type:
+

bool

+
+
+

The world coordinate is transformed and the status of the occupancy +grid cell is returned. If the point lies outside the bounds of +the occupancy grid return True (obstacle)

+

If there is no occupancy grid this function always returns False (free).

+
+
Seealso:
+

occgrid() validate_endpoint() BinaryOccGrid.isoccupied()

+
+
+
+ +
+
+message(s, color=None)
+

Print message to message channel

+
+
Parameters:
+
    +
  • s (str) – message to print

  • +
  • color (str, optional) – color to print it, defaults to color specified at +constructor time.

  • +
+
+
+
+ +
+
+property occgrid
+

Occupancy grid

+
+
Returns:
+

occupancy grid used for planning

+
+
Return type:
+

OccGrid or subclass or None

+
+
+

Returns the occupancy grid that was optionally inflated at constructor time.

+
+
Seealso:
+

validate_endpoint() isoccupied()

+
+
+
+ +
+
+plot(path=None, line=None, line_r=None, configspace=False, unwrap=True, direction=None, background=True, path_marker=None, path_marker_reverse=None, start_marker=None, goal_marker=None, start_vehicle=None, goal_vehicle=None, start=None, goal=None, ax=None, block=None, bgargs={}, **unused)
+

Plot vehicle path (superclass)

+
+
Parameters:
+
    +
  • path ((N, 2) or ndarray(N, 3)) – path, defaults to None

  • +
  • direction (array_like(N), optional) – travel direction associated with each point on path, is either >0 or <0, defaults to None

  • +
  • line (sequence of dict of arguments for plot) – line style for forward motion, default is striped yellow on black

  • +
  • line_r (sequence of dict of arguments for plot) – line style for reverse motion, default is striped red on black

  • +
  • configspace (bool, optional) – plot the path in 3D configuration space, input must be 3xN. +Start and goal style will be given by qstart_marker and qgoal_marker, defaults to False

  • +
  • unwrap (bool, optional) – for configuration space plot unwrap \(\theta\) so +there are no discontinuities at \(\pm \pi\), defaults to True

  • +
  • background (bool, optional) – plot occupancy grid if present, default True

  • +
  • start_marker (dict, optional) – style for marking start point

  • +
  • goal_marker (dict, optional) – style for marking goal point

  • +
  • start_vehicle (dict) – style for vehicle animation object at start configuration

  • +
  • goal_vehicle (dict) – style for vehicle animation object at goal configuration

  • +
  • start (array_like(2) or array_like(3), optional) – start position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value previously set

  • +
  • goal (array_like(2) or array_like(3), optional) – goal position \((x, y)\) or configuration \((x, y, \theta)\), defaults to value previously set

  • +
  • bgargs (dict, optional) – arguments passed to plot_bg(), defaults to None

  • +
  • ax (matplotlib axes) – axes to plot into

  • +
  • block (bool, optional) – block after displaying the plot

  • +
+
+
+

Plots the start and goal location/pose if they are specified by +start or goal or were set by the object constructor or +plan or query method.

+

If the start and goal have length 2, planning in +\(\mathbb{R}^2\), then markers are drawn using styles specified by +start_marker and end_marker which are dicts using Matplotlib +keywords, for example:

+
planner.plot(path, start=dict(marker='s', color='b'))
+
+
+

If the start and goal have length 3, planning in \(\SE{2}\), +and configspace is False, then direction-indicating markers are used +to display start and goal configuration. These are also given as dicts +but have two items: 'shape' which is the shape of the polygonal +marker and is either 'triangle' or 'car'. The second item +'args' is passed to base.plot_poly() and Matplotlib and could +have values such as filled=True or color.

+

If configspace is False then a 3D plot is created and the start and +goal are indicated by Matplotlib markers specified by the dicts +start_marker and end_marker

+

Default values are provided for all markers:

+
+
    +
  • the start point is a circle

  • +
  • the goal point is a star

  • +
  • the start vehicle style is a VehiclePolygon(shape='car') as +an unfilled outline

  • +
  • the goal vehicle style is a VehiclePolygon(shape='car') as +a transparent filled shape

  • +
+
+

If background is True then the background of the plot is either or +both of:

+
    +
  • the occupancy grid

  • +
  • the distance field of the planner

  • +
+

Additional arguments bgargs can be passed through to plot_bg()

+

If path is specified it has one column per point and either 2 or 3 rows:

+
    +
  • 2 rows describes motion in the \(xy\)-plane and a 2D plot is created

  • +
  • 3 rows describes motion in the \(xy\theta\)-configuration space. By +default only the \(xy\)-plane is plotted unless configspace +is True in which case motion in \(xy\theta\)-configuration space +is shown as a 3D plot.

  • +
+

If the planner supports bi-directional motion then the direction +option gives the direction for each point on the path.

+

Forward motion segments are drawn using style information from line +while reverse motion segments are drawn using style information from +line_r. These are each a sequence of dicts of Matplotlib plot +options which can draw an arbitrary number of lines on top of each +other. The default:

+
line = (
+        {color:'black', linewidth:4},
+        {color:'yellow', linewidth:3, dashes:(5,5)}
+    )
+
+
+

will draw a blackline of width 4 with a dashed yellow line of width 3 +plotted on top, giving a line of alternating black and yellow dashes.

+
+
Seealso:
+

plot_bg() base.plot_poly()

+
+
+
+ +
+
+plot_bg(distance=None, cmap='gray', ax=None, inflated=True, colorbar=True, block=None, **unused)
+

Plot background (superclass)

+
+
Parameters:
+
    +
  • distance (ndarray(N,M), optional) – override distance field, defaults to None

  • +
  • cmap (str or Colormap, optional) – Specify a colormap for the distance field, defaults to ‘gray’

  • +
+
+
+

Displays the background which is either the occupancy grid or a distance +field. The distance field encodes the distance of a point from the goal, small +distance is dark, a large distance is bright.

+
+
If the planner has an occupancy grid then that will be displayed with:
    +
  • free cells in white

  • +
  • occupied cells in red

  • +
  • inflated occupied cells in pink

  • +
+
+
+

If distance is provided, or the planner has a distancemap attribute +the the distance field will be used as the background and obstacle cells +(actual or inflated) will be shown in red. A colorbar is added.

+
+ +
+
+progress_end()
+

Finalize a progress bar (superclass)

+

Remove/cleanip a progress bar, for +example:

+
planner.progress_start(100)
+for i in range(100):
+    # ...
+    planner.progress_next()
+planner.progress_end()
+
+
+
+
Seealso:
+

progress_start() progress_next()

+
+
+
+ +
+
+progress_next()
+

Increment a progress bar (superclass)

+

Create a progress bar for an operation that has n steps, for +example:

+
planner.progress_start(100)
+for i in range(100):
+    # ...
+    planner.progress_next()
+planner.progress_end()
+
+
+
+
Seealso:
+

progress_start() progress_end()

+
+
+
+ +
+
+progress_start(n)
+

Initialize a progress bar (superclass)

+
+
Parameters:
+

n (int) – Number of iterations in the operation

+
+
+

Create a progress bar for an operation that has n steps, for +example:

+
planner.progress_start(100)
+for i in range(100):
+    # ...
+    planner.progress_next()
+planner.progress_end()
+
+
+
+
Seealso:
+

progress_next() progress_end()

+
+
+
+ +
+
+property random
+

Get private random number generator

+
+
Returns:
+

NumPy random number generator

+
+
Return type:
+

numpy.random.Generator

+
+
+

Has methods including:

+ +

The generator is initialized with the seed provided at constructor +time.

+
+
Seealso:
+

numpy.random.default_rng()

+
+
+
+ +
+
+random_init(seed=None)
+

Initialize private random number generator

+
+
Parameters:
+

seed (int) – random number seed, defaults to value given to constructor

+
+
+

The private random number generator is initialized. The seed is seed +or the value given to the constructor. If None, the generator will +be randomly seeded using a seed from the operating system.

+
+ +
+
+property start
+

Set/get start point or configuration (superclass)

+
+
Getter:
+

Return start point or configuration

+
+
Return type:
+

ndarray(2) or ndarray(3)

+
+
Setter:
+

Set start point or configuration

+
+
Param:
+

array_like(2) or array_like(3)

+
+
+

The start is either a point \((x, y)\) or a configuration \((x, y, \theta)\).

+
+
Seealso:
+

goal()

+
+
+
+ +
+
+validate_endpoint(p, dtype=None)
+

Validate start or goal point

+
+
Parameters:
+
    +
  • p (array_like(2)) – the point

  • +
  • dtype (str, optional) – data type for point coordinates, defaults to None

  • +
+
+
Raises:
+

ValueError – point is inside obstacle

+
+
Returns:
+

point as a NumPy array of specified dtype

+
+
Return type:
+

ndarray(2)

+
+
+

The coordinate is tested to be a free cell in the occupancy grid.

+
+
Seealso:
+

isoccupied() occgrid()

+
+
+
+ +
+
+property verbose
+

Get verbosity

+
+
Returns:
+

verbosity

+
+
Return type:
+

bool

+
+
+

If verbosity print more diagnostic messages to the planner’s +message channel.

+
+ +
+ +
+ + +
+
+
+ +
+ +
+

© Copyright 2023, Jesse Haviland and Peter Corke. + Last updated on 11-May-2024. +

+
+ + + +
+
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/mobile-vehicle-animation.html b/mobile-vehicle-animation.html new file mode 100644 index 00000000..945cf361 --- /dev/null +++ b/mobile-vehicle-animation.html @@ -0,0 +1,570 @@ + + + + + + + + + + + Animations — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

Animations

+

These classes create a graphical object that can be animated to show vehicle +position or pose.

+
Inheritance diagram of roboticstoolbox.VehicleMarker, roboticstoolbox.VehiclePolygon, roboticstoolbox.VehicleIcon
+ + + + + +
+
+class roboticstoolbox.mobile.VehicleAnimationBase[source]
+

Abstract base class to support animation of a vehicle in a Matplotlib plot

+

There are three concrete subclasses:

+
    +
  • VehicleMarker animates a Matplotlib marker (shows position only)

  • +
  • VehiclePolygon animates a polygon shape (outline or filled), including predefined shapes (shows position and orientation)

  • +
  • VehicleIcon animates an image (shows position and orientation)

  • +
+

An instance a of these classes can be used in three different ways, firstly:

+
a = VehiclePolygon("car", color="red")
+a.add()
+
+
+

adds an instance of the animation shape to the plot and subsequent calls +to:

+
a.update(q)
+
+
+

will animate it with the configuration given by q.

+

Secondly, an instance can be passed to a Vehicle subclass object to make an animation +during simulation:

+
a = VehiclePolygon("car", color="red")
+veh = Bicycle(animation=a)
+
+
+

Thirdly:

+
a = VehiclePolygon("car", color="red")
+a.plot(q)
+
+
+

adds an instance of the animation shape to the plot with the specified +configuration. It cannot be moved, but the method does return a reference +to the Matplotlib object added to the plot.

+
+ +
+

Simple marker

+
+
+class roboticstoolbox.mobile.VehicleMarker(**kwargs)[source]
+

Bases: VehicleAnimationBase

+
+
+__init__(**kwargs)[source]
+

Create graphical animation of vehicle as a Matplotlib marker

+
+
Parameters:
+

kwargs – additional arguments passed to Matplotlib plot().

+
+
Returns:
+

animation object

+
+
Return type:
+

VehicleAnimation

+
+
+

Creates an object that can be passed to a Vehicle subclass to depict +the moving robot as a simple Matplotlib marker during simulation.

+

The default marker is a red filled circle with a white outline.

+

For example, to animate a simulation with a blue square marker:

+
a = VehicleMarker(marker="s", markerfacecolor="b")
+veh = Bicycle(driver=RandomPath(10), animation=a)
+veh.run()
+
+
+
+

Note

+

A marker can only indicate vehicle position, not orientation.

+
+
+
Seealso:
+

Vehicle()

+
+
+
+ +
+
+add(ax=None, **kwargs)
+

Add vehicle animation to the current plot

+
+
Parameters:
+
    +
  • ax (Axes, optional) – Axis to add to, defaults to current axis

  • +
  • kwargs – additional arguments passed to Matplotlib plot(), which +override arguments given to the constructor.

  • +
+
+
+

A reference to the animation object is kept, and it will be deleted +from the plot when the VehicleAnimation object is garbage collected.

+

The animation is not displayed until update() is called.

+
+
Seealso:
+

update()

+
+
+
+ +
+
+plot(q, **kwargs)
+

Add vehicle to the current plot (superclass)

+
+
Parameters:
+
    +
  • q (array_like(2) or array_like(3)) – vehicle position or configuration

  • +
  • kwargs – additional arguments passed to Matplotlib plot(), which +override arguments given to the constructor.

  • +
+
+
Returns:
+

reference to Matplotlib object

+
+
+

The animation object is rendered into the current axes.

+
+ +
+
+update(q)
+

Update the vehicle animation (superclass)

+
+
Parameters:
+

q (array_like(2) or array_like(3)) – vehicle position or configuration

+
+
+

The graphical depiction of the vehicle position or configuration is updated.

+
+
Seealso:
+

add()

+
+
+
+ +
+ +
+
+

Polygon shape

+
+
+class roboticstoolbox.mobile.VehiclePolygon(shape='car', scale=1, **kwargs)[source]
+

Bases: VehicleAnimationBase

+
+
+__init__(shape='car', scale=1, **kwargs)[source]
+

Create graphical animation of vehicle as a polygon

+
+
Parameters:
+
    +
  • shape (ndarray(2,n) or str) – polygon shape as vertices or a predefined shape, defaults to “car”

  • +
  • scale (float) – Length of the vehicle on the plot, defaults to 1

  • +
  • kwargs – additional arguments passed to Matplotlib Polygon such as +color (face+edge), alpha, facecolor, edgecolor, +linewidth etc.

  • +
+
+
Raises:
+
+
+
Returns:
+

animation object

+
+
Return type:
+

VehiclePolygon

+
+
+

Creates an object that can be passed to a Vehicle subclass to +depict the moving robot as a polygon during simulation.

+

For example, to animate a simulation with a red filled car-shaped polygon:

+
a = VehiclePolygon("car", color="r")
+veh = Bicycle(driver=RandomPath(10), animation=a)
+veh.run()
+
+
+

shape can be:

+
+
    +
  • "car" a rectangle with chamfered front corners

  • +
  • "triangle" an isocles triangle pointing in the forward direction

  • +
  • an 2xN NumPy array of vertices, does not have to be closed.

  • +
+
+

The polygon is scaled to an image with a length of scale in the +vehicle x-direction, in the units of the plot.

+
+
Seealso:
+

Vehicle() matplotlib.patches.Polygon

+
+
+
+ +
+
+add(ax=None, **kwargs)
+

Add vehicle animation to the current plot

+
+
Parameters:
+
    +
  • ax (Axes, optional) – Axis to add to, defaults to current axis

  • +
  • kwargs – additional arguments passed to Matplotlib plot(), which +override arguments given to the constructor.

  • +
+
+
+

A reference to the animation object is kept, and it will be deleted +from the plot when the VehicleAnimation object is garbage collected.

+

The animation is not displayed until update() is called.

+
+
Seealso:
+

update()

+
+
+
+ +
+
+plot(q, **kwargs)
+

Add vehicle to the current plot (superclass)

+
+
Parameters:
+
    +
  • q (array_like(2) or array_like(3)) – vehicle position or configuration

  • +
  • kwargs – additional arguments passed to Matplotlib plot(), which +override arguments given to the constructor.

  • +
+
+
Returns:
+

reference to Matplotlib object

+
+
+

The animation object is rendered into the current axes.

+
+ +
+
+update(q)
+

Update the vehicle animation (superclass)

+
+
Parameters:
+

q (array_like(2) or array_like(3)) – vehicle position or configuration

+
+
+

The graphical depiction of the vehicle position or configuration is updated.

+
+
Seealso:
+

add()

+
+
+
+ +
+ +
+
+

Image icon

+
+
+class roboticstoolbox.mobile.VehicleIcon(filename, origin=None, scale=1, rotation=0)[source]
+

Bases: VehicleAnimationBase

+
+
+__init__(filename, origin=None, scale=1, rotation=0)[source]
+

Create graphical animation of vehicle as an image icon

+
+
Parameters:
+
    +
  • filename (str) – Standard icon name or a path to an image

  • +
  • origin (array_like(2)) – Origin of the vehicle coordinate frame, defaults to centre

  • +
  • scale (float) – Length of the vehicle on the plot, defaults to 1

  • +
  • rotation (float) – Vehicle icon heading in degrees, defaults to 0

  • +
+
+
Raises:
+

ValueError – Icon file not found

+
+
Returns:
+

animation object

+
+
Return type:
+

VehicleAnimation

+
+
+

Creates an object that can be passed to a Vehicle subclass to +depict the moving robot as an image icon during simulation. The image +is translated and rotated to represent the vehicle configuration.

+

The car is scaled to an image with a horizontal length (width) of +scale in the units of the plot. By default the image is assumed to +contain a car parallel to the x-axis and facing right. If the vehicle +is facing upward set rotation to 90.

+

The vehicle rotates about its origin which is expressed in terms of +normalized coordinates in the range 0 to 1. By default it is in the +middle of the icon image, (0.2, 0.5) moves it toward the back of the +vehicle, (0.8, 0.5) moves it toward the front of the vehicle.

+

filename can be an included image:

+
+
    +
  • "greycar" a grey and white car (top view)

  • +
  • "redcar" a red car (top view)

  • +
  • "piano" a piano (top view)

  • +
+
+

or the path to an image file, including extension.

+

The included images are:

+"greycar" +"redcar" +"piano" +

For example, to animate a simulation with the red car icon:

+
a = VehicleIcon("redcar", scale=2)
+veh = Bicycle(driver=RandomPath(10), animation=a)
+veh.run(animation=a)
+
+
+
+

Note

+

The standard icons are provided in the package rtb-data

+
+
+
Seealso:
+

Vehicle

+
+
+
+ +
+
+add(ax=None, **kwargs)
+

Add vehicle animation to the current plot

+
+
Parameters:
+
    +
  • ax (Axes, optional) – Axis to add to, defaults to current axis

  • +
  • kwargs – additional arguments passed to Matplotlib plot(), which +override arguments given to the constructor.

  • +
+
+
+

A reference to the animation object is kept, and it will be deleted +from the plot when the VehicleAnimation object is garbage collected.

+

The animation is not displayed until update() is called.

+
+
Seealso:
+

update()

+
+
+
+ +
+
+plot(q, **kwargs)
+

Add vehicle to the current plot (superclass)

+
+
Parameters:
+
    +
  • q (array_like(2) or array_like(3)) – vehicle position or configuration

  • +
  • kwargs – additional arguments passed to Matplotlib plot(), which +override arguments given to the constructor.

  • +
+
+
Returns:
+

reference to Matplotlib object

+
+
+

The animation object is rendered into the current axes.

+
+ +
+
+update(q)
+

Update the vehicle animation (superclass)

+
+
Parameters:
+

q (array_like(2) or array_like(3)) – vehicle position or configuration

+
+
+

The graphical depiction of the vehicle position or configuration is updated.

+
+
Seealso:
+

add()

+
+
+
+ +
+ +
+
+ + +
+
+
+ +
+ +
+

© Copyright 2023, Jesse Haviland and Peter Corke. + Last updated on 11-May-2024. +

+
+ + + +
+
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/mobile-vehicle-base.html b/mobile-vehicle-base.html new file mode 100644 index 00000000..e0aed693 --- /dev/null +++ b/mobile-vehicle-base.html @@ -0,0 +1,962 @@ + + + + + + + + + + + Vehicle abstract baseclass — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

Vehicle abstract baseclass

+
+
+class roboticstoolbox.mobile.VehicleBase(covar=None, speed_max=inf, accel_max=inf, x0=[0, 0, 0], dt=0.1, control=None, seed=0, animation=None, verbose=False, plot=False, workspace=None, polygon=None)[source]
+
+
+__init__(covar=None, speed_max=inf, accel_max=inf, x0=[0, 0, 0], dt=0.1, control=None, seed=0, animation=None, verbose=False, plot=False, workspace=None, polygon=None)[source]
+

Superclass for vehicle kinematic models

+
+
Parameters:
+
    +
  • covar (ndarray(2,2), optional) – odometry covariance, defaults to zero

  • +
  • speed_max (float, optional) – maximum speed, defaults to \(\infty\)

  • +
  • accel_max (float, optional) – maximum acceleration, defaults to \(\infty\)

  • +
  • x0 (array_like(3), optional) – Initial state, defaults to (0,0,0)

  • +
  • dt (float, optional) – sample update interval, defaults to 0.1

  • +
  • control (array_like(2), interp1d, VehicleDriverBase) – vehicle control inputs, defaults to None

  • +
  • animation (VehicleAnimationBase subclass, optional) – Graphical animation of vehicle, defaults to None

  • +
  • verbose (bool, optional) – print lots of info, defaults to False

  • +
  • workspace (float, array_like(2), array_like(4)) – dimensions of 2D plot area, defaults to (-10:10) x (-10:10), +see plotvol2()

  • +
  • polygon (Polygon2) – bounding polygon of vehicle

  • +
+
+
+

This is an abstract superclass that simulates the motion of a mobile +robot under the action of a controller. The controller provides +control inputs to the vehicle, and the output odometry is returned. +The true state, effectively unknowable in practice, is computed +and accessible.

+

The workspace can be specified in several ways:

+ + + + + + + + + + + + + + + + + + + + + +

workspace

x-range

y-range

A (scalar)

-A:A

-A:A

[A, B]

A:B

A:B

[A, B, C, D]

A:B

C:D

+
+
Note:
+

Set seed=None to have it randomly initialized from the +operating system.

+
+
Seealso:
+

Bicycle Unicycle VehicleAnimationBase

+
+
+
+ +
+
+__str__()[source]
+

String representation of vehicle (superclass)

+
+
Returns:
+

String representation of vehicle object

+
+
Return type:
+

str

+
+
+
+ +
+
+f(x, odo, v=None)[source]
+

State transition function

+
+
Parameters:
+
    +
  • x (array_like(3), ndarray(n,3)) – vehicle state \((x, y, \theta)\)

  • +
  • odo (array_like(2)) – vehicle odometry \((\delta_d, \delta_\theta)\)

  • +
  • v (array_like(2), optional) – additive odometry noise, defaults to (0,0)

  • +
+
+
Returns:
+

predicted vehicle state

+
+
Return type:
+

ndarray(3)

+
+
+

Predict the next state based on current state and odometry +value. v is a random variable that represents additive +odometry noise for simulation purposes.

+
+\[f: \vec{x}_k, \delta_d, \delta_\theta \mapsto \vec{x}_{k+1}\]
+

For particle filters it is useful to apply the odometry to the +states of all particles and this can be achieved if x is a 2D +array with one state per row. v is ignored in this case.

+
+

Note

+

This is the state update equation used for EKF localization.

+
+
+
Seealso:
+

deriv() Fx() Fv()

+
+
+
+ +
+
+Fx(x, odo)[source]
+

Jacobian of state transition function df/dx

+
+
Parameters:
+
    +
  • x (array_like(3)) – vehicle state \((x, y, \theta)\)

  • +
  • odo (array_like(2)) – vehicle odometry \((\delta_d, \delta_\theta)\)

  • +
+
+
Returns:
+

Jacobian matrix

+
+
Return type:
+

ndarray(3,3)

+
+
+

Returns the Jacobian matrix \(\frac{\partial \vec{f}}{\partial \vec{x}}\) for +the given state and odometry.

+
+
Seealso:
+

f() deriv() Fv()

+
+
+
+ +
+
+Fv(x, odo)[source]
+

Jacobian of state transition function df/dv

+
+
Parameters:
+
    +
  • x (array_like(3)) – vehicle state \((x, y, \theta)\)

  • +
  • odo (array_like(2)) – vehicle odometry \((\delta_d, \delta_\theta)\)

  • +
+
+
Returns:
+

Jacobian matrix

+
+
Return type:
+

ndarray(3,2)

+
+
+

Returns the Jacobian matrix \(\frac{\partial \vec{f}}{\partial \vec{v}}\) for +the given state and odometry.

+
+
Seealso:
+

f() deriv() Fx()

+
+
+
+ +
+
+property control
+

Get/set vehicle control (superclass)

+
+
Getter:
+

Returns vehicle’s control

+
+
Setter:
+

Sets the vehicle’s control

+
+
Type:
+

2-tuple, callable, interp1d or VehicleDriver

+
+
+

This is the input to the vehicle during a simulation performed +by run(). The control input can be:

+
+
    +
  • a constant tuple as the control inputs to the vehicle

  • +
  • a function called as f(vehicle, t, x) that returns a tuple

  • +
  • an interpolator called as f(t) that returns a tuple

  • +
  • a driver agent, subclass of driversVehicleDriverBase

  • +
+
+

Example:

+
>>> from roboticstoolbox import Bicycle, RandomPath
+>>> bike = Bicycle()
+>>> bike.control = RandomPath(10)
+>>> print(bike)
+Bicycle: x = [ 0, 0, 0 ]
+  L=1, steer_max=1.41372, speed_max=inf, accel_max=inf
+
+
+
+
Seealso:
+

run() eval_control() scipy.interpolate.interp1d VehicleDriverBase

+
+
+
+ +
+
+eval_control(control, x)[source]
+

Evaluate vehicle control input (superclass method)

+
+
Parameters:
+
    +
  • control ([type]) – vehicle control

  • +
  • x (array_like(3)) – vehicle state \((x, y, heta)\)

  • +
+
+
Raises:
+

ValueError – bad control

+
+
Returns:
+

vehicle control inputs

+
+
Return type:
+

ndarray(2)

+
+
+

Evaluates the control for this time step and state. Control is set +by the control() property to:

+
+
    +
  • a constant tuple as the control inputs to the vehicle

  • +
  • a function called as f(vehicle, t, x) that returns a tuple

  • +
  • an interpolator called as f(t) that returns a tuple

  • +
  • a driver agent, subclass of VehicleDriverBase

  • +
+
+

Vehicle steering, speed and acceleration limits are applied to the +result before being input to the vehicle model.

+
+
Seealso:
+

run() control() scipy.interpolate.interp1d()

+
+
+
+ +
+
+limits_va(v, v_prev)[source]
+

Apply velocity and acceleration limits (superclass)

+
+
Parameters:
+
    +
  • v (float) – desired velocity

  • +
  • v_prev (list with single element) – previous velocity, reference to list

  • +
+
+
Returns:
+

allowed velocity

+
+
Return type:
+

float

+
+
+

Determine allowable velocity given desired velocity, speed and +acceleration limits.

+
+

Note

+

This function requires previous velocity, v_prev to enable +acceleration limiting. This is passed as a reference to a mutable value, +a single-element list. This is reset to zero at the start of each simulation.

+
+
+ +
+
+polygon(q)[source]
+

Bounding polygon at vehicle configuration

+
+
Parameters:
+

q (array_like(3)) – vehicle configuration \((x, y, heta)\)

+
+
Returns:
+

bounding polygon of vehicle at configuration q

+
+
Return type:
+

Polygon2

+
+
+

The bounding polygon of the vehicle is returned for the configuration +q. Can be used for collision checking using the intersects() +method.

+
+
Seealso:
+

Polygon2

+
+
+
+ +
+
+abstract deriv(x, u)[source]
+
+ +
+
+add_driver(driver)[source]
+

Add a driver agent (superclass)

+
+
Parameters:
+

driver (VehicleDriverBase subclass) – a driver agent object

+
+
+
+

Warning

+

Deprecated. Use vehicle.control = driver instead.

+
+
+
Seealso:
+

VehicleDriverBase control()

+
+
+
+ +
+
+run(T=10, x0=None, control=None, animate=True)[source]
+

Simulate motion of vehicle (superclass)

+
+
Parameters:
+
    +
  • T (float, optional) – Simulation time in seconds, defaults to 10

  • +
  • x0 (array_like(3) or array_like(2)) – Initial state, defaults to value given to Vehicle constructor

  • +
  • control (array_like(2), callable, driving agent) – vehicle control inputs, defaults to None

  • +
  • animate (bool, optional) – Enable graphical animation of vehicle, defaults to False

  • +
+
+
Returns:
+

State trajectory, each row is \((x,y,\theta)\).

+
+
Return type:
+

ndarray(n,3)

+
+
+

Runs the vehicle simulation for T seconds and optionally plots +an animation. The method step() performs each time step.

+

The control inputs are provided by control which can be:

+
+
    +
  • a constant tuple as the control inputs to the vehicle

  • +
  • a function called as f(vehicle, t, x) that returns a tuple

  • +
  • an interpolator called as f(t) that returns a tuple, see +SciPy interp1d

  • +
  • a driver agent, subclass of VehicleDriverBase

  • +
+
+

This is evaluated by eval_control() which applies velocity, +acceleration and steering limits.

+

The simulation can be stopped prematurely by the control function +calling stopsim().

+
+

Note

+
    +
  • the simulation is fixed-time step with the step given by the dt +attribute set by the constructor.

  • +
  • integration uses rectangular integration.

  • +
+
+
+
Seealso:
+

init() step() control() run_animation()

+
+
+
+ +
+
+run_animation(T=10, x0=None, control=None, format=None, file=None)[source]
+

Simulate motion of vehicle (superclass)

+
+
Parameters:
+
    +
  • T (float, optional) – Simulation time in seconds, defaults to 10

  • +
  • x0 (array_like(3) or array_like(2)) – Initial state, defaults to value given to Vehicle constructor

  • +
  • control (array_like(2), callable, driving agent) – vehicle control inputs, defaults to None

  • +
  • format (str, optional) – Output format

  • +
  • file (str, optional) – File name

  • +
+
+
Returns:
+

Matplotlib animation object

+
+
Return type:
+

matplotlib.animation.FuncAnimation()

+
+
+

Runs the vehicle simulation for T seconds and returns an animation +in various formats:

+
``format``    ``file``   description
+============  =========  ============================
+``"html"``    str, None  return HTML5 video
+``"jshtml"``  str, None  return JS+HTML video
+``"gif"``     str        return animated GIF
+``"mp4"``     str        return MP4/H264 video
+``None``                 return a ``FuncAnimation`` object
+
+
+

The allowables types for file are given in the second column. A str +value is the file name. If None is an option then return the video as a string.

+

For the last case, a reference to the animation object must be held if used for +animation in a Jupyter cell:

+
anim = robot.run_animation(T=20)
+
+
+

The control inputs are provided by control which can be:

+
+
    +
  • a constant tuple as the control inputs to the vehicle

  • +
  • a function called as f(vehicle, t, x) that returns a tuple

  • +
  • an interpolator called as f(t) that returns a tuple, see +SciPy interp1d

  • +
  • a driver agent, subclass of VehicleDriverBase

  • +
+
+

This is evaluated by eval_control() which applies velocity, +acceleration and steering limits.

+

The simulation can be stopped prematurely by the control function +calling stopsim().

+
+

Note

+
    +
  • the simulation is fixed-time step with the step given by the dt +attribute set by the constructor.

  • +
  • integration uses rectangular integration.

  • +
+
+
+
Seealso:
+

init() step() control() run_animation()

+
+
+
+ +
+
+init(x0=None, control=None, animate=True)[source]
+

Initialize for simulation (superclass)

+
+
Parameters:
+

x0 (array_like(3) or array_like(2)) – Initial state, defaults to value given to Vehicle constructor

+
+
+

Performs the following initializations:

+
+
    +
  1. Clears the state history

  2. +
  3. Set the private random number generator to initial value

  4. +
  5. Sets state \(x = x_0\)

  6. +
  7. Sets previous velocity to zero

  8. +
  9. If a driver is attached, initialize it

  10. +
  11. If animation is enabled, initialize that

  12. +
+
+

If animation is enabled by constructor and no animation object is given, +use a default VehiclePolygon("car")

+
+
Seealso:
+

VehicleAnimationBase run()

+
+
+
+ +
+
+step(u=None, animate=True, pause=True)[source]
+

Step simulator by one time step (superclass)

+
+
Returns:
+

odometry \((\delta_d, \delta_\theta)\)

+
+
Return type:
+

ndarray(2)

+
+
+
    +
  1. Obtain the vehicle control inputs

  2. +
  3. Integrate the vehicle state forward one timestep

  4. +
  5. Updates the stored state and state history

  6. +
  7. Update animation if enabled.

  8. +
  9. Returns the odometry

  10. +
+
    +
  • veh.step((vel, steer)) for a Bicycle vehicle model

  • +
  • veh.step((vel, vel_diff)) for a Unicycle vehicle model

  • +
  • veh.step() as above but control is taken from the control +attribute which might be a function or driver agent.

  • +
+

Example:

+
>>> from roboticstoolbox import Bicycle
+>>> bike = Bicycle()  # default bicycle model
+>>> bike.step((1, 0.2))  # one step: v=1, γ=0.2
+array([0.1   , 0.0203])
+>>> bike.x
+array([0.1   , 0.    , 0.0203])
+>>> bike.step((1, 0.2))  # another step: v=1, γ=0.2
+array([0.1   , 0.0203])
+>>> bike.x
+array([0.2   , 0.002 , 0.0405])
+
+
+
+

Note

+

Vehicle control input limits are applied.

+
+
+
Seealso:
+

control(), update(), run()

+
+
+
+ +
+
+property workspace
+

Size of robot workspace (superclass)

+
+
Returns:
+

workspace bounds [xmin, xmax, ymin, ymax]

+
+
Return type:
+

ndarray(4)

+
+
+

Returns the bounds of the workspace as specified by constructor +option workspace

+
+ +
+
+property x
+

Get vehicle state/configuration (superclass)

+
+
Returns:
+

Vehicle state \((x, y, \theta)\)

+
+
Return type:
+

ndarray(3)

+
+
Seealso:
+

q()

+
+
+
+ +
+
+property q
+

Get vehicle state/configuration (superclass)

+
+
Returns:
+

Vehicle state \((x, y, \theta)\)

+
+
Return type:
+

ndarray(3)

+
+
Seealso:
+

x()

+
+
+
+ +
+
+property x0
+

Get vehicle initial state/configuration (superclass)

+
+
Returns:
+

Vehicle state \((x, y, \theta)\)

+
+
Return type:
+

ndarray(3)

+
+
+

Set by VehicleBase subclass constructor. The state is set to this value +at the beginning of each simulation run.

+
+
Seealso:
+

run()

+
+
+
+ +
+
+property random
+

Get private random number generator

+
+
Returns:
+

NumPy random number generator

+
+
Return type:
+

numpy.random.Generator

+
+
+

Has methods including:

+
+
+

The generator is initialized with the seed provided at constructor +time every time init() is called.

+
+
Seealso:
+

init() numpy.random.default_rng()

+
+
+
+ +
+
+property x_hist
+

Get vehicle state/configuration history (superclass)

+
+
Returns:
+

Vehicle state history

+
+
Return type:
+

ndarray(n,3)

+
+
+

The state \((x, y, heta)\) at each time step resulting from a +simulation run, one row per timestep.

+
+
Seealso:
+

run()

+
+
+
+ +
+
+property speed_max
+

Get maximum speed of vehicle (superclass)

+
+
Returns:
+

maximum speed

+
+
Return type:
+

float

+
+
+

Set by VehicleBase() subclass constructor.

+
+
Seealso:
+

accel_max() steer_max() u_limited()

+
+
+
+ +
+
+property accel_max
+

Get maximum acceleration of vehicle (superclass)

+
+
Returns:
+

maximum acceleration

+
+
Return type:
+

float

+
+
+

Set by VehicleBase() subclass constructor.

+
+
Seealso:
+

speed_max() steer_max() u_limited()

+
+
+
+ +
+
+property dt
+

Get sample time (superclass)

+
+
Returns:
+

discrete time step for simulation

+
+
Return type:
+

float

+
+
+

Set by VehicleBase() subclass constructor. Used by +step() to update state.

+
+
Seealso:
+

run()

+
+
+
+ +
+
+property verbose
+

Get verbosity (superclass)

+
+
Returns:
+

verbosity level

+
+
Return type:
+

bool

+
+
+

Set by VehicleBase subclass constructor.

+
+ +
+
+stopsim()[source]
+

Stop the simulation (superclass)

+

A control function can stop the simulation initated by the run +method.

+
+
Seealso:
+

run()

+
+
+
+ +
+
+plot_xy(*args, block=None, **kwargs)[source]
+

Plot xy-path from history

+
+
Parameters:
+
    +
  • block (bool, optional) – block until plot dismissed, defaults to None

  • +
  • args – positional arguments passed to plot()

  • +
  • kwargs – keyword arguments passed to plot()

  • +
+
+
+

The \((x,y)\) trajectory from the simulation history is plotted as +\(x\) vs :math:`y.

+
+
Seealso:
+

run() plot_xyt()

+
+
+
+ +
+
+plot_xyt(block=None, **kwargs)[source]
+

Plot configuration vs time from history

+
+
Parameters:
+
    +
  • block (bool, optional) – block until plot dismissed, defaults to False

  • +
  • args – positional arguments passed to plot()

  • +
  • kwargs – keyword arguments passed to plot()

  • +
+
+
+

The \((x,y, heta)\) trajectory from the simulation history is plotted +against time.

+
+
Seealso:
+

run() plot_xy()

+
+
+
+ +
+ +
+ + +
+
+
+ +
+ +
+

© Copyright 2023, Jesse Haviland and Peter Corke. + Last updated on 11-May-2024. +

+
+ + + +
+
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/mobile-vehicle-bicycle.html b/mobile-vehicle-bicycle.html new file mode 100644 index 00000000..d2c65c44 --- /dev/null +++ b/mobile-vehicle-bicycle.html @@ -0,0 +1,1041 @@ + + + + + + + + + + + Bicycle — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

Bicycle

+
+
+class roboticstoolbox.mobile.Bicycle(L=1, steer_max=1.413716694115407, **kwargs)[source]
+

Bases: VehicleBase

+
+
+__init__(L=1, steer_max=1.413716694115407, **kwargs)[source]
+

Create bicycle kinematic model

+
+
Parameters:
+
    +
  • L (float, optional) – wheel base, defaults to 1

  • +
  • steer_max (float, optional) – maximum steering angle, defaults to \(0.45\pi\)

  • +
  • kwargs – additional arguments passed to VehicleBase +constructor

  • +
+
+
+

Model the motion of a bicycle model with equations of motion given by:

+
+\[\begin{split}\dot{x} &= v \cos \theta \\ +\dot{y} &= v \sin \theta \\ +\dot{\theta} &= \frac{v}{L} \tan \gamma\end{split}\]
+

where \(v\) is the velocity in body frame x-direction, and +\(\gamma\) is the angle of the steered wheel.

+
+
Seealso:
+

f() deriv() Fx() meth:Fv VehicleBase

+
+
+
+ +
+
+property l
+

Vehicle wheelbase

+
+
Returns:
+

vehicle wheelbase

+
+
Return type:
+

float

+
+
+
+ +
+
+property radius_min
+

Vehicle turning radius

+
+
Returns:
+

radius of minimum possible turning circle

+
+
Return type:
+

float

+
+
+

Minimum turning radius based on maximum steered wheel angle and wheelbase.

+
+\[\frac{l}{\tan \gamma_{max}}\]
+
+
Seealso:
+

curvature_max()

+
+
+
+ +
+
+property curvature_max
+

Vehicle maximum path curvature

+
+
Returns:
+

maximum curvature

+
+
Return type:
+

float

+
+
+

Maximum path curvature based on maximum steered wheel angle and wheelbase.

+
+\[\frac{\tan \gamma_{max}}{l}\]
+
+
Seealso:
+

radius_min()

+
+
+
+ +
+
+property steer_max
+

Vehicle maximum steered wheel angle

+
+
Returns:
+

maximum angle

+
+
Return type:
+

float

+
+
+
+ +
+
+deriv(x, u, limits=True)[source]
+

Time derivative of state

+
+
Parameters:
+
    +
  • x (array_like(3)) – vehicle state \((x, y, \theta)\)

  • +
  • u (array_like(2)) – control input \((v, \gamma)\)

  • +
  • limits (bool) – limits are applied to input, default True

  • +
+
+
Returns:
+

state derivative \((\dot{x}, \dot{y}, \dot{\theta})\)

+
+
Return type:
+

ndarray(3)

+
+
+

Returns the time derivative of state (3x1) at the state x with velocity \(v\) +and steered wheel angle \(\gamma\)

+
+\[\begin{split}\dot{x} &= v \cos \theta \\ +\dot{y} &= v \sin \theta \\ +\dot{\theta} &= \frac{v}{L} \tan \gamma\end{split}\]
+

If limits is True then speed, acceleration and steer-angle limits are +applied to u.

+
+
Seealso:
+

f()

+
+
+
+ +
+
+u_limited(u)[source]
+

Apply vehicle velocity, acceleration and steering limits

+
+
Parameters:
+

u (array_like(2)) – Desired vehicle inputs \((v, \gamma)\)

+
+
Returns:
+

Allowable vehicle inputs \((v, \gamma)\)

+
+
Return type:
+

ndarray(2)

+
+
+

Velocity and acceleration limits are applied to \(v\) and +steered wheel angle limits are applied to \(\gamma\).

+
+ +
+
+Fv(x, odo)
+

Jacobian of state transition function df/dv

+
+
Parameters:
+
    +
  • x (array_like(3)) – vehicle state \((x, y, \theta)\)

  • +
  • odo (array_like(2)) – vehicle odometry \((\delta_d, \delta_\theta)\)

  • +
+
+
Returns:
+

Jacobian matrix

+
+
Return type:
+

ndarray(3,2)

+
+
+

Returns the Jacobian matrix \(\frac{\partial \vec{f}}{\partial \vec{v}}\) for +the given state and odometry.

+
+
Seealso:
+

f() deriv() Fx()

+
+
+
+ +
+
+Fx(x, odo)
+

Jacobian of state transition function df/dx

+
+
Parameters:
+
    +
  • x (array_like(3)) – vehicle state \((x, y, \theta)\)

  • +
  • odo (array_like(2)) – vehicle odometry \((\delta_d, \delta_\theta)\)

  • +
+
+
Returns:
+

Jacobian matrix

+
+
Return type:
+

ndarray(3,3)

+
+
+

Returns the Jacobian matrix \(\frac{\partial \vec{f}}{\partial \vec{x}}\) for +the given state and odometry.

+
+
Seealso:
+

f() deriv() Fv()

+
+
+
+ +
+
+property accel_max
+

Get maximum acceleration of vehicle (superclass)

+
+
Returns:
+

maximum acceleration

+
+
Return type:
+

float

+
+
+

Set by VehicleBase() subclass constructor.

+
+
Seealso:
+

speed_max() steer_max() u_limited()

+
+
+
+ +
+
+add_driver(driver)
+

Add a driver agent (superclass)

+
+
Parameters:
+

driver (VehicleDriverBase subclass) – a driver agent object

+
+
+
+

Warning

+

Deprecated. Use vehicle.control = driver instead.

+
+
+
Seealso:
+

VehicleDriverBase control()

+
+
+
+ +
+
+property control
+

Get/set vehicle control (superclass)

+
+
Getter:
+

Returns vehicle’s control

+
+
Setter:
+

Sets the vehicle’s control

+
+
Type:
+

2-tuple, callable, interp1d or VehicleDriver

+
+
+

This is the input to the vehicle during a simulation performed +by run(). The control input can be:

+
+
    +
  • a constant tuple as the control inputs to the vehicle

  • +
  • a function called as f(vehicle, t, x) that returns a tuple

  • +
  • an interpolator called as f(t) that returns a tuple

  • +
  • a driver agent, subclass of driversVehicleDriverBase

  • +
+
+

Example:

+
>>> from roboticstoolbox import Bicycle, RandomPath
+>>> bike = Bicycle()
+>>> bike.control = RandomPath(10)
+>>> print(bike)
+Bicycle: x = [ 0, 0, 0 ]
+  L=1, steer_max=1.41372, speed_max=inf, accel_max=inf
+
+
+
+
Seealso:
+

run() eval_control() scipy.interpolate.interp1d VehicleDriverBase

+
+
+
+ +
+
+property dt
+

Get sample time (superclass)

+
+
Returns:
+

discrete time step for simulation

+
+
Return type:
+

float

+
+
+

Set by VehicleBase() subclass constructor. Used by +step() to update state.

+
+
Seealso:
+

run()

+
+
+
+ +
+
+eval_control(control, x)
+

Evaluate vehicle control input (superclass method)

+
+
Parameters:
+
    +
  • control ([type]) – vehicle control

  • +
  • x (array_like(3)) – vehicle state \((x, y, heta)\)

  • +
+
+
Raises:
+

ValueError – bad control

+
+
Returns:
+

vehicle control inputs

+
+
Return type:
+

ndarray(2)

+
+
+

Evaluates the control for this time step and state. Control is set +by the control() property to:

+
+
    +
  • a constant tuple as the control inputs to the vehicle

  • +
  • a function called as f(vehicle, t, x) that returns a tuple

  • +
  • an interpolator called as f(t) that returns a tuple

  • +
  • a driver agent, subclass of VehicleDriverBase

  • +
+
+

Vehicle steering, speed and acceleration limits are applied to the +result before being input to the vehicle model.

+
+
Seealso:
+

run() control() scipy.interpolate.interp1d()

+
+
+
+ +
+
+f(x, odo, v=None)
+

State transition function

+
+
Parameters:
+
    +
  • x (array_like(3), ndarray(n,3)) – vehicle state \((x, y, \theta)\)

  • +
  • odo (array_like(2)) – vehicle odometry \((\delta_d, \delta_\theta)\)

  • +
  • v (array_like(2), optional) – additive odometry noise, defaults to (0,0)

  • +
+
+
Returns:
+

predicted vehicle state

+
+
Return type:
+

ndarray(3)

+
+
+

Predict the next state based on current state and odometry +value. v is a random variable that represents additive +odometry noise for simulation purposes.

+
+\[f: \vec{x}_k, \delta_d, \delta_\theta \mapsto \vec{x}_{k+1}\]
+

For particle filters it is useful to apply the odometry to the +states of all particles and this can be achieved if x is a 2D +array with one state per row. v is ignored in this case.

+
+

Note

+

This is the state update equation used for EKF localization.

+
+
+
Seealso:
+

deriv() Fx() Fv()

+
+
+
+ +
+
+init(x0=None, control=None, animate=True)
+

Initialize for simulation (superclass)

+
+
Parameters:
+

x0 (array_like(3) or array_like(2)) – Initial state, defaults to value given to Vehicle constructor

+
+
+

Performs the following initializations:

+
+
    +
  1. Clears the state history

  2. +
  3. Set the private random number generator to initial value

  4. +
  5. Sets state \(x = x_0\)

  6. +
  7. Sets previous velocity to zero

  8. +
  9. If a driver is attached, initialize it

  10. +
  11. If animation is enabled, initialize that

  12. +
+
+

If animation is enabled by constructor and no animation object is given, +use a default VehiclePolygon("car")

+
+
Seealso:
+

VehicleAnimationBase run()

+
+
+
+ +
+
+limits_va(v, v_prev)
+

Apply velocity and acceleration limits (superclass)

+
+
Parameters:
+
    +
  • v (float) – desired velocity

  • +
  • v_prev (list with single element) – previous velocity, reference to list

  • +
+
+
Returns:
+

allowed velocity

+
+
Return type:
+

float

+
+
+

Determine allowable velocity given desired velocity, speed and +acceleration limits.

+
+

Note

+

This function requires previous velocity, v_prev to enable +acceleration limiting. This is passed as a reference to a mutable value, +a single-element list. This is reset to zero at the start of each simulation.

+
+
+ +
+
+plot_xy(*args, block=None, **kwargs)
+

Plot xy-path from history

+
+
Parameters:
+
    +
  • block (bool, optional) – block until plot dismissed, defaults to None

  • +
  • args – positional arguments passed to plot()

  • +
  • kwargs – keyword arguments passed to plot()

  • +
+
+
+

The \((x,y)\) trajectory from the simulation history is plotted as +\(x\) vs :math:`y.

+
+
Seealso:
+

run() plot_xyt()

+
+
+
+ +
+
+plot_xyt(block=None, **kwargs)
+

Plot configuration vs time from history

+
+
Parameters:
+
    +
  • block (bool, optional) – block until plot dismissed, defaults to False

  • +
  • args – positional arguments passed to plot()

  • +
  • kwargs – keyword arguments passed to plot()

  • +
+
+
+

The \((x,y, heta)\) trajectory from the simulation history is plotted +against time.

+
+
Seealso:
+

run() plot_xy()

+
+
+
+ +
+
+polygon(q)
+

Bounding polygon at vehicle configuration

+
+
Parameters:
+

q (array_like(3)) – vehicle configuration \((x, y, heta)\)

+
+
Returns:
+

bounding polygon of vehicle at configuration q

+
+
Return type:
+

Polygon2

+
+
+

The bounding polygon of the vehicle is returned for the configuration +q. Can be used for collision checking using the intersects() +method.

+
+
Seealso:
+

Polygon2

+
+
+
+ +
+
+property q
+

Get vehicle state/configuration (superclass)

+
+
Returns:
+

Vehicle state \((x, y, \theta)\)

+
+
Return type:
+

ndarray(3)

+
+
Seealso:
+

x()

+
+
+
+ +
+
+property random
+

Get private random number generator

+
+
Returns:
+

NumPy random number generator

+
+
Return type:
+

numpy.random.Generator

+
+
+

Has methods including:

+
+
+

The generator is initialized with the seed provided at constructor +time every time init() is called.

+
+
Seealso:
+

init() numpy.random.default_rng()

+
+
+
+ +
+
+run(T=10, x0=None, control=None, animate=True)
+

Simulate motion of vehicle (superclass)

+
+
Parameters:
+
    +
  • T (float, optional) – Simulation time in seconds, defaults to 10

  • +
  • x0 (array_like(3) or array_like(2)) – Initial state, defaults to value given to Vehicle constructor

  • +
  • control (array_like(2), callable, driving agent) – vehicle control inputs, defaults to None

  • +
  • animate (bool, optional) – Enable graphical animation of vehicle, defaults to False

  • +
+
+
Returns:
+

State trajectory, each row is \((x,y,\theta)\).

+
+
Return type:
+

ndarray(n,3)

+
+
+

Runs the vehicle simulation for T seconds and optionally plots +an animation. The method step() performs each time step.

+

The control inputs are provided by control which can be:

+
+
    +
  • a constant tuple as the control inputs to the vehicle

  • +
  • a function called as f(vehicle, t, x) that returns a tuple

  • +
  • an interpolator called as f(t) that returns a tuple, see +SciPy interp1d

  • +
  • a driver agent, subclass of VehicleDriverBase

  • +
+
+

This is evaluated by eval_control() which applies velocity, +acceleration and steering limits.

+

The simulation can be stopped prematurely by the control function +calling stopsim().

+
+

Note

+
    +
  • the simulation is fixed-time step with the step given by the dt +attribute set by the constructor.

  • +
  • integration uses rectangular integration.

  • +
+
+
+
Seealso:
+

init() step() control() run_animation()

+
+
+
+ +
+
+run_animation(T=10, x0=None, control=None, format=None, file=None)
+

Simulate motion of vehicle (superclass)

+
+
Parameters:
+
    +
  • T (float, optional) – Simulation time in seconds, defaults to 10

  • +
  • x0 (array_like(3) or array_like(2)) – Initial state, defaults to value given to Vehicle constructor

  • +
  • control (array_like(2), callable, driving agent) – vehicle control inputs, defaults to None

  • +
  • format (str, optional) – Output format

  • +
  • file (str, optional) – File name

  • +
+
+
Returns:
+

Matplotlib animation object

+
+
Return type:
+

matplotlib.animation.FuncAnimation()

+
+
+

Runs the vehicle simulation for T seconds and returns an animation +in various formats:

+
``format``    ``file``   description
+============  =========  ============================
+``"html"``    str, None  return HTML5 video
+``"jshtml"``  str, None  return JS+HTML video
+``"gif"``     str        return animated GIF
+``"mp4"``     str        return MP4/H264 video
+``None``                 return a ``FuncAnimation`` object
+
+
+

The allowables types for file are given in the second column. A str +value is the file name. If None is an option then return the video as a string.

+

For the last case, a reference to the animation object must be held if used for +animation in a Jupyter cell:

+
anim = robot.run_animation(T=20)
+
+
+

The control inputs are provided by control which can be:

+
+
    +
  • a constant tuple as the control inputs to the vehicle

  • +
  • a function called as f(vehicle, t, x) that returns a tuple

  • +
  • an interpolator called as f(t) that returns a tuple, see +SciPy interp1d

  • +
  • a driver agent, subclass of VehicleDriverBase

  • +
+
+

This is evaluated by eval_control() which applies velocity, +acceleration and steering limits.

+

The simulation can be stopped prematurely by the control function +calling stopsim().

+
+

Note

+
    +
  • the simulation is fixed-time step with the step given by the dt +attribute set by the constructor.

  • +
  • integration uses rectangular integration.

  • +
+
+
+
Seealso:
+

init() step() control() run_animation()

+
+
+
+ +
+
+property speed_max
+

Get maximum speed of vehicle (superclass)

+
+
Returns:
+

maximum speed

+
+
Return type:
+

float

+
+
+

Set by VehicleBase() subclass constructor.

+
+
Seealso:
+

accel_max() steer_max() u_limited()

+
+
+
+ +
+
+step(u=None, animate=True, pause=True)
+

Step simulator by one time step (superclass)

+
+
Returns:
+

odometry \((\delta_d, \delta_\theta)\)

+
+
Return type:
+

ndarray(2)

+
+
+
    +
  1. Obtain the vehicle control inputs

  2. +
  3. Integrate the vehicle state forward one timestep

  4. +
  5. Updates the stored state and state history

  6. +
  7. Update animation if enabled.

  8. +
  9. Returns the odometry

  10. +
+
    +
  • veh.step((vel, steer)) for a Bicycle vehicle model

  • +
  • veh.step((vel, vel_diff)) for a Unicycle vehicle model

  • +
  • veh.step() as above but control is taken from the control +attribute which might be a function or driver agent.

  • +
+

Example:

+
>>> from roboticstoolbox import Bicycle
+>>> bike = Bicycle()  # default bicycle model
+>>> bike.step((1, 0.2))  # one step: v=1, γ=0.2
+array([0.1   , 0.0203])
+>>> bike.x
+array([0.1   , 0.    , 0.0203])
+>>> bike.step((1, 0.2))  # another step: v=1, γ=0.2
+array([0.1   , 0.0203])
+>>> bike.x
+array([0.2   , 0.002 , 0.0405])
+
+
+
+

Note

+

Vehicle control input limits are applied.

+
+
+
Seealso:
+

control(), update(), run()

+
+
+
+ +
+
+stopsim()
+

Stop the simulation (superclass)

+

A control function can stop the simulation initated by the run +method.

+
+
Seealso:
+

run()

+
+
+
+ +
+
+property verbose
+

Get verbosity (superclass)

+
+
Returns:
+

verbosity level

+
+
Return type:
+

bool

+
+
+

Set by VehicleBase subclass constructor.

+
+ +
+
+property workspace
+

Size of robot workspace (superclass)

+
+
Returns:
+

workspace bounds [xmin, xmax, ymin, ymax]

+
+
Return type:
+

ndarray(4)

+
+
+

Returns the bounds of the workspace as specified by constructor +option workspace

+
+ +
+
+property x
+

Get vehicle state/configuration (superclass)

+
+
Returns:
+

Vehicle state \((x, y, \theta)\)

+
+
Return type:
+

ndarray(3)

+
+
Seealso:
+

q()

+
+
+
+ +
+
+property x0
+

Get vehicle initial state/configuration (superclass)

+
+
Returns:
+

Vehicle state \((x, y, \theta)\)

+
+
Return type:
+

ndarray(3)

+
+
+

Set by VehicleBase subclass constructor. The state is set to this value +at the beginning of each simulation run.

+
+
Seealso:
+

run()

+
+
+
+ +
+
+property x_hist
+

Get vehicle state/configuration history (superclass)

+
+
Returns:
+

Vehicle state history

+
+
Return type:
+

ndarray(n,3)

+
+
+

The state \((x, y, heta)\) at each time step resulting from a +simulation run, one row per timestep.

+
+
Seealso:
+

run()

+
+
+
+ +
+ +
+ + +
+
+
+ +
+ +
+

© Copyright 2023, Jesse Haviland and Peter Corke. + Last updated on 11-May-2024. +

+
+ + + +
+
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/mobile-vehicle-diffsteer.html b/mobile-vehicle-diffsteer.html new file mode 100644 index 00000000..5db28ab8 --- /dev/null +++ b/mobile-vehicle-diffsteer.html @@ -0,0 +1,966 @@ + + + + + + + + + + + Differential steering — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

Differential steering

+
+
+class roboticstoolbox.mobile.DiffSteer(W=1, **kwargs)[source]
+

Bases: Unicycle

+
+
+__init__(W=1, **kwargs)[source]
+

Create differential steering kinematic model

+
+
Parameters:
+
    +
  • W (float, optional) – vehicle width, defaults to 1

  • +
  • kwargs – additional arguments passed to VehicleBase +constructor

  • +
+
+
+

Model the motion of a unicycle model with equations of motion given by:

+
+\[\begin{split}\dot{x} &= v \cos \theta \\ +\dot{y} &= v \sin \theta \\ +\dot{\theta} &= \omega\end{split}\]
+

where \(v = (v_R + v_L)/2\) is the velocity in body frame x-direction, and +\(\omega = (v_R - v_L)/W\) is the turn rate.

+
+
Seealso:
+

f() deriv() Fx() meth:Fv Vehicle

+
+
+
+ +
+
+init()[source]
+

Initialize for simulation (superclass)

+
+
Parameters:
+

x0 (array_like(3) or array_like(2)) – Initial state, defaults to value given to Vehicle constructor

+
+
+

Performs the following initializations:

+
+
    +
  1. Clears the state history

  2. +
  3. Set the private random number generator to initial value

  4. +
  5. Sets state \(x = x_0\)

  6. +
  7. Sets previous velocity to zero

  8. +
  9. If a driver is attached, initialize it

  10. +
  11. If animation is enabled, initialize that

  12. +
+
+

If animation is enabled by constructor and no animation object is given, +use a default VehiclePolygon("car")

+
+
Seealso:
+

VehicleAnimationBase run()

+
+
+
+ +
+
+u_limited(u)[source]
+

Apply vehicle velocity and acceleration limits

+
+
Parameters:
+

u (array_like(2)) – Desired vehicle inputs \((v_L, v_R)\)

+
+
Returns:
+

Allowable vehicle inputs \((v_L, v_R)\)

+
+
Return type:
+

ndarray(2)

+
+
+

Velocity and acceleration limits are applied to \(v\) and +turn rate limits are applied to \(\omega\).

+
+ +
+
+Fv(x, odo)
+

Jacobian of state transition function df/dv

+
+
Parameters:
+
    +
  • x (array_like(3)) – vehicle state \((x, y, \theta)\)

  • +
  • odo (array_like(2)) – vehicle odometry \((\delta_d, \delta_\theta)\)

  • +
+
+
Returns:
+

Jacobian matrix

+
+
Return type:
+

ndarray(3,2)

+
+
+

Returns the Jacobian matrix \(\frac{\partial \vec{f}}{\partial \vec{v}}\) for +the given state and odometry.

+
+
Seealso:
+

f() deriv() Fx()

+
+
+
+ +
+
+Fx(x, odo)
+

Jacobian of state transition function df/dx

+
+
Parameters:
+
    +
  • x (array_like(3)) – vehicle state \((x, y, \theta)\)

  • +
  • odo (array_like(2)) – vehicle odometry \((\delta_d, \delta_\theta)\)

  • +
+
+
Returns:
+

Jacobian matrix

+
+
Return type:
+

ndarray(3,3)

+
+
+

Returns the Jacobian matrix \(\frac{\partial \vec{f}}{\partial \vec{x}}\) for +the given state and odometry.

+
+
Seealso:
+

f() deriv() Fv()

+
+
+
+ +
+
+property accel_max
+

Get maximum acceleration of vehicle (superclass)

+
+
Returns:
+

maximum acceleration

+
+
Return type:
+

float

+
+
+

Set by VehicleBase() subclass constructor.

+
+
Seealso:
+

speed_max() steer_max() u_limited()

+
+
+
+ +
+
+add_driver(driver)
+

Add a driver agent (superclass)

+
+
Parameters:
+

driver (VehicleDriverBase subclass) – a driver agent object

+
+
+
+

Warning

+

Deprecated. Use vehicle.control = driver instead.

+
+
+
Seealso:
+

VehicleDriverBase control()

+
+
+
+ +
+
+property control
+

Get/set vehicle control (superclass)

+
+
Getter:
+

Returns vehicle’s control

+
+
Setter:
+

Sets the vehicle’s control

+
+
Type:
+

2-tuple, callable, interp1d or VehicleDriver

+
+
+

This is the input to the vehicle during a simulation performed +by run(). The control input can be:

+
+
    +
  • a constant tuple as the control inputs to the vehicle

  • +
  • a function called as f(vehicle, t, x) that returns a tuple

  • +
  • an interpolator called as f(t) that returns a tuple

  • +
  • a driver agent, subclass of driversVehicleDriverBase

  • +
+
+

Example:

+
>>> from roboticstoolbox import Bicycle, RandomPath
+>>> bike = Bicycle()
+>>> bike.control = RandomPath(10)
+>>> print(bike)
+Bicycle: x = [ 0, 0, 0 ]
+  L=1, steer_max=1.41372, speed_max=inf, accel_max=inf
+
+
+
+
Seealso:
+

run() eval_control() scipy.interpolate.interp1d VehicleDriverBase

+
+
+
+ +
+
+deriv(x, u, limits=True)[source]
+

Time derivative of state

+
+
Parameters:
+
    +
  • x (array_like(3)) – vehicle state \((x, y, \theta)\)

  • +
  • u (array_like(2)) – Desired vehicle inputs \((v_L, v_R)\)

  • +
  • limits (bool) – limits are applied to input, default True

  • +
+
+
Returns:
+

state derivative \((\dot{x}, \dot{y}, \dot{\theta})\)

+
+
Return type:
+

ndarray(3)

+
+
+

Returns the time derivative of state (3x1) at the state x with left and +right wheel speeds u.

+
+\[\begin{split}\dot{x} &= v \cos \theta \\ +\dot{y} &= v \sin \theta \\ +\dot{\theta} &= \omega\end{split}\]
+

where \(v = (v_R + v_L)/2\) is the velocity in body frame x-direction, and +\(\omega = (v_R - v_L)/W\) is the turn rate.

+

If limits is True then speed and acceleration limits are applied to the +wheel speeds u.

+
+
Seealso:
+

f()

+
+
+
+ +
+
+property dt
+

Get sample time (superclass)

+
+
Returns:
+

discrete time step for simulation

+
+
Return type:
+

float

+
+
+

Set by VehicleBase() subclass constructor. Used by +step() to update state.

+
+
Seealso:
+

run()

+
+
+
+ +
+
+eval_control(control, x)
+

Evaluate vehicle control input (superclass method)

+
+
Parameters:
+
    +
  • control ([type]) – vehicle control

  • +
  • x (array_like(3)) – vehicle state \((x, y, heta)\)

  • +
+
+
Raises:
+

ValueError – bad control

+
+
Returns:
+

vehicle control inputs

+
+
Return type:
+

ndarray(2)

+
+
+

Evaluates the control for this time step and state. Control is set +by the control() property to:

+
+
    +
  • a constant tuple as the control inputs to the vehicle

  • +
  • a function called as f(vehicle, t, x) that returns a tuple

  • +
  • an interpolator called as f(t) that returns a tuple

  • +
  • a driver agent, subclass of VehicleDriverBase

  • +
+
+

Vehicle steering, speed and acceleration limits are applied to the +result before being input to the vehicle model.

+
+
Seealso:
+

run() control() scipy.interpolate.interp1d()

+
+
+
+ +
+
+f(x, odo, v=None)
+

State transition function

+
+
Parameters:
+
    +
  • x (array_like(3), ndarray(n,3)) – vehicle state \((x, y, \theta)\)

  • +
  • odo (array_like(2)) – vehicle odometry \((\delta_d, \delta_\theta)\)

  • +
  • v (array_like(2), optional) – additive odometry noise, defaults to (0,0)

  • +
+
+
Returns:
+

predicted vehicle state

+
+
Return type:
+

ndarray(3)

+
+
+

Predict the next state based on current state and odometry +value. v is a random variable that represents additive +odometry noise for simulation purposes.

+
+\[f: \vec{x}_k, \delta_d, \delta_\theta \mapsto \vec{x}_{k+1}\]
+

For particle filters it is useful to apply the odometry to the +states of all particles and this can be achieved if x is a 2D +array with one state per row. v is ignored in this case.

+
+

Note

+

This is the state update equation used for EKF localization.

+
+
+
Seealso:
+

deriv() Fx() Fv()

+
+
+
+ +
+
+limits_va(v, v_prev)
+

Apply velocity and acceleration limits (superclass)

+
+
Parameters:
+
    +
  • v (float) – desired velocity

  • +
  • v_prev (list with single element) – previous velocity, reference to list

  • +
+
+
Returns:
+

allowed velocity

+
+
Return type:
+

float

+
+
+

Determine allowable velocity given desired velocity, speed and +acceleration limits.

+
+

Note

+

This function requires previous velocity, v_prev to enable +acceleration limiting. This is passed as a reference to a mutable value, +a single-element list. This is reset to zero at the start of each simulation.

+
+
+ +
+
+plot_xy(*args, block=None, **kwargs)
+

Plot xy-path from history

+
+
Parameters:
+
    +
  • block (bool, optional) – block until plot dismissed, defaults to None

  • +
  • args – positional arguments passed to plot()

  • +
  • kwargs – keyword arguments passed to plot()

  • +
+
+
+

The \((x,y)\) trajectory from the simulation history is plotted as +\(x\) vs :math:`y.

+
+
Seealso:
+

run() plot_xyt()

+
+
+
+ +
+
+plot_xyt(block=None, **kwargs)
+

Plot configuration vs time from history

+
+
Parameters:
+
    +
  • block (bool, optional) – block until plot dismissed, defaults to False

  • +
  • args – positional arguments passed to plot()

  • +
  • kwargs – keyword arguments passed to plot()

  • +
+
+
+

The \((x,y, heta)\) trajectory from the simulation history is plotted +against time.

+
+
Seealso:
+

run() plot_xy()

+
+
+
+ +
+
+polygon(q)
+

Bounding polygon at vehicle configuration

+
+
Parameters:
+

q (array_like(3)) – vehicle configuration \((x, y, heta)\)

+
+
Returns:
+

bounding polygon of vehicle at configuration q

+
+
Return type:
+

Polygon2

+
+
+

The bounding polygon of the vehicle is returned for the configuration +q. Can be used for collision checking using the intersects() +method.

+
+
Seealso:
+

Polygon2

+
+
+
+ +
+
+property q
+

Get vehicle state/configuration (superclass)

+
+
Returns:
+

Vehicle state \((x, y, \theta)\)

+
+
Return type:
+

ndarray(3)

+
+
Seealso:
+

x()

+
+
+
+ +
+
+property random
+

Get private random number generator

+
+
Returns:
+

NumPy random number generator

+
+
Return type:
+

numpy.random.Generator

+
+
+

Has methods including:

+
+
+

The generator is initialized with the seed provided at constructor +time every time init() is called.

+
+
Seealso:
+

init() numpy.random.default_rng()

+
+
+
+ +
+
+run(T=10, x0=None, control=None, animate=True)
+

Simulate motion of vehicle (superclass)

+
+
Parameters:
+
    +
  • T (float, optional) – Simulation time in seconds, defaults to 10

  • +
  • x0 (array_like(3) or array_like(2)) – Initial state, defaults to value given to Vehicle constructor

  • +
  • control (array_like(2), callable, driving agent) – vehicle control inputs, defaults to None

  • +
  • animate (bool, optional) – Enable graphical animation of vehicle, defaults to False

  • +
+
+
Returns:
+

State trajectory, each row is \((x,y,\theta)\).

+
+
Return type:
+

ndarray(n,3)

+
+
+

Runs the vehicle simulation for T seconds and optionally plots +an animation. The method step() performs each time step.

+

The control inputs are provided by control which can be:

+
+
    +
  • a constant tuple as the control inputs to the vehicle

  • +
  • a function called as f(vehicle, t, x) that returns a tuple

  • +
  • an interpolator called as f(t) that returns a tuple, see +SciPy interp1d

  • +
  • a driver agent, subclass of VehicleDriverBase

  • +
+
+

This is evaluated by eval_control() which applies velocity, +acceleration and steering limits.

+

The simulation can be stopped prematurely by the control function +calling stopsim().

+
+

Note

+
    +
  • the simulation is fixed-time step with the step given by the dt +attribute set by the constructor.

  • +
  • integration uses rectangular integration.

  • +
+
+
+
Seealso:
+

init() step() control() run_animation()

+
+
+
+ +
+
+run_animation(T=10, x0=None, control=None, format=None, file=None)
+

Simulate motion of vehicle (superclass)

+
+
Parameters:
+
    +
  • T (float, optional) – Simulation time in seconds, defaults to 10

  • +
  • x0 (array_like(3) or array_like(2)) – Initial state, defaults to value given to Vehicle constructor

  • +
  • control (array_like(2), callable, driving agent) – vehicle control inputs, defaults to None

  • +
  • format (str, optional) – Output format

  • +
  • file (str, optional) – File name

  • +
+
+
Returns:
+

Matplotlib animation object

+
+
Return type:
+

matplotlib.animation.FuncAnimation()

+
+
+

Runs the vehicle simulation for T seconds and returns an animation +in various formats:

+
``format``    ``file``   description
+============  =========  ============================
+``"html"``    str, None  return HTML5 video
+``"jshtml"``  str, None  return JS+HTML video
+``"gif"``     str        return animated GIF
+``"mp4"``     str        return MP4/H264 video
+``None``                 return a ``FuncAnimation`` object
+
+
+

The allowables types for file are given in the second column. A str +value is the file name. If None is an option then return the video as a string.

+

For the last case, a reference to the animation object must be held if used for +animation in a Jupyter cell:

+
anim = robot.run_animation(T=20)
+
+
+

The control inputs are provided by control which can be:

+
+
    +
  • a constant tuple as the control inputs to the vehicle

  • +
  • a function called as f(vehicle, t, x) that returns a tuple

  • +
  • an interpolator called as f(t) that returns a tuple, see +SciPy interp1d

  • +
  • a driver agent, subclass of VehicleDriverBase

  • +
+
+

This is evaluated by eval_control() which applies velocity, +acceleration and steering limits.

+

The simulation can be stopped prematurely by the control function +calling stopsim().

+
+

Note

+
    +
  • the simulation is fixed-time step with the step given by the dt +attribute set by the constructor.

  • +
  • integration uses rectangular integration.

  • +
+
+
+
Seealso:
+

init() step() control() run_animation()

+
+
+
+ +
+
+property speed_max
+

Get maximum speed of vehicle (superclass)

+
+
Returns:
+

maximum speed

+
+
Return type:
+

float

+
+
+

Set by VehicleBase() subclass constructor.

+
+
Seealso:
+

accel_max() steer_max() u_limited()

+
+
+
+ +
+
+step(u=None, animate=True, pause=True)
+

Step simulator by one time step (superclass)

+
+
Returns:
+

odometry \((\delta_d, \delta_\theta)\)

+
+
Return type:
+

ndarray(2)

+
+
+
    +
  1. Obtain the vehicle control inputs

  2. +
  3. Integrate the vehicle state forward one timestep

  4. +
  5. Updates the stored state and state history

  6. +
  7. Update animation if enabled.

  8. +
  9. Returns the odometry

  10. +
+
    +
  • veh.step((vel, steer)) for a Bicycle vehicle model

  • +
  • veh.step((vel, vel_diff)) for a Unicycle vehicle model

  • +
  • veh.step() as above but control is taken from the control +attribute which might be a function or driver agent.

  • +
+

Example:

+
>>> from roboticstoolbox import Bicycle
+>>> bike = Bicycle()  # default bicycle model
+>>> bike.step((1, 0.2))  # one step: v=1, γ=0.2
+array([0.1   , 0.0203])
+>>> bike.x
+array([0.1   , 0.    , 0.0203])
+>>> bike.step((1, 0.2))  # another step: v=1, γ=0.2
+array([0.1   , 0.0203])
+>>> bike.x
+array([0.2   , 0.002 , 0.0405])
+
+
+
+

Note

+

Vehicle control input limits are applied.

+
+
+
Seealso:
+

control(), update(), run()

+
+
+
+ +
+
+stopsim()
+

Stop the simulation (superclass)

+

A control function can stop the simulation initated by the run +method.

+
+
Seealso:
+

run()

+
+
+
+ +
+
+property verbose
+

Get verbosity (superclass)

+
+
Returns:
+

verbosity level

+
+
Return type:
+

bool

+
+
+

Set by VehicleBase subclass constructor.

+
+ +
+
+property workspace
+

Size of robot workspace (superclass)

+
+
Returns:
+

workspace bounds [xmin, xmax, ymin, ymax]

+
+
Return type:
+

ndarray(4)

+
+
+

Returns the bounds of the workspace as specified by constructor +option workspace

+
+ +
+
+property x
+

Get vehicle state/configuration (superclass)

+
+
Returns:
+

Vehicle state \((x, y, \theta)\)

+
+
Return type:
+

ndarray(3)

+
+
Seealso:
+

q()

+
+
+
+ +
+
+property x0
+

Get vehicle initial state/configuration (superclass)

+
+
Returns:
+

Vehicle state \((x, y, \theta)\)

+
+
Return type:
+

ndarray(3)

+
+
+

Set by VehicleBase subclass constructor. The state is set to this value +at the beginning of each simulation run.

+
+
Seealso:
+

run()

+
+
+
+ +
+
+property x_hist
+

Get vehicle state/configuration history (superclass)

+
+
Returns:
+

Vehicle state history

+
+
Return type:
+

ndarray(n,3)

+
+
+

The state \((x, y, heta)\) at each time step resulting from a +simulation run, one row per timestep.

+
+
Seealso:
+

run()

+
+
+
+ +
+ +
+ + +
+
+
+ +
+ +
+

© Copyright 2023, Jesse Haviland and Peter Corke. + Last updated on 11-May-2024. +

+
+ + + +
+
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/mobile-vehicle-unicycle.html b/mobile-vehicle-unicycle.html new file mode 100644 index 00000000..afe48b16 --- /dev/null +++ b/mobile-vehicle-unicycle.html @@ -0,0 +1,965 @@ + + + + + + + + + + + Unicycle — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

Unicycle

+
+
+class roboticstoolbox.mobile.Unicycle(W=1, steer_max=inf, **kwargs)[source]
+

Bases: VehicleBase

+
+
+__init__(W=1, steer_max=inf, **kwargs)[source]
+

Create unicycle kinematic model

+
+
Parameters:
+
    +
  • W (float, optional) – vehicle width, defaults to 1

  • +
  • steer_max (float) – maximum turn rate

  • +
  • kwargs – additional arguments passed to VehicleBase +constructor

  • +
+
+
+

Model the motion of a unicycle model with equations of motion given by:

+
+\[\begin{split}\dot{x} &= v \cos \theta \\ +\dot{y} &= v \sin \theta \\ +\dot{\theta} &= \omega\end{split}\]
+

where \(v\) is the velocity in body frame x-direction, and +\(\omega\) is the turn rate.

+
+
Seealso:
+

f() deriv() Fx() meth:Fv Vehicle

+
+
+
+ +
+
+deriv(x, u, limits=True)[source]
+

Time derivative of state

+
+
Parameters:
+
    +
  • x (array_like(3)) – vehicle state \((x, y, \theta)\)

  • +
  • u (array_like(2)) – control input \((v, \omega)\)

  • +
  • limits (bool) – limits are applied to input, default True

  • +
+
+
Returns:
+

state derivative \((\dot{x}, \dot{y}, \dot{\theta})\)

+
+
Return type:
+

ndarray(3)

+
+
+

Returns the time derivative of state (3x1) at the state x with velocity \(v\) +and turn rate \(\omega\)

+
+\[\begin{split}\dot{x} &= v \cos \theta \\ +\dot{y} &= v \sin \theta \\ +\dot{\theta} &= \omega\end{split}\]
+

If limits is True then speed, acceleration and steer-angle limits are +applied to u.

+
+
Seealso:
+

f()

+
+
+
+ +
+
+u_limited(u)[source]
+

Apply vehicle velocity, acceleration and steering limits

+
+
Parameters:
+

u (array_like(2)) – Desired vehicle inputs \((v, \omega)\)

+
+
Returns:
+

Allowable vehicle inputs \((v, \omega)\)

+
+
Return type:
+

ndarray(2)

+
+
+

Velocity and acceleration limits are applied to \(v\) and +turn rate limits are applied to \(\omega\).

+
+ +
+
+Fv(x, odo)
+

Jacobian of state transition function df/dv

+
+
Parameters:
+
    +
  • x (array_like(3)) – vehicle state \((x, y, \theta)\)

  • +
  • odo (array_like(2)) – vehicle odometry \((\delta_d, \delta_\theta)\)

  • +
+
+
Returns:
+

Jacobian matrix

+
+
Return type:
+

ndarray(3,2)

+
+
+

Returns the Jacobian matrix \(\frac{\partial \vec{f}}{\partial \vec{v}}\) for +the given state and odometry.

+
+
Seealso:
+

f() deriv() Fx()

+
+
+
+ +
+
+Fx(x, odo)
+

Jacobian of state transition function df/dx

+
+
Parameters:
+
    +
  • x (array_like(3)) – vehicle state \((x, y, \theta)\)

  • +
  • odo (array_like(2)) – vehicle odometry \((\delta_d, \delta_\theta)\)

  • +
+
+
Returns:
+

Jacobian matrix

+
+
Return type:
+

ndarray(3,3)

+
+
+

Returns the Jacobian matrix \(\frac{\partial \vec{f}}{\partial \vec{x}}\) for +the given state and odometry.

+
+
Seealso:
+

f() deriv() Fv()

+
+
+
+ +
+
+property accel_max
+

Get maximum acceleration of vehicle (superclass)

+
+
Returns:
+

maximum acceleration

+
+
Return type:
+

float

+
+
+

Set by VehicleBase() subclass constructor.

+
+
Seealso:
+

speed_max() steer_max() u_limited()

+
+
+
+ +
+
+add_driver(driver)
+

Add a driver agent (superclass)

+
+
Parameters:
+

driver (VehicleDriverBase subclass) – a driver agent object

+
+
+
+

Warning

+

Deprecated. Use vehicle.control = driver instead.

+
+
+
Seealso:
+

VehicleDriverBase control()

+
+
+
+ +
+
+property control
+

Get/set vehicle control (superclass)

+
+
Getter:
+

Returns vehicle’s control

+
+
Setter:
+

Sets the vehicle’s control

+
+
Type:
+

2-tuple, callable, interp1d or VehicleDriver

+
+
+

This is the input to the vehicle during a simulation performed +by run(). The control input can be:

+
+
    +
  • a constant tuple as the control inputs to the vehicle

  • +
  • a function called as f(vehicle, t, x) that returns a tuple

  • +
  • an interpolator called as f(t) that returns a tuple

  • +
  • a driver agent, subclass of driversVehicleDriverBase

  • +
+
+

Example:

+
>>> from roboticstoolbox import Bicycle, RandomPath
+>>> bike = Bicycle()
+>>> bike.control = RandomPath(10)
+>>> print(bike)
+Bicycle: x = [ 0, 0, 0 ]
+  L=1, steer_max=1.41372, speed_max=inf, accel_max=inf
+
+
+
+
Seealso:
+

run() eval_control() scipy.interpolate.interp1d VehicleDriverBase

+
+
+
+ +
+
+property dt
+

Get sample time (superclass)

+
+
Returns:
+

discrete time step for simulation

+
+
Return type:
+

float

+
+
+

Set by VehicleBase() subclass constructor. Used by +step() to update state.

+
+
Seealso:
+

run()

+
+
+
+ +
+
+eval_control(control, x)
+

Evaluate vehicle control input (superclass method)

+
+
Parameters:
+
    +
  • control ([type]) – vehicle control

  • +
  • x (array_like(3)) – vehicle state \((x, y, heta)\)

  • +
+
+
Raises:
+

ValueError – bad control

+
+
Returns:
+

vehicle control inputs

+
+
Return type:
+

ndarray(2)

+
+
+

Evaluates the control for this time step and state. Control is set +by the control() property to:

+
+
    +
  • a constant tuple as the control inputs to the vehicle

  • +
  • a function called as f(vehicle, t, x) that returns a tuple

  • +
  • an interpolator called as f(t) that returns a tuple

  • +
  • a driver agent, subclass of VehicleDriverBase

  • +
+
+

Vehicle steering, speed and acceleration limits are applied to the +result before being input to the vehicle model.

+
+
Seealso:
+

run() control() scipy.interpolate.interp1d()

+
+
+
+ +
+
+f(x, odo, v=None)
+

State transition function

+
+
Parameters:
+
    +
  • x (array_like(3), ndarray(n,3)) – vehicle state \((x, y, \theta)\)

  • +
  • odo (array_like(2)) – vehicle odometry \((\delta_d, \delta_\theta)\)

  • +
  • v (array_like(2), optional) – additive odometry noise, defaults to (0,0)

  • +
+
+
Returns:
+

predicted vehicle state

+
+
Return type:
+

ndarray(3)

+
+
+

Predict the next state based on current state and odometry +value. v is a random variable that represents additive +odometry noise for simulation purposes.

+
+\[f: \vec{x}_k, \delta_d, \delta_\theta \mapsto \vec{x}_{k+1}\]
+

For particle filters it is useful to apply the odometry to the +states of all particles and this can be achieved if x is a 2D +array with one state per row. v is ignored in this case.

+
+

Note

+

This is the state update equation used for EKF localization.

+
+
+
Seealso:
+

deriv() Fx() Fv()

+
+
+
+ +
+
+init(x0=None, control=None, animate=True)
+

Initialize for simulation (superclass)

+
+
Parameters:
+

x0 (array_like(3) or array_like(2)) – Initial state, defaults to value given to Vehicle constructor

+
+
+

Performs the following initializations:

+
+
    +
  1. Clears the state history

  2. +
  3. Set the private random number generator to initial value

  4. +
  5. Sets state \(x = x_0\)

  6. +
  7. Sets previous velocity to zero

  8. +
  9. If a driver is attached, initialize it

  10. +
  11. If animation is enabled, initialize that

  12. +
+
+

If animation is enabled by constructor and no animation object is given, +use a default VehiclePolygon("car")

+
+
Seealso:
+

VehicleAnimationBase run()

+
+
+
+ +
+
+limits_va(v, v_prev)
+

Apply velocity and acceleration limits (superclass)

+
+
Parameters:
+
    +
  • v (float) – desired velocity

  • +
  • v_prev (list with single element) – previous velocity, reference to list

  • +
+
+
Returns:
+

allowed velocity

+
+
Return type:
+

float

+
+
+

Determine allowable velocity given desired velocity, speed and +acceleration limits.

+
+

Note

+

This function requires previous velocity, v_prev to enable +acceleration limiting. This is passed as a reference to a mutable value, +a single-element list. This is reset to zero at the start of each simulation.

+
+
+ +
+
+plot_xy(*args, block=None, **kwargs)
+

Plot xy-path from history

+
+
Parameters:
+
    +
  • block (bool, optional) – block until plot dismissed, defaults to None

  • +
  • args – positional arguments passed to plot()

  • +
  • kwargs – keyword arguments passed to plot()

  • +
+
+
+

The \((x,y)\) trajectory from the simulation history is plotted as +\(x\) vs :math:`y.

+
+
Seealso:
+

run() plot_xyt()

+
+
+
+ +
+
+plot_xyt(block=None, **kwargs)
+

Plot configuration vs time from history

+
+
Parameters:
+
    +
  • block (bool, optional) – block until plot dismissed, defaults to False

  • +
  • args – positional arguments passed to plot()

  • +
  • kwargs – keyword arguments passed to plot()

  • +
+
+
+

The \((x,y, heta)\) trajectory from the simulation history is plotted +against time.

+
+
Seealso:
+

run() plot_xy()

+
+
+
+ +
+
+polygon(q)
+

Bounding polygon at vehicle configuration

+
+
Parameters:
+

q (array_like(3)) – vehicle configuration \((x, y, heta)\)

+
+
Returns:
+

bounding polygon of vehicle at configuration q

+
+
Return type:
+

Polygon2

+
+
+

The bounding polygon of the vehicle is returned for the configuration +q. Can be used for collision checking using the intersects() +method.

+
+
Seealso:
+

Polygon2

+
+
+
+ +
+
+property q
+

Get vehicle state/configuration (superclass)

+
+
Returns:
+

Vehicle state \((x, y, \theta)\)

+
+
Return type:
+

ndarray(3)

+
+
Seealso:
+

x()

+
+
+
+ +
+
+property random
+

Get private random number generator

+
+
Returns:
+

NumPy random number generator

+
+
Return type:
+

numpy.random.Generator

+
+
+

Has methods including:

+
+
+

The generator is initialized with the seed provided at constructor +time every time init() is called.

+
+
Seealso:
+

init() numpy.random.default_rng()

+
+
+
+ +
+
+run(T=10, x0=None, control=None, animate=True)
+

Simulate motion of vehicle (superclass)

+
+
Parameters:
+
    +
  • T (float, optional) – Simulation time in seconds, defaults to 10

  • +
  • x0 (array_like(3) or array_like(2)) – Initial state, defaults to value given to Vehicle constructor

  • +
  • control (array_like(2), callable, driving agent) – vehicle control inputs, defaults to None

  • +
  • animate (bool, optional) – Enable graphical animation of vehicle, defaults to False

  • +
+
+
Returns:
+

State trajectory, each row is \((x,y,\theta)\).

+
+
Return type:
+

ndarray(n,3)

+
+
+

Runs the vehicle simulation for T seconds and optionally plots +an animation. The method step() performs each time step.

+

The control inputs are provided by control which can be:

+
+
    +
  • a constant tuple as the control inputs to the vehicle

  • +
  • a function called as f(vehicle, t, x) that returns a tuple

  • +
  • an interpolator called as f(t) that returns a tuple, see +SciPy interp1d

  • +
  • a driver agent, subclass of VehicleDriverBase

  • +
+
+

This is evaluated by eval_control() which applies velocity, +acceleration and steering limits.

+

The simulation can be stopped prematurely by the control function +calling stopsim().

+
+

Note

+
    +
  • the simulation is fixed-time step with the step given by the dt +attribute set by the constructor.

  • +
  • integration uses rectangular integration.

  • +
+
+
+
Seealso:
+

init() step() control() run_animation()

+
+
+
+ +
+
+run_animation(T=10, x0=None, control=None, format=None, file=None)
+

Simulate motion of vehicle (superclass)

+
+
Parameters:
+
    +
  • T (float, optional) – Simulation time in seconds, defaults to 10

  • +
  • x0 (array_like(3) or array_like(2)) – Initial state, defaults to value given to Vehicle constructor

  • +
  • control (array_like(2), callable, driving agent) – vehicle control inputs, defaults to None

  • +
  • format (str, optional) – Output format

  • +
  • file (str, optional) – File name

  • +
+
+
Returns:
+

Matplotlib animation object

+
+
Return type:
+

matplotlib.animation.FuncAnimation()

+
+
+

Runs the vehicle simulation for T seconds and returns an animation +in various formats:

+
``format``    ``file``   description
+============  =========  ============================
+``"html"``    str, None  return HTML5 video
+``"jshtml"``  str, None  return JS+HTML video
+``"gif"``     str        return animated GIF
+``"mp4"``     str        return MP4/H264 video
+``None``                 return a ``FuncAnimation`` object
+
+
+

The allowables types for file are given in the second column. A str +value is the file name. If None is an option then return the video as a string.

+

For the last case, a reference to the animation object must be held if used for +animation in a Jupyter cell:

+
anim = robot.run_animation(T=20)
+
+
+

The control inputs are provided by control which can be:

+
+
    +
  • a constant tuple as the control inputs to the vehicle

  • +
  • a function called as f(vehicle, t, x) that returns a tuple

  • +
  • an interpolator called as f(t) that returns a tuple, see +SciPy interp1d

  • +
  • a driver agent, subclass of VehicleDriverBase

  • +
+
+

This is evaluated by eval_control() which applies velocity, +acceleration and steering limits.

+

The simulation can be stopped prematurely by the control function +calling stopsim().

+
+

Note

+
    +
  • the simulation is fixed-time step with the step given by the dt +attribute set by the constructor.

  • +
  • integration uses rectangular integration.

  • +
+
+
+
Seealso:
+

init() step() control() run_animation()

+
+
+
+ +
+
+property speed_max
+

Get maximum speed of vehicle (superclass)

+
+
Returns:
+

maximum speed

+
+
Return type:
+

float

+
+
+

Set by VehicleBase() subclass constructor.

+
+
Seealso:
+

accel_max() steer_max() u_limited()

+
+
+
+ +
+
+step(u=None, animate=True, pause=True)
+

Step simulator by one time step (superclass)

+
+
Returns:
+

odometry \((\delta_d, \delta_\theta)\)

+
+
Return type:
+

ndarray(2)

+
+
+
    +
  1. Obtain the vehicle control inputs

  2. +
  3. Integrate the vehicle state forward one timestep

  4. +
  5. Updates the stored state and state history

  6. +
  7. Update animation if enabled.

  8. +
  9. Returns the odometry

  10. +
+
    +
  • veh.step((vel, steer)) for a Bicycle vehicle model

  • +
  • veh.step((vel, vel_diff)) for a Unicycle vehicle model

  • +
  • veh.step() as above but control is taken from the control +attribute which might be a function or driver agent.

  • +
+

Example:

+
>>> from roboticstoolbox import Bicycle
+>>> bike = Bicycle()  # default bicycle model
+>>> bike.step((1, 0.2))  # one step: v=1, γ=0.2
+array([0.1   , 0.0203])
+>>> bike.x
+array([0.1   , 0.    , 0.0203])
+>>> bike.step((1, 0.2))  # another step: v=1, γ=0.2
+array([0.1   , 0.0203])
+>>> bike.x
+array([0.2   , 0.002 , 0.0405])
+
+
+
+

Note

+

Vehicle control input limits are applied.

+
+
+
Seealso:
+

control(), update(), run()

+
+
+
+ +
+
+stopsim()
+

Stop the simulation (superclass)

+

A control function can stop the simulation initated by the run +method.

+
+
Seealso:
+

run()

+
+
+
+ +
+
+property verbose
+

Get verbosity (superclass)

+
+
Returns:
+

verbosity level

+
+
Return type:
+

bool

+
+
+

Set by VehicleBase subclass constructor.

+
+ +
+
+property workspace
+

Size of robot workspace (superclass)

+
+
Returns:
+

workspace bounds [xmin, xmax, ymin, ymax]

+
+
Return type:
+

ndarray(4)

+
+
+

Returns the bounds of the workspace as specified by constructor +option workspace

+
+ +
+
+property x
+

Get vehicle state/configuration (superclass)

+
+
Returns:
+

Vehicle state \((x, y, \theta)\)

+
+
Return type:
+

ndarray(3)

+
+
Seealso:
+

q()

+
+
+
+ +
+
+property x0
+

Get vehicle initial state/configuration (superclass)

+
+
Returns:
+

Vehicle state \((x, y, \theta)\)

+
+
Return type:
+

ndarray(3)

+
+
+

Set by VehicleBase subclass constructor. The state is set to this value +at the beginning of each simulation run.

+
+
Seealso:
+

run()

+
+
+
+ +
+
+property x_hist
+

Get vehicle state/configuration history (superclass)

+
+
Returns:
+

Vehicle state history

+
+
Return type:
+

ndarray(n,3)

+
+
+

The state \((x, y, heta)\) at each time step resulting from a +simulation run, one row per timestep.

+
+
Seealso:
+

run()

+
+
+
+ +
+ +
+ + +
+
+
+ +
+ +
+

© Copyright 2023, Jesse Haviland and Peter Corke. + Last updated on 11-May-2024. +

+
+ + + +
+
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/mobile-vehicle-vehicle.html b/mobile-vehicle-vehicle.html new file mode 100644 index 00000000..d97347c8 --- /dev/null +++ b/mobile-vehicle-vehicle.html @@ -0,0 +1,133 @@ + + + + + + + + + + + Vehicle superclass — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+
    +
  • + +
  • +
  • +
+
+
+
+
+ +
+

Vehicle superclass

+

The various vehicle models all subclass this class.

+
+
+roboticstoolbox.mobile.Vehicle
+

alias of <module ‘roboticstoolbox.mobile.Vehicle’ from ‘/opt/hostedtoolcache/Python/3.7.17/x64/lib/python3.7/site-packages/roboticstoolbox/mobile/Vehicle.py’>

+
+ +
+ + +
+
+
+ +
+ +
+

© Copyright 2023, Jesse Haviland and Peter Corke. + Last updated on 11-May-2024. +

+
+ + + +
+
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/mobile-vehicle.html b/mobile-vehicle.html new file mode 100644 index 00000000..d8450961 --- /dev/null +++ b/mobile-vehicle.html @@ -0,0 +1,295 @@ + + + + + + + + + + + Motion models — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

Motion models

+

These classes describe vehicle motion models, sometimes referred to as +kinematic or kino-dynamic models. They provide methods to:

+
    +
  • predict new configuration based on odometry

  • +
  • compute configuration derivative

  • +
  • simulate and animate motion

  • +
  • compute Jacobians

  • +
+
+ +
+
+ + +
+
+
+ +
+ +
+

© Copyright 2023, Jesse Haviland and Peter Corke. + Last updated on 11-May-2024. +

+
+ + + +
+
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/mobile.html b/mobile.html new file mode 100644 index 00000000..62d58e18 --- /dev/null +++ b/mobile.html @@ -0,0 +1,179 @@ + + + + + + + + + + + Mobile robots — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+
    +
  • + +
  • +
  • +
+
+
+ +
+ +
+ +
+

© Copyright 2023, Jesse Haviland and Peter Corke. + Last updated on 11-May-2024. +

+
+ + + +
+
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/mobile_SLAM.html b/mobile_SLAM.html new file mode 100644 index 00000000..cf631f20 --- /dev/null +++ b/mobile_SLAM.html @@ -0,0 +1,1985 @@ + + + + + + + Localization and Mapping — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+ +
+
+
+ + + + + + + \ No newline at end of file diff --git a/mobile_drivers.html b/mobile_drivers.html new file mode 100644 index 00000000..61725990 --- /dev/null +++ b/mobile_drivers.html @@ -0,0 +1,356 @@ + + + + + + + Mobile robot drivers — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+ +
+
+
+ + + + + + + \ No newline at end of file diff --git a/mobile_planner.html b/mobile_planner.html new file mode 100644 index 00000000..02fc8de6 --- /dev/null +++ b/mobile_planner.html @@ -0,0 +1,5187 @@ + + + + + + + Mobile robot path planning — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+ +
+
+
+ + + + + + + \ No newline at end of file diff --git a/mobile_reactive.html b/mobile_reactive.html new file mode 100644 index 00000000..f4819f07 --- /dev/null +++ b/mobile_reactive.html @@ -0,0 +1,659 @@ + + + + + + + Reactive navigation — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+ +
+
+
+ + + + + + + \ No newline at end of file diff --git a/mobile_vehicle.html b/mobile_vehicle.html new file mode 100644 index 00000000..1b2200ab --- /dev/null +++ b/mobile_vehicle.html @@ -0,0 +1,2346 @@ + + + + + + + Mobile robot kinematic models — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+ +
+
+
+ + + + + + + \ No newline at end of file diff --git a/mobile_vehicle_animation.html b/mobile_vehicle_animation.html new file mode 100644 index 00000000..fe5fdc79 --- /dev/null +++ b/mobile_vehicle_animation.html @@ -0,0 +1,551 @@ + + + + + + + Mobile robot animations — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+ +
+
+
+ + + + + + + \ No newline at end of file diff --git a/mobile_vehicle_bicycle.html b/mobile_vehicle_bicycle.html new file mode 100644 index 00000000..560fb6b8 --- /dev/null +++ b/mobile_vehicle_bicycle.html @@ -0,0 +1,896 @@ + + + + + + + Bicycle model — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+ +
+
+
+ + + + + + + \ No newline at end of file diff --git a/mobile_vehicle_unicycle.html b/mobile_vehicle_unicycle.html new file mode 100644 index 00000000..44eb3304 --- /dev/null +++ b/mobile_vehicle_unicycle.html @@ -0,0 +1,829 @@ + + + + + + + Unicycle model — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+ +
+
+
+ + + + + + + \ No newline at end of file diff --git a/mobile_vehicle_vehicle.html b/mobile_vehicle_vehicle.html new file mode 100644 index 00000000..31077f8c --- /dev/null +++ b/mobile_vehicle_vehicle.html @@ -0,0 +1,126 @@ + + + + + + + Vehicle superclass — Robotics Toolbox for Python documentation + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+ +
+
+
+ + + + + + + \ No newline at end of file diff --git a/modules.html b/modules.html new file mode 100644 index 00000000..4cad0b03 --- /dev/null +++ b/modules.html @@ -0,0 +1,209 @@ + + + + + + + + + pgraph — Simple graph functionality for Python documentation + + + + + + + + + + + + + + + + + +
+ + +
+ + +
+
+ + + + + + + \ No newline at end of file diff --git a/objects.inv b/objects.inv new file mode 100644 index 00000000..ffcb1acb Binary files /dev/null and b/objects.inv differ diff --git a/pgraph/PGraph.py b/pgraph/PGraph.py deleted file mode 100644 index 7ddb9202..00000000 --- a/pgraph/PGraph.py +++ /dev/null @@ -1,1900 +0,0 @@ -from abc import ABC -import sys -import numpy as np -import matplotlib.pyplot as plt -import copy -from collections.abc import Iterable -import tempfile -import subprocess -import webbrowser - -from spatialmath.base.graphics import axes_logic - - -class PGraph(ABC): - - def __init__(self, arg=None, metric=None, heuristic=None, verbose=False): - # we use a list and a dict, the list respects the order of adding - self._vertexlist = [] - self._vertexdict = {} - self._edgelist = set() - self._verbose = verbose - self._ncomponents = 0 - self._connectivitychange = False - if metric is None: - self.metric = "L2" - else: - self.metric = metric - if heuristic is None: - self.heuristic = self.metric - else: - self.heuristic = heuristic - - def __str__(self): - s = f"{self.__class__.__name__}: {self.n} {'vertex' if self.n==1 else 'vertices'}, {self.ne} edge{'s'[:self.ne^1]}, {self.nc} component{'s'[:self.nc^1]}" - return s - - def __repr__(self): - return str(self) - - @classmethod - def Dict(cls, d, reverse=False): - """ - Create graph from parent/child dictionary - - :param d: dictionary that maps from ``Vertex`` subclass to ``Vertex`` subclass - :type d: dict - :param reverse: reverse link direction, defaults to False - :type reverse: bool, optional - :return: graph - :rtype: UGraph or DGraph - - Behaves like a constructor for a ``DGraph`` or ``UGraph`` from a - dictionary that maps vertices to parents. From this information it - can create a tree graph. - - By default parent vertices are linked their children. If ``reverse`` is - True then children are linked to their parents. - """ - - g = cls() - - for vertex, parent in d.items(): - if isinstance(vertex, str): - vertex_name = vertex - else: - vertex_name = vertex.name - - if vertex_name in g: - vertex = g[vertex_name] - else: - vertex = g.add_vertex(UVertex(), name=vertex_name) - - if isinstance(parent, str): - parent_name = parent - else: - parent_name = parent.name - if parent_name in g: - parent = g[parent_name] - else: - parent = g.add_vertex(UVertex(), name=parent_name) - - if reverse: - g.add_edge(vertex, parent) - else: - g.add_edge(parent, vertex) - - return g - - @classmethod - def Adjacency(cls, A, coords=None, names=None): - """ - Create graph from adjacency matrix - - :param A: adjacency matrix - :type A: ndarray(N,N) - :param coords: coordinates of vertices, defaults to None - :type coords: ndarray(N,M), optional - :param names: names of vertices, defaults to None - :type names: list(N) of str, optional - - :return: [description] - :rtype: [type] - - Create a directed or undirected graph where non-zero elements ``A[i,j]`` - correspond to edges from vertex ``i`` to vertex ``j``. - - .. warning:: For undirected graph ``A`` should be symmetric but this - is not checked. Only the upper triangular part is used. - """ - - if A.shape[0] != A.shape[1]: - raise ValueError("Adjacency matrix must be square") - if names is not None and len(names) != A.shape[0]: - raise ValueError("length of names must match dimension of adjacency matrix") - if coords is not None and coords.shape[0] != A.shape[0]: - raise ValueError("coords must have same number of rows as adjacency matrix") - - g = cls() - - name = None - coord = None - for i in range(A.shape[0]): - if names is not None: - name = names[i] - if coords is not None: - coord = coords[i, :] - g.add_vertex(name=name, coord=coord) - - if isinstance(g, UGraph): - # undirected graph - for i in range(A.shape[0]): - for j in range(i + 1, A.shape[1]): - if A[i, j] > 0: - g[i].connect(g[j], cost=A[i, j]) - else: - # directed graph - for i in range(A.shape[0]): - for j in range(A.shape[1]): - if A[i, j] > 0: - if i == j: - raise ValueError("loops in graph not supported") - g[i].connect(g[j], cost=A[i, j]) - - return g - - def copy(self): - """ - Deepcopy of graph - - :param g: A graph - :type g: PGraph - :return: deep copy - :rtype: PGraph - """ - return copy.deepcopy(self) - - def add_vertex(self, vertex, name=None): - """ - Add a vertex to the graph (superclass method) - - :param vertex: vertex to add - :type vertex: Vertex subclass - :param name: name of vertex - :type name: str - - ``G.add_vertex(v)`` add vertex ``v`` to the graph ``G``. - - If the vertex has no name and ``name`` is None give it a default name - ``#N`` where ``N`` is a consecutive integer. - - The vertex is placed into a dictionary with a key equal to its name. - """ - if name is None: - name = vertex.name - if name is None: - name = f"#{len(self._vertexlist)}" - vertex.name = name - self._vertexlist.append(vertex) - self._vertexdict[vertex.name] = vertex - if self._verbose: - print(f"New vertex {vertex.name}: {vertex.coord}") - vertex._graph = self - self._connectivitychange = True - return vertex - - def add_edge(self, v1, v2, **kwargs): - """ - Add an edge to the graph (superclass method) - - :param v1: first vertex (start if a directed graph) - :type v1: Vertex subclass - :param v2: second vertex (end if a directed graph) - :type v2: Vertex subclass - :param kwargs: optional arguments to pass to ``Vertex.connect`` - :return: edge - :rtype: Edge - - Create an edge between a vertex pair and adds it to the graph. - - This is a graph centric way of creating an edge. The - alternative is the ``connect`` method of a vertex. - - :seealso: :meth:`Edge.connect` :meth:`Vertex.connect` - """ - if isinstance(v1, str): - v1 = self[v1] - elif not isinstance(v1, Vertex): - raise TypeError("v1 must be Vertex subclass or string name") - if isinstance(v2, str): - v2 = self[v2] - elif not isinstance(v2, Vertex): - raise TypeError("v2 must be Vertex subclass or string name") - - if self._verbose: - print(f"New edge from {v1.name} to {v2.name}") - return v1.connect(v2, **kwargs) - - def remove(self, x): - """ - Remove element from graph (superclass method) - - :param x: element to remove from graph - :type x: Edge or Vertex subclass - :raises TypeError: unknown type - - The edge or vertex is removed, and all references and lists are - updated. - - .. warning:: The connectivity of the network may be changed. - """ - if isinstance(x, Edge): - # remove an edge - - # remove edge from the edgelist of connected vertices - x.v1._edgelist.remove(x) - x.v2._edgelist.remove(x) - - # indicate that connectivity has changed - x.v1._connectivitychange = True - x.v2._connectivitychange = True - self._connectivitychange = True - - # remove references to the vertices - x.v1 = None - x.v2 = None - - # remove from list of all edges - self._edgelist.remove(x) - - elif isinstance(x, Vertex): - # remove a vertex - - # remove all edges of this vertex - for edge in copy.copy(x._edgelist): - self.remove(edge) - - # remove from list and dict of all edges - self._vertexlist.remove(x) - del self._vertexdict[x.name] - else: - raise TypeError("expecting Edge or Vertex") - - def show(self): - print("vertices:") - for v in self._vertexlist: - print(" " + str(v)) - print("edges:") - for e in self._edgelist: - print(" " + str(e)) - - @property - def n(self): - """ - Number of vertices - - :return: Number of vertices - :rtype: int - """ - return len(self._vertexdict) - - @property - def ne(self): - """ - Number of edges - - :return: Number of vertices - :rtype: int - """ - return len(self._edgelist) - - @property - def nc(self): - """ - Number of components - - :return: Number of components - :rtype: int - - .. note:: - - - Components are labeled from 0 to ``g.nc-1``. - - A graph coloring algorithm is run if the graph connectivity - has changed. - - .. note:: A lazy approach is used, and if a connectivity changing - operation has been performed since the last call, the graph - coloring algorithm is run which is potentially expensive for - a large graph. - """ - self._graphcolor() - return self._ncomponents - - def _metricfunc(self, metric): - - def L1(v): - return np.linalg.norm(v, 1) - - def L2(v): - return np.linalg.norm(v) - - def SE2(v): - # wrap angle to range [-pi, pi) - v[2] = (v[2] + np.pi) % (2 * np.pi) - np.pi - return np.linalg.norm(v) - - if callable(metric): - return metric - elif isinstance(metric, str): - if metric == "L1": - return L1 - elif metric == "L2": - return L2 - elif metric == "SE2": - return SE2 - else: - raise ValueError("unknown metric") - - @property - def metric(self): - """ - Get the distance metric for graph - - :return: distance metric - :rtype: callable - - This is a function of a vector and returns a scalar. - """ - return self._metric - - @metric.setter - def metric(self, metric): - r""" - Set the distance metric for graph - - :param metric: distance metric - :type metric: callable or str - - This is a function of a vector and returns a scalar. It can be - user defined function or a string: - - - 'L1' is the norm :math:`L_1 = \Sigma_i | v_i |` - - 'L2' is the norm :math:`L_2 = \sqrt{ \Sigma_i v_i^2}` - - 'SE2' is a mixed norm for vectors :math:`(x, y, \theta)` and - is :math:`\sqrt{x^2 + y^2 + \bar{\theta}^2}` where :math:`\bar{\theta}` - is :math:`\theta` wrapped to the interval :math:`[-\pi, \pi)` - - The metric is used by :meth:`closest` and :meth:`distance` - """ - self._metric = self._metricfunc(metric) - - @property - def heuristic(self): - """ - Get the heuristic distance metric for graph - - :return: heuristic distance metric - :rtype: callable - - This is a function of a vector and returns a scalar. - """ - return self._heuristic - - @heuristic.setter - def heuristic(self, heuristic): - r""" - Set the heuristic distance metric for graph - - :param metric: heuristic distance metric - :type metric: callable or str - - This is a function of a vector and returns a scalar. It can be - user defined function or a string: - - - 'L1' is the norm :math:`L_1 = \Sigma_i | v_i |` - - 'L2' is the norm :math:`L_2 = \sqrt{ \Sigma_i v_i^2}` - - 'SE2' is a mixed norm for vectors :math:`(x, y, \theta)` and - is :math:`\sqrt{x^2 + y^2 + \bar{\theta}^2}` where :math:`\bar{\theta}` - is :math:`\theta` wrapped to the interval :math:`[-\pi, \pi)` - - The heuristic distance is only used by the A* planner :meth:`path_Astar`. - """ - self._heuristic = self._metricfunc(heuristic) - - def __repr__(self): - s = [] - for vertex in self: - ss = f"{vertex.name} at {vertex.coord}" - if vertex.label is not None: - ss += " component={vertex.label}" - s.append(ss) - return "\n".join(s) - - def __getitem__(self, i): - """ - Get vertex (superclass method) - - :param i: vertex description - :type i: int or str - :return: the referenced vertex - :rtype: Vertex subclass - - Retrieve a vertex by index or name: - - -``g[i]`` is the i'th vertex in the graph. This reflects the order of - addition to the graph. - -``g[s]`` is vertex named ``s`` - -``g[v]`` is ``v`` where ``v`` is a ``Vertex`` subclass - - This method also supports iteration over the vertices in a graph:: - - for v in g: - print(v) - - will iterate over all the vertices. - """ - if isinstance(i, int): - return self._vertexlist[i] - elif isinstance(i, str): - return self._vertexdict[i] - elif isinstance(i, Vertex): - return i - - def __contains__(self, item): - """ - Test if vertex in graph - - :param item: vertex or name of vertex - :type item: Vertex subclass or str - :return: true if vertex exists in the graph - :rtype: bool - - - ``'name' in graph`` is true if a vertex named ``'name'`` exists in the - graph. - - ``v in graph`` is true if the vertex reference ``v`` exists in the - graph. - - """ - if isinstance(item, str): - return item in self._vertexdict - elif isinstance(item, Vertex): - return item in self._vertexdict.values() - - def closest(self, coord): - """ - Vertex closest to point - - :param coord: coordinates of a point - :type coord: ndarray(n) - :return: closest vertex - :rtype: Vertex subclass - - Returns the vertex closest to the given point. Distance is computed - according to the graph's metric. - - :seealso: :meth:`metric` - """ - min_dist = np.inf - min_which = None - - for vertex in self: - d = self.metric(vertex.coord - coord) - if d < min_dist: - min_dist = d - min_which = vertex - - return min_which, min_dist - - def edges(self): - """ - Get all edges in graph (superclass method) - - :return: All edges in the graph - :rtype: list of Edge references - - We can iterate over all edges in the graph by:: - - for e in g.edges(): - print(e) - - .. note:: The ``edges()`` of a Vertex is a list of all edges connected - to that vertex. - - :seealso: :meth:`Vertex.edges` - """ - return self._edgelist - - def plot( - self, - colorcomponents=True, - force2d=False, - vopt={}, - eopt={}, - text={}, - block=False, - grid=True, - ax=None, - ): - """ - Plot the graph - - :param vopt: vertex format, defaults to 12pt o-marker - :type vopt: dict, optional - :param eopt: edge format, defaults to None - :type eopt: dict, optional - :param text: text label format, defaults to None - :type text: False or dict, optional - :param colorcomponents: color vertices and edges by component, defaults to None - :type color: bool, optional - :param block: block until figure is dismissed, defaults to True - :type block: bool, optional - - The graph is plotted using matplotlib. - - If ``colorcomponents`` is True then each component is assigned a unique - color. ``vertex`` and ``edge`` cannot include a color keyword. - - If ``text`` is a dict it is used to format text labels for the vertices - which are the vertex names. If ``text`` is None default formatting is - used. If ``text`` is False no labels are added. - """ - vopt = {**dict(marker="o", markersize=12), **vopt} - eopt = {**dict(linewidth=3), **eopt} - - if colorcomponents: - color = plt.cm.coolwarm(np.linspace(0, 1, self.nc)) - - if len(self[0].coord) == 2 or force2d: - # 2D plotting - if ax is None: - ax = axes_logic(ax, 2) - for c in range(self.nc): - # for each component - for vertex in self.component(c): - if text is not False: - ax.text(vertex.x, vertex.y, " " + vertex.name, **text) - if colorcomponents: - ax.plot(vertex.x, vertex.y, color=color[c, :], **vopt) - for v in vertex.neighbours(): - ax.plot( - [vertex.x, v.x], - [vertex.y, v.y], - color=color[c, :], - **eopt, - ) - else: - ax.plot(vertex.x, vertex.y, **vopt) - for v in vertex.neighbours(): - ax.plot([vertex.x, v.x], [vertex.y, v.y], **eopt) - else: - # 3D or higher plotting, just do (x, y, z) - if ax is None: - ax = axes_logic(ax, 3) - for c in range(self.nc): - # for each component - for vertex in self.component(c): - if text is not False: - ax.text( - vertex.x, vertex.y, vertex.z, " " + vertex.name, **text - ) - if colorcomponents: - ax.plot( - vertex.x, - vertex.y, - vertex.z, - **{**dict(color=color[c, :]), **vopt}, - ) - for v in vertex.neighbours(): - ax.plot( - [vertex.x, v.x], - [vertex.y, v.y], - [vertex.z, v.z], - **{**dict(color=color[c, :]), **eopt}, - ) - else: - ax.plot(vertex.x, vertex.y, **vopt) - for v in vertex.neighbours(): - ax.plot( - [vertex.x, v.x], - [vertex.y, v.y], - [vertex.z, v.z], - **eopt, - ) - # if nc > 1: - # # add a colorbar - # plt.colorbar() - ax.grid(grid) - plt.show(block=block) - - def highlight_path(self, path, block=False, **kwargs): - """ - Highlight a path through the graph - - :param path: [description] - :type path: [type] - :param block: [description], defaults to True - :type block: bool, optional - - The vertices and edges along the path are overwritten with a different - size/width and color. - - :seealso: :meth:`highlight_vertex` :meth:`highlight_edge` - """ - for i in range(len(path)): - if i < len(path) - 1: - e = path[i].edgeto(path[i + 1]) - self.highlight_edge(e, **kwargs) - self.highlight_vertex(path[i], **kwargs) - plt.show(block=block) - - def highlight_edge(self, edge, scale=2, color="r", alpha=0.5): - """ - Highlight an edge in the graph - - :param edge: The edge to highlight - :type edge: Edge subclass - :param scale: Overwrite with a line this much bigger than the original, - defaults to 1.5 - :type scale: float, optional - :param color: Overwrite with a line in this color, defaults to 'r' - :type color: str, optional - """ - p1 = edge.v1 - p2 = edge.v2 - plt.plot( - [p1.x, p2.x], [p1.y, p2.y], color=color, linewidth=3 * scale, alpha=alpha - ) - - def highlight_vertex(self, vertex, scale=2, color="r", alpha=0.5): - """ - Highlight a vertex in the graph - - :param edge: The vertex to highlight - :type edge: Vertex subclass - :param scale: Overwrite with a line this much bigger than the original, - defaults to 1.5 - :type scale: float, optional - :param color: Overwrite with a line in this color, defaults to 'r' - :type color: str, optional - """ - if isinstance(vertex, Iterable): - for n in vertex: - if isinstance(n, str): - n = self[n] - plt.plot(n.x, n.y, "o", color=color, markersize=12 * scale, alpha=alpha) - else: - plt.plot( - vertex.x, vertex.y, "o", color=color, markersize=12 * scale, alpha=alpha - ) - - def dotfile(self, filename=None, direction=None): - """ - Export graph as a GraphViz dot file - - :param filename: filename to save graph to, defaults to None - :type filename: str, optional - - ``g.dotfile()`` creates the specified file which contains the - `GraphViz `_ code to represent the embedded graph. By default output - is to the console. - - .. note:: - - - The graph is undirected if it is a subclass of ``UGraph`` - - The graph is directed if it is a subclass of ``DGraph`` - - Use ``neato`` rather than dot to get the embedded layout - - .. note:: If ``filename`` is a file object then the file will *not* - be closed after the GraphViz model is written. - - :seealso: :func:`showgraph` - """ - - if filename is None: - f = sys.stdout - elif isinstance(filename, str): - f = open(filename, "w") - else: - f = filename - - if isinstance(self, DGraph): - print("digraph {", file=f) - else: - print("graph {", file=f) - - if direction is not None: - print(f"rankdir = {direction}", file=f) - - # add the vertices including name and position - for vertex in self: - if vertex.coord is None: - print(' "{:s}"'.format(vertex.name), file=f) - else: - print( - ' "{:s}" [pos="{:.5g},{:.5g}"]'.format( - vertex.name, vertex.coord[0], vertex.coord[1] - ), - file=f, - ) - print(file=f) - # add the edges - for e in self.edges(): - if isinstance(self, DGraph): - print(' "{:s}" -> "{:s}"'.format(e.v1.name, e.v2.name), file=f) - else: - print(' "{:s}" -- "{:s}"'.format(e.v1.name, e.v2.name), file=f) - - print("}", file=f) - - if filename is None or isinstance(filename, str): - f.close() # noqa - - def showgraph(self, **kwargs): - """ - Display graph in a browser tab - - :param kwargs: arguments passed to :meth:`dotfile` - - ``g.showgraph()`` renders and displays the graph in a browser tab. The - graph is exported in `GraphViz `_ format, rendered to - PDF using ``dot`` and then displayed in a browser tab. - - :seealso: :func:`dotfile` - """ - # create the temporary dotfile - dotfile = tempfile.TemporaryFile(mode="w") - self.dotfile(dotfile, **kwargs) - - # rewind the dot file, create PDF file in the filesystem, run dot - dotfile.seek(0) - pdffile = tempfile.NamedTemporaryFile(suffix=".pdf", delete=False) - result = subprocess.run("dot -Tpdf", shell=True, stdin=dotfile, stdout=pdffile) - - if result.returncode == 0: - # dot ran happily - # open the PDF file in browser (hopefully portable), then cleanup - webbrowser.open(f"file://{pdffile.name}") - # time.sleep(1) - # os.remove(pdffile.name) - - def iscyclic(self): - pass - - def average_degree(self): - r""" - Average degree of the graph - - :return: average degree - :rtype: float - - Average degree is :math:`2 E / N` for an undirected graph and - :math:`E / N` for a directed graph where :math:`E` is the total number of - edges and :math:`N` is the number of vertices. - - """ - if isinstance(self, DGraph): - return len(self.edges()) / self.n - elif isinstance(self, UGraph): - return 2 * len(self.edges()) / self.n - - # --------------------------------------------------------------------------- # - - # MATRIX REPRESENTATIONS - - def Laplacian(self): - """ - Laplacian matrix for the graph - - :return: Laplacian matrix - :rtype: NumPy ndarray - - ``g.Laplacian()`` is the Laplacian matrix (NxN) of the graph where N - is the number of vertices. - - .. note:: - - - Laplacian is always positive-semidefinite. - - Laplacian has at least one zero eigenvalue. - - The number of zero-valued eigenvalues is the number of connected - components in the graph. - - :seealso: :meth:`adjacency` :meth:`incidence` :meth:`degree` - """ - return self.degree() - (self.adjacency() > 0) - - def connectivity(self, vertices=None): - """ - Graph connectivity - - :return: a list with the number of edges per vertex - :rtype: list - - The average vertex connectivity is:: - - mean(g.connectivity()) - - and the minimum vertex connectivity is:: - - min(g.connectivity()) - """ - - c = [] - if vertices is None: - vertices = self - for n in vertices: - c.append(len(n._edgelist)) - return c - - def degree(self): - """ - Degree matrix of graph - - :return: degree matrix - :rtype: ndarray(N,N) - - This is a diagonal matrix where element ``[i,i]`` is the number - of edges connected to vertex id ``i``. - - :seealso: :meth:`adjacency` :meth:`incidence` :meth:`laplacian` - """ - - return np.diag(self.connectivity()) - - def adjacency(self): - """ - Adjacency matrix of graph - - :returns: adjacency matrix - :rtype: ndarray(N,N) - - The elements of the adjacency matrix ``[i,j]`` are 1 if vertex ``i`` is - connected to vertex ``j``, else 0. - - .. note:: - - - vertices are numbered in their order of creation. A vertex index - can be resolved to a vertex reference by ``graph[i]``. - - for an undirected graph the matrix is symmetric - - Eigenvalues of ``A`` are real and are known as the spectrum of the graph. - - The element ``A[i,j]`` can be considered the number of walks of length one - edge from vertex ``i`` to vertex ``j`` (either zero or one). - - If ``Ak = A ** k`` the element ``Ak[i,j]`` is the number of - walks of length ``k`` from vertex ``i`` to vertex ``j``. - - :seealso: :meth:`Laplacian` :meth:`incidence` :meth:`degree` - """ - # create a dict mapping vertex to an id - vdict = {} - for i, vert in enumerate(self): - vdict[vert] = i - - A = np.zeros((self.n, self.n)) - for vertex in self: - for n in vertex.neighbours(): - A[vdict[vertex], vdict[n]] = 1 - return A - - def incidence(self): - """ - Incidence matrix of graph - - :returns: incidence matrix - :rtype: ndarray(n,ne) - - The elements of the incidence matrix ``I[i,j]`` are 1 if vertex ``i`` is - connected to edge ``j``, else 0. - - .. note:: - - - vertices are numbered in their order of creation. A vertex index - can be resolved to a vertex reference by ``graph[i]``. - - edges are numbered in the order they appear in ``graph.edges()``. - - :seealso: :meth:`Laplacian` :meth:`adjacency` :meth:`degree` - """ - edges = self.edges() - I = np.zeros((self.n, len(edges))) - - # create a dict mapping edge to an id - edict = {} - for i, edge in enumerate(edges): - edict[edge] = i - - for i, vertex in enumerate(self): - for i, e in enumerate(vertex.edges()): - I[i, edict[e]] = 1 - - return I - - def distance(self): - """ - Distance matrix of graph - - :return: distance matrix - :rtype: ndarray(n,n) - - The elements of the distance matrix ``D[i,j]`` is the edge cost of moving - from vertex ``i`` to vertex ``j``. It is zero if the vertices are not - connected. - """ - # create a dict mapping vertex to an id - vdict = {} - for i, vert in enumerate(self): - vdict[vert] = i - - A = np.zeros((self.n, self.n)) - for v1 in self: - for v2, edge in v1.incidences(): - A[vdict[v1], vdict[v2]] = edge.cost - return A - - # GRAPH COMPONENTS - - def component(self, c): - """ - All vertices in specified graph component - - ``graph.component(c)`` is a list of all vertices in graph component ``c``. - """ - self._graphcolor() # ensure labels are uptodate - return [v for v in self if v.label == c] - - def samecomponent(self, v1, v2): - """ - Test if vertices belong to same graph component - - :param v1: vertex - :type v1: Vertex subclass - :param v2: vertex - :type v2: Vertex subclass - :return: true if vertices belong to same graph component - :rtype: bool - - Test whether vertices belong to the same component. For a: - - - directed graph this implies a path between them - - undirected graph there is not necessarily a path between them - """ - self._graphcolor() # ensure labels are uptodate - - return v1.label == v2.label - - # def remove(self, v): - # # remove edges from neighbour's edge list - # for e in v.edges(): - # next = e.next(v) - # next._edgelist.remove(e) - # next._connectivitychange = True - - # # remove references from the graph - # self._vertexlist.remove(v) - # for key, value in self._vertexdict.items(): - # if value is v: - # del self._vertexdict[key] - # break - - # v._edgelist = [] # remove all references to edges - # --------------------------------------------------------------------------- # - - def path_BFS(self, S, G, verbose=False, summary=False): - """ - Breadth-first search for path - - :param S: start vertex - :type S: Vertex subclass - :param G: goal vertex - :type G: Vertex subclass - :return: list of vertices from S to G inclusive, path length - :rtype: list of Vertex subclass, float - - Returns a list of vertices that form a path from vertex ``S`` to - vertex ``G`` if possible, otherwise return None. - - """ - if isinstance(S, str): - S = self[S] - elif not isinstance(S, Vertex): - raise TypeError("start must be Vertex subclass or string name") - if isinstance(G, str): - G = self[G] - elif not isinstance(S, Vertex): - raise TypeError("goal must be Vertex subclass or string name") - - # we use lists not sets since the order is instructive in verbose - # mode, really need ordered sets... - frontier = [S] - explored = [] - parent = {} - done = False - - while frontier: - if verbose: - print() - print("FRONTIER:", ", ".join([v.name for v in frontier])) - print("EXPLORED:", ", ".join([v.name for v in explored])) - - x = frontier.pop(0) - if verbose: - print(" expand", x.name) - - # expand the vertex - for n in x.neighbours(): - if n is G: - if verbose: - print(" goal", n.name, "reached") - parent[n] = x - done = True - break - if n not in frontier and n not in explored: - # add it to the frontier - frontier.append(n) - if verbose: - print(" add", n.name, "to the frontier") - parent[n] = x - if done: - break - explored.append(x) - if verbose: - print(" move", x.name, "to the explored list") - else: - # no path - return None - - # reconstruct the path from start to goal - x = G - path = [x] - length = 0 - - while x is not S: - p = parent[x] - length += x.edgeto(p).cost - path.insert(0, p) - x = p - - if summary or verbose: - print( - f"{len(explored)} vertices explored, {len(frontier)} remaining on the frontier" - ) - - return path, length - - def path_UCS(self, S, G, verbose=False, summary=False): - """ - Uniform cost search for path - - :param S: start vertex - :type S: Vertex subclass - :param G: goal vertex - :type G: Vertex subclass - :return: list of vertices from S to G inclusive, path length, tree - :rtype: list of Vertex subclass, float, dict - - Returns a list of vertices that form a path from vertex ``S`` to - vertex ``G`` if possible, otherwise return None. - - The search tree is returned as dict that maps a vertex to its parent. - - The heuristic is the distance metric of the graph, which defaults to - Euclidean distance. - """ - if isinstance(S, str): - S = self[S] - elif not isinstance(S, Vertex): - raise TypeError("start must be Vertex subclass or string name") - if isinstance(G, str): - G = self[G] - elif not isinstance(S, Vertex): - raise TypeError("goal must be Vertex subclass or string name") - - frontier = [S] - explored = [] - parent = {} - f = {S: 0} # evaluation function - - while frontier: - if verbose: - print() - print( - "FRONTIER:", ", ".join([f"{v.name}({f[v]:.0f})" for v in frontier]) - ) - print("EXPLORED:", ", ".join([v.name for v in explored])) - - i = np.argmin([f[n] for n in frontier]) # minimum f in frontier - x = frontier.pop(i) - if verbose: - print(" expand", x.name) - if x is G: - break - # expand the vertex - for n, e in x.incidences(): - fnew = f[x] + e.cost - if n not in frontier and n not in explored: - # add it to the frontier - parent[n] = x - f[n] = fnew - frontier.append(n) - if verbose: - print(" add", n.name, "to the frontier") - - elif n in frontier: - # neighbour is already in the frontier - # cost of path via x is lower that previous, reparent it - if fnew < f[n]: - if verbose: - print( - f" reparent {n.name}: cost {fnew} via {x.name} is less than cost {f[n]} via {parent[n].name}, change parent from {parent[n].name} to {x.name} " - ) - f[n] = fnew - parent[n] = x - - explored.append(x) - if verbose: - print(" move", x.name, "to the explored list") - else: - # no path - return None - - # reconstruct the path from start to goal - x = G - path = [x] - length = 0 - - while x is not S: - p = parent[x] - length += p.edgeto(x).cost - path.insert(0, p) - x = p - - parent_names = {} - for v, p in parent.items(): - parent_names[v.name] = p.name - - if summary or verbose: - print( - f"{len(explored)} vertices explored, {len(frontier)} remaining on the frontier" - ) - - return path, length, parent_names - - def path_Astar(self, S, G, verbose=False, summary=False): - """ - A* search for path - - :param S: start vertex - :type S: Vertex subclass - :param G: goal vertex - :type G: Vertex subclass - :return: list of vertices from S to G inclusive, path length, tree - :rtype: list of Vertex subclass, float, dict - - Returns a list of vertices that form a path from vertex ``S`` to - vertex ``G`` if possible, otherwise return None. - - The search tree is returned as dict that maps a vertex to its parent. - - The heuristic is the distance metric of the graph, which defaults to - Euclidean distance. - - :seealso: :meth:`heuristic` - """ - if isinstance(S, str): - S = self[S] - elif not isinstance(S, Vertex): - raise TypeError("start must be Vertex subclass or string name") - if isinstance(G, str): - G = self[G] - elif not isinstance(S, Vertex): - raise TypeError("goal must be Vertex subclass or string name") - - frontier = [S] - explored = [] - parent = {} - g = {S: 0} # cost to come - f = {S: 0} # evaluation function - - while frontier: - if verbose: - print() - print( - "FRONTIER:", ", ".join([f"{v.name}({f[v]:.0f})" for v in frontier]) - ) - print("EXPLORED:", ", ".join([v.name for v in explored])) - - i = np.argmin([f[n] for n in frontier]) # minimum f in frontier - x = frontier.pop(i) - if verbose: - print(" expand", x.name) - if x is G: - break - # expand the vertex - for n, e in x.incidences(): - if n not in frontier and n not in explored: - # add it to the frontier - frontier.append(n) - parent[n] = x - g[n] = g[x] + e.cost # update cost to come - f[n] = g[n] + n.heuristic_distance(G) # heuristic - if verbose: - print(" add", n.name, "to the frontier") - elif n in frontier: - # neighbour is already in the frontier - gnew = g[x] + e.cost - if gnew < g[n]: - # cost of path via x is lower that previous, reparent it - if verbose: - print( - f" reparent {n.name}: cost {gnew} via {x.name} is less than cost {g[n]} via {parent[n].name}, change parent from {parent[n].name} to {x.name} " - ) - g[n] = gnew - f[n] = g[n] + n.heuristic_distance(G) # heuristic - - parent[n] = x # reparent - - explored.append(x) - if verbose: - print(" move", x.name, "to the explored list") - - else: - # no path - return None - - # reconstruct the path from start to goal - x = G - path = [x] - length = 0 - - while x is not S: - p = parent[x] - length += p.edgeto(x).cost - path.insert(0, p) - x = p - - parent_names = {} - for v, p in parent.items(): - parent_names[v.name] = p.name - - if summary or verbose: - print( - f"{len(explored)} vertices explored, {len(frontier)} remaining on the frontier" - ) - - return path, length, parent_names - - -# -------------------------------------------------------------------------- # - - -class UGraph(PGraph): - """ - Class for undirected graphs - - .. inheritance-diagram:: UGraph - """ - - def add_vertex(self, coord=None, name=None): - """ - Add vertex to undirected graph - - :param coord: coordinate for an embedded graph, defaults to None - :type coord: array-like, optional - :param name: vertex name, defaults to "#i" - :type name: str, optional - :return: new vertex - :rtype: UVertex - - - ``g.add_vertex()`` creates a new vertex with optional ``coord`` and - ``name``. - - ``g.add_vertex(v)`` takes an instance or subclass of UVertex and adds - it to the graph - """ - if isinstance(coord, UVertex): - vertex = coord - else: - vertex = UVertex(coord) - super().add_vertex(vertex, name=name) - return vertex - - @classmethod - def vertex_copy(self, vertex): - return DVertex(coord=vertex.coord, name=vertex.name) - - def _graphcolor(self): - """ - Color the graph - - Performs a depth-first labeling operation, assigning the ``label`` - attribute of every vertex with a sequential integer starting from 0. - - This method checks the ``_connectivitychange`` attribute of all vertices - and if any are True it will perform the coloring operation. This flag - is set True by any operation that adds or removes a vertex or edge. - - :seealso: :meth:`nc` - """ - if self._connectivitychange or any([n._connectivitychange for n in self]): - - # color the graph - - # clear all the labels - for vertex in self: - vertex.label = None - vertex._connectivitychange = False - - lastlabel = None - for label in range(self.n): - assignment = False - for v in self: - # find first vertex with no label - if v.label is None: - # do BFS - q = [v] # initialize frontier - while len(q) > 0: - v = q.pop() # expand v - v.label = label - for n in v.neighbours(): - if n.label is None: - q.append(n) - lastlabel = label - assignment = True - break - if not assignment: - break - - self._ncomponents = lastlabel + 1 - - -class DGraph(PGraph): - """ - Class for directed graphs - - .. inheritance-diagram:: DGraph - """ - - def add_vertex(self, coord=None, name=None): - """ - Add vertex to directed graph - - :param coord: coordinate for an embedded graph, defaults to None - :type coord: array-like, optional - :param name: vertex name, defaults to "#i" - :type name: str, optional - :return: new vertex - :rtype: DVertex - - - ``g.add_vertex()`` creates a new vertex with optional ``coord`` and - ``name``. - - ``g.add_vertex(v)`` takes an instance or subclass of DVertex and adds - it to the graph - """ - if isinstance(coord, Vertex): - vertex = coord - else: - vertex = DVertex(coord=coord, name=name) - super().add_vertex(vertex, name=name) - return vertex - - @classmethod - def vertex_copy(self, vertex): - return DVertex(coord=vertex.coord, name=vertex.name) - - def _graphcolor(self): - """ - Color the graph - - Performs a depth-first labeling operation, assigning the ``label`` - attribute of every vertex with a sequential integer starting from 0. - - This method checks the ``_connectivitychange`` attribute of all vertices - and if any are True it will perform the coloring operation. This flag - is set True by any operation that adds or removes a vertex or edge. - - :seealso: :meth:`nc` - """ - if self._connectivitychange or any([n._connectivitychange for n in self]): - - # color the graph - - # clear all the labels - for vertex in self: - vertex.label = None - vertex._connectivitychange = False - - # initial labeling pass - merge = {} - nextlabel = 1 - for v in self: - if v.label is None: - # no label, try to inherit one from a neighbour - for n in v.neighbours(): - if n.label is not None: - # neighbour has a label - v.label = n.label - break - - if v.label is None: - # still not labeled, assign a new label - v.label = nextlabel - nextlabel += 1 - - # now look for clashes - for n in v.neighbours(): - if n.label is None: - # neighbour has no label, give it this one - n.label = v.label - elif v.label != n.label: - # label clash, note it for merging - merge[n.label] = v.label - - # merge labels and find unique labels - unique = set() - for v in self: - while v.label in merge: - v.label = merge[v.label] - unique.add(v.label) - - final = {u: i for i, u in enumerate(unique)} - for v in self: - v.label = final[v.label] - - self._ncomponents = len(unique) - - -# ========================================================================== # - - -class Edge: - """ - Edge class - - Is used to represent directed directed and undirected edges. - - Each edge has: - - ``cost`` cost of traversing this edge, required for planning methods - - ``data`` reference to arbitrary data associated with the edge - - ``v1`` first vertex, start vertex for a directed edge - - ``v2`` second vertex, end vertex for a directed edge - - .. note:: - - - An undirected graph is created by having a single edge object in the - edgelist of _each_ vertex. - - This class can be inherited to provide user objects with graph capability. - - Inheritance is an alternative to providing arbitrary user data. - - An Edge points to a pair of vertices. At ``connect`` time the vertices - get references back to the Edge object. - - ``graph.add_edge(v1, v2)`` calls ``v1.connect(v2)`` - """ - - def __init__(self, v1=None, v2=None, cost=None, data=None): - """ - Create an edge object - - :param v1: start of the edge, defaults to None - :type v1: Vertex subclass, optional - :param v2: end of the edge, defaults to None - :type v2: Vertex subclass, optional - :param cost: edge cost, defaults to None - :type cost: any, optional - :param data: edge data, defaults to None - :type data: any, optional - - Creates an edge but does not connect it to the vertices or add it to the - graph. - - If vertices are given, and have associated coordinates, the edge cost - will be computed according to the distance measure associated with the - graph. - - ``data`` is a way of associating any object with the edge, its value - can be found as the ``.data`` attribute of the edge. An alternative - approach is to subclass the ``Edge`` class. - - .. note:: To compute edge cost from the vertices, the vertices must have - been added to the graph. - - :seealso: :meth:`Edge.connect` :meth:`Vertex.connect` - """ - self.v1 = v1 - self.v2 = v2 - - self.data = data - - # try to compute edge cost as metric distance if not given - if cost is not None: - self.cost = cost - elif not (v1 is None or v1.coord is None or v2 is None or v2.coord is None): - self.cost = v1._graph.metric(v1.coord - v2.coord) - else: - self.cost = None - - def __repr__(self): - return str(self) - - def __str__(self): - - s = f"Edge{{{self.v1} -- {self.v2}, cost={self.cost:.4g}}}" - if self.data is not None: - s += f" data={self.data}" - return s - - def connect(self, v1, v2): - """ - Add edge to the graph - - :param v1: start of the edge - :type v1: Vertex subclass - :param v2: end of the edge - :type v2: Vertex subclass - - The edge is added to the graph and connects vertices ``v1`` and ``v2``. - - .. note:: The vertices must already be added to the graph. - """ - - if v1._graph is None: - raise ValueError("vertex v1 does not belong to a graph") - if v2._graph is None: - raise ValueError("vertex v2 does not belong to a graph") - if not v1._graph is v2._graph: - raise ValueError("vertices must belong to the same graph") - - # connect edge to its vertices - self.v1 = v1 - self.v2 = v2 - - # tell the vertices to add edge to their edgelists as appropriate for - # DGraph or UGraph - v1.connect(v2, edge=self) - - def next(self, vertex): - """ - Return other end of an edge - - :param vertex: one vertex on the edge - :type vertex: Vertex subclass - :raises ValueError: ``vertex`` is not on the edge - :return: the other vertex on the edge - :rtype: Vertex subclass - - ``e.next(v1)`` is the vertex at the other end of edge ``e``, ie. the - vertex that is not ``v1``. - """ - - if self.v1 is vertex: - return self.v2 - elif self.v2 is vertex: - return self.v1 - else: - raise ValueError("shouldnt happen") - - def vertices(self): - raise DeprecationWarning("use endpoints instead") - - @property - def endpoints(self): - return [self.v1, self.v2] - - # def remove(self): - # """ - # Remove edge from graph - - # ``e.remove()`` removes ``e`` from the graph, but does not delete the - # edge object. - # """ - # # remove this edge from the edge list of both end vertices - # if self in self.v1._edgelist: - # self.v1._edgelist.remove(self) - # if self in self.v2._edgelist: - # self.v2._edgelist.remove(self) - - # # indicate that connectivity has changed - # self.v1._connectivitychange = True - # self.v2._connectivitychange = True - - # # remove references to the vertices - # self.v1 = None - # self.v2 = None - - -# ========================================================================== # - - -class Vertex: - """ - Superclass for vertices of directed and non-directed graphs. - - Each vertex has: - - ``name`` - - ``label`` an int indicating which graph component contains it - - ``_edgelist`` a list of edge objects that connect this vertex to others - - ``coord`` the coordinate in an embedded graph (optional) - """ - - def __init__(self, coord=None, name=None): - self._edgelist = [] - if coord is None: - self.coord = None - else: - self.coord = np.r_[coord] - self.name = name - self.label = None - self._connectivitychange = True - self._edgelist = [] - self._graph = None # reference to owning graph - # print('Vertex init', type(self)) - - def __str__(self): - return f"[{self.name:s}]" - - def __repr__(self): - if self.coord is None: - coord = "?" - else: - coord = ", ".join([f"{x:.4g}" for x in self.coord]) - return f"{self.__class__.__name__}[{self.name:s}, coord=({coord})]" - - def copy(self, cls=None): - if cls is not None: - return cls.vertex_copy(self) - else: - return self.__class__(coord=self.coord, name=self.name) - - def neighbours(self): - """ - Neighbours of a vertex - - ``v.neighbours()`` is a list of neighbours of this vertex. - - .. note:: For a directed graph the neighbours are those on edges leaving this vertex - """ - return [e.next(self) for e in self._edgelist] - - def neighbors(self): - """ - Neighbors of a vertex - - ``v.neighbors()`` is a list of neighbors of this vertex. - - .. note:: For a directed graph the neighbours are those on edges leaving this vertex - """ - return [e.next(self) for e in self._edgelist] - - def isneighbour(self, vertex): - """ - Test if vertex is a neigbour - - :param vertex: vertex reference - :type vertex: Vertex subclass - :return: true if a neighbour - :rtype: bool - - For a directed graph this is true only if the edge is from ``self`` to - ``vertex``. - """ - return vertex in [e.next(self) for e in self._edgelist] - - def incidences(self): - """ - Neighbours and edges of a vertex - - ``v.incidences()`` is a generator that returns a list of incidences, - tuples of (vertex, edge) for all neighbours of the vertex ``v``. - - .. note:: For a directed graph the edges are those leaving this vertex - """ - return [(e.next(self), e) for e in self._edgelist] - - def connect(self, dest, edge=None, cost=None, data=None): - """ - Connect two vertices with an edge - - :param dest: The vertex to connect to - :type dest: ``Vertex`` subclass - :param edge: Use this as the edge object, otherwise a new ``Edge`` - object is created from the vertices being connected, - and the ``cost`` and ``edge`` parameters, defaults to None - :type edge: ``Edge`` subclass, optional - :param cost: the cost to traverse this edge, defaults to None - :type cost: float, optional - :param data: reference to arbitrary data associated with the edge, - defaults to None - :type data: Any, optional - :raises TypeError: vertex types are different subclasses - :return: the edge connecting the vertices - :rtype: Edge - - ``v1.connect(v2)`` connects vertex ``v1`` to vertex ``v2``. - - .. note:: - - - If the vertices subclass ``UVertex`` the edge is undirected, and if - they subclass ``DVertex`` the edge is directed. - - Vertices must both be of the same ``Vertex`` subclass - - :seealso: :meth:`Edge` - """ - - if not dest.__class__.__bases__[0] is self.__class__.__bases__[0]: - raise TypeError("must connect vertices of same type") - elif isinstance(edge, Edge): - e = edge - else: - e = Edge(self, dest, cost=cost, data=data) - - self._graph._edgelist.add(e) - self._graph._connectivitychange = True - self._connectivitychange = True - - return e - - def edgeto(self, dest): - """ - Get edge connecting vertex to specific neighbour - - :param dest: a neigbouring vertex - :type dest: ``Vertex`` subclass - :raises ValueError: ``dest`` is not a neighbour - :return: the edge from this vertex to ``dest`` - :rtype: Edge - - .. note:: - - - For a directed graph ``dest`` must be at the arrow end of the edge - """ - for n, e in self.incidences(): - if n is dest: - return e - raise ValueError("dest is not a neighbour") - - def edges(self): - """ - All outgoing edges of vertex - - :return: List of all edges leaving this vertex - :rtype: list of Edge - - .. note:: - - - For a directed graph the edges are those leaving this vertex - - For a non-directed graph the edges are those leaving or entering - this vertex - """ - return self._edgelist - - def heuristic_distance(self, v2): - return self._graph.heuristic(self.coord - v2.coord) - - def distance(self, coord): - """ - Distance from vertex to point - - :param coord: coordinates of the point - :type coord: ndarray(n) or Vertex - :return: distance - :rtype: float - - Distance is computed according to the graph's metric. - - :seealso: :meth:`metric` - """ - if isinstance(coord, Vertex): - coord = coord.coord - return self._graph.metric(self.coord - coord) - - @property - def degree(self): - """ - Degree of vertex - - :return: degree of the vertex - :rtype: int - - Returns the number of edges connected to the vertex. - - .. note:: For a ``DGraph`` only outgoing edges are considered. - - :seealso: :meth:`edges` - """ - return len(self.edges()) - - @property - def x(self): - """ - The x-coordinate of an embedded vertex - - :return: The x-coordinate - :rtype: float - """ - return self.coord[0] - - @property - def y(self): - """ - The y-coordinate of an embedded vertex - - :return: The y-coordinate - :rtype: float - """ - return self.coord[1] - - @property - def z(self): - """ - The z-coordinate of an embedded vertex - - :return: The z-coordinate - :rtype: float - """ - return self.coord[2] - - def closest(self): - return self._graph.closest(self.coord) - - -class UVertex(Vertex): - """ - Vertex subclass for undirected graphs - - This class can be inherited to provide user objects with graph capability. - - - .. inheritance-diagram:: UVertex - - """ - - def connect(self, other, **kwargs): - - if isinstance(other, Vertex): - e = super().connect(other, **kwargs) - elif isinstance(other, Edge): - e = super().connect(edge=other) - else: - raise TypeError("bad argument") - - # e = super().connect(other, **kwargs) - - self._edgelist.append(e) - other._edgelist.append(e) - self._graph._edgelist.add(e) - - return e - - -class DVertex(Vertex): - """ - Vertex subclass for directed graphs - - This class can be inherited to provide user objects with graph capability. - - .. inheritance-diagram:: DVertex - - """ - - def connect(self, other, **kwargs): - if isinstance(other, Vertex): - e = super().connect(other, **kwargs) - elif isinstance(other, Edge): - e = super().connect(edge=other) - else: - raise TypeError("bad argument") - - self._edgelist.append(e) - return e - - def remove(self): - self._edgelist = None # remove all references to edges - - -if __name__ == "__main__": - - g = UGraph() - print(g) - - for i in range(10): - g.add_vertex() - - g.add_edge(g[0], g[1]) - - print(g) diff --git a/pgraph/__init__.py b/pgraph/__init__.py deleted file mode 100644 index 011d41c7..00000000 --- a/pgraph/__init__.py +++ /dev/null @@ -1 +0,0 @@ -from pgraph.PGraph import UGraph, DGraph, UVertex, DVertex, Edge diff --git a/planning.html b/planning.html new file mode 100644 index 00000000..712d9559 --- /dev/null +++ b/planning.html @@ -0,0 +1,220 @@ + + + + + + + + + Path planning — Simple graph functionality for Python documentation + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

Path planning

+

These methods find optimal paths from a start vertex to a goal vertex.

+
+
+class PGraph.PGraph(arg=None, metric=None, heuristic=None, verbose=False)[source]
+
+
+path_Astar(S, G, verbose=False, summary=False)[source]
+

A* search for path

+
+
Parameters:
+
    +
  • S (Vertex subclass) – start vertex

  • +
  • G (Vertex subclass) – goal vertex

  • +
+
+
Returns:
+

list of vertices from S to G inclusive, path length, tree

+
+
Return type:
+

list of Vertex subclass, float, dict

+
+
+

Returns a list of vertices that form a path from vertex S to +vertex G if possible, otherwise return None.

+

The search tree is returned as dict that maps a vertex to its parent.

+

The heuristic is the distance metric of the graph, which defaults to +Euclidean distance.

+
+
Seealso:
+

heuristic()

+
+
+
+ +
+
+path_BFS(S, G, verbose=False, summary=False)[source]
+

Breadth-first search for path

+
+
Parameters:
+
    +
  • S (Vertex subclass) – start vertex

  • +
  • G (Vertex subclass) – goal vertex

  • +
+
+
Returns:
+

list of vertices from S to G inclusive, path length

+
+
Return type:
+

list of Vertex subclass, float

+
+
+

Returns a list of vertices that form a path from vertex S to +vertex G if possible, otherwise return None.

+
+ +
+
+path_UCS(S, G, verbose=False, summary=False)[source]
+

Uniform cost search for path

+
+
Parameters:
+
    +
  • S (Vertex subclass) – start vertex

  • +
  • G (Vertex subclass) – goal vertex

  • +
+
+
Returns:
+

list of vertices from S to G inclusive, path length, tree

+
+
Return type:
+

list of Vertex subclass, float, dict

+
+
+

Returns a list of vertices that form a path from vertex S to +vertex G if possible, otherwise return None.

+

The search tree is returned as dict that maps a vertex to its parent.

+

The heuristic is the distance metric of the graph, which defaults to +Euclidean distance.

+
+ +
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/py-modindex.html b/py-modindex.html new file mode 100644 index 00000000..b680217a --- /dev/null +++ b/py-modindex.html @@ -0,0 +1,140 @@ + + + + + + + + Python Module Index — Simple graph functionality for Python documentation + + + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
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+ + +

Python Module Index

+ +
+ p +
+ + + + + + + +
 
+ p
+ PGraph +
+ + +
+
+
+ +
+ +
+

© Copyright 2020, Peter Corke. + Last updated on 08-Jan-2025. +

+
+ + Built with Sphinx using a + theme + provided by Read the Docs. + + +
+
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/pyproject.toml b/pyproject.toml deleted file mode 100644 index db85ddf5..00000000 --- a/pyproject.toml +++ /dev/null @@ -1,73 +0,0 @@ -[project] -name = "pgraph-python" -version = "0.6.3" -authors = [{ name = "Peter Corke", email = "rvc@petercorke.com" }] -description = "Mathematical graphs for Python" -readme = "README.md" -requires-python = ">=3.7" -classifiers = [ - "Development Status :: 5 - Production/Stable", - # Indicate who your project is intended for - "Intended Audience :: Developers", - # Specify the Python versions you support here. - "Programming Language :: Python :: 3.9", - "Programming Language :: Python :: 3.10", - "Programming Language :: Python :: 3.11", - "Programming Language :: Python :: 3.12", - "Programming Language :: Python :: 3.13", - - "License :: OSI Approved :: MIT License", - "Operating System :: OS Independent", -] -keywords = [ - "python", - "graph", - "directed graph", - "undirected graph", - "a* search", - "adjacency matrix", - "Laplacian matrix", - "plotting", - "optimal path", -] - -dependencies = ["spatialmath-python", "numpy>=1.18.0", "scipy", "matplotlib"] - -[project.urls] - -"Homepage" = "https://github.com/petercorke/pgraph-python" -"Bug Tracker" = "https://github.com/pypa/sampleproject/issues" -"Documentation" = "https://petercorke.github.io/pgraph-python" -"GitHub Source" = "https://github.com/petercorke/pgraph-python" -"Changelog" = "https://github.com/petercorke/pgraph-python/blob/master/CHANGELOG.md" -"Coverage" = "https://codecov.io/gh/petercorke/spatialmath-python" - -[project.optional-dependencies] - -dev = ["pytest", "pytest-cov"] - -docs = [ - "sphinx", - "recommonmark", - "sphinx-rtd-theme", - "sphinx-autorun", - "sphinx-favicon", -] - -[tool.pytest.ini_options] -pythonpath = [".", "pgraph"] - - -[build-system] -requires = ["setuptools", "oldest-supported-numpy"] -build-backend = "setuptools.build_meta" - -[tool.setuptools] -packages = ["pgraph"] - -[tool.black] -line-length = 88 -target-version = ['py311'] - -[tool.coverage.run] -omit = [] diff --git a/search.html b/search.html new file mode 100644 index 00000000..9e3873f8 --- /dev/null +++ b/search.html @@ -0,0 +1,137 @@ + + + + + + + + Search — Simple graph functionality for Python documentation + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
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  • + +
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  • +
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+ + + + +
+ +
+ +
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+ +
+ +
+

© Copyright 2020, Peter Corke. + Last updated on 08-Jan-2025. +

+
+ + Built with Sphinx using a + theme + provided by Read the Docs. + + +
+
+
+
+
+ + + + + + + + + + + + \ No newline at end of file diff --git a/searchindex.js b/searchindex.js new file mode 100644 index 00000000..1cdede34 --- /dev/null +++ b/searchindex.js @@ -0,0 +1 @@ +Search.setIndex({"alltitles": {"Convert graph to matrix form": [[6, null]], "Directed graph": [[1, null]], "Directed graph vertex": [[3, null]], "Directed graphs": [[2, null]], "Edge": [[4, null]], "Graphs for Python": [[5, null]], "Indices and tables": [[5, "indices-and-tables"]], "PGraph module": [[0, null]], "Path planning": [[8, null]], "Undirected graph": [[9, null]], "Undirected graph vertex": [[11, null]], "Undirected graphs": [[10, null]], "pgraph": [[7, null]]}, "docnames": ["PGraph", "dgraph", "directed", "dvertex", "edge", "index", "matrices", "modules", "planning", "ugraph", "undirected", "uvertex"], "envversion": {"sphinx": 64, "sphinx.domains.c": 3, "sphinx.domains.changeset": 1, "sphinx.domains.citation": 1, "sphinx.domains.cpp": 9, "sphinx.domains.index": 1, 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"modul": 0, "path": 8, "pgraph": [0, 7], "plan": 8, "python": 5, "tabl": 5, "undirect": [9, 10, 11], "vertex": [3, 11]}}) \ No newline at end of file diff --git a/tests/test_graph.py b/tests/test_graph.py deleted file mode 100644 index f0541b24..00000000 --- a/tests/test_graph.py +++ /dev/null @@ -1,737 +0,0 @@ - -import unittest -import numpy as np -import numpy.testing as nt - -from pgraph import * - -class TestUGraph(unittest.TestCase): - - def test_constructor(self): - - g = UGraph() - - v1 = g.add_vertex() - v2 = g.add_vertex() - self.assertEqual(g.n, 2) - self.assertIsInstance(v1, UVertex) - self.assertIsInstance(v2, UVertex) - - def test_constructor2(self): - - g = UGraph() - - v = g.add_vertex([1,2,3]) - self.assertIsInstance(v, UVertex) - self.assertEqual(v.x, 1) - self.assertEqual(v.y, 2) - self.assertEqual(v.z, 3) - - def test_str(self): - g = UGraph() - - v0 = g.add_vertex([1,2,3]) - v1 = g.add_vertex([1,2,3]) - v0.connect(v1) - - self.assertEqual(str(g), "UGraph: 2 vertices, 1 edge, 1 component") - - g = DGraph() - - v0 = g.add_vertex([1,2,3]) - v1 = g.add_vertex([1,2,3]) - v0.connect(v1) - - self.assertEqual(str(g), "DGraph: 2 vertices, 1 edge, 1 component") - - - s = repr(g) - self.assertIsInstance(s, str) - self.assertEqual(len(s.split('\n')), 2) - - def test_attr(self): - - g = UGraph() - - v1 = g.add_vertex(name='v1') - self.assertEqual(v1.name, 'v1') - - v1 = g.add_vertex(coord=[1,2,3]) - self.assertIsInstance(v1.coord, np.ndarray) - self.assertEqual(v1.coord.shape, (3,)) - self.assertEqual(list(v1.coord), [1,2,3]) - - def test_constructor3(self): - - g = UGraph() - v1 = g.add_vertex() - v2 = g.add_vertex() - v3 = g.add_vertex() - - self.assertIs(g[0], v1) - self.assertIs(g[0], v1) - self.assertIs(g[0], v1) - - class MyNode(UVertex): - def __init__(self, a): - super().__init__() - self.a = a - - v1 = g.add_vertex(MyNode(1)) - v2 = g.add_vertex(MyNode(2)) - - self.assertIsInstance(v1, MyNode) - v1.connect(v2) - self.assertEqual(v1.neighbours()[0].a, 2) - - def test_neighbours(self): - g = UGraph() - v1 = g.add_vertex(name='v1') - v2 = g.add_vertex(name='v2') - v3 = g.add_vertex(name='v3') - v4 = g.add_vertex(name='v4') - v1.connect(v2) - v1.connect(v3) - - n = v1.neighbours() - self.assertTrue(len(n) == 2) - self.assertTrue(v2 in n) - self.assertTrue(v3 in n) - self.assertFalse(v1 in n) - self.assertFalse(v4 in n) - - n = v2.neighbours() - self.assertTrue(len(n) == 1) - self.assertTrue(v1 in n) - self.assertFalse(v2 in n) - self.assertFalse(v3 in n) - self.assertFalse(v4 in n) - - - g = UGraph() - v1 = g.add_vertex(name='v1') - v2 = g.add_vertex(name='v2') - v3 = g.add_vertex(name='v3') - v1.connect(v2) - - self.assertTrue(v1.isneighbour(v2)) - self.assertTrue(v2.isneighbour(v1)) - self.assertFalse(v1.isneighbour(v3)) - self.assertFalse(v3.isneighbour(v1)) - - g = DGraph() - v1 = g.add_vertex(name='v1') - v2 = g.add_vertex(name='v2') - v3 = g.add_vertex(name='v3') - v4 = g.add_vertex(name='v4') - v1.connect(v2) - v1.connect(v3) - - n = v1.neighbours() - self.assertTrue(len(n) == 2) - self.assertTrue(v2 in n) - self.assertTrue(v3 in n) - self.assertFalse(v1 in n) - self.assertFalse(v4 in n) - - n = v2.neighbours() - self.assertTrue(len(n) == 0) - self.assertFalse(v1 in n) - self.assertFalse(v2 in n) - self.assertFalse(v3 in n) - self.assertFalse(v4 in n) - - g = DGraph() - v1 = g.add_vertex(name='v1') - v2 = g.add_vertex(name='v2') - v3 = g.add_vertex(name='v3') - v1.connect(v2) - self.assertTrue(v1.isneighbour(v2)) - self.assertFalse(v2.isneighbour(v1)) - self.assertFalse(v1.isneighbour(v3)) - self.assertFalse(v3.isneighbour(v1)) - - def test_getitem(self): - g = UGraph() - v1 = g.add_vertex(name='v1') - v2 = g.add_vertex(name='v2') - v3 = g.add_vertex(name='v3') - - self.assertIs(g[0], v1) - self.assertIs(g[1], v2) - self.assertIs(g[2], v3) - - self.assertIs(g['v1'], v1) - self.assertIs(g['v2'], v2) - self.assertIs(g['v3'], v3) - - self.assertIs(g[v1], v1) - self.assertIs(g[v2], v2) - self.assertIs(g[v3], v3) - - v = [v for v in g] - self.assertEqual(len(v), 3) - self.assertEqual(v, [v1, v2, v3]) - - def test_connect(self): - - g = UGraph() - v1 = g.add_vertex() - v2 = g.add_vertex() - v3 = g.add_vertex() - e12 = v1.connect(v2) - e13 = v1.connect(v3) - - self.assertEqual(g.n, 3) - self.assertEqual(g.ne, 2) - - self.assertIsInstance(e12, Edge) - self.assertIsInstance(e12, Edge) - - self.assertTrue(e12 in v1.edges()) - self.assertTrue(e12 in v2.edges()) - self.assertFalse(e12 in v3.edges()) - - self.assertTrue(e13 in v1.edges()) - self.assertTrue(e13 in v3.edges()) - self.assertFalse(e13 in v2.edges()) - - - def test_remove_node(self): - - g = UGraph() - v1 = g.add_vertex() - v2 = g.add_vertex() - v3 = g.add_vertex() - e12 = v1.connect(v2) - e13 = v1.connect(v3) - - g.remove(v1) - self.assertEqual(g.n, 2) - self.assertEqual(g.ne, 0) - self.assertEqual(g.nc, 2) - - self.assertFalse(e12 in v2.edges()) - self.assertFalse(e12 in v3.edges()) - - self.assertFalse(e13 in v3.edges()) - self.assertFalse(e13 in v2.edges()) - - def test_remove_edge(self): - - g = UGraph() - v1 = g.add_vertex() - v2 = g.add_vertex() - v3 = g.add_vertex() - e12 = v1.connect(v2) - e13 = v1.connect(v3) - - g.remove(e12) - self.assertEqual(g.n, 3) - self.assertEqual(g.ne, 1) - self.assertEqual(g.nc, 2) - - self.assertFalse(e12 in v1.edges()) - self.assertFalse(e12 in v2.edges()) - - self.assertTrue(e13 in v1.edges()) - self.assertTrue(e13 in v3.edges()) - - def test_edge1(self): - - g = UGraph() - v1 = g.add_vertex() - v2 = g.add_vertex() - v3 = g.add_vertex() - v4 = g.add_vertex() - v1.connect(v2) - v1.connect(v3) - - self.assertEqual(len(v1.edges()), 2) - self.assertEqual(len(v2.edges()), 1) - self.assertEqual(len(v3.edges()), 1) - - self.assertEqual(len(v1.neighbours()), 2) - self.assertEqual(len(v2.neighbours()), 1) - self.assertEqual(len(v3.neighbours()), 1) - - self.assertTrue(v2 in v1.neighbours()) - self.assertTrue(v3 in v1.neighbours()) - self.assertFalse(v4 in v1.neighbours()) - - def test_edge2(self): - - g = UGraph() - v1 = g.add_vertex(name='n1') - v2 = g.add_vertex(name='n2') - v3 = g.add_vertex(name='n3') - v4 = g.add_vertex(name='n4') - - g.add_edge('n1', 'n2') - g.add_edge('n1', 'n3') - - self.assertEqual(len(v1.edges()), 2) - self.assertEqual(len(v2.edges()), 1) - self.assertEqual(len(v3.edges()), 1) - - self.assertEqual(len(v1.neighbours()), 2) - self.assertEqual(len(v2.neighbours()), 1) - self.assertEqual(len(v3.neighbours()), 1) - - self.assertTrue(v2 in v1.neighbours()) - self.assertTrue(v3 in v1.neighbours()) - self.assertFalse(v4 in v1.neighbours()) - - def test_edge3(self): - - g = UGraph() - v1 = g.add_vertex(name='n1') - v2 = g.add_vertex(name='n2') - v3 = g.add_vertex(name='n3') - v4 = g.add_vertex(name='n4') - - g.add_edge(v1, v2) - g.add_edge(v1, v3) - - self.assertEqual(len(v1.edges()), 2) - self.assertEqual(len(v2.edges()), 1) - self.assertEqual(len(v3.edges()), 1) - - self.assertEqual(len(v1.neighbours()), 2) - self.assertEqual(len(v2.neighbours()), 1) - self.assertEqual(len(v3.neighbours()), 1) - - self.assertTrue(v2 in v1.neighbours()) - self.assertTrue(v3 in v1.neighbours()) - self.assertFalse(v4 in v1.neighbours()) - - - def test_edgeto(self): - - g = UGraph() - v1 = g.add_vertex() - v2 = g.add_vertex() - v3 = g.add_vertex() - e12 = v1.connect(v2) - e13 = v1.connect(v3) - - self.assertIsInstance(v1.edgeto(v2), Edge) - self.assertIs(v1.edgeto(v2), e12) - self.assertIs(v1.edgeto(v2), e12) - - def test_add_vertex(self): - - g = UGraph() - - v = UVertex() - - g.add_vertex(v) - self.assertEqual(v.name, '#0') - self.assertTrue(v in g) - self.assertTrue(v._graph, g) - - def test_properties(self): - - g = UGraph() - v1 = g.add_vertex() - v2 = g.add_vertex() - v3 = g.add_vertex() - v4 = g.add_vertex() - e12 = v1.connect(v2) - e13 = v1.connect(v3) - - self.assertEqual(g.n, 4) - self.assertEqual(g.ne, 2) - self.assertEqual(g.average_degree(), 1.0) - - g = DGraph() - v1 = g.add_vertex() - v2 = g.add_vertex() - v3 = g.add_vertex() - v4 = g.add_vertex() - e12 = v1.connect(v2) - e13 = v1.connect(v3) - - self.assertEqual(g.n, 4) - self.assertEqual(g.ne, 2) - self.assertEqual(g.average_degree(), 0.5) - - def test_contains(self): - - g = UGraph() - v1 = g.add_vertex() - v2 = g.add_vertex() - - self.assertTrue('#0' in g) - self.assertFalse('#2' in g) - - self.assertTrue(v1 in g) - g2 = UGraph() - self.assertFalse(v1 in g2) - - def test_Dict(self): - - v1 = UVertex(name='v1') - v2 = UVertex(name='v2') - v3 = UVertex(name='v3') - v4 = UVertex(name='v4') - - parent = {} - parent[v2] = v1 - parent[v3] = v1 - parent[v4] = v3 - - g = UGraph.Dict(parent) - - self.assertIsInstance(g, UGraph) - self.assertEqual(g.n, 4) - self.assertEqual(g.ne, 3) - self.assertTrue('v1' in g) - self.assertTrue('v2' in g) - self.assertTrue('v3' in g) - self.assertTrue('v4' in g) - - self.assertTrue(g['v2'] in g['v1'].neighbours()) - self.assertTrue(g['v3'] in g['v1'].neighbours()) - self.assertTrue(g['v4'] in g['v3'].neighbours()) - - def test_Adjacency(self): - A = np.zeros((5, 5)) - A[1,2] = 5 # 1 <--> 2 - A[3,4] = 2 # 3 <--> 4 - - coords = np.random.rand(5, 3) - names = "zero one two three four".split(" ") - - g = UGraph.Adjacency(A, coords, names) - self.assertIsInstance(g, UGraph) - self.assertEqual(g.n, 5) - self.assertEqual(g.ne, 2) - self.assertTrue(g['two'] in g['one'].neighbours()) - self.assertTrue(g['three'] in g['four'].neighbours()) - e = g['two'].edgeto(g['one']) - self.assertEqual(e.cost, 5) - e = g['three'].edgeto(g['four']) - self.assertEqual(e.cost, 2) - nt.assert_almost_equal(g['two'].coord, coords[2,:]) - - A = np.zeros((5, 5)) - A[1,2] = 5 # 1 --> 2 - A[3,4] = 2 # 3 --> 4 - - coords = np.random.rand(5, 3) - names = "zero one two three four".split(" ") - - g = DGraph.Adjacency(A, coords, names) - self.assertIsInstance(g, DGraph) - self.assertEqual(g.n, 5) - self.assertEqual(g.ne, 2) - self.assertTrue(g['two'] in g['one'].neighbours()) - self.assertTrue(g['four'] in g['three'].neighbours()) - e = g['one'].edgeto(g['two']) - self.assertEqual(e.cost, 5) - e = g['three'].edgeto(g['four']) - self.assertEqual(e.cost, 2) - nt.assert_almost_equal(g['two'].coord, coords[2,:]) - - - # def test_remove_edge(self): - - # g = UGraph() - # v1 = g.add_vertex() - # v2 = g.add_vertex() - # v3 = g.add_vertex() - # e12 = v1.connect(v2) - # e13 = v1.connect(v3) - - # self.assertEqual(g.nc, 1) - # e12.remove() - - # self.assertEqual(g.nc, 2) - - # self.assertEqual(len(v1.edges()), 1) - # self.assertEqual(len(v2.edges()), 0) - # self.assertEqual(len(v3.edges()), 1) - - # self.assertEqual(len(v1.neighbours()), 1) - # self.assertEqual(len(v2.neighbours()), 0) - # self.assertEqual(len(v3.neighbours()), 1) - - # def test_remove_vertex(self): - - # g = UGraph() - # v1 = g.add_vertex() - # v2 = g.add_vertex() - # v3 = g.add_vertex() - # e12 = v1.connect(v2) - # e13 = v1.connect(v3) - - # self.assertEqual(g.n, 3) - # self.assertEqual(g.nc, 1) - # g.remove(v1) - - # self.assertEqual(g.n, 2) - # self.assertEqual(g.nc, 2) - - # self.assertEqual(len(v1.edges()), 0) - # self.assertEqual(len(v2.edges()), 0) - # self.assertEqual(len(v3.edges()), 0) - - # self.assertEqual(len(v1.neighbours()), 0) - # self.assertEqual(len(v2.neighbours()), 0) - # self.assertEqual(len(v3.neighbours()), 0) - - def test_components(self): - - g = UGraph() - v1 = g.add_vertex() - v2 = g.add_vertex() - v3 = g.add_vertex() - - self.assertEqual(g.nc, 3) - v1.connect(v2) - self.assertEqual(g.nc, 2) - v1.connect(v3) - self.assertEqual(g.nc, 1) - - def test_matrices(self): - g = UGraph() - v1 = g.add_vertex() - v2 = g.add_vertex() - v3 = g.add_vertex() - v4 = g.add_vertex() - e12 = v1.connect(v2) - e13 = v1.connect(v3) - - A = g.adjacency() - self.assertIsInstance(A, np.ndarray) - self.assertEqual(A.shape, (g.n, g.n)) - - A = g.Laplacian() - self.assertIsInstance(A, np.ndarray) - self.assertEqual(A.shape, (g.n, g.n)) - - A = g.incidence() - self.assertIsInstance(A, np.ndarray) - self.assertEqual(A.shape, (g.n, g.ne)) - - A = g.distance() - self.assertIsInstance(A, np.ndarray) - self.assertEqual(A.shape, (g.n, g.n)) - - def test_metric(self): - g = UGraph() - v1 = g.add_vertex([1,2,3]) - p = [7,6,6] - self.assertAlmostEqual(v1.distance(p), np.sqrt(61)) - - g = UGraph(metric='L2') - v1 = g.add_vertex([1,2,3]) - p = [7,6,6] - self.assertAlmostEqual(v1.distance(p), np.sqrt(61)) - - g = UGraph(metric='L1') - v1 = g.add_vertex([1,2,3]) - p = [7,6,6] - self.assertEqual(v1.distance(p), 13) - - g = UGraph(metric='SE2') - v1 = g.add_vertex([1,2,0]) - p = [7,6,0] - self.assertAlmostEqual(v1.distance(p), np.sqrt(52)) - p = [7,6,2*np.pi] - self.assertAlmostEqual(v1.distance(p), np.sqrt(52)) - p = [7,6,-2*np.pi] - self.assertAlmostEqual(v1.distance(p), np.sqrt(52)) - p = [7,6,4*np.pi] - self.assertAlmostEqual(v1.distance(p), np.sqrt(52)) - p = [7,6,-4*np.pi] - self.assertAlmostEqual(v1.distance(p), np.sqrt(52)) - - p = [7,6,np.pi] - self.assertAlmostEqual(v1.distance(p), np.sqrt(52+np.pi**2)) - - v2 = g.add_vertex([1,2,np.pi/2]) - p = [7,6,np.pi/2] - self.assertAlmostEqual(v2.distance(p), np.sqrt(52)) - p = [7,6,-np.pi/2] - self.assertAlmostEqual(v2.distance(p), np.sqrt(52+np.pi**2)) - - def test_heuristic(self): - g = UGraph() - p = [2, 3, 4] - self.assertAlmostEqual(g.heuristic(p), np.sqrt(29)) - - g = UGraph(heuristic='L2') - p = [2, 3, 4] - self.assertAlmostEqual(g.heuristic(p), np.sqrt(29)) - - g = UGraph(heuristic='L1') - p = [2, 3, 4] - self.assertAlmostEqual(g.heuristic(p), 9) - - - def test_closest(self): - g = UGraph() - v1 = g.add_vertex([1,2,3]) - v2 = g.add_vertex([4,5,6]) - v, d = g.closest([4, 5, 7]) - self.assertIs(v, v2) - self.assertEqual(d, 1) - - def test_BFS(self): - g = UGraph() - v1 = g.add_vertex(coord=[0,0], name='v1') - v2 = g.add_vertex(coord=[1,1], name='v2') - v3 = g.add_vertex(coord=[2,2], name='v3') - v4 = g.add_vertex(coord=[1,3], name='v4') - v5 = g.add_vertex(coord=[0,4], name='v5') - v6 = g.add_vertex(coord=[-5,2], name='v6') - v7 = g.add_vertex(coord=[0,6], name='v7') - - v1.connect(v2) - v2.connect(v3) - v3.connect(v4) - v4.connect(v5) - v1.connect(v6) - e = v6.connect(v5) - - p = g.path_UCS(v1, v7) - self.assertIsNone(p) - - p, length = g.path_BFS(v1, v5, verbose=True, summary=True) - self.assertIsInstance(p, list) - self.assertEqual(len(p), 3) - self.assertEqual(p, [v1, v6, v5]) - - def test_UCS(self): - g = UGraph() - v1 = g.add_vertex(coord=[0,0], name='v1') - v2 = g.add_vertex(coord=[1,1], name='v2') - v3 = g.add_vertex(coord=[2,2], name='v3') - v4 = g.add_vertex(coord=[1,3], name='v4') - v5 = g.add_vertex(coord=[0,4], name='v5') - v6 = g.add_vertex(coord=[-5,2], name='v6') - v7 = g.add_vertex(coord=[0,6], name='v7') - - v1.connect(v2) - v2.connect(v3) - v3.connect(v4) - v4.connect(v5) - v1.connect(v6) - e = v6.connect(v5) - - p = g.path_UCS(v1, v7) - self.assertIsNone(p) - - p, length, parent = g.path_UCS(v1, v5, verbose=True, summary=True) - self.assertIsInstance(p, list) - self.assertEqual(len(p), 5) - self.assertEqual(p, [v1, v2, v3, v4, v5]) - self.assertIsInstance(length, float) - self.assertAlmostEqual(length, 5.656854249492381) - self.assertIsInstance(parent, dict) - self.assertEqual(parent[v2.name], v1.name) - self.assertEqual(parent[v3.name], v2.name) - self.assertEqual(parent[v4.name], v3.name) - self.assertEqual(parent[v5.name], v4.name) - - def test_Astar(self): - g = UGraph() - v1 = g.add_vertex(coord=[0,0], name='v1') - v2 = g.add_vertex(coord=[1,1], name='v2') - v3 = g.add_vertex(coord=[2,2], name='v3') - v4 = g.add_vertex(coord=[1,3], name='v4') - v5 = g.add_vertex(coord=[0,4], name='v5') - v6 = g.add_vertex(coord=[-5,2], name='v6') - v7 = g.add_vertex(coord=[0,6], name='v7') - - v1.connect(v2) - v2.connect(v3) - v3.connect(v4) - v4.connect(v5) - v1.connect(v6) - e = v6.connect(v5) - - p = g.path_Astar(v1, v7) - self.assertIsNone(p) - - p, length, parent = g.path_Astar(v1, v5, verbose=True, summary=True) - self.assertIsInstance(p, list) - self.assertEqual(len(p), 5) - self.assertEqual(p, [v1, v2, v3, v4, v5]) - self.assertIsInstance(length, float) - self.assertAlmostEqual(length, 5.656854249492381) - - self.assertIsInstance(parent, dict) - self.assertEqual(parent[v2.name], v1.name) - self.assertEqual(parent[v3.name], v2.name) - self.assertEqual(parent[v4.name], v3.name) - self.assertEqual(parent[v5.name], v4.name) - - def test_plot(self): - - g = UGraph() - v1 = g.add_vertex(coord=[0,0], name='v1') - v2 = g.add_vertex(coord=[1,1], name='v2') - v3 = g.add_vertex(coord=[2,2], name='v3') - v4 = g.add_vertex(coord=[1,3], name='v4') - v5 = g.add_vertex(coord=[0,4], name='v5') - v6 = g.add_vertex(coord=[-5,2], name='v6') - v7 = g.add_vertex(coord=[0,6], name='v7') - - v1.connect(v2) - v2.connect(v3) - v3.connect(v4) - v4.connect(v5) - v1.connect(v6) - e = v6.connect(v5) - - g.plot() - - p, length, parent = g.path_Astar(v1, v5, verbose=True, summary=True) - g.highlight_path(p) - - def test_dotfile(self): - import pathlib - - g = UGraph() - v1 = g.add_vertex(coord=[0,0], name='v1') - v2 = g.add_vertex(coord=[1,1], name='v2') - v3 = g.add_vertex(coord=[2,2], name='v3') - v4 = g.add_vertex(coord=[1,3], name='v4') - v5 = g.add_vertex(coord=[0,4], name='v5') - v6 = g.add_vertex(coord=[-5,2], name='v6') - v7 = g.add_vertex(coord=[0,6], name='v7') - - path = pathlib.Path('./dotfile.dot') - g.dotfile(str(path)) - self.assertTrue(path.is_file()) - - g.showgraph() - -class TestDGraph(unittest.TestCase): - - def test_constructor(self): - - g = DGraph() - - v1 = g.add_vertex() - v2 = g.add_vertex() - self.assertEqual(g.n, 2) - self.assertIsInstance(v1, DVertex) - self.assertIsInstance(v2, DVertex) - -class TestGraph(unittest.TestCase): - - def test_print(self): - pass - - def test_plot(self): - pass - - -# ========================================================================== # - -if __name__ == "__main__": - unittest.main() \ No newline at end of file diff --git a/ugraph.html b/ugraph.html new file mode 100644 index 00000000..d70d83f3 --- /dev/null +++ b/ugraph.html @@ -0,0 +1,900 @@ + + + + + + + + + Undirected graph — Simple graph functionality for Python documentation + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

Undirected graph

+
+
+class PGraph.UGraph(arg=None, metric=None, heuristic=None, verbose=False)[source]
+

Bases: PGraph

+

Class for undirected graphs

+
Inheritance diagram of UGraph
+ + + +
+
+classmethod Adjacency(A, coords=None, names=None)
+

Create graph from adjacency matrix

+
+
Parameters:
+
    +
  • A (ndarray(N,N)) – adjacency matrix

  • +
  • coords (ndarray(N,M), optional) – coordinates of vertices, defaults to None

  • +
  • names (list(N) of str, optional) – names of vertices, defaults to None

  • +
+
+
Returns:
+

[description]

+
+
Return type:
+

[type]

+
+
+

Create a directed or undirected graph where non-zero elements A[i,j] +correspond to edges from vertex i to vertex j.

+
+

Warning

+

For undirected graph A should be symmetric but this +is not checked. Only the upper triangular part is used.

+
+
+ +
+
+classmethod Dict(d, reverse=False)
+

Create graph from parent/child dictionary

+
+
Parameters:
+
    +
  • d (dict) – dictionary that maps from Vertex subclass to Vertex subclass

  • +
  • reverse (bool, optional) – reverse link direction, defaults to False

  • +
+
+
Returns:
+

graph

+
+
Return type:
+

UGraph or DGraph

+
+
+

Behaves like a constructor for a DGraph or UGraph from a +dictionary that maps vertices to parents. From this information it +can create a tree graph.

+

By default parent vertices are linked their children. If reverse is +True then children are linked to their parents.

+
+ +
+
+Laplacian()
+

Laplacian matrix for the graph

+
+
Returns:
+

Laplacian matrix

+
+
Return type:
+

NumPy ndarray

+
+
+

g.Laplacian() is the Laplacian matrix (NxN) of the graph where N +is the number of vertices.

+
+

Note

+
    +
  • Laplacian is always positive-semidefinite.

  • +
  • Laplacian has at least one zero eigenvalue.

  • +
  • +
    The number of zero-valued eigenvalues is the number of connected

    components in the graph.

    +
    +
    +
  • +
+
+
+
Seealso:
+

adjacency() incidence() degree()

+
+
+
+ +
+
+__init__(arg=None, metric=None, heuristic=None, verbose=False)
+
+ +
+
+add_edge(v1, v2, **kwargs)
+

Add an edge to the graph (superclass method)

+
+
Parameters:
+
    +
  • v1 (Vertex subclass) – first vertex (start if a directed graph)

  • +
  • v2 (Vertex subclass) – second vertex (end if a directed graph)

  • +
  • kwargs – optional arguments to pass to Vertex.connect

  • +
+
+
Returns:
+

edge

+
+
Return type:
+

Edge

+
+
+

Create an edge between a vertex pair and adds it to the graph.

+

This is a graph centric way of creating an edge. The +alternative is the connect method of a vertex.

+
+
Seealso:
+

Edge.connect() Vertex.connect()

+
+
+
+ +
+
+add_vertex(coord=None, name=None)[source]
+

Add vertex to undirected graph

+
+
Parameters:
+
    +
  • coord (array-like, optional) – coordinate for an embedded graph, defaults to None

  • +
  • name (str, optional) – vertex name, defaults to “#i”

  • +
+
+
Returns:
+

new vertex

+
+
Return type:
+

UVertex

+
+
+
    +
  • g.add_vertex() creates a new vertex with optional coord and +name.

  • +
  • g.add_vertex(v) takes an instance or subclass of UVertex and adds +it to the graph

  • +
+
+ +
+
+adjacency()
+

Adjacency matrix of graph

+
+
Returns:
+

adjacency matrix

+
+
Return type:
+

ndarray(N,N)

+
+
+

The elements of the adjacency matrix [i,j] are 1 if vertex i is +connected to vertex j, else 0.

+
+

Note

+
    +
  • vertices are numbered in their order of creation. A vertex index +can be resolved to a vertex reference by graph[i].

  • +
  • for an undirected graph the matrix is symmetric

  • +
  • Eigenvalues of A are real and are known as the spectrum of the graph.

  • +
  • The element A[i,j] can be considered the number of walks of length one +edge from vertex i to vertex j (either zero or one).

  • +
  • If Ak = A ** k the element Ak[i,j] is the number of +walks of length k from vertex i to vertex j.

  • +
+
+
+
Seealso:
+

Laplacian() incidence() degree()

+
+
+
+ +
+
+average_degree()
+

Average degree of the graph

+
+
Returns:
+

average degree

+
+
Return type:
+

float

+
+
+

Average degree is \(2 E / N\) for an undirected graph and +\(E / N\) for a directed graph where \(E\) is the total number of +edges and \(N\) is the number of vertices.

+
+ +
+
+closest(coord)
+

Vertex closest to point

+
+
Parameters:
+

coord (ndarray(n)) – coordinates of a point

+
+
Returns:
+

closest vertex

+
+
Return type:
+

Vertex subclass

+
+
+

Returns the vertex closest to the given point. Distance is computed +according to the graph’s metric.

+
+
Seealso:
+

metric()

+
+
+
+ +
+
+component(c)
+

All vertices in specified graph component

+

graph.component(c) is a list of all vertices in graph component c.

+
+ +
+
+connectivity(vertices=None)
+

Graph connectivity

+
+
Returns:
+

a list with the number of edges per vertex

+
+
Return type:
+

list

+
+
+

The average vertex connectivity is:

+
mean(g.connectivity())
+
+
+

and the minimum vertex connectivity is:

+
min(g.connectivity())
+
+
+
+ +
+
+copy()
+

Deepcopy of graph

+
+
Parameters:
+

g (PGraph) – A graph

+
+
Returns:
+

deep copy

+
+
Return type:
+

PGraph

+
+
+
+ +
+
+degree()
+

Degree matrix of graph

+
+
Returns:
+

degree matrix

+
+
Return type:
+

ndarray(N,N)

+
+
+

This is a diagonal matrix where element [i,i] is the number +of edges connected to vertex id i.

+
+
Seealso:
+

adjacency() incidence() laplacian()

+
+
+
+ +
+
+distance()
+

Distance matrix of graph

+
+
Returns:
+

distance matrix

+
+
Return type:
+

ndarray(n,n)

+
+
+

The elements of the distance matrix D[i,j] is the edge cost of moving +from vertex i to vertex j. It is zero if the vertices are not +connected.

+
+ +
+
+dotfile(filename=None, direction=None)
+

Export graph as a GraphViz dot file

+
+
Parameters:
+

filename (str, optional) – filename to save graph to, defaults to None

+
+
+

g.dotfile() creates the specified file which contains the +GraphViz code to represent the embedded graph. By default output +is to the console.

+
+

Note

+
    +
  • The graph is undirected if it is a subclass of UGraph

  • +
  • The graph is directed if it is a subclass of DGraph

  • +
  • Use neato rather than dot to get the embedded layout

  • +
+
+
+

Note

+

If filename is a file object then the file will not +be closed after the GraphViz model is written.

+
+
+
Seealso:
+

showgraph()

+
+
+
+ +
+
+edges()
+

Get all edges in graph (superclass method)

+
+
Returns:
+

All edges in the graph

+
+
Return type:
+

list of Edge references

+
+
+

We can iterate over all edges in the graph by:

+
for e in g.edges():
+    print(e)
+
+
+
+

Note

+

The edges() of a Vertex is a list of all edges connected +to that vertex.

+
+
+
Seealso:
+

Vertex.edges()

+
+
+
+ +
+
+property heuristic
+

Get the heuristic distance metric for graph

+
+
Returns:
+

heuristic distance metric

+
+
Return type:
+

callable

+
+
+

This is a function of a vector and returns a scalar.

+
+ +
+
+highlight_edge(edge, scale=2, color='r', alpha=0.5)
+

Highlight an edge in the graph

+
+
Parameters:
+
    +
  • edge (Edge subclass) – The edge to highlight

  • +
  • scale (float, optional) – Overwrite with a line this much bigger than the original, +defaults to 1.5

  • +
  • color (str, optional) – Overwrite with a line in this color, defaults to ‘r’

  • +
+
+
+
+ +
+
+highlight_path(path, block=False, **kwargs)
+

Highlight a path through the graph

+
+
Parameters:
+
    +
  • path ([type]) – [description]

  • +
  • block (bool, optional) – [description], defaults to True

  • +
+
+
+

The vertices and edges along the path are overwritten with a different +size/width and color.

+
+
Seealso:
+

highlight_vertex() highlight_edge()

+
+
+
+ +
+
+highlight_vertex(vertex, scale=2, color='r', alpha=0.5)
+

Highlight a vertex in the graph

+
+
Parameters:
+
    +
  • edge (Vertex subclass) – The vertex to highlight

  • +
  • scale (float, optional) – Overwrite with a line this much bigger than the original, +defaults to 1.5

  • +
  • color (str, optional) – Overwrite with a line in this color, defaults to ‘r’

  • +
+
+
+
+ +
+
+incidence()
+

Incidence matrix of graph

+
+
Returns:
+

incidence matrix

+
+
Return type:
+

ndarray(n,ne)

+
+
+

The elements of the incidence matrix I[i,j] are 1 if vertex i is +connected to edge j, else 0.

+
+

Note

+
    +
  • vertices are numbered in their order of creation. A vertex index +can be resolved to a vertex reference by graph[i].

  • +
  • edges are numbered in the order they appear in graph.edges().

  • +
+
+
+
Seealso:
+

Laplacian() adjacency() degree()

+
+
+
+ +
+
+iscyclic()
+
+ +
+
+property metric
+

Get the distance metric for graph

+
+
Returns:
+

distance metric

+
+
Return type:
+

callable

+
+
+

This is a function of a vector and returns a scalar.

+
+ +
+
+property n
+

Number of vertices

+
+
Returns:
+

Number of vertices

+
+
Return type:
+

int

+
+
+
+ +
+
+property nc
+

Number of components

+
+
Returns:
+

Number of components

+
+
Return type:
+

int

+
+
+
+

Note

+
    +
  • Components are labeled from 0 to g.nc-1.

  • +
  • A graph coloring algorithm is run if the graph connectivity +has changed.

  • +
+
+
+

Note

+

A lazy approach is used, and if a connectivity changing +operation has been performed since the last call, the graph +coloring algorithm is run which is potentially expensive for +a large graph.

+
+
+ +
+
+property ne
+

Number of edges

+
+
Returns:
+

Number of vertices

+
+
Return type:
+

int

+
+
+
+ +
+
+path_Astar(S, G, verbose=False, summary=False)
+

A* search for path

+
+
Parameters:
+
    +
  • S (Vertex subclass) – start vertex

  • +
  • G (Vertex subclass) – goal vertex

  • +
+
+
Returns:
+

list of vertices from S to G inclusive, path length, tree

+
+
Return type:
+

list of Vertex subclass, float, dict

+
+
+

Returns a list of vertices that form a path from vertex S to +vertex G if possible, otherwise return None.

+

The search tree is returned as dict that maps a vertex to its parent.

+

The heuristic is the distance metric of the graph, which defaults to +Euclidean distance.

+
+
Seealso:
+

heuristic()

+
+
+
+ +
+
+path_BFS(S, G, verbose=False, summary=False)
+

Breadth-first search for path

+
+
Parameters:
+
    +
  • S (Vertex subclass) – start vertex

  • +
  • G (Vertex subclass) – goal vertex

  • +
+
+
Returns:
+

list of vertices from S to G inclusive, path length

+
+
Return type:
+

list of Vertex subclass, float

+
+
+

Returns a list of vertices that form a path from vertex S to +vertex G if possible, otherwise return None.

+
+ +
+
+path_UCS(S, G, verbose=False, summary=False)
+

Uniform cost search for path

+
+
Parameters:
+
    +
  • S (Vertex subclass) – start vertex

  • +
  • G (Vertex subclass) – goal vertex

  • +
+
+
Returns:
+

list of vertices from S to G inclusive, path length, tree

+
+
Return type:
+

list of Vertex subclass, float, dict

+
+
+

Returns a list of vertices that form a path from vertex S to +vertex G if possible, otherwise return None.

+

The search tree is returned as dict that maps a vertex to its parent.

+

The heuristic is the distance metric of the graph, which defaults to +Euclidean distance.

+
+ +
+
+plot(colorcomponents=True, force2d=False, vopt={}, eopt={}, text={}, block=False, grid=True, ax=None)
+

Plot the graph

+
+
Parameters:
+
    +
  • vopt (dict, optional) – vertex format, defaults to 12pt o-marker

  • +
  • eopt (dict, optional) – edge format, defaults to None

  • +
  • text (False or dict, optional) – text label format, defaults to None

  • +
  • colorcomponents – color vertices and edges by component, defaults to None

  • +
  • block (bool, optional) – block until figure is dismissed, defaults to True

  • +
+
+
+

The graph is plotted using matplotlib.

+

If colorcomponents is True then each component is assigned a unique +color. vertex and edge cannot include a color keyword.

+

If text is a dict it is used to format text labels for the vertices +which are the vertex names. If text is None default formatting is +used. If text is False no labels are added.

+
+ +
+
+remove(x)
+

Remove element from graph (superclass method)

+
+
Parameters:
+

x (Edge or Vertex subclass) – element to remove from graph

+
+
Raises:
+

TypeError – unknown type

+
+
+

The edge or vertex is removed, and all references and lists are +updated.

+
+

Warning

+

The connectivity of the network may be changed.

+
+
+ +
+
+samecomponent(v1, v2)
+

Test if vertices belong to same graph component

+
+
Parameters:
+
    +
  • v1 (Vertex subclass) – vertex

  • +
  • v2 (Vertex subclass) – vertex

  • +
+
+
Returns:
+

true if vertices belong to same graph component

+
+
Return type:
+

bool

+
+
+

Test whether vertices belong to the same component. For a:

+
    +
  • directed graph this implies a path between them

  • +
  • undirected graph there is not necessarily a path between them

  • +
+
+ +
+
+show()
+
+ +
+
+showgraph(**kwargs)
+

Display graph in a browser tab

+
+
Parameters:
+

kwargs – arguments passed to dotfile()

+
+
+

g.showgraph() renders and displays the graph in a browser tab. The +graph is exported in GraphViz format, rendered to +PDF using dot and then displayed in a browser tab.

+
+
Seealso:
+

dotfile()

+
+
+
+ +
+
+classmethod vertex_copy(vertex)[source]
+
+ +
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/undirected.html b/undirected.html new file mode 100644 index 00000000..089afa95 --- /dev/null +++ b/undirected.html @@ -0,0 +1,146 @@ + + + + + + + + + Undirected graphs — Simple graph functionality for Python documentation + + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

Undirected graphs

+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file diff --git a/uvertex.html b/uvertex.html new file mode 100644 index 00000000..23a72e3c --- /dev/null +++ b/uvertex.html @@ -0,0 +1,422 @@ + + + + + + + + + Undirected graph vertex — Simple graph functionality for Python documentation + + + + + + + + + + + + + + + + + + +
+ + +
+ +
+
+
+ +
+
+
+
+ +
+

Undirected graph vertex

+
+
+class PGraph.UVertex(coord=None, name=None)[source]
+

Bases: Vertex

+

Vertex subclass for undirected graphs

+

This class can be inherited to provide user objects with graph capability.

+
Inheritance diagram of UVertex
+ + + +
+
+__init__(coord=None, name=None)
+
+ +
+
+closest()
+
+ +
+
+connect(other, **kwargs)[source]
+

Connect two vertices with an edge

+
+
Parameters:
+
    +
  • dest (Vertex subclass) – The vertex to connect to

  • +
  • edge (Edge subclass, optional) – Use this as the edge object, otherwise a new Edge +object is created from the vertices being connected, +and the cost and edge parameters, defaults to None

  • +
  • cost (float, optional) – the cost to traverse this edge, defaults to None

  • +
  • data (Any, optional) – reference to arbitrary data associated with the edge, +defaults to None

  • +
+
+
Raises:
+

TypeError – vertex types are different subclasses

+
+
Returns:
+

the edge connecting the vertices

+
+
Return type:
+

Edge

+
+
+

v1.connect(v2) connects vertex v1 to vertex v2.

+
+

Note

+
    +
  • If the vertices subclass UVertex the edge is undirected, and if +they subclass DVertex the edge is directed.

  • +
  • Vertices must both be of the same Vertex subclass

  • +
+
+
+
Seealso:
+

Edge()

+
+
+
+ +
+
+copy(cls=None)
+
+ +
+
+property degree
+

Degree of vertex

+
+
Returns:
+

degree of the vertex

+
+
Return type:
+

int

+
+
+

Returns the number of edges connected to the vertex.

+
+

Note

+

For a DGraph only outgoing edges are considered.

+
+
+
Seealso:
+

edges()

+
+
+
+ +
+
+distance(coord)
+

Distance from vertex to point

+
+
Parameters:
+

coord (ndarray(n) or Vertex) – coordinates of the point

+
+
Returns:
+

distance

+
+
Return type:
+

float

+
+
+

Distance is computed according to the graph’s metric.

+
+
Seealso:
+

metric()

+
+
+
+ +
+
+edges()
+

All outgoing edges of vertex

+
+
Returns:
+

List of all edges leaving this vertex

+
+
Return type:
+

list of Edge

+
+
+
+

Note

+
    +
  • For a directed graph the edges are those leaving this vertex

  • +
  • +
    For a non-directed graph the edges are those leaving or entering

    this vertex

    +
    +
    +
  • +
+
+
+ +
+
+edgeto(dest)
+

Get edge connecting vertex to specific neighbour

+
+
Parameters:
+

dest (Vertex subclass) – a neigbouring vertex

+
+
Raises:
+

ValueErrordest is not a neighbour

+
+
Returns:
+

the edge from this vertex to dest

+
+
Return type:
+

Edge

+
+
+
+

Note

+
    +
  • For a directed graph dest must be at the arrow end of the edge

  • +
+
+
+ +
+
+heuristic_distance(v2)
+
+ +
+
+incidences()
+

Neighbours and edges of a vertex

+

v.incidences() is a generator that returns a list of incidences, +tuples of (vertex, edge) for all neighbours of the vertex v.

+
+

Note

+

For a directed graph the edges are those leaving this vertex

+
+
+ +
+
+isneighbour(vertex)
+

Test if vertex is a neigbour

+
+
Parameters:
+

vertex (Vertex subclass) – vertex reference

+
+
Returns:
+

true if a neighbour

+
+
Return type:
+

bool

+
+
+

For a directed graph this is true only if the edge is from self to +vertex.

+
+ +
+
+neighbors()
+

Neighbors of a vertex

+

v.neighbors() is a list of neighbors of this vertex.

+
+

Note

+

For a directed graph the neighbours are those on edges leaving this vertex

+
+
+ +
+
+neighbours()
+

Neighbours of a vertex

+

v.neighbours() is a list of neighbours of this vertex.

+
+

Note

+

For a directed graph the neighbours are those on edges leaving this vertex

+
+
+ +
+
+property x
+

The x-coordinate of an embedded vertex

+
+
Returns:
+

The x-coordinate

+
+
Return type:
+

float

+
+
+
+ +
+
+property y
+

The y-coordinate of an embedded vertex

+
+
Returns:
+

The y-coordinate

+
+
Return type:
+

float

+
+
+
+ +
+
+property z
+

The z-coordinate of an embedded vertex

+
+
Returns:
+

The z-coordinate

+
+
Return type:
+

float

+
+
+
+ +
+ +
+ + +
+
+ +
+
+
+
+ + + + + + + \ No newline at end of file