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:

:linenos:

>>> from roboticstoolbox import ETS

>>> ETS.rx(45, 'deg')

>>> ETS.tz(0.75)

>>> e = ETS.rx(0.3) * ETS.tz(0.75)

>>> print(e)

>>> e.eval()

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 sequence, substituting in values, and the result is an

SE(3) matrix encapsulated in an ``SE3`` object.

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 ETS

>>> e = ETS.rz() * ETS.tx(1) * ETS.rz() * ETS.tx(1)

>>> print(e)

>>> len(e)

>>> e[1:3]

>>> e.eval([0, 0])

>>> e.eval([90, -90], 'deg')

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.

**Reference:**

- `A simple and systematic approach to assigning Denavit-Hartenberg parameters <https://petercorke.com/robotics/a-simple-and-systematic-approach-to-assigning-denavit-hartenberg-parameters>`_.

Peter I. Corke, IEEE Transactions on Robotics, 23(3), pp 590-594, June 2007.

.. autoclass:: roboticstoolbox.robot.ETS.ETS

:members: rx, ry, rz, tx, ty, tz, eta, eval, T, joints, axis, jtype, config, __getitem__, __mul__, __repr__, __str__

.. autoclass:: roboticstoolbox.robot.ETS2.ETS

:members: rx, ry, rz, tx, ty, tz, eta, eval, T, joints, axis, jtype, config, __getitem__, __mul__, __repr__, __str__

ETS Robot models

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")