A Trajectory Tracking Steer-by-Wire Control System for Ground Vehicles

76 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 55, NO. 1, JANUARY 2006 A Trajectory Tracking Steer-by-Wire Control System for Ground Vehicles Pradeep Setlur, John R. Wagner, Darren M. Dawson, and David Braganza Abstract—The application of multi-disciplinary automotive technologies to hybrid vehicles has resulted in the integration of alternative propulsion sources and drive-by-wire components for enhanced ground vehicle performance, fuel economy, and occupant safety. The integration of steer-by-wire systems in vehicles facilitates autonomous and semi-autonomous operations, better lateral vehicle behavior, an adjustable steering “feel,” and elimination of problems arising due to potential engine cycling. In this paper, a continuous time-varying tracking controller is designed for the vehicle’s position/orientation using a simplified vehicle description and reference model for tracking. The tracking error is globally, exponentially forced to a neighborhood of about zero by transforming the system into a flat input-state system and then fusing a filtered tracking error transformation with the dynamic oscillator design. Mathematical models are presented for a steerby-wire rack and pinion unit, vehicle chassis, and tire/road interface dynamics. Representative numerical results are discussed to demonstrate the vehicle’s transient response for a prescribed trajectory profile. Index Terms—Control system, nonlinear trajectory control, steer-by-wire, vehicle tracking controller. NOMENCLATURE BM 1,M 2 Brack Bsc Bx Caf , Car Cα CG FD Ffr,rack Fxf , Fxr Fyf , Fyr Fxwf , Fxwr Fywf , Fywr g Motor damping (N-m-s/rad). Rack damping (N-m-sec/rad). Steering column damping (N-m-sec/rad). Suspension roll damping (N-m-rad/s). Front, rear total tire cornering stiffness (N/rad). Lateral tire stiffness (N/rad). Center of gravity. Aerodynamic drag force (N). Rack/piston friction (N-m). Longitudinal tire force along the vehicle axis from front (rear) wheels (N). Lateral tire force along the vehicle axis from front (rear) wheels (N). Longitudinal tire force in wheel plane (N). Lateral tire force in the wheel plane (N). Acceleration constant (m/s 2 ). Manuscript received January 16, 2004; revised September 29, 2004 and February 8, 2005. This work was supported in part by two DOC Grants, an ARO Automotive Center Grant, a DOE Contract, a Honda Corporation Grant, and a DARPA Contract. The review of this paper was coordinated by Dr. M. S. Ahmed. P. Setlur is with the Electrical and Electronic Engineering Department, California State University, Sacramento, CA 95819 USA. J. R. Wagner is with the Department of Mechanical Engineering, Clemson University, Clemson, SC 29634-0915 USA (e-mail: jwagner@clemson.edu). D. M. Dawson and D. Braganza are with the Department of Electrical and Computer Engineering, Clemson University, Clemson, SC 29634 USA. Digital Object Identifier 10.1109/TVT.2005.861189 h ia1,a2 IM 1,M 2 IRoll Isw Izz kb1,b2 KS 1 KS 2 kτ 1,τ 2 Kx L1,2 lf lr m mrack ms N r R R1,2 rL rp Tdriver Tfr,c TM 1,M 2 u, ui v, vi VS 1,S 2 x, y xr , yr x1 , y1 xr 1 , yr 1 yrack α δ δF θM 1,M 2 θsw µ φ φr 0018-9545/$20.00 © 2006 IEEE Center of gravity height (m). Armature current (A). Lumped inertia of motor (kg-m 2 ). Vehicle moment of inertia about the longitudinal axis (kg-m 2 ). Steering wheel lumped inertia (kg-m 2 ). Vehicle moment of inertia about the yaw axis (kg-m 2 ). Motor emf constant (V-s/rad). Lumped compliance due to steering column and torque sensor stiffness (N-m/rad). Lumped compliance due to motor shaft and torque sensor stiffness (N-m/rad). Motor torque constant (N-m/A). Suspension roll stiffness (N-m/rad). Motor electrical inductance (H). Distance from CG to front axle (m). Distance from CG to rear axle (m). Total vehicle mass (kg). Rack/pinion lumped mass (kg). Sprung mass (kg). Normal tire load (N). Vehicle yaw velocity (rad/s). Radius of the tire (m). Motor electrical resistance (Ω). Offset of king pin axis at applied force (m). Pinion gear radius (m). Torque produced by driver (N-m). Steering column friction (N-m). Torque produced by motors (N-m). Longitudinal vehicle, wheel center speeds (m/s). Lateral vehicle, wheel center speeds (m/s). Source voltage (V). Global vehicle Cartesian position (m). Reference vehicle Cartesian position (m). Vehicle “shifted” Cartesian position (m). Reference vehicle “shifted” Cartesian position (m). Rack lateral displacement (m). Tire side slip angle (rad). Steer angle of the wheel (rad). Actual front wheel steer angle (rad). Motor angular displacement (rad). Steering wheel and column deflection (rad). Coefficient of friction. Vehicle yaw angle (rad). Reference vehicle yaw angle (rad). SETLUR et al.: TRAJECTORY TRACKING STEER-BY-WIRE CONTROL SYSTEM FOR GROUND VEHICLES ψ ω Subscripts: i f, r Vehicle roll angle (rad). Wheel angular velocity (rad/s). ith wheel and axle location. Front, rear axles. I. INTRODUCTION HE MECHANICAL and hydraulic subsystems in passenger and light-duty ground vehicles are being upgraded with drive-by-wire components in order to boost overall performance, reduce power consumption, and enhance passenger safety. For example, the electrical equivalents for traditional mechanical linkages and/or hydraulic power assist systems include “brake-by-wire,” “steer-by-wire,” and “throttle-by-wire” [11], [23], which offer less environmental concerns due to the removal of hydraulic fluids and continual engine parasitic losses. A wealth of research has been conducted on hydraulic power steering systems (e.g., [20], [21], and [26]), electric power steering systems, which maintain a mechanical linkage between the driver and steering mechanism but replace the hydraulics with an electric motor (e.g., [12], [13], and [17]), and most recently steer-by-wire (e.g., [8] and [16]). In a steer-by-wire system, dual servo-motors are introduced to control the driver interface and the steering mechanism; the direct connection between the driver and wheel assembly is removed. A variety of control architectures have been proposed for enhanced vehicle steering characteristics and automatic tracking in steer-by-wire vehicles. Furthermore, the development of reliable positioning systems has made accurate trajectory generation very practical. Vehicle steering systems translate the driver’s steering commands into the rotation of the front wheels about their kingpin axes. The effort required to steer the vehicle must be balanced between power assistance to facilitate vehicle turning and road “feel” for driver feedback. Servo-motor-based steering systems offer improved lateral vehicle responsiveness, weight reduction, and occupant safety (e.g., [18]). To effectively analyze a steering system, mathematical models that consider the steering and various chassis subsystems must exist. In [16], the authors presented a detailed steer-by-wire system model with accompanying platform dynamics to describe the vehicle’s motion. A vehicle following a prescribed path (or trajectory) is a challenging problem due to the fact that the system is under-actuated. In [2], the authors developed linear and nonlinear controllers for the steering system. However, the controller required the forward velocity to be constant and non-zero. In [14], the authors attempted to predict the vehicle’s path for prescribed driving conditions in the presence of disturbances. However, they did not address the issue of vehicle control. In [19], the vehicle’s cornering instability was studied and a robust controller to protect the vehicle from spin was proposed. Their work did not discuss the trajectory-following problem. On similar lines, in [1] the authors proposed a controller to stabilize disturbance yaw forces in the vehicle. As in previous works, it was assumed that the vehicle’s longitudinal velocity is constant so that the trajectory following has to be performed as a separate control task. T 77 In this paper, the problem of tracking a reference trajectory with respect to a fixed, global coordinate frame is considered. The model used in the control development is a simplified form of the vehicle dynamics. A set of transformations based on differential flatness is applied to the simplified, vehicle and reference model. This manipulation of the system into a suitable form allows a Lyapunov-based nonlinear controller to be designed for the transformed Cartesian dynamics. The controller ensures globally uniformly ultimately bounded (GUUB) tracking (i.e., state tracking with respect to a reference signal generator) to be designed in a similar manner to [3] and [5]. The controller uses an embedded dynamic oscillator to provide additional design flexibility. The paper is organized as follows: In Section II, the low-order vehicle dynamics are presented along with the controller model and reference signal generator. The tire-road interface reaction forces are also discussed in this section. The open-loop tracking dynamics are then transformed to facilitate the subsequent controller development and stability analysis. Section III defines the control problem. In Section IV, the design of the proposed GUUB tracking controller is presented. The corresponding closed-loop error system and the stability analysis are presented. Numerical results are presented in Section V. Section VI contains the concluding remarks. II. MATHEMATICAL MODELS The mathematical modeling of the steering system components permits the vehicle’s lateral responsiveness for various design configurations and control algorithms to be studied. A series of analytical models will be presented for the steering system and chassis dynamics, as well as the tracking controller and its trajectory generator. The steering and chassis dynamics offer insight into the controller derivation for the simplified vehicle model. The dynamic models presented in this section are used in the numerical simulations to estimate the vehicle’s behavior. A. Steering System The steer-by-wire system is comprised of the driver interface and the directional control assembly. The driver interface includes the steering wheel and column, a torque sensor, and a low torque dc servo-motor. The differential equation for the haptic interface unit [16] becomes θ̈sw = 1 [Tdriver − Bsc (θ̇sw − θ̇M 1 ) Isw − Ks1 (θsw − θM 1 ) − Tfr,c ]. (1) The motor shaft rotational dynamics are expressed as θ̈M 1 = (1/IM 1 )[−BM 1 θ̇M 1 − Bsc (θ̇sw − θ̇M 1 ) − Ks1 (θM 1 − θsw ) + TM 1 ] (2) where the motor torque is TM 1 = kτ 1 ia1 . The electrical current differential equation becomes dia1 = (1/L1 )(−R1 ia1 − kb1 θ̇M 1 + Vs1 ). (3) dt The applied voltage for the servo-motor Vs1 , is a function of both the driver interface and the directional control assembly. 78 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 55, NO. 1, JANUARY 2006 The directional control assembly uses a high torque servomotor to displace the rack. The rack dynamics reflect the direct application of the motor torque, so that ÿrack = (1/mrack )[−2KL (yrack − rL δF ) − Brack ẏrack − (Ks2 /rp )((yrack /rp ) − θM 2 ) − Ffr,rack ]. (4) Then, the motor displacement θM 2 differential equation becomes θ̈M 2 = (1/IM 2 )[−BM 2 θ̇M 2 − Ks2 (θM 2 − yrack /rp ) + TM 2 ] (5) where, TM 2 = kτ 2 ia2 . The armature current ia2 may be expressed as dia2 = (1/L2 )(−R2 ia2 − kb 2 θ̇M 2 + Vs2 ) dt where Vs2 is the supply voltage for this motor. (6) B. Chassis and Tire/Road Interface Dynamics The vehicle model used for the numerical simulation within the study has four degrees of freedom: longitudinal velocity u, lateral velocity v, yaw rate r, and body roll ψ. Heydinger et al. [10] and Wong [25] provide a detailed derivation of the equations of motion. The longitudinal acceleration for the vehicle may be expressed as (u̇ − vr) = (1/m)(2Fxf + 2Fxr − ms hrψ̇). (7) The vehicle’s lateral acceleration is influenced by the longitudinal velocity, the yaw, and the roll in the following manner: (v̇ + ru) = (1/m)(2Fyf + 2Fyr − ms hψ̈). (8) The vehicle’s yaw rate is a function of the distance from the vehicle’s center of gravity (CG) to the tires, as well as the front and rear tire lateral forces, such that ṙ = (1/Izz )(2lf Fyf − 2lr Fyr ). (9) The vehicle’s roll angular acceleration may be expressed as ψ̈ = (1/IRoll )[−Bx ψ̇ − (Kx − ms gh)ψ − ms h(v̇ + ru)]. (10) The differential equations for the lateral and roll accelerations are coupled; however, they are inherently more useful if stated explicitly. If (8) is inserted into (10), then the roll expression becomes 1 ψ̈ = IRoll − m2s h2 /m × [−Bx ψ̇ − (Kx − ms gh)ψ − (ms h/m)(2Fyf + 2Fyr )]. (11) Conversely, if (10) is substituted into (8), then the lateral acceleration maybe expressed as v̇ = 1 m− m2s h2 /IRoll [2Fyf + 2Fyr + (ms h/IRoll ) ·(Bx ψ̇ + (Kx − ms gh)ψ) − ru] (12) Fig. 1. Vehicle with local and global coordinates. The dynamics of the tires interaction with the road can be defined using the front and rear tire slip angles. The tire slip angle is given by αi = tan−1 (vi /ui ) − δi . (13) If the longitudinal velocity is assumed quasi-constant, then the longitudinal tire forces may be neglected. Although a series of tire models exist to describe the longitudinal and lateral forces, as well as the aligning torques for various road surfaces and operating conditions, the tire dynamics are simplified to a front and rear cornering stiffness. Consequentially, the front and rear tire lateral forces are Fyf = Caf αf , Fyr = Car αr . (14) C. Controller Model Development The kinematic equations of motion of the vehicle’s center of mass (COM) can be written as (refer to Fig. 1) q̇ = S(q)V (15) where q̇(t) = [ẋ(t) ẏ(t) φ̇(t)]T ∈ ℜ3 represents the time derivative of q(t) = [x(t)y(t)φ(t)]T ∈ ℜ3 . The variables, x(t), y(t), and φ(t) ∈ ℜ1 denote the Cartesian position and orientation, respectively, of the vehicle’s COM. The transformation matrix S(q) ∈ ℜ3×3 is defined as   cos(φ) − sin(φ) 0 (16) S(q) =  sin(φ) cos(φ) 0  0 0 1 the velocity vector V (t) ∈ ℜ3 is V = [u v r]T , where u(t), v(t), and r(t) ∈ ℜ1 denote the vehicle’s longitudinal, lateral, and yaw velocities, respectively. Under the assumptions SETLUR et al.: TRAJECTORY TRACKING STEER-BY-WIRE CONTROL SYSTEM FOR GROUND VEHICLES that i) the body-fixed coordinate axis coincides with the CG; ii) the mass distribution is homogeneous; iii) the heave, pitch, and roll modes can be neglected since the controller should be robust enough to handle these effects; and iv) the half tread of the vehicle is small (i.e., bicycle model), the vehicle model can be expressed similar to [9], as u̇ = (1/m)[2(Fxf + Fxr ) + mvr − FD (u)] and ẏ(t) given in (15) may be used with (16) and (24) to obtain ẍ = ut cos(φ) − ε sin(φ)um (25) ÿ = ut sin(φ) + ε cos(φ)um (26) φ̇ = r (27) ṙ = (m/Izz )(um − g(·)/ε). (28) The function g(u, v, r) ∈ ℜ1 and the parameter ε ∈ ℜ1 are defined as v̇ = (1/m)[2(Fyf + Fyr ) − mur] ṙ = (1/Izz )[2(lf Fyf − lr Fyr )] 79 (17) g(u, v, r) = 2[(lf + lr )/m]εFyr (29) where FD represents the retarding force due to aerodynamic drag. The front wheel longitudinal and lateral forces are related to the tire forces by As presented in [15] and [22], a set of transformations based on differential flatness are defined as Fxf = Fxwf cos(δf ) − Fywf sin(δf ) x1 = x − (Izz /m)ε cos(φ) (31) y1 = y − (Izz /m)ε sin(φ) (32) Fyf = Fxwf sin(δf ) + Fywf cos(δf ). (18) The front and rear tire forces, Fxwf (t), Fywf (t), Fxwr (t), and Fywr (t) ∈ ℜ1 may be explicitly written as Fxr = Fxwr = −µN tanh[(u − Rωr )/Rωr ] (19) Fyr = Fywr = −Cα tanh{arctan[(v − lr r)/u]} (20) Fxwf = −µN tanh{(u − Rωf )/Rωf } (21) Fywf = −Cα tanh{arctan[(v + lf r)/u] − δf }. (22) Remark 1: It has been assumed that the vehicle is front-wheel driven and steered. Remark 2: The tire model represented in (19)–(22) was observed to closely fit the experimental data given in [7]. However, other models may be used to calculate these tire forces. Remark 3: For purposes of this paper, we consider Fxf (t) and Fyf (t) to be the control inputs. In an actual system, the desired trajectory signals for ωf (t) and δf (t) must be generated on-line and supplied to low level wheel speed and steering controllers, respectively. The system of nonlinear equations representing the tire forces (19)–(22), together with the designed control signals for Fxf (t) and Fyf (t), can be solved by numerical, on-line, iterative methods to obtain desired trajectory signals for ωf (t) and δf (t). The system dynamics may be rewritten in a more convenient form if the control inputs Fxf (t) and Fyf (t) are designed in the following manner: Fxf = 1 m FD (u) − Fxr + ut 2 2 Fyf = m um − Fyr 2lf (23) where ut (t) and um (t) ∈ ℜ1 are subsequently designed control inputs. After substituting the control given in (23) into (17), the system dynamics can be rewritten as u̇ = ut + vr, v̇ = um /lf − ur,    m (lf + lr ) Fyr . ṙ = um − 2 Izz m (24) To develop a system model using the Cartesian position and orientation of the vehicle’s COM, the time derivative of ẋ(t) ε = 1/lf . (30) where x1 (t), y1 (t) ∈ ℜ1 represents the “shifted” Cartesian position. After taking the second time derivatives of (31) and (32), substituting (25)–(28), and canceling common terms, the shifted Cartesian dynamics become ẍ1 = (ut + (Izz /m)εφ̇2 ) cos(φ) − g(·) sin(φ) 2 ÿ1 = (ut + (Izz /m)εφ̇ ) sin(φ) + g(·) cos(φ). (33) (34) D. Reference Model Development A reference model based on the structure of the system dynamics given by (25) and (26) will be designed to facilitate the tracking controller with ẍr = urt cos(φr ) − ε sin(φr )urm (35) ÿr = urt sin(φr ) + ε cos(φr )urm (36) φ̇r = rr (37) ṙr = (m/Izz )urm (38) where xr (t), yr (t), φr (t) ∈ ℜ1 denote the reference Cartesian position and orientation, respectively. The signal rr (t) ∈ ℜ1 is an auxiliary reference state variable used to facilitate the analysis, urt (t), urm (t) ∈ ℜ1 denote the reference input signals. It is assumed that the reference signals urt (t) and urm (t) are selected such that urt (t), urm (t), u̇rt (t), and u̇rm (t) are bounded and xr (t), yr (t), φr (t) and their first two time derivatives remain bounded at all times (i.e., xr (t), yr (t), φr (t), ẋr (t), ẏr (t), φ̇r (t), ẍr (t), ÿr (t) and φ̈r (t) ∈ L∞ ). A transformation similar to the one given in (31) and (32) is applied to the reference system, (35)–(38), as follows: xr 1 = xr − (Izz /m)ε cos(φr ) (39) yr 1 = yr − (Izz /m)ε sin(φr ) (40) where xr 1 (t), yr 1 (t) ∈ ℜ1 represents the “shifted” reference Cartesian position. After taking the second time derivatives of (39) and (40), substituting (35) and (36), and then canceling common terms, the shifted reference Cartesian dynamics can be 80 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 55, NO. 1, JANUARY 2006 written as,   ẍr 1 = urt + (Izz /m)εφ̇2r cos(φr )   ÿr 1 = urt + (Izz /m)εφ̇2r sin(φr ). (41) In addition, the orientation tracking error φ̃(t) ∈ ℜ1 is defined as φ̃ = φ − φr . (42) Remark 4: The inputs to the reference generator, urt (t) and urm (t) are analogous to the throttle and steering torque inputs, respectively, for a vehicle equipped with a conventional steering system. These reference inputs can be selected appropriately to generate the position and orientation reference signals. (47) To write the open-loop error system in a more convenient form, the following globally invertible transformation is introduced      sx φ̃ cos φ − 2 sin φ φ̃ sin φ + 2 cos φ 0 w  z1  =  0 0 −1   sy  cos φ sin φ 0 z2 φ̃ (48) III. CONTROL PROBLEM STATEMENT To address vehicle platooning and trajectory-tracking issues, it is essential for the independent operation of the forward thrust and steering mechanism. However, the control algorithm must holistically integrate these two functions. Thus, a nonlinear control algorithm, which simultaneously adjusts the steer angle δf (t) and the rotational speed of the wheel ωf (t) while monitoring the position velocity variables with reference to the global coordinate frame, will be designed for a front-wheel driven and steered vehicle with a steer-by-wire mechanism. The controller must force the vehicle position coordinates [i.e., x(t), y(t), and φ(t)] to track the corresponding coordinates of a reference vehicle (i.e., xr (t), yr (t), and φr (t)). To validate the control algorithm’s performance, a standard vehicle maneuver (e.g., “J-turn”) will be executed and the performance analyzed. where w(t) ∈ ℜ1 , z(t) = [z1 (t) z2 (t)]T ∈ ℜ2 are auxiliary tracking-error variables. After taking the time derivative of (54) and using (27), (28), (33), (34), (37), (38), and (41)–(47), the open-loop tracking-error dynamics can be rewritten as ẇ = uTc J T z + f + 2g(·) (49) ż = uc (50) u̇1 = (m/Izz )[−um + urm + (1/ε)g(·)] (51) where the auxiliary control signal uc (t) = [u1 (t) u2 (t)]T ∈ ℜ2 is related to the open-loop signals ut (t) and r(t) according to the following globally invertible transformation: uc = T −1 ut r −Π ut r = T (uc + Π). (52) The matrix T (·) ∈ ℜ2×2 and the auxiliary measurable signal Π(·) ∈ ℜ2 are defined as IV. CONTROLLER DESIGN PROCESS The power-train control module (PCM) regulates the engine and transmission operations so that the torque applied to the drive wheels permits the desired vehicle motion. The steering control unit (SCU) provides the steering control authority to the vehicle via a high-torque motor that displaces the rack. These two controllers facilitate semi-autonomous and autonomous operating scenarios. From a hardware perspective, global positioning sensors (GPS) can provide the necessary position and velocity signals with respect to the “world” frame. Similarly, on-board sensors will provide the velocities in the “body-fixed” frame. The nonlinear control algorithm will now be designed to ensure trajectory tracking. A detailed stability analysis will be presented to validate the design. −sx sin(φ) + sy cos(φ) 1 −1 0 T =  −rr Π =  −ε(Izz /m)r2 + ẍr 1 cos(φ) + ÿr 1 sin(φ)  (54) −µr (x̃˙ 1 cos(φ) + ỹ˙ 1 sin(φ))  with f (·) ∈ ℜ1 denoting an auxiliary signal defined as f = 2[ẍr 1 sin(φ) − ÿr 1 cos(φ) + rr z2 ] − 2µr (x̃˙ 1 sin(φ) − ỹ˙ 1 cos(φ)). A. Open-Loop Tracking Error System Formulation sx = x̃˙ 1 + µr x̃1 (43) sy = ỹ˙ 1 + µr ỹ1 (44) 1 where µr ∈ ℜ represents a constant positive control gain. The variables x̃1 (t), ỹ1 (t) ∈ ℜ1 denote the shifted Cartesian tracking error signals, which are x̃1 = x1 − xr 1 (45) ỹ1 = y1 − yr 1 . (46) (55) The skew symmetric matrix J ∈ ℜ2×2 is defined as J= To quantify the control objective, the filtered tracking error signal s(t) = [sx sy ]T ∈ ℜ2 is defined as (53) 0 1 −1 . 0 (56) Remark 5: Based on the definition of sx (t) and sy (t) given in (43) and (44), standard arguments [4] can be made to prove that i) if sx (t), sy (t) ∈ L∞ , then x̃1 (t), ỹ1 (t), x̃˙ 1 (t), ỹ˙ 1 (t) ∈ L∞ , and ii) if sx (t) and sy (t) are GUUB, then x̃1 (t) and ỹ1 (t) are GUUB. Remark 6: As illustrated above, the open-loop error system has been formulated for the shifted Cartesian signals denoted by, x1 (t), y1 (t), xr 1 (t), and yr 1 (t), which were originally introduced in (31), (39), and (40). To illustrate the significance of the shifted tracking-error signals, the actual Cartesian tracking error, denoted by x̃(t) and ỹ(t), are defined as x̃ = x − xr ỹ = y − yr . (57) SETLUR et al.: TRAJECTORY TRACKING STEER-BY-WIRE CONTROL SYSTEM FOR GROUND VEHICLES Equation (57) along with (31), (32), (39), (40), (45), and (46) can now be used to show that actual Cartesian tracking error can be written in terms of the shifted Cartesian tracking error as x̃ = x̃1 + ε[sin(φ) − sin(φr )] ỹ = ỹ1 − ε[cos(φ) − cos(φr )]. In addition, it may be noted that as a direct consequence of the mean-value theorem [24], it can be shown that | sin(φ) − sin(φr )| ≤ |φ̃| (60) | cos(φ) − cos(φr )| ≤ |φ̃|. (61) Hence, (58)–(61), can be used to show that |x̃| ≤ |x̃1 | + ε|φ̃| |ỹ| ≤ |ỹ1 | + ε|φ̃|. B. Control Development The control objective is to design a controller that exponentially forces the Cartesian/orientation tracking error to a neighborhood of about zero that can be made arbitrarily small (i.e., GUUB). It will be assumed that the signals x(t), y(t), φ(t), ẋ(t), ẏ(t), and r(t) are available for measurement. To achieve this objective, we define an auxiliary error signal z̃(t) ∈ ℜ2 as the difference between the subsequently designed auxiliary signal zd (t) ∈ ℜ2 and the transformed variable z(t) defined in (54) as (63) In addition, an auxiliary error signal η(t) ∈ ℜ1 is defined as the difference between the subsequently designed auxiliary signal, ud1 (t) ∈ ℜ1 and the auxiliary signal u(t) defined in (52) as η = ud1 − u1 . δd = γ0 exp(−γ1 t) + ε1 (64) u2 ]T = ua − k2 z (65) where k2 ∈ ℜ1 is a positive, constant control gain, and the auxiliary control signal ua (t) ∈ ℜ2 is  ua = (k1 w + f )/δd2 Jzd + Ω1 zd . (66) The dynamics of the auxiliary control signal denoted by zd (t) ∈ ℜ2 are defined by the following oscillator-like relationship [3]:  żd = (δ̇d /δd )zd + (k1 w + f )/δd2 + wΩ1 Jzd (67) zdT (0)zd (0) = δd2 (0). (68) (70) with k1 , γ0 , γ1 , ε1 ∈ ℜ being positive, constant design parameters, and f (·) defined in (55). C. Closed-Loop Error System Development To facilitate the closed-loop error system development for w(t), the auxiliary control input ud1 (t) is injected by adding and subtracting the term ud1 z2 to the right-side of the open-loop dynamic expression for w(t) given in (49) and then utilizing (64) to obtain ẇ = [ud1 u2 ]J T z − ηz2 + f + 2g(·). (71) After substituting (65) for [ud1 u2 ], adding and subtracting uTa Jzd to the resulting expression, and utilizing (63), the dynamics for w(t) can be rewritten as ẇ = −uTa Jzd + uTa J z̃ − ηz2 + f + 2g(·) (72) where the fact that J T = −J has been applied. Finally, by substituting (66) for only the first occurrence of ua (t) in (72), exploiting the skew symmetry of J defined in (56), and observing that J T J = I2 (Note that I2 denotes the standard 2 × 2 identity matrix), the final expression for the closed-loop error system for w(t) becomes ẇ = −k1p w − 4kn 1 w + uTa J z̃ − ηz2 + 2g(·) (73) 1 where k1p , kn 1 ∈ ℜ are chosen to satisfy k1 = k1p + 4kn 1 . To determine the closed-loop error system for z̃(t), the time derivative of (63) is taken, (67) is substituted for żd (t), and (50) is substituted for ż(t), so that   z̃˙ = (δ̇d /δd )zd + (k1 w + f )/δd2 + wΩ1 Jzd − [ud1 u2 ]T + [η 0]T (74) where the auxiliary control input ud1 (t) was injected by adding and subtracting [ud1 0]T to the right-side of (74) and then (64) was applied. The final expression for the closed-loop error system for z̃(t) is obtained by following a similar procedure to [3] as Based on the subsequent stability analysis and the structure of the open-loop error system given in (49) and (50), the auxiliary signal ud (t) is designed as [ud1 (69) 1 (62) From (62), it is clear that if x̃1 (t), ỹ1 (t), φ̃(t) are GUUB, then x̃(t) and ỹ(t) are GUUB. Remark 7: During the subsequent section, the open-loop tracking-error system given in (49)–(51) will be utilized to design the control inputs for the signals u2 (t) and um (t). It may be noted that the actual control input ut (t) can then be calculated by using (52). z̃ = zd − z. The auxiliary terms Ω1 (t), δd (t) ∈ ℜ1 are defined as   Ω1 = k2 + (δ̇d /δd ) + (k1 w2 + wf )/δd2 (58) (59) 81 z̃˙ = −k2 z̃ + wJua + [η 0]T . (75) Based on the subsequent stability analysis, um (t) is designed as um = −k3 η − z̃1 + wz2 − (Izz /m)u̇d1 + urm + (1/ε)g(·) (76) where k3 ∈ ℜ1 is a positive constant control gain, and the explicit definition for u̇d1 (t) is given in the Appendix. To develop the closed-loop error system for η(t), the time derivative of (64) is taken and then the control input is substituted as defined in (76) such that (Izz /m)η̇ = −k3 η + wz2 − z̃1 where (51) has been substituted for u̇1 (t). (77) 82 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 55, NO. 1, JANUARY 2006 D. Stability Analysis Theorem 1: Given the closed-loop system of (73), (75), and (77), the position/orientation tracking error signals defined in (47) and (57) are GUUB in the sense that |x̃(t)|, |ỹ(t)|, |φ̃(t)| ≤ β4 exp(−λ4 t) + ε4 (78) 1 where β4 , λ4 , and ε4 ∈ ℜ are positive constants that are explicitly defined in the subsequent stability proof. Proof: To prove Theorem 1, a non-negative, scalar function, denoted by V (t) ∈ ℜ1 , is defined as 1 2 Izz 2 1 T w + η + z̃ z̃. (79) 2 2m 2 After taking the time derivative of (79) and making the appropriate substitutions from (73), (75), and (77), the following expression may be obtained:   V̇ = w −k1p w + uTa J z̃ − ηz2   + z̃ T −k2 z̃ + wJua + [η 0]T   + η[−k3 η + wz2 − z̃1 ] + −4kn 1 w2 + 2g(·)w . (80) V = After utilizing the fact that J T = −J, canceling common terms, and utilizing (79), V̇ (t) can be upper bounded as   V̇ ≤ −2 min{k1p , k2 , k3 }V + −4kn 1 w2 + 2g(·)w . (81) Applying the nonlinear damping argument to the bracketed terms in (81), the following upper bound for V̇ (t) is |g(·)|2 . (82) kn 1 After completing the squares and utilizing (79), the solution for the differential inequality given by (82) can be upper bounded as V̇ ≤ −2 min{k1p , k2 , k3 }V + V (t) ≤ V (0) exp(−ζ4 t)  t + (1/kn 1 ) exp(−ζ4 (t − τ ))(|g(·)|2 ) dτ (83) 0 where ζ4 is some positive scalar constant. It is easy to see that (83) can be rewritten in the following manner by using (79):   (84) Ψ(t)2 ≤ −β02 exp(−2ζ4 t) + |g(·)|2∞ ζ4 kn 1 where Ψ(t) ∈ ℜ4 is defined as Ψ = [w η z̃ T ]T . (85) 1 The constants β0 , ζ4 ∈ ℜ are defined explicitly as follows: β0 = Ψ(0), ζ4 = min{k1p , k2 , k3 }. to conclude that sx (t), sy (t), φ̃(t) ∈ L∞ . Based on the fact that sx (t), sy (t), φ̃(t) ∈ L∞ and the fact that the reference trajectory is selected so that xr 1 (t), yr 1 (t), φr (t), ẋr 1 (t), ẏr 1 (t), φ̇r (t), ẍr 1 (t), ÿr 1 (t) ∈ L∞ , (43)–(47) can be utilized to conclude that x̃1 (t), ỹ1 (t), ẋ1 (t), ẏ1 (t), x1 (t), y1 (t), φ(t) ∈ L∞ , as mentioned in Remark 6. Using the fact that sx (t), sy (t), φ(t), ẋ(t), ẏ(t) ∈ L∞ , and from the fact that g(·) ∈ L∞ , it is concluded that f (·), T (·) ∈ L∞ from (55) and (53). Based on these facts, (54), (64), (65), (66), (69), and (70) can now be utilized, to show that ud1 (t), ua (t), żd (t), Ω1 (t), u1 (t), u2 (t), Π(·) ∈ L∞ . From (52), it can now be concluded that ut (t), r(t) ∈ L∞ . Based on the previous facts, it is easy to show that u̇d1 (t) ∈ L∞ (see the Appendix for the explicit expression for u̇d1 (t)). After utilizing (76), it can now be shown that the control input, um (t) ∈ L∞ . Standard signal-chasing arguments can now be employed to conclude that all of the remaining signals in the control and the system remain bounded during closed-loop operation. To prove (78), it is first shown that z(t) defined in (54) is GUUB by applying the triangle inequality to (63), and hence, obtain the following bound for z(t): z ≤ z̃ + zd  ≤ β1 exp(−λ1 t) + ε2 (88) where (70), (84), and (85) have been utilized, and the positive constants β1 , λ1 , ε2 ∈ ℜ1 are some constants of analysis. Equations (43), (44), (84), (85), (87), (88), and Remark 6 may now be employed to obtain the following tracking-error bounds: |x̃1 (t)|, |ỹ1 (t)|, |φ̃(t)| ≤ β3 exp(−λ3 t) + ε3 (89) where β3 , λ3 , and ε3 ∈ ℜ1 are constants of analysis. Utilizing the observation made in Remark 6, and making use of (89), it can easily be seen that (78) is valid, where β4 , λ4 , and ε4 ∈ ℜ1 are positive constants. Remark 8: From (78), it is clear that the tracking error variables, x̃(t), ỹ(t), and φ̃(t), can be made arbitrarily small by reducing the design parameter, ε1 . Further, the rate of convergence of the errors to this arbitrarily small neighborhood around zero, can be controlled by proper choice of the design parameters, k1 , k2 , k3 , γ1 , and µr . Remark 9: It is easy to show that the proposed controller can be used to develop a set-point control strategy for the Cartesian position and orientation of the COM of the vehicle (see [3], for more details). (86) From (84) and (85), it is clear to see that w(t), η(t), z̃(t) ∈ L∞ . After utilizing (63) and the fact that z̃(t), δd (t) ∈ L∞ , it is concluded that z(t), zd (t) ∈ L∞ . From the fact that z(t), w(t) ∈ L∞ , the inverse transformation of (54) can be used, which is explicitly given as      1 1 sx w − 2 sin φ 0 2 φ sin φ + 2 cos φ 1   z1   sy  =  1 cos φ 0 − φ cos φ − 2 sin φ 2 2 z2 0 −1 0 φ̃ (87) V. NUMERICAL RESULTS The controller design presented in Section IV was implemented on a passenger/light-duty vehicle model [16], per the governing dynamics discussed in Section II. In this case, the reference signals that generated the required maneuver were determined by trial and error. The “path-planning” problem is not addressed in this work. In practical applications, the desired trajectories may be obtained from a lead vehicle or operator. To demonstrate the general tracking performance, a representative “J-turn” was chosen as the desired maneuver and the reference SETLUR et al.: TRAJECTORY TRACKING STEER-BY-WIRE CONTROL SYSTEM FOR GROUND VEHICLES Fig. 2. 83 Vehicle reference path for a “J-turn” maneuver. Fig. 4. Vehicle position and angular tracking errors. The control gains that resulted in the best performance were k1 = 0.02, γ0 = 5, k2 = 2000, γ1 = 0.1, k3 = 5000, ε1 = 0.1 µr = 0.1. All the reference vehicle states were initialized to zero for the purpose of this simulation. The initial states of the vehicle were also initialized to zero with the exception of x(0) = −2 m, Fig. 3. Computed reference trajectories. trajectory was generated by defining urt (t) and urm (t) as urt = 10t exp(−t) urm = 10 exp(−0.05t)(1 − exp(−0.00905t)). The generated reference path is displayed in Fig. 2. The reference trajectories xr (t), yr (t), and φr (t) that are calculated and supplied to the controller are presented in Fig. 3. The auxiliary signal zd (t) was initialized as zd (0) = [0 − 1.1]T . The required function g(·) in the control signal (76) is calculated from (20), (29), and (30). The vehicle parameters used for the simulation were m = 788 kg, lf = 1 m, Izz = 1126 kg − m2 , lr = 2 m, IRoll = 171 kg − m2 Cα = 6510 N/rad. φ(0) = −0.3 rad, u(0) = 0.1 m/s. The vehicle dynamics, with integrated steering tracking algorithms, were numerically simulated for t = 150 s. The position and orientation tracking errors for the maneuver are shown in Fig. 4. Clearly, the vehicle path closely follows the reference trajectory. As shown in Fig. 4, the controller quickly compensated for the initial errors and during the transient phase, the maximum errors were ±4.5 m and ±1.2 m along the x and y directions, respectively. The orientation error initially increased to ±0.03 radians (≈ 1.7◦ ) before reaching a steady state value close to zero. The steady state position errors x̃(t) and ỹ(t) were within ±0.09 m and ±0.002 m, respectively. All errors were within 0.1% of the reference trajectory generated, thus validating the performance of the control algorithm. The vehicle’s longitudinal, lateral, and yaw velocities are shown in Fig. 5. Remark 10: The controller designed in (76) requires exact model knowledge (i.e., the exact vehicle parameters such as the sprung mass of the vehicle, inertia of the vehicle about the yaw axis, distance from the front axle to the COM and distance from the rear axle to the COM must be known a priori). In the actual operating environment, the calculation of these parameters may be difficult which could result in vehicle model uncertainties. In such scenarios, robust high gain, high frequency, or adaptive 84 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 55, NO. 1, JANUARY 2006 and    ˙ f˙ = 2 −φ̃ urt + Imεφ̇2r cos(φ̃)    − sin(φ̃) u̇rt + 2Imεφ̈r φ̇r + µ urt + Imεφ̇2r − µr φ̇(x̃˙ 1 cos(φ) + ỹ˙ 1 sin(φ)) + rr ż2  + mIurm z2 + µr g(·) . REFERENCES Fig. 5. Vehicle velocities. Longitudinal u, lateral v, and yaw r. control techniques similar to those discussed by Dixon et al. [6] could be employed to deal with parametric uncertainty. VI. CONCLUSION In this paper, an exact model knowledge nonlinear tracking controller has been presented to force a vehicle’s trajectory to follow a given reference path (or trajectory). A complete stability analysis, using Lyapunov-based techniques, has been presented to demonstrate that i) the position and orientation tracking errors are globally, exponentially forced to a neighborhood of about zero, which can be made arbitrarily small (this type of stability result is often referred to as practical tracking or GUUB), and ii) a unified framework is developed that solves the regulation problem and the tracking problem. Representative numerical results were presented to demonstrate the efficacy of the controller in enabling the vehicle to track a given trajectory. APPENDIX CALCULATION OF u̇d1 To calculate u̇d1 (t), the time derivative of (65) is computed and then the time derivative of ua (t) defined in (66) is substituted to obtain   u̇d1 = − k1 ẇ + f˙/δd2 zd2 + 2((k1 w + f )δ̇d /δd3 )zd2 + Ω̇1 zd1  + Ω1 żd1 − k1 w + f /δd2 żd2 − k2 ż1 where the time derivatives of Ω1 (t) and f (t) are explicitly given by the following expressions:  k1 w2 + wf δ̇d δ̈d δ̇d2 (2k1 w + f )ẇ + wf˙ Ω̇1 = − 2+ −2 δd δd δd2 δd3 [1] J. Ackermann and W. Sienel, “Robust yaw and damping of cars with front and rear wheel steering,” IEEE Trans. Contr. Syst. Technol, vol. 1, no. 1, pp. 15–20, Mar. 1993. [2] J. Ackermann, J. Guldner, W. Sienel, R. Steinhauser, and V. Utkin, “Linear and nonlinear controller design for robust automotive steering,” IEEE Trans. Contr. Syst. Technol., vol. 3, no. 1, pp. 132–143, Mar. 1995. [3] A. Behal, W. Dixon, D. Dawson, and Y. Fang, “Tracking and regulation control of an underactuated surface vessel with nonintegrable dynamics,” IEEE Trans. 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Law, “Modeling, characterization and simulation of automobile power steering systems for the prediction of on-center handling,” SAE, Paper No. 960178, 1996. SETLUR et al.: TRAJECTORY TRACKING STEER-BY-WIRE CONTROL SYSTEM FOR GROUND VEHICLES [21] L. Segel, “On the lateral stability and control of the automobile as influenced by the dynamics of the steering system,” J. Eng. Ind., vol. 88, no. 3, pp. 283–295, 1966. [22] P. Setlur, D. Dawson, Y. Fang, and B. Costic, “Nonlinear tracking control of the VTOL aircraft,” in Proc. 39th IEEE Conf. Decision and Control, vol. 5, Orlando, FL, 2001, pp. 4592–4597. [23] N. Stanton and P. Marsden, “Drive-by-wire systems: Some reflections on the trend to automate the driver role,” in Proc. Institut. Mechanical Engineers, Part D: J. Automobile Eng., vol. 211, no. D4, 1997, pp. 267– 276. [24] M. Stoll, Introduction to Real Analysis. Reading, MA: Addison-Wesley, 1997. [25] J. Wong, Theory of Ground Vehicles. New York: Wiley, 1978. [26] T. Wong, “Hydraulic power steering system design and optimization simulation,” SAE, Paper No. 2001-01-0479, 2001. Pradeep Setlur received the B.E. degree in instrumentation technology from the University of Mysore, India, in 1995 and the M.S. and Ph.D. degrees in electrical engineering from Clemson University, Clemson, SC, in 1999 and 2003, respectively. He worked as a Post-Doctoral Researcher at the Biomimetic and Cognitive Robotics Laboratory, Brooklyn College, The City University of New York, in 2004. His research interests include modeling and nonlinear control of robotic and automotive systems. Dr. Setlur is currently an Assistant Professor in the Electrical and Electronic Engineering Department, the California State University, Sacramento. John R. Wagner received the B.S. and M.S. degrees in mechanical engineering from the State University of New York , Buffalo, and the Ph.D. degree in mechanical engineering from Purdue University, West Lafayette, IN, in 1983, 1985, and 1989, respectively. He worked for Delco Electronics and Delphi Automotive Systems from 1989 to 1998. In August 1998, he joined the Department of Mechanical Engineering at Clemson University, Clemson, SC, where he currently is an Associate Professor. His research interests include nonlinear control theory, behavioral modeling, diagnostic/prognostic strategies, and mechatronic system design with 85 application to transportation systems. He has established the multi-disciplinary Automotive Research Laboratory and started the Rockwell Automation Mechatronics Educational Laboratory. Darren M. Dawson received the B.S. degree in electrical engineering from the Georgia Institute of Technology, Atlanta, in 1984. He then worked for Westinghouse as a control engineer from 1985 to 1987. In 1987, he returned to the Georgia Institute of Technology, where he received the Ph.D. degree in electrical engineering in March 1990. In July 1990, he joined the Electrical and Computer Engineering Department at Clemson University, Clemson, SC, where he currently holds the position of McQueen Quattlebaum Professor. His research interests are nonlinear control techniques for mechatronic applications such as electric machinery, robotic systems, aerospace systems, acoustic noise, underactuated systems, magnetic bearings, mechanical friction, paper handling/textile machines, flexible beams/robots/rotors, cable structures, and vision-based systems. He also focuses on the development of real-time hardware and software systems for control implementation. David Braganza received the B.E. degree from the University of Pune, India, in 2002 and the M.S. degree, from Clemson University, Clemson, SC, in 2004, both in electrical engineering. He is currently pursuing the Ph.D. degree in electrical engineering at Clemson University. His research interests include modeling and nonlinear control of robotic and autonomous systems and the development of real-time software for control applications.