Cross-Coupling Control for Slippage Minimization of a Four-Wheel-Steering Mobile Robot Maxim Makatchev Dept. of Manufacturing Engineering and Engineering Management City University of Hong Kong Hong Kong Email: maxim.makatchev@cityu.edu.hk Sherman Y. T. Lang Integrated Manufacturing Technologies Institute National Research Council of Canada London Ontario Canada N6G 4X8 Email: Sherman.Lang@nrc.ca S. K. Tso John J. McPhee Centre for Intelligent Design Automation and Manufacturing City University of Hong Kong Hong Kong Email: mesktso@cityu.edu.hk Dept. of Systems Design Engineering University of Waterloo Ontario Canada N2L 3G1 Email: mcphee@real.uwaterloo.ca Abstract The paper presents a cross-coupling controller for a four-wheel steering mobile robot with independently actuated steering and driving of each wheel. The aim of the controller is to minimize slippage during transient states of the vehicle control. Linear approximation of the criterion for no slippage is adopted for the design of the LQR-based cross-coupling controller of steering actuators. Driving actuators are coupled with steering actuators in pairs corresponding to each wheel so that their transient errors are kept mutually proportional. The experiments performed on a dynamic model of the vehicle demonstrate significant reduction of slippage with moderate vehicle posture errors. Keywords: cross-coupling control four-wheel steering wheeled mobile robots. 1 Introduction Control strategies for wheeled mobile robots (WMRs) can be divided into those based on the kinematics and on the dynamics of the system. Kinematics-based control of WMRs has been investigated for particular prototypes e. g. 11 14 as well as for generic WMR models 13 17 18. The basic assumption in most of the existing WMR kinematic models for instance in 3 12 18 is so called pure rolling condition i. e. no tire slippage occurs. Incidentally wheel plants are not incorporated into the WMR kinematic models. The omission of wheel plants from the kinematic models and the respective controllers leads to independent wheel plant errors during the transient states of vehicle control with a number of negative implications. First kinematic model-based dead reckoning can produce significant errors even if inertial forces applied to the vehicle are negligible 5. Second in redundant WMRs independent errors of the wheel plants generally cause tire slippage. Tire slippage itself is a source of dead reckoning errors. It can also affect controllability and stability of the vehicle due to highly nonlinear behaviour of the lateral friction force between the tire and the ground for lateral slip angle values above a certain limit 19. Dynamic models of wheeled vehicles usually incorporate slippage and thus lead to a more adequate controller design. Extensive study has been made for automobile dynamics and control where the simplified two-wheel (single-track) model is often utilized for conventional and four-wheel steering (4WS) systems 1 2. However while the single-track model may be adequate for description of 4WS automobiles where wheels are mechanically coupled in pairs WMRs with independent steering actuators require a more complex dynamic model. Dynamics and control of generic WMRs is investigated in 4 7. Controllers that are based on detailed dynamic models are normally computa-
tionally demanding and their robustness is difficult to investigate analytically. There is a need for controllers that preserve the simplicity of kinematics-based control and take into account important dynamic effects of WMRs. One possible way is to use an approximation of vehicle dynamics that targets particular dynamic effects. As examples we cite (a) the approximated dynamic model that is developed in 3 which incorporates slippage due to misalignment of the wheels via minimization of a friction functional and (b) the least-squares estimation of the WMR frame posture combining the redundant sensed wheel parameters that is proposed in 6. Another possible way is to augment a vehicle model and the respective controller so that a particular dynamic effect is included into the control objectives. In work 1 for example the single-track dynamic model of a four-wheel-steering vehicle is augmented by steering actuator dynamics allowing a dedicated controller to be used for synchronization of wheel steering angles. This paper follows the latter approach. The paper considers a WMR equipped with four independently steered and driven wheels. Such a kind of WMR design with a redundant number of actuated wheels is often beneficial in applications involving heavy load transfer and requiring high maneuverability and reliability. Due to redundancy of the actuators of the vehicle tire slippage caused by errors of steering and driving actuators is practically unavoidable. To reduce tire slippage in the special case when steering angles are fixed a combined controller of driving actuators of a multiple-wheel WMR has been developed in 16. Works 5 8 9 propose a cross-coupling controller for a differential drive mobile robot to reduce the robot s body orientation errors. A cross-coupling LQR-based controller to steer wheels in pairs at the same angle has been proposed in 1 for a four-wheel WMR with independent steering and driving of each wheel. Similarly to 1 we augment a vehicle model (in our case a kinematic model) with wheel actuator dynamics. We extend the approach of 1 by: (a) incorporating not only steering but also driving actuator dynamics (b) cross-coupling all four wheels of the vehicle instead of coupling them pairwise (c) generalizing the control objectives so that the wheels are able to track a linear approximation of the vehicle kinematic constraints during transient states of control which allows the four wheels to be steered at arbitrary angles (not necessarily the same angles as in 1). The LQRbased cross-coupling controller of four steering actuators is utilized with the control goal of steering all the wheels at their respective required angles keeping the corresponding errors mutually proportional. The driv- Y O W 3 X y W 4 C W 1 v 1 W 2 Slip angle 1 ψ 1 Figure 1: Kinematics of the WMR. ing actuators are coupled with their respective steering actuators so that their errors are kept mutually proportional as well. Simulation studies performed on a dynamic model especially developed for the WMR show significant reduction of wheel slip angles for typical turning maneuvers. The paper is organized as follows. In section 2 we describe the vehicle kinematics. Section 3 presents the design of the cross-coupling controller. Section 4 includes the simulation results for a turning maneuver. Conclusion remarks are given in section 5. 2 Kinematic model A diagram of the vehicle is presented in Fig. 1. Assuming that the vehicle frame is a rigid body the velocities v i of the wheel base points W i are related to the velocity of the reference point of the frame C as v i = v c θz CW i i = 1... 4 (1) where θ is the angular velocity of the vehicle s frame coordinate system {C x y} with respect to the fixed coordinate system of the plane {O X Y } z is the unit vertical vector. Assuming natural embedding of the vector CW i into three-dimensional space the crossproduct is well-defined. Pure rolling condition implies that the wheel s rotation rate φ i and steering angle ψ i with respect to the frame coordinate system are such that φ i = v i r θ x v c tan ψ i = v iy v ix i = 1... 4 (2)
where r is the radius of the wheel v ix and v iy are the components of the vector v i in the frame coordinate system {C x y}. 3 Controller design One of the known kinematics-based control schemes can be applied to control the WMR s frame posture (e. g. 18). The trajectory (path) controller output in this case can be presented as the desired velocity v c of the frame reference point and the desired angular velocity θ of the frame which are used to calculate the respective desired wheel control inputs φ id ψ id. Further we consider φ i d ψ id i = 1... 4 available as inputs of our wheel cross-coupling controller. Steering actuator dynamics is modeled as F i = b i /s(s a i ) with the steering motor current I i (% of maximal value signed depending on the polarity of the applied voltage) as an input variable and the steering angle ψ i as an output variable. Let the discrete-time state space model of the steering actuator i with the continuous-time transfer function F i and the sampling time T samp be x2i1 (k 1) x 2i (k 1) = G i x2i1 (k) x 2i (k) H i u i (k) where x 2i1 (k) is equal to the steering angle ψ i at the step k so that x 1 (k) = ψ 1... x 7 (k) = ψ 4 x 2i (k) is an appropriate state variable depending on the representation the plant input u i (k) is equal to I i at the step k. The discrete-time state space model of the vehicle steering system can be presented as x(k 1) y(k) = Gx(k) Hu(k) = Cx(k) (3) with the state vector and the control inputs x = x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 T u = I 1 I 2 I 3 I 4 T and G 1 G = G 2 G 3 G 4 H = H 1 H 2 H 3 H 4. Matrix C can be chosen so that regulating outputs y at the reference value y d will reflect our control goals. For an independent control C = 1 1 1 1 (4) y d = ψ 2d ψ 3 d ψ 4d. (5) To choose an appropriate matrix C for a crosscoupling controller we consider the particular relationship between the wheel steering actuators. Denoting the initial steering angles as ψ is the initial errors δ i = ψ i s ψ id. The case when some of the initial errors are equal to zero can be easily handled by excluding the respective actuators from the cross-coupling controller or by modifying the resultant C and y d so that those actuators are controlled independently. Further in the paper we will assume that δ i i = 1... 4. The condition ψ i ψ i d δ i = ψ j ψ jd δ j δ i δ j (6) will be referred to as the condition of proportional errors between the steering actuators i and j. As is easy to see the relation defined by (6) is transitive: if the condition is satisfied between the actuator of wheel 1 and the actuator of each of the other wheels then it is satisfied for any pair of them. Assuming that ψ is and ψ id i = 1... 4 satisfy the kinematic constraints implied by (1) and (2) (i. e. all the wheel axes intersect at a common point) the condition of proportional errors defines a linear approximation of the kinematic constraints during the transition from ψ is to ψ id. The overall control architecture then can be interpreted as follows: the high-level controller of the vehicle frame posture defines the nodes in the space of the controlled variables which satisfy the kinematic constraints and the low-level steering controller tracks the piecewise linear trajectory that connects the nodes. Similarly the condition of proportional errors can relate steering and driving actuators of the wheels. For simplicity of the controller design first we cross-couple the steering actuators while each driving actuator is coupled with its respective steering actuator. The output matrix and the reference value 1 1 C = δ 1 δ 2 1 δ 1 δ 3 (7) 1 1 δ 4 y d = ψ2d δ 2 ψ3d δ 3 ψ4d δ 4 (8)
implement the condition of proportional errors for the system (3) so that the steering actuator errors of wheels 2 3 and 4 are kept proportional to the steering actuator error of wheel 1. Thus similarly to 1 we treat the cross-coupling problem as a regulation problem and apply linear quadratic optimal regulator theory for the controller design. The control law u(k) = Kx(k) Kν(k) ν(k) = ν(k 1) y d (k) y(k) minimizes the performance index J = 1 2 k= (ξ Qξ η Rη) where ξ(k) = x(k) x( ) ν(k) ν( ) T η(k) = u(k) u( ) ν(k) = ν 1 (k) ν 2 (k) ν 3 (k) ν 4 (k) T and matrices K K are derived from the respective Riccati equation 15. We consider the state variables x available for feedback as the measured outputs of the system as defined by C in (4) and the practical values of G allow state estimation. As this study is primarily concerned with the crosscoupling of the steering actuators we assume that the driving actuators have fast enough response so that they can effectively track the reference inputs that are directly fed from the respective steering actuator control loops. Denote the initial error of driving actuator velocities as ε i = φ is φ id. The driving actuator reference inputs φ ir = φ is ε i (ψ i ψ is ) (9) δ i implement the condition of proportional errors between the steering actuator and the driving actuator of the wheel i. The overall control architecture is presented in Fig. 2. 4 Simulation results Simulation studies have been carried out with the dynamic model of the vehicle. We compare the performance of three schemes of wheel actuator coupling in the experiments with the turning maneuver. The reference trajectory consists of a straight-line interval and a circular arc of the radius R = 2 m with the center closer to wheels 1 and 4. The reference frame linear velocity is.6 m/s the reference yaw angle is equal to the reference path slope angle at every instant. These settings and constraints (1) (2) imply that during the line-interval following all reference steering angles are equal to zero and during the circular-arc following the reference steering angles are related as ψ 1 d = ψ 3d = 38.8 ψ 2 d = ψ 4d = 2.1. Steering actuator 1 is set to have a slower response than the others and the driving actuators are assumed to have much faster response than the steering actuators. The settings of the dynamics of the wheel steering actuators that correspond to a typical operation conditions of the WMR prototype developed at City University of Hong Kong and some of the parameters of the vehicle are as follows: G 1 = G 2 = G 3 = G 4 = 1.968 1.9675 1.785 1.7851 1.82 1.817 1.768 1.768 H 1 =.1533.1516.1751 H 2 =.1615.165 H 3 =.1491.184 H 4 =.1685 =.1 s sampling time T samp CW iy =.75 m r =.12 m. Three control schemes are tested: uncoupled control (LQR-based controller for the steering motors that corresponds to (4) (5) with the driving motors controlled independently); partially coupled control (uncoupled controller for the steering motors with the driving motors coupled with the respective steering motors according to (9)); cross-coupling control (LQRbased controller for the steering motors that corresponds to (7) (8) with the driving motors coupled with the respective steering motors according to (9)). The results are shown in Fig. 3. The path following error is larger for the coupling controllers as the fast wheels are slowed down to synchronize with the slow ones. For the cross-coupling controller the linear displacement from the reference circular arc is about three times larger than for the respective displacement for the uncoupled controller. The maximum lateral slip angles are at least two times smaller for the inner steering wheels that are subject to the most severe slippage conditions. 5 Conclusion CW ix = 1 m The study investigates the efficiency of the linear approximation of kinematic constraints during transient states of control of the four-wheel steering and driving WMR. The trade-off between the wheel lateral slip angles and accuracy of trajectory tracking is investigated experimentally. It is shown that LQR-based cross-coupling controller reduces lateral slip angles to the limits where many conventional types of tires are subject to close-to-linear lateral friction force which
implies better controllability and stability of the vehicle. At the same time vehicle frame posture errors are kept moderate compared to uncoupled control. These effects of the proposed control scheme can be particularly useful for applications where high priority is given to smooth motion stability and high-precision dead reckoning of the vehicle rather than to accuracy of trajectory tracking for example for some applications of behaviour-based robot control. Acknowledgments The authors would like to acknowledge the assistance of Nigel Lee. The project is supported by the Research Grant Council of Hong Kong and a Strategic Research Grant of City University of Hong Kong. References 1 J. Ackermann Robust Decoupling Ideal Steering Dynamics and Yaw Stabilization of 4WS Cars Automatica vol. 3 no. 11 pp.1761 1768 1994. 2 J. Ackermann A. Bartlett D. Kaesbauer W. Seinel and R. Steinhauser Robust Control. Springer-Verlag 1993. 3 J. C. Alexander and J. H. Maddocks On the kinematics of wheeled mobile robots International Journal of Robotics Research vol. 8 no. 5 pp. 15 27 1989. 4 A. Bétourné G. Campion Dynamic Modelling and Control Design of a Class of Omnidirectional Mobile Robots Proc. of IEEE Int. Conf. on Robotics and Automation 1996 pp. 281 2815. 5 J. Borenstein Y. Koren Motion Control Analysis of a Mobile Robot Journal of Dynamic Systems Measurement and Control vol. 19 pp. 73 79 1987. 6 T. Burke H. F. Durrant-Whyte Kinematics for Modular Wheeled Mobile Robots Proc. of the 1993 IEEE/RSJ Int. Conf. on Intelligent Robots and Systems Yokohama Japan 1993 pp. 1279 1286. 7 G. Campion. G. Bastin B. D Andréa-Novel Structural Properties and Classification of Kinematic and Dynamic Models of Wheeled Mobile Robots IEEE Trans. on Robotics and Automation vol. 12 no. 1 pp. 47 61 1996. 8 L. Feng Y. Koren J. Borenstein Cross-coupling motion controller for mobile robots IEEE Control Systems Magazine pp. 35 43 December 1993. 9 L. Feng Y. Koren J. Borenstein A modelreference adaptive motion controller for a differential-drive mobile robot Proc. of IEEE Int. Conf. on Robotics and Automation 1994 vol. 4 pp. 391 396. 1 N. Matsumoto H. Kuraoka M. Ohba An experimental study on vehicle lateral and yaw motion control Proc. of Int. Conf. on Industrial Electronics Control and Instrumentation (IECON) 1991 vol. 1 pp. 113 118. 11 P. F. Muir C.P. Neuman Kinematic Modeling for Feedback Control of an Omnidirectional Wheeled Mobile Robot Proc. of IEEE Int. Conf. on Robotics and Automation 1987 pp. 1772 1778. 12 P. F. Muir C.P. Neuman Kinematic Modeling of Wheeled Mobile Robots Journal of Robotic Systems vol. 4 no. 2 pp. 281 329 1987. 13 R. M. Murray S. S. Sastry Steering Nonholonomic Systems Using Sinusoids Proc. of the 29th IEEE Conf. on Decision and Control 199 vol. 4 pp. 297 211. 14 W. L. Nelson I. J. Cox Local Path Control for an Autonomous Vehicle Proc. of IEEE Int. Conf. on Robotics and Automation 1998 vol. 3 pp. 154 151. 15 K. Ogata Discrete-Time Control Systems Prentice-Hall 1995. 16 D. B. Reister M. A. Urensen Position and constraint force control of a vehicle with two or more steerable drive wheels IEEE Trans. on Robotics and Automation vol. 9 no. 6 pp. 723 731 1993. 17 C. Samson Control of Chained Systems Application to Path Following and Time-Varying Point- Stabilization of Mobile Robots IEEE Trans. on Automatic Control vol. 4 no. 1 pp. 64 76 1995. 18 B. Thuilot B. d Andréa-Novel A. Micaeli Modeling and Feedback Control of Mobile Robots Equipped with Several Steering Wheels IEEE Trans. on Robotics and Automation vol. 12 no. 3 pp. 375 39 1996. 19 J. Y. Wong Theory of Ground Vehicles. New York: Wiley 1993.
y 1d ν 1 (k) z 1 Steering motor 1 x 1 (k)x 2 (k) ψ 1 (k) = x 1 (k) motor 1 y 2d ν 2 (k) z 1 K Steering motor 2 K x 3 (k) x 4 (k) C ψ 2 (k) = x 3 (k) motor 2 y 3d ν 3 (k) z 1 Steering motor 3 x 5 (k) x 6 (k) ψ 3 (k) = x 5 (k) motor 3 y 4d ν 4 (k) z 1 Steering motor 4 x 7 (k)x 8 (k) ψ 4 (k) = x 7 (k) motor 4 Figure 2: Cross-coupling control system. 1 5-5 -1-15 -2 Lateral Slip Angle 1-25 1 2 3 Lateral Slip Angle 3 2 1 5-5 Lateral Slip Angle 2-1 1 2 3 Lateral Slip Angle 4 15 Y (m).6.5.4.3.2 Path of Frame Reference Point C Reference path Uncoupled control Partially coupled control Cross-coupling control 1-1 1 5-5.1-2 1 2 3-1 1 2 3 -.1.2.4.6.8 1 1.2 1.4 1.6 1.8 X (m) (a) Lateral slip angles. (b) Path of frame reference point. Figure 3: Performance of the different control schemes.