Model predictive control for simultaneous robot tracking and regulation

Model Predictive Control (IMPC) is used in this paper for tracking and regulation of a nonholonomic mobile robot. To guarantee MPC tracking stability, a control Lyapunov function is employed. Based on a control Lyapunov function, the cost function of MPC is added a terminal state penalty. And a terminall state region is found to constrain terminal state. The analysis results show that the MPC tracking control has simultaineous tracking and regulation capability. Simulation results are provided to verify the proposed control strategy. It is shown that the control strategy is fedsible. Keywords-Receding horizon control, Model predictive control, nonholonomic mobile robots I. INTRODUCTtON Tracking control of nonholonomic mobile robots is to control the robots to move along a given time vary trajectory (reference trajectory). It is fundamental motion control problem and has been intensively investigated in robotic community using different approaches [2][4][6] [9][10][15][16][24][26]. Recently, controllers with simultaneous tracking and regulation capability have been explored. A differential kinematic cortroller was developed in [8], which provides a global exponential stable property for both tracking control and regulation. A unifying framework for tracking and regulating control was presented in [21] by using dynamic feedback linearisation. However, these two research results were not using a single controller. The switch between their controllers for tracking and regulation is required. In [ 181, a single global stable controller with simultaneous tracking and regulating capability was reported by using backstepping technique while it includes saturation constraints of control signals. Besides the stability and saturation constraints consideration in these research for tracking control, tracking performance, such as tracking time, trajectory length or control energy, are also crucial. Model Predictive Control (MPC) is one of the frequently applied optimisation control techniques in industry. It is designed to handle a constrained optimisation problem [20]. Due to the use of predictive control horizon in MPC, the control stability becomes one of the main problems [l]. It was shown that using infinite receding horizon can guarantee the control stability for even nonlinear systems [ 171, but it is computational intractable in practice. For finite receding horizon, it was proved that forcing the terminal state to equal zero can guarantee the stability [23]. However, the terminal state equality constraint costs long time for the optimisation. Further work shows that the terminal state equality constraint can be relaxed as a terminal state inequality, i.e. a terminal state region, by adding a terminal state penalty to the optimised cost function if a linear state feedback controller exists in the terminal state region [5][7]. Furthermore, the linear feedback control is never applied to the system since it is only used to find the terminal state region to insure that the system will move into this region after finite predictive control horizon. Recent researches in [ 11][19][22][25] show that the local linear feedback controller is not necessary. Any other controllers can be used to find the terminal state region as long as a stability condition is met. And the researches in [ 14][22] show that the terminal state penalty can be a control Lyapunov function, which will guarantee the stability once the terminal state is within the terminal state region. This paper investigates application of the stabilising MPC using a control Lyapunov function in robot tracking control. A control Lyapunov function is developed to guarantee the stability. Based on this control Lyapwnov function, a terminal state region and its corresponding virtual controller are found. The using of the terminal state region constrains the MPC optimisation, while the virtual controller provides a feasible solution in the terminal state region for the MPC optimisation. This stabilising MPC generates a single controller with simultaneous tracking and regulating capability. The switch between tracking control and regulation is not necessary. Moreover, the control signal constraints are explicitly imposed on the controller. And it is feasible to select different optimisation objectives to improve the tracking performance. Another problem in MPC is the heavy computation, which should never under-estimated even for finite horizon. In this paper, the proposed controller pursuits a suboptimal solution rather than an optimal solution to reduce the computation time. Applying to MPC to regulation problem was reported in [11][27]. The stability was not discussed in [27]. The terminal state region in [ 1 I ] is a group of equations, which costs long time for the optimisation. The paper is organised as follows. Section I1 introduces the tracking control problem of a nonholonomic mobile robot. The MPC approach is described in section III. The terminal state controller and terminal state region are found in section IV. The simulation results are provided in section VI. Finally, our conclusion and future works are discussed in section VII. 212 0-7803-8748-11041%20.00 02004 IEEE. 11. KINEMATIC TRACKING CONTROL A differential driving mobile robot is a typical nonholonomic mobile robot, which has two real driving wheels and a front castor for body support. Speed control of the two real wheels (v, and v,) leads to control of the robot linear speed (v=(v,+v,)L?) and the angular speed (w=(v,-v,)B) in which B is the wheelbase. The motion state of the robot can be described by its position (x, y ) , the midpoint of the rear axis of the robot, and its orientation (e). The kinematics equation is as follows: The state and control sign4 vector are denoted as x = (x, y, e)T and U = (v, w ) ~ . In tracking control, reference trajectories should be described by a reference state vector x, = (xn yn QJT and a reference control signal vector U, = (vn w,lT. The reference trajectory should have the same kinematics as (1): To control (1) to track (2), an error state x, can be defined as follows [ 161: And the error state dynamic model is derived as: i, = wy, + v, case, e, = w, w ye = -wx, + v, sine, Redefining the control signals as: Then, the error state dynamic model (4) becomes as: x, = [; ] = [-: ;: I [ ; : I + [ Irr 0 0 0 6 , (5) To analysis the local stability for the error state dynamic model (4), a linearised error state model of (6) can be obtained as follows: x,= [ -w l r ; 0 v, :] x,+ [a 0 0 ;] U, (7) Since the linearised error state model (7) is controllable, the local asymptotic stable controllers can be found [16]. However, this local linear controllable property is lost when the linear speed or angular speed converges to zero ( lim(v,(tl2 -+ w,(t)' = o 1. Therefore, many controllers developed so far require this persistent excitation condition, i.e. the controlled robot can not be stopped; otherwise the control stability will be lost. Due to the requirement for the persistent excitation condition, the motion control of nonholonomic mobile robots has been divided into two independent problems to handle: regulation (parking) and tracking. For the regulation problem ( lim(v,(t)* + w, ( t )* = 0 ), the local linearised model is not controllable and there does not exist a time invariant feedback control law [3]. So, independent controllers have been proposed for the two problems. The switch of two independent controllers is needed when a tracking robot is required to stop. To avoid the switching, investigation on a single controller with simultaneous tracking and regulation capability is necessary. r+m