Ring-Shaped Wheeled Mobile Robot Circulation with Modified Van der Pol Limit-Cycle Reference

Abstract

Defining and tracking trajectories in complex environments for nonholonomic mobile robots are challenging due to the underactuated dynamics and nonintegrable velocity constraints of these robots, which preclude smooth, time-invariant feedback stabilization and yield uncontrollable linearizations around equilibrium points. As a result, maintaining structured motions such as ring-shaped limit cycles becomes particularly difficult under large initial deviations or external disturbances. In this paper, a control framework based on a dynamically generated reference trajectory is proposed, where the desired motion is defined by a modified Van der Pol oscillator. Unlike conventional approaches relying on predefined geometric paths, the proposed method embeds the target orbit into a dynamic auxiliary nonlinear system whose trajectories converge to a stable limit cycle, enabling local asymptotic convergence to the desired motion. A discontinuous robust control law is designed for a perturbed wheeled mobile robot, and the resulting closed-loop system is analyzed within the framework of solutions of systems with discontinuous right-hand sides. It is shown that the tracking error dynamics are uniformly and ultimately bounded with respect to matched disturbances and that, in the disturbance-free case, the tracking errors converge asymptotically to the origin. As a consequence, the robot’s trajectory converges to the invariant limit cycle of the reference dynamics, therebydriving the robot’s trajectory toward the invariant limit cycle of the reference dynamics. The simulation results demonstrate an improvement in the transient response relative to standard circular reference tracking. The experimental results further corroborate these findings, showing that the modified Van der Pol reference keeps the position tracking errors tightly bounded, while mitigating the large initial overshoot associated with the circular reference.