A Stabilizing Time-switching Control Strategy for the Rolling Sphere

The problem of the asymptotic stabilization of a five dimensional nonholonomic systems, namely the "ball and plate" or "rolling sphere" system, is discussed and solved by means of a hybrid control law relying on a suitable finite state machine. A control law is associated to each state of the machine and, by using a simple switching strategy, the origin is proven to be globally asymptotically stable in the sense of Lyapunov. Moreover, a particular function is proven to be a Lyapunov function for the considered hybrid system. The chosen control law takes naturally into account the presence of possible control saturations. Simulations are presented showing the effectiveness of the proposed control scheme.