Controlled synchronization of mechanical systems with a unilateral constraint

This paper addresses the controlled synchronization problem of mechanical systems subjected to a geometric unilateral constraint as well as the design of a switching coupling law to obtain synchronization. To define the synchronization problem, we propose a distance function induced by the quotient metric, which is based on an equivalence relation using the impact map. A Lyapunov function is constructed to investigate the synchronization problem for two identical one-dimensional mechanical systems. Sufficient conditions for the individual systems and their controlled interaction are provided under which synchronization can be ensured. We present a (coupling) control law which ensures global synchronization, also in the presence of grazing trajectories and accumulation points (Zeno behavior). The results are illustrated using a numerical example