A Hybrid Automaton for a Class of Multi-Contact Rigid-Body Systems with Friction and Impacts*

Hybrid automata is a powerful framework for representing the dynamics of rigid-body mechanical systems. Thus far however, work has mainly focused on examples with only single points of contact. Consider for instance, the ubiquitous bouncing-ball example. The case of multiple contacts is much more complex. An event at one contact can affect the other contacts even when there is no apparent transmission mechanism. In this paper, we present a general hybrid automaton for systems consisting of multiple rigid-bodies with convex differentiable surfaces. The hybrid automaton assumes that Newton's impact law (restitution) and Coulomb friction are acting at the points of contact. Multiple contacts are addressed by introducing computational nodes which can be considered as the well-known mythical modes in the discontinuous systems' literature. These nodes consist of non-dynamical discrete states and reset operations which find valid combinations of contact forces. In this manner, they will guide the executions of the hybrid automaton to the relevant hybrid discrete locations.