  We consider the feedback design for stabilizing a rigid body system by making
and breaking multiple contacts with the environment without prespecifying the
timing or the number of occurrence of the contacts. We model such a system as a
discrete-time hybrid polynomial system, where the state-input space is
partitioned into several polytopic regions with each region associated with a
different polynomial dynamics equation. Based on the notion of occupation
measures, we present a novel controller synthesis approach that solves
finite-dimensional semidefinite programs as approximations to an
infinite-dimensional linear program to stabilize the system. The optimization
formulation is simple and convex, and for any fixed degree of approximations
the computational complexity is polynomial in the state and control input
dimensions. We illustrate our approach on some robotics examples.
