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We consider the feedback design for stabilizing a rigid body system by makingand breaking multiple contacts with the environment without prespecifying thetiming or the number of occurrence of the contacts. We model such a system as adiscrete-time hybrid polynomial system, where the state-input space ispartitioned into several polytopic regions with each region associated with adifferent polynomial dynamics equation. Based on the notion of occupationmeasures, we present a novel controller synthesis approach that solvesfinite-dimensional semidefinite programs as approximations to aninfinite-dimensional linear program to stabilize the system. The optimizationformulation is simple and convex, and for any fixed degree of approximationsthe computational complexity is polynomial in the state and control inputdimensions. We illustrate our approach on some robotics examples.

Controller Synthesis for Discrete-time Hybrid Polynomial Systems via Occupation Measures