Safe Control of Hopping in Uneven Terrain

Using a classic problem from robotics of a vertical hopping machine, we demonstrate an approach for investigating the safety of a hybrid discrete/continuous dynamic system operating in an uncertain environment. The challenges imposed by the environment are expressed in terms of constraints imposed in the phase space of the system as it undergoes periodic motion. The approach is demonstrated first with a hopper that has state feedback to govern the timing of thrust and subsequently for a timer-based hopper. The latter case increases the dimensionality of the problem and must be treated numerically. However, the use of a multiresolution surface representation of the feasible regions in state space reduces the computational burden of the approach.