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This paper illustrates the application of recent research inregion-of-attraction analysis for nonlinear hybrid limit cycles. Three examplesystems are analyzed in detail: the van der Pol oscillator, the "rimlesswheel", and the "compass gait", the latter two being simplified models ofunderactuated walking robots. The method used involves decomposition of thedynamics about the target cycle into tangential and transverse components, anda search for a Lyapunov function in the transverse dynamics usingsum-of-squares analysis (semidefinite programming). Each example illuminatesdifferent aspects of the procedure, including optimization of transversalsurfaces, the handling of impact maps, optimization of the Lyapunov function,and orbitally-stabilizing control design.

Regions of Attraction for Hybrid Limit Cycles of Walking Robots