This paper illustrates the application of recent research in
region-of-attraction analysis for nonlinear hybrid limit cycles. Three example
systems are analyzed in detail: the van der Pol oscillator, the "rimless
wheel", and the "compass gait", the latter two being simplified models of
underactuated walking robots. The method used involves decomposition of the
dynamics about the target cycle into tangential and transverse components, and
a search for a Lyapunov function in the transverse dynamics using
sum-of-squares analysis (semidefinite programming).