Regions of Attraction for Hybrid Limit Cycles of Walking Robots

TL;DR

This paper applies advanced region-of-attraction analysis techniques to hybrid limit cycles in walking robot models, demonstrating the method on several examples including the rimless wheel and compass gait.

Contribution

It introduces a decomposition-based Lyapunov function search method using sum-of-squares analysis for hybrid limit cycles in walking robots.

Findings

Effective analysis of stability regions for hybrid limit cycles

Optimization of transversal surfaces and impact handling

Design of orbitally-stabilizing control strategies

Abstract

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). Each example illuminates different aspects of the procedure, including optimization of transversal surfaces, the handling of impact maps, optimization of the Lyapunov function, and orbitally-stabilizing control design.