Lyapunov stability analysis of rigid body systems with multiple contacts by sums-of-squares programming

TL;DR

This paper extends Lyapunov stability analysis for multi-contact rigid body systems using sums-of-squares programming by introducing multiple Lyapunov functions that permit temporary increases, improving stability verification.

Contribution

It proposes a novel Lyapunov-based method with multiple functions allowing temporary increases, compatible with SOS programming, for analyzing multi-contact rigid body stability.

Findings

Successfully applied to a rigid body with two point contacts

Improves over conservative single Lyapunov function methods

Enables stability verification when exact conditions are unknown

Abstract

Reliable quasi-static object manuipulation and robotic locomotion require verification of the stability of equilibria under rigid contacts and friction. In a recent paper, M. Posa, M. Tobenkin, and R. Tedrake demonstrated that sums-of-squares (SOS) programming can be used to verify Lyapunov stability via Lyapunov's direct method. This test was successfully applied to several simple problems with a single point contact. At the same time it has been found that this method is too conservative for several multi-contact systems. In this paper, an extension of Lyapunov's direct method is proposed, which makes use of several Lyapunov functions, and which allows \emph{temporary} increase of those Lyapunov function along a motion trajectory. The proposed method remains compatible with SOS programming techniques. The improved stability test is successfully applied to a rigid body with 2 point contacts, for which the exact conditions of Lyapunov stability are unknown.