Lyapunov stability of a rigid body with two frictional contacts

TL;DR

This paper develops a method to analyze the Lyapunov stability of a planar rigid body with two frictional contacts, using a reduced Poincaré map to determine stability or instability of equilibrium states.

Contribution

It introduces a semi-analytic approach to assess stability of systems with hybrid dynamics and multiple contacts, which was previously very challenging.

Findings

The method can determine stability or instability for most equilibrium configurations.

Simulation examples validate the approach and illustrate stability regions.

The approach simplifies complex hybrid dynamics into scalar functions for analysis.

Abstract

Lyapunov stability of a mechanical system means that the dynamic response stays bounded in an arbitrarily small neighborhood of a static equilibrium configuration under small perturbations in positions and velocities. This type of stability is highly desired in robotic applications that involve multiple unilateral contacts. Nevertheless, Lyapunov stability analysis of such systems is extremely difficult, because even small perturbations may result in hybrid dynamics where the solution involves many nonsmooth transitions between different contact states. This paper concerns with Lyapunov stability analysis of a planar rigid body with two frictional unilateral contacts under inelastic impacts, for a general class of equilibrium configurations under a constant external load. The hybrid dynamics of the system under contact transitions and impacts is formulated, and a \Poincare map at two-contact states is introduced. Using invariance relations, this \Poincare map is reduced into two semi-analytic scalar functions that entirely encode the dynamic behavior of solutions under any small initial perturbation. These two functions enable determination of Lyapunov stability or instability for almost any equilibrium state. The results are demonstrated via simulation examples and by plotting stability and instability regions in two-dimensional parameter spaces that describe the contact geometry and external load.