An Improved Lyapunov Stability Test of Equilibria Under Frictional Unilateral Contact by Sums of Squares Programming

Reliable quasi-static object manipulation and robotic locomotion require the verification of the stability of equilibria under unilateral contacts and friction. In a recent paper, Posa et al. (2016) demonstrated that sums-of-squares (SOS) programming can be used to verify the Lyapunov stability of planar systems via Lyapunov’s direct method if impacts are inelastic. However, this method appears to be too conservative to verify the stability of some remarkably simple examples. In this article, an extension of Lyapunov’s direct method is proposed, which makes use of a piecewise smooth Lyapunov function and allows a temporary increase of the Lyapunov function along a motion trajectory. In addition, a modified SOS formulation enables the investigation of planar systems with partially elastic contacts. The proposed method remains compatible with SOS programming techniques. The improved stability test is successfully applied to a point mass on a slope and to a rigid body with two contact points. For the latter, several mechanisms of instability have been demonstrated experimentally, but the exact conditions of Lyapunov stability are unknown.