On the hybrid dynamics of planar mechanisms supported by frictional contacts. I: necessary conditions for stability

This paper is concerned with the stability of planar mechanisms supported by multiple frictional contacts against gravity. Stability of equilibrium postures is investigated under initial perturbations which may involve sliding or separation at the contacts. The frictional dynamics is formulated using the notion of contact modes, and the related problems of solution ambiguity and inconsistency are reviewed. The paper then uses the condition of strong equilibrium to eliminate ambiguities, and defines a new condition of kinematic-strong equilibrium which additionally eliminates frictional inconsistencies. It is then proven that strong equilibrium is necessary for stability of frictional equilibrium postures, and that kinematic-strong equilibrium guarantees finite-time recovery of an initially perturbed contact. The results are demonstrated on a reduced model of a rigid body having a variable center-of-mass and supported by two frictional contacts. A companion paper completes the analysis of this reduced problem by investigating the overall hybrid dynamics and deriving sufficient conditions for its stability.