Finding All Stable Orientations of Assemblies with Friction

Abstract

In this paper, we include Coulomb friction at contacts between bodies and give a characterization of the entire set of stable orientations of an assembly under uniform gravity. Our characterization is based on the concept of potential stability, which describes a necessary but not sufficient condition for the stability of an assembly. Orientations that are computed as being unstable, however, are guaranteed to fall apart. Our characterization reveals that the set of stable orientations maps out a convex region on the unit-sphere of directions and corresponds to a spherical analog of a planar polygon-the region is bounded by a sequence of vertices joined by great arcs. Linear programming techniques are used to automatically find this set of vertices, yielding a description of the range of stable orientations for any assembly. For frictionless assemblies, our characterization of stable orientations is exact. For assemblies with friction, some conservative approximations associated with the use of a linearized Coulomb law are made.