FINDING ALL STABLE ORIENTATIONS OF ASSEMBLIES WITH FRICTION

Previous work by Mattikalli et al, [1] considered the stability of ass emblies of frictionless contacting bodies under uniform gravity, A lin ear programming-based technique was described that would automatically determine a single stable orientation for an assembly (if such an ori entation existed), In this paper, we include Coulomb friction at conta cts between bodies and give a characterization of the entire set of st able orientations of an assembly under uniform gravity, Our characteri zation is based on the concept of potential stability, which describes a necessary but not sufficient condition for the stability of an asse mbly, Orientations that are computed as being unstable, however, are g uaranteed to fall apart. Our characterization reveals that the set of stable orientations maps out a convex region on the unit-sphere of dir ections and corresponds to a spherical analog of a planar polygon-the region is bounded by a sequence of vertices joined by great arcs. Line ar programming techniques are used to automatically find this set of v ertices, yielding a description of the range of stable orientations fo r any assembly, For frictionless assemblies, our characterization of s table orientations is exact, For assemblies with friction, some conser vative approximations associated with the use of a linearized Coulomb law are made.