Previous work by Mattikalli et al. (1993) considered the stability of assemblies of frictionless contacting bodies with uniform gravity. A linear programming-based technique was described that would automatically determine a single stable orientation for an assembly (if such an orientation existed). In this paper, we give an exact characterization of the entire set of stable orientations of any assembly under uniform gravity. Our characterization reveals that the set of stable orientations maps out a convex region on the unit-sphere of directions. The region is bounded by a sequence of vertices adjoined with great arcs. Linear programming techniques are used to automatically find this set of vertices, yielding a precise description of the range of stable orientations for any frictionless assembly.
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