Let Them Fall Where They May: Capture Regions of Curved Objects and Polyhedra

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Abstract
When a three-dimensional object is placed in contact with a supporting plane, gravitational forces move it to one of a finite set of stable poses. For each stable pose, there is a region in the part's configuration space called a capture region; for any initial configuration within the region, the object is guaranteed to converge to that pose. The problem of computing maximal capture regions from an object model is analyzed assuming only that the dynamics are dissipative; the precise equations governing the system are unnecessary. An algorithm, based on Morse theory, is first developed for objects with smooth, convex hulls. The formulation is then extended to objects with piecewise-smooth hulls using a catalog of critical points de rived from stratified Morse theory. Algorithms have been fully implemented for objects with smooth and polyhedral convex hulls. As examples from the implementation demonstrate, calcu lating these regions from a geometric model is computationally practical.

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