On the geometry of contact formation cells for systems of polygons

The efficient planning of contact tasks for intelligent robotic systems requires a thorough understanding of the kinematic constraints imposed on the system by the contacts. In this paper, we derive closed-form analytic solutions for the position and orientation of a passive polygon moving in sliding and rolling contact with two or three active polygons whose positions and orientations are independently controlled. This is accomplished by applying a simple elimination procedure to solve the appropriate system of contact constraint equations. We also prove that the set of solutions to the contact constraint equations is a smooth submanifold of the system's configuration space and we study its projection onto the configuration space of the active polygons. By relating these results to the wrench matrices commonly used in grasp analysis, we discover a previously unknown and highly nonintuitive class of nongeneric contact situations. In these situations, for a specific fixed configuration of the active polygons, the passive polygon can maintain three contacts on three mutually nonparallel edges while retaining one degree of freedom of motion. >