ON THE GEOMETRY OF CONTACT FORMATION CELLS FOR SYSTEMS OF POLYGONS

The efficient planning of contact tasks: for intelligent robotic syste ms requires a thorough understanding of the kinematic constraints impo sed on the system by the contacts, In this paper, we derive closed-for m analytic solutions for the position and orientation of a passive pol ygon moving in sliding and rolling contact with two or three active po lygons whose positions and orientations are independently controlled. This is accomplished by applying a simple elimination procedure to sol ve the appropriate system of contact constraint equations, The benefit s of having analytic solutions are numerous. For example, they elimina te the need for iterative nonlinear equation solving algorithms to det ermine the position and orientation of the passive polygon given the p ositions and orientations of the active ones, Also, because they conta in the configuration variables of the active polygons and the relevant geometric parameters, models of geometric and control uncertainty can be readily incorporated into the solutions, This will facilitate the analysis of the effects of these uncertainties on the kinematic constr aints. We also prove that the set of solutions to the contact constrai nt equations is a smooth submanifold of the system's configuration spa ce and we study its projection onto the configuration space of the act ive polygons (i.e., the lower-dimensional configuration space of contr ollable parameters), 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 s ituations, for a specific fixed configuration of the active polygons, the passive polygon can maintain three contacts on three mutually nonp arallel edges while retaining one degree of freedom of motion.