On the relation between friction and part shape in parallel-jaw grasping

Previously, the authors gave planning algorithms for the problem of grasping and orienting planar parts using a parallel-jaw gripper, with analysis simplified by assuming zero friction between the part and the gripper. Their analysis is generalized to allow Coulomb friction. Coulomb friction introduces ranges of orientations where a part can become wedged between two points on its boundary. It is shown that for any planar part operating under Coulomb friction, there is a part (linear/circular in boundary) operating under zero friction that has the same mechanical behavior. Based on this observation, a bound on the coefficient of friction is given that permits the existence of a plan to orient a given part. For friction below this bound, a new planning algorithm that will find a plan for orienting the part is described. It is shown that this plan works under the weaker assumption that friction introduces nondeterminism into the transfer function, i.e., there can be more than one possible outcome for certain ranges of grasp angles