On the passive force closure set of planar grasps and fixtures

In this paper we consider grasps and fixtures whose contacts react according to force-displacement laws consistent with friction constraints at the contacts. The passive force closure set of such grasps and fixtures is the set of external wrenches (forces and torques) that can act on the grasped object and be stably balanced by the contacts. An external wrench belongs to this set if it induces a feasible equilibrium whose basin of attraction contains the initial unperturbed grasp configuration. The paper focuses on planar grasps and fixtures having sharp-tipped fingers or fixels that satisfy linear force displacement laws. Using Morse theory, we characterize the number and stability type of k-contact equilibria induced by a given external wrench. Based on this analysis, we provide closed-form expressions for the passive force closure set of k-contact grasps and fixtures. Computation of the allowed external wrenches is illustrated with examples. Finally, the paper describes experiments verifying the passive force closure set on a spring-loaded grasping system.