Grasp Analysis as Linear Matrix Inequality Problems

Three important problems in the study of grasping and manipulation by multi ngered robotic hands are: (a) Given a grasp characterized by a set of contact points and the associated contact models, determine if the grasp has force closure; (b) If the grasp does not have force closure, determine if the ngers are able to apply a speci ed resultant wrench on the object; and (c) Compute \optimal" contact forces if the answer to problem (b) is a rmative. In this paper, based on an early result by Buss, Hashimoto and Moore, which transforms the nonlinear friction cone constraints into positive de niteness of certain symmetric matrices, we further cast the friction cone constraints into linear matrix inequalities (LMIs) and formulate all three of the problems stated above as a set of convex optimization problems involving LMIs. The latter problems have been extensively studied in optimization and control community and highly e cient algorithms with polynomial time complexity are now available for their solutions. We perform simulation studies to show the simplicity and e ciency of the LMI formulation to the three problems.