Three important problems in the study of grasping and manipulation by multifingered 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 specified resultant wrench on the object; and (c) Compute "optimal" contact forces if the answer to problem (b) is affirmative. In this paper, based on an early result by Buss, Hashimoto and Moore, which transforms the nonlinear friction cone constraints into positive definiteness 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 efficient algorithms with polynomial time complexity are now available for their solutions. We perform simulation studies to show the simplicity and efficiency of the LMI formulation to the three problems.

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Abstract — We present a novel Weighted Barrier Function (WBF) method of efficiently computing optimal grasping force distributions for multifingered hands. Second-order conic friction constraints are not linearized, as in many previous works. The force distributions are smooth and rapidly computable, and they enable flexibility in selecting between firm, stable grasps or looser, more efficient grasps. Furthermore, fingers can be disengaged and re-engaged in a smooth manner, which is a critical capability for a large number of manipulation tasks. We present efficient solution methods that do not incur the increased computational complexity associated with solving the Semi-Definite Programming formulations presented in previous works. We present results from static and dynamic simulations which demonstrate the flexibility and computational efficiency associated with WBF force distributions. I.