Grasp analysis as linear matrix inequality problems

Three fundamental problems in the study of grasping and dextrous manipulation with multifingered robotic hands are as follows, a) Given a robotic hand and a grasp characterized by a set of contact points and the associated contact models, determine if the grasp has force closure, b) Given a grasp along with robotic hand kinematic structure and joint effort limit constraints, determine if the fingers are able to apply a specified resultant wrench on the object, c) Compute "optimal" contact forces if the answer to problem b) is affirmative. In this paper, based on an early result by Buss et al., which transforms the nonlinear friction cone constraints into positive definiteness constraints imposed on certainty symmetric matrices, we further cast the friction cone constraints into linear matrix inequalities (LMI) and formulate all three of the problems stated above as a set of convex optimization problems involving LMI. The latter problems have been extensively studied in optimization and control communities. Currently highly efficient algorithms with polynomial time complexity have been developed and made available. We perform numerical studies to show the simplicity and efficiency of the LMI formulation to the three grasp analysis problems.