Dikin-type algorithms for dextrous grasping force optimization

One of the central issues in dextrous robotic hand grasping is to balance external forces acting on the object and at the same time achieve grasp stability and minimum grasping effort. A companion paper shows that the nonlinear function-force limit constraints on grasping forces are equivalent to the positive definiteness of a certain matrix subject to linear constraints. Further compensation of the external object force is also a linear constraint on this matrix. Consequently, the task of grasping force optimization can be formulated as a problem with semidefinite constraints. In this paper, two versions of strictly convex cost functions, one of them self-concordant, are considered. These are twice-continuously differentiable functions that tend to infinity at the boundary of positive definiteness. For the general class of such cost functions, Dikin-type algorithms are presented. it is shown that the proposed algorithms guarantee convergence to the unique solution of the semidefinite programming problem associated with dextrous grasping force optimization. Numerical examples demonstrate the simplicity of implementation, the good numerical properties, and the optimality of the approach.