On the convergence of fixed-point iteration in solving complementarity problems arising in robot locomotion and manipulation

Model-based approaches to the planning or control of robot locomotion or manipulation requires the solution of complementarity problems that model intermittent contact. Fixed-point iteration is a method of computing fixed points of functions and there are several fixed-point theorems to guarantee the existence of fixed points. With the help of proximal point functions, the complementarity problems that arise in multibody dynamics can be rewritten in a form suitable for solution by a fixed-point iteration method. This fixed-point “prox method” has been popular over the last decades. However, the tuning of the iteration parameter r is difficult, because r affects the convergence of the fixed-point iteration method in ways not understood by current theoretical results. In this paper, we first investigate some factors that affect the choice of r, which further determines the convergence rate. Also we study the loss of accuracy caused by a commonly used relaxation parameter, which is known as “constraint force mixing”.