Solving nonconvex problems of multibody dynamics with joints, contact, and small friction by successive convex relaxation

Time-stepping methods using impulse-velocity approaches are guaranteed to have a solution for any friction coefficient, but they may have nonconvex solution sets. We present an example of a configuration with a nonconvex solution set for any nonzero value of the friction coefficient. We construct an iterative algorithm that solves convex subproblems and that is guaranteed, for sufficiently small friction coefficients, to retrieve, at a linear convergence rate, the velocity solution of the nonconvex linear complementarity problem whenever the frictionless configuration can be disassembled. In addition, we show that one step of the iterative algorithm provides an excellent approximation to the velocity solution of the original, possibly nonconvex, problem if the product between the friction coefficient and the slip velocity is small.