Contact-Implicit Differential Dynamic Programming for Model Predictive Control with Relaxed Complementarity Constraints

Abstract

In this work, we propose a novel differential dynamic programming (DDP) framework for systems involving contact with the ground. The approach converts a general constrained differential dynamic programming into contact-implicit one by incorporating contact dynamics in a linear complementarity problem (LCP) formulation. Analytical gradients of the contact dynamics are obtained through a relaxed complementarity condition in the LCP formulation that helps the search directions of optimization avoid stalling in bad local minima or saddle points. Incorporation of contact dynamics and its analytical gradients into DDP enables an online discovery of not only dynamically-feasible trajectories of states, control inputs, and contact forces but also contact mode sequences. We demonstrate that our Contact-Implicit Differential Dynamic Programming framework successfully finds totally new dynamic motions with contact mode sequences in a variety of robotic systems including an one-legged hopping robot and planar quadrupedal robot in simulation environment.