Contact-Implicit Direct Collocation With a Discontinuous Velocity State

Collisions between rigid bodies are impulsive events, meaning they create near-instantaneous, finite discontinuities in the velocity state of the dynamic system. If velocity is assumed to be continuous, that requirement may come into conflict with the desire for bodies to not interpenetrate, or with Coulomb friction (creating a condition known as the Painlevé paradox). This letter presents a contact-implicit framework for trajectory optimization using direct collocation that combines an unplanned impulsive contact model with an implicit, high-order numerical integration scheme. We demonstrate that this framework can resolve problems that other implicit formulations cannot, namely elastic or partially elastic collisions, and impacts without collision – the established resolution of the Painlevé paradox. We then evaluate its applicability to legged robotics problems and test its ability to discover the intricate sequence of varied collision types necessary to execute a skateboarding trick.

I. Introduction

Contact allows robots to interact with objects and terrain, making it an essential component of almost any useful task they might be expected to perform. Robots are often modelled as tree-like systems of rigid bodies, which allows their dynamics to be described using a finite set of generalized coordinates and associated ordinary differential equations (ODEs) [1]. However, contact introduces discontinuities in the directions of allowable motion, corresponding to unilateral inequality constraints that preclude the use of numerical methods for simulating ODEs. For the assumption of rigidity to hold, collisions must be impulsive: relative motion must cease at the instant of contact, under the action of infinitely-large forces.