A geometrically implicit time-stepping method for multibody systems with intermittent contact

Accurate dynamic simulation with robust handling of intermittent contact is necessary for a wide range of robotics problems, including the design of parts feeding devices, manipulation and kinodynamic planning, and designing grasp strategies. In this paper we present an implicit time-stepping scheme for dynamic simulation of multibody systems with intermittent contact by incorporating the contact constraints as a set of complementarity and algebraic equations within the dynamics model. We model each body as an intersection of convex inequalities and write the contact constraints as complementarity constraints between the contact force and a distance function dependent on the closest points on the bodies. The closest points satisfy a set of algebraic constraints obtained from the Karush-Kuhn-Tucker (KKT) conditions of the minimum distance problem. We prove that these algebraic equations and the complementarity constraints taken together ensure satisfaction of the contact constraints. This enables us to formulate a geometrically implicit time-stepping scheme (i.e. we do not need to approximate the distance function) as a nonlinear complementarity problem. The resulting time-stepper is therefore more accurate and does not rely on a closed-form distance function. We demonstrate through example simulations the fidelity of this approach to analytical solutions and previously described simulation and experimental results.