Variational collision integrators in forward dynamics and optimal control

The numerical simulation of multibody dynamics often involves constraints of various forms. First of all, we present a structure preserving integrator for mechanical systems with holonomic (bilateral) and unilateral contact constraints, the latter being in the form of a non-penetration condition. The scheme is based on a discrete variant of Hamilton’s principle in which both the discrete trajectory and the unknown collision time are varied (cf. [Fete 03]). As a consequence, the collision event enters the discrete equations of motion as an unknown that has to be computed on-the-fly whenever a collision is imminent. The additional bilateral constraints are efficiently dealt with employing a discrete null space reduction (including a projection and a local reparametrisation step) which considerably reduces the number of unknowns and improves the condition number during each time-step as compared to a standard treatment with Lagrange multipliers (cf. [Leye 12]). In previous works, discrete mechanics and optimal control for constrained systems (DMOCC) has been introduced for the structure preserving simulation of optimal control problems for rigid multibody systems, whereby possible contacts or collisions between the bodies have been disregarded see [Leye 10]. In the formulation presented here, both collision avoidance as well as explicitly planned collisions between non-smooth bodies are included. To this end, a subdifferentiable global contact detection algorithm, the supporting separating hyperplane linear program (SSHLP), based on the signed distance between supporting hyperplanes of two convex sets, is used in the simulation of optimal control problems.