Efficient Collision Modelling for Numerical Optimal Control

Efficiently solving optimal control problems (OCP) is a prerequisite for using methods such as model predictive control (MPC) for real-time control of complex robotic systems. For many applications, it is mandatory to guarantee that the OCP solutions correspond to collision-free motions. However, establishing appropriate constraints for OCP is not straightforward. A main reason is that the function determining the Euclidean distance between two arbitrary convex shapes is often non-differentiable and non-convex. Moreover, this function can generally only be defined implicitly, for example, by an optimization problem. Recently, it was proposed to derive collision constraints by utilizing duality theory of convex optimization as the resulting constraints are differentiable and allow to check the exact distance between arbitrary compact convex sets. These constraints naturally depend on the chosen representation of the sets for which collision avoidance should be established. In this paper, we show that the computational effort of solving OCP including the proposed constraints can be reduced by explicitly considering the type of sets involved and choosing their representation properly. To this end, we consider the two prominent cases where collision avoidance has to be established between two polytopic sets or between a polytopic and an ellipsoidal set. The claim that our proposed constraints allow improved computational performance is supported by numerical experiments with a car model.