Parametrized hyperplane arrangements for control design with collision avoidance constraints

Abstract : This paper concerns the problem of collision avoidance in a dynamical multi-obstacle environment. The design problem is commonly stated in the literature in terms of a receding-horizon constrained optimization problem over a non-convex domain. Preliminary results based on hyper-plane arrangements lead to a mixed-integer formulation of the problem. This formalism is adequate for a static multi-obstacle environment, but may be impractical in a dynamical context. Nevertheless, this shortcoming can be alleviated by considering an additional analysis step and by an appropriate proper choice of the representation of the environment in a time-varying framework. The present paper tackles this issue by using zonotopes and discussing their parametrization in a collision avoidance type of application.