Efficient algorithms for collision avoidance at intersections

We consider the problem of synthesising the least restrictive controller for collision avoidance of multiple vehicles at an intersection. The largest set of states for which there exists a control that avoids collisions is known as the maximal controlled invariant set. Exploiting results from the scheduling literature we prove that, for a general model of vehicle dynamics at an intersection, the problem of checking membership in the maximal controlled invariant set is NP-hard. We then describe an algorithm that solves this problem approximately and with provable error bounds. The approximate solution is used to design a supervisor for collision avoidance whose complexity scales polynomially with the number of vehicles. The supervisor is based on a hybrid algorithm that employs a dynamic model of the vehicles and periodically solves a scheduling problem.