Trajectory optimization in convex underapproximations of safe regions

This paper discusses a computationally efficient method for optimizing aircraft trajectories in a two-aircraft conflict scenario, under a noncooperative setting. It is assumed that the future trajectory of the uncontrolled aircraft is unknown, but that deterministic input bounds are given. Unsafe reachable sets are computed in a game theoretic framework to account for worst case behaviors. Overapproximations of the reachable sets are used as constraints in a convex trajectory optimization program at each time step. We prove the safety property of the optimization program, and address the errors introduced by model linearization through a robustness analysis. Simulation results demonstrate that the algorithm generates collision free conflict resolution trajectories even as the adversary aircraft inputs are drawn randomly from within input bounds.