Under-Approximating Reach-Avoid Sets for Space Vehicle Maneuvering in the Presence of Debris

We investigate the safety verification of a satellite attempting to maneuver to a new position while avoiding multiple debris. We assume that the satellite, desired rendezvous point, and debris are near the same circular orbit, with dynamics modeled by Clohessy-Wiltshire-Hill equations. The resulting high dimensional reach-avoid problem precludes direct application of existing reachability tools. We propose two assured underapproximations of the reach-avoid set that are computable with existing LTI reachability tools. These methods exploit invariant sets, and use recent results in decomposition-based reachability to conduct lower-dimensional reachability calculations that exactly result in the true higher-dimensional reachable set. We demonstrate our approach on a scenario with multiple debris.