Reachability-Based Safe-Start Regions for Approach to a Tumbling Target with Rotating LOS Constraints
This paper presents a closed-form test that decides, before the maneuver begins, whether a chaser can reach and hold station at the hold point of a tumbling, uncooperative target inside a line-of-sight (LOS) corridor that turns with the target. A closed-loop controller cannot answer this question: a receding-horizon controller checks the corridor only over its prediction horizon, and near a tumbling target the corridor sweeps past faster than bounded thrust can follow, so a start that is feasible at the first step can become unrecoverable later. Today the only ways to know are to fly the controller in simulation or to compute a Hamilton--Jacobi reachable set, both too slow onboard. The test combines two criteria derived from bounded-thrust relative orbital dynamics: a directional erosion margin, the corridor margin that rotation-induced drift consumes before the thruster arrests it, and a synchronization radius, beyond which the apparent rotational velocity cannot be cancelled. Guidance pairs a three-regime tracking law with a receding-horizon quadratic program. Benchmarked against polytopic backward and forward reachable sets, Hamilton--Jacobi level sets and closed-loop Monte Carlo simulation, the test runs over two orders of magnitude faster than Hamilton--Jacobi and, over 500 closed-loop cases, predicts feasibility with 91% recall and 80% precision. The gap to Hamilton--Jacobi is structural, not a method error: reaching the hold point and co-rotating with it is a stronger requirement than arriving with arbitrary velocity, and the gap widens with tumble rate. The test therefore gives an onboard go/no-go answer where Hamilton--Jacobi reachability is too expensive.