Reachability-Based Safe-Start Regions for Approach to a Tumbling Target with Rotating LOS Constraints

This paper presents a reachability-aware guidance architecture for autonomous approach to a tumbling, uncooperative target under a rotating line-of-sight (LOS) docking corridor. The LOS admissible set rotates with the target body frame, producing time-varying polyhedral constraints in the chaser's relative coordinates. A safe-start region is constructed via two conservative criteria: (i) directional per-constraint erosion, the margin consumed by rotation-induced drift before thrust can arrest it, and (ii) a synchronization range bound $r < 2a_{\max}/\omega_t^2$ ensuring the chaser can cancel the apparent rotational velocity without overshooting the hold point. Closed-loop guidance uses a receding-horizon MPC controller with Clohessy-Wiltshire-Hill (CWH) prediction dynamics and explicit LOS corridor constraints in the quadratic program. Truth propagation uses the exact discrete CWH state-transition matrix with sub-stepping, so feasibility claims are physically honest: no reference blending or state projection is applied. A three-regime tracking law manages the transition from long-range inertial approach to body-frame co-rotation and synchronized hold. The analytical safe-start region is benchmarked against four standard reachability engines (backward and forward polytopic reachable sets, Hamilton-Jacobi level sets, and closed-loop Monte Carlo): the closed-form criteria are 250x faster than Hamilton-Jacobi reachability while predicting closed-loop feasibility with precision 0.80 and recall 0.91 on a 500-case sweep. The residual 6% false-positive rate and the IoU gap against Hamilton-Jacobi quantify a structural property: the synchronization set (reach and co-rotate) is a strict subset of the positional reachable set, the gap widening with tumble rate. The analytical bound is thus a sound inner certificate for onboard go/no-go decisions where Hamilton-Jacobi is prohibitively expensive.