Passively Safe Spacecraft Motion Using Reachable Sets and Orbital Element Differences

The problem of passive safety between spacecraft is studied in this paper by means of backward reachable sets and orbital element differences. These reachable sets characterize the initial states that enter a swept region around a designated spacecraft within some time horizon, leading to collision or conjunction events. We provide an in-depth passive safety analysis using such representations and discuss how the motion of spacecraft can be constrained to satisfy such safety constraints. The work herein is particularly relevant for future formation-flying missions but can be extended to certain constellations, for example, Walker constellations. We demonstrate how to compute these sets for general dynamics models, which typically require numerical methods, but provide analytical and closed-form solutions for the Keplerian case as well as when first-order secular [Formula: see text] perturbations are considered with a mean orbital element difference description. Additionally, we show how the passive safety constraint based on such sets can be further simplified (or reduced) by considering projections or slices of the derived reachable sets. Finally, we propose an algorithm that finds passively safe configurations when considering multiple spacecraft. Simulations verify the passive safety constraints derived in this paper and their utility for a number of design cases in formation flying and a [Formula: see text]-perturbed low-Earth-orbit Walker constellation. In the latter case, long-term passive safety is shown even when subjected to unmodeled perturbations.