Trajectory Synthesis for the Coordinated Inspection of a Spacecraft with Safety Guarantees

As outer space becomes increasingly congested, there exists a growing need for auxiliary spacecraft to perform support missions for existing satellites with guarantees for safety and mission success. We focus on a multispacecraft inspection mission, wherein a team of "deputy" spacecraft inspect a passive "chief" spacecraft by traveling to a set of inspection points while satisfying a set of safety constraints, namely, that they avoid aligning themselves with the sun, that they avoid colliding with one another, and that they avoid colliding with the chief. We model the deputy dynamics using the Clohessy-Wiltshire-Hill equations, and subsequently discretize the environment by exploiting elliptical natural motion trajectories. Using this finite state space, we construct a Markov decision process (MDP) model of the environment and determine the optimal sequence of inspection points for each deputy to visit by solving a vehicle routing problem. To ensure that the deputies satisfy the safety constraints, we form the product MDP of the original MDP and a nondeterministic Buchi automaton that encodes the sensing task and safety constraints. Using this product MDP, we propose a pair of decentralized algorithms that each seeks to minimize the weighted combination of the time and fuel required to safely complete the mission. The first is an offline algorithm that synthesizes a safe trajectory for each deputy that requires no communication at runtime, while the second is an online algorithm that enforces safety at runtime by leveraging communication between the deputies. We provide numerical examples demonstrating the efficacy of both proposed algorithms.