Fast Fuel-Optimal Constrained Impulsive Control with Application to Distributed Spacecraft

This work extends a reachable set theory approach for fuel-optimal control to incorporate convex and non-convex state, time, and magnitude constraints. Reachable set theory considers problems with norm-like cost functions and time-varying linear dynamics over a fixed time horizon and can produce optimality conditions for control input profiles. These optimality conditions enable the identification of optimal maneuver times and directions without needing to explicitly solve for the maneuvers themselves. While previous works have leveraged this approach to generate unconstrained, numerically-efficient optimal control solutions, this work proposes a novel impulsive control solver that finds provably-optimal trajectories under non-convex constraints. The solver is designed from a new set of optimality conditions, derived by reformulating a single reachable set into multiple time-ordered, step-wise reachable set waypoint sub-problems, where a waypoint is the state reached after a given maneuver. Non-convex constraints are satisfied at each waypoint, preserving convexity at each sub-problem without needing to convexify the full control problem, enabling the computationally efficient identification of constrained optimal maneuvers, and guaranteeing a user-defined bound on constrained cost. To produce an optimal maneuver plan, the solver rolls out constrained trajectories until the remaining unconstrained solution adheres to the problem constraints. The fuel-optimal control problem considered by this work is broadly applicable to many fields and particularly significant to Distributed Space Systems (DSS), which incorporate multiple spacecraft to overcome the limitations and risks posed by a single spacecraft. Multi-agent guidance and propulsive control techniques, often for small spacecraft at close separations, require guarantees of collision avoidance, efficient maneuvering to preserve a restricted fuel payload, and numerically-simple algorithms functional under on-board computational processing limitations. The proposed solver offers both user-defined bounds on constrained maneuver optimality and enhanced computational efficiency to directly address these challenges, and these capabilities are demonstrated comparatively against baseline solvers from the literature through Monte Carlo analysis on a variety of relevant DSS scenarios. Overall, this work enables the benefits of reachable set theory, namely computational efficiency and provable optimality, to be extended to control problems of higher fidelity.