Application of Indirect Multi-Stage Reachable Set Determination Algorithm for Low-Thrust Spacecraft Trajectory Optimization

The reachable set of controlled dynamical systems consist of the set of all possible reachable states from a given initial condition, over a certain period of time, under the influence of its dynamics and external inputs. The ability to generate reachable sets is significant for analyzing system behavior, planning trajectories, verifying system properties, optimizing control strategies, and handling uncertainties. In this paper, we revisit a recent indirect multi-stage optimal control formulation for rapid estimation of reachable sets. Implementation details applied to a low-thrust spacecraft application are presented. We use a minimum-fuel formulation to estimate both the reachable and feasible sets. Using the so-called thrust switching function, one can scale per-stage costate vectors to produce minimum-time trajectories (i.e., those that lie on the boundary of the reachable set). Reachable set analyses are presented for a low-thrust Earth-Mars rendezvous problem. A minimum-time Earth-Mars rendezvous problem is also solved and the results indicate that the minimum-time solution lies within the feasible set of position vectors, but on the boundary of the reachable set of velocity vectors of the spacecraft.