Reachable set computation for spacecraft relative motion with energy-limited low-thrust

For spacecraft in formation flight, the knowledge on a reachable set for their relative motion is crucial as it can be used to 1) assess their operational capability and plan tasks accordingly, and 2) predict the operational boundary of neighboring spacecraft, from which the collision probability can be effectively monitored, improving the level of situational awareness. In this paper, a new approach for reachable set computation is proposed that computes accurate inner and outer approximations of the reachable set for a spacecraft's relative motion with energy-limited low thrust. Finding the exact boundary of the reachable set requires solving an optimal control problem with infinitely many sets of initial and terminal conditions, which is intractable. To overcome this difficulty, an analytical solution to the optimal control problem is introduced, and an ellipsoidal approximation method is applied to the solution to find two inner and outer ellipsoids that approximate the exact boundary of the reachable set. The effectiveness of the proposed approach is demonstrated with illustrative numerical examples.