Formation Establishment and Reconfiguration Using Differential Elements inJ2-Perturbed Orbits

A computationally efficient algorithm is developed for onboard planning of $n$-impulse fuel-optimal maneuvers for establishment and reconfiguration of spacecraft formations. The method is valid in circular and elliptic orbits and includes first-order secular $J_2$ effects. The dynamics are expressed in terms of differential mean orbital elements, and relations are provided to allow the formation designer to transform these into intuitive geometric quantities for visualization and analysis. The maneuver targeting problem is formulated as an optimal control problem in both continuous and discrete time. The continuous-time formulation cannot be solved directly in an efficient manner, and the discrete-time formulation, which has an analytical solution, does not directly yield the optimal thrust times. Therefore, a new flight-suitable algorithm is designed by iteratively solving the discrete-time formulation while using the continuous-time necessary conditions to refine the thrust times until they converge to the optimal values. Simulation results illustrate the performance for a variety of reconfiguration maneuvers and reference orbits, including examples for the NASA CubeSat Proximity Operations Demonstration mission.