Formation establishment and reconfiguration using differential elements in J2-perturbed orbits

A practical algorithm is developed for on-board 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 J₂ 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 efficientmanner, and the discrete-time formulation, which has an analytical solution, does not directly yield the optimal thrust times. Therefore, a practical 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 are shown for a variety of reconfiguration maneuvers and reference orbits, including simulations with and without navigation errors for the NASA CubeSat Proximity Operations Demonstration mission.