Optimal Relocation of Satellites Flying in Near-Circular-Orbit Formations

In this paper we present a general analytic formulation for optimal transfer paths for satellites flying in formation based upon the circular Hill's problem. Optimization is performed to minimize the transfer energy required from the thruster. We consider the optimization problem as the choice of trajectory the satellite should follow during the maneuver, whereas the time taken is fixed as are the boundary conditions. We show that this optimal control problem has simple analytic solutions that provide a powerful basis from which to develop formation control strategies. In formation flying, maneuvers require low levels of thrust, and we assume that the thruster will be firing throughout the maneuver and the optimization scheme solves for the magnitude and direction of thrust as functions of time. We illustrate how we can exploit these analytic results and present examples of maneuvers using them. We also present a discussion of a docking problem, where we reverse the analysis and solve for initial conditions fixing the thruster characteristics in the optimal solutions found. Finally, we present a discussion of how we might exploit the natural dynamics to gain further propellant savings by adjusting the boundary conditions. This is illustrated by considering a cross-track maneuver for plane change.