Optimal Configuration of Tetrahedral Spacecraft Formations

The problem of determining minimum-fuel maneuver sequences for a four-spacecraft formation is considered. The objective of this paper is to find fuel-optimal spacecraft trajectories that transfer four spacecraft from an initial parking orbit to a desired terminal reference orbit while satisfying a set of constraints on the formation at the terminal time. Trajectories involving both one and two allowable maneuvers per spacecraft are considered. The resulting nonlinear optimal control problem is solved numerically using a recently developed direct transcription method called the Gauss pseudospectral method. The results presented in this paper highlight interesting features of the fuel-optimal formation and control. Furthermore, by showing that the discretized first-order optimality conditions from an indirect formulation are satisfied, a post-optimality analysis of the results demonstrates the accuracy and usefulness of the Gauss pseudospectral method.