Delta-V-optimal centralized guidance strategy for under-actuated N-satellite formations

This paper addresses the computation of Delta-V-optimal, safe, relative orbit reconfigurations for satellite formations in a centralized fashion. The formations under consideration comprise an uncontrolled chief spacecraft flying with an arbitrary number, N, of deputy satellites, where each deputy is equipped with a single electric thruster. Indeed, this represents a technological solution that is becoming widely employed by the producers of small-satellite platforms. While adopting a single electric thruster does reduce the required power, weight, and size of the orbit control system, it comes at the cost of rendering the satellite under-actuated. In this setting, the satellite can provide a desired thrust vector only after an attitude maneuver is carried out to redirect the thruster nozzle opposite to the desired thrust direction. In order to further extend the applicability range of such under-actuated platforms, guidance strategies are developed to support different reconfiguration scenarios for N-satellite formations. This paper starts from a classical non-convex quadratically constrained trajectory optimization formulation, which passes through multiple simplifications and approximations to arrive to two novel convex formulations, namely a second-order cone programming formulation, and a linear programming one. Out of five guidance formulations proposed in this article, the most promising three were compared through an extensive benchmark analysis that is applied to fifteen of the most widely-used solvers. This benchmark experiment provides information about the key distinctions between the different problem formulations, and under which conditions each one of them can be recommended.