Minimum-Fuel Deployment for Spacecraft Formations via Optimal Control

Abstract

This paper focuses on the issue of minimum-fuel deployment for satellite formation flying. We address it as an optimal control problem, the necessary optimality conditions of which are derived from Pontryagin's maximum principle. These are numerically enforced by finding the root of a so-called shooting function. However, optimal control laws for minimum-fuel problems are discontinuous and produce shooting functions with singular Jacobian matrices. The resulting problems cannot be solved easily and require the use of a regularization technique. We extend our previously developed continuation-smoothing method to the multisatellite context by using an adapted initialization procedure. Because realistic mission scenarios may require it, our approach additionally offers the ability to slightly modify a given maneuver strategy to balance the fuel consumption among the satellites to a certain degree. A number of tests concerning low-Earth orbits are carried out in the paper as examples. Our model includes the J 2 effect, which leads to numerical difficulties. We show the efficiency of our method in this challenging context: several maneuver strategies are detected and analyzed from the space dynamics angle. We finally point out that beyond this application, a whole class of deployment/reconfiguration problems may be handled through this approach.