Fuel-Balanced Formation Flight Control of Underactuated Satellites

We consider a continuous-time optimal control problem for a swarm of single thruster, single reaction wheel spacecraft aiming to reach a target formation. The dynamic model of each spacecraft is obtained by augmenting the Hill-Clohessy-Wiltshire equations with the coupled dynamics of the reaction wheel and thruster. For the optimal control problem, we penalize the deviation from the target formation, the overall fuel usage, and the fuel imbalance between agents. The optimal control law is then obtained by using the minimum principle to formulate a split-boundary-value ODE, which is then solved numerically. Numerical simulations for a simple swarm of three satellites show that the proposed approach successfully reduces the fuel consumption of the most fuel-intensive spacecraft, thus extending the overall lifetime of the formation system.