Fuel-Optimal Stationkeeping via Differential Inclusions

Stationkeeping of one spacecraft in low Earth orbit with respect to another, or with respect to a reference point in space, is a common orbit maintenance and guidance requirement. This paper deals with formulating a infinite time fuel-optimal control problem using the Hill equations (also known as the Clohessy-Wiltshire equations) and solving it via a direct approach using concepts of hodograph space and differential inclusions. The differential inclusion based direct method has been selected due to its excellent convergence robustness. Using this methodology, numerous optimal solutions corresponding to various differential drag profiles and Stationkeeping error tolerances were easily obtained from trivial initial guesses. The major contribution of this paper is the interesting observation made regarding the structure of the fuel-optimal solutions as a function of the differential drag profiles and Stationkeeping error tolerances. Results from this study can be used for estimating fuel budgets and developing fuel-optimal Stationkeeping guidance laws.