Coplanar Low-Thrust Transfer with Eclipses Using Analytical Costate Guess

Low-thrust orbital transfers are difficult to optimize by indirect methods. The main issues come from the costate guess and from the numerical propagation accuracy required by the shooting method. In the case of a coplanar minimum-time low-thrust transfer with eclipses, an analytical costate guess is proposed. The optimal control problem reduces to an unconstrained minimization problem with two unknowns. A derivative-free algorithm yields a quasi-optimal solution from scratch in a few minutes. No specific guess is necessary and the algorithm properties ensure finding the global minimum of the unconstrained problem. The method is applicable to thrust levels from large to very low and to any eclipse configuration, as exemplified on a transfer towards the geostationary orbit.