Low Thrust Minimum-Fuel Orbital Transfer: A Homotopic Approach

We describe the study of an Earth orbital transfer with a low-thrust (typically electro-ionic) propulsion system. The objective is the maximization of the final mass, which leads to a discontinuous control with a huge number of thrust arcs. The resolution method is based on single shooting, combined to a homotopic approach in order to cope with the problem of the initial guess, which is actually critical for nontrivial problems. An important aspect of this choice is that we make no assumptions on the control structure and in particular do not set the number of thrust arcs. This strategy allowed us to solve our problem (a transfer from low Earth orbit to geosynchronous equatorial orbit, for a spacecraft with mass of 1500 kg, either with or without a rendezvous) for thrusts as low as 0.1 N, which corresponds to a one-year transfer involving several hundreds of revolutions and thrust arcs. The numerical results obtained also revealed strong regularity in the optimal control structure, as well as some practically interesting empiric laws concerning the dependency of the final mass with respect to the transfer time and maximal thrust.