Low Thrust Minimum-Fuel Orbital Transfer: A Homotopic Approach

Abstract

We describe in this paper 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 non-trivial 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 kgs, either with or without a rendezvous) for thrusts as low as 0.1N, 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. * PhD student,

Constructive Methods for Initialization and Handling Mixed State-Input Constraints in Optimal Control

“…On the other hand, the differential method generally converges faster due to the prediction behavior. More details on this topic can, for example, be found in [19,20].…”

“…The model equations and the optimal control objective are recapitulated before the homotopy approach is used to ‡ A special case occurs if the initial trajectory x 0 t; u 0 t, t 2 0; T 0 directly satisfies Eq. (3), that is, x 0 T 0 ; T 0 0, such that the boundary conditions (20) reduce to the original ones in Eq. (3) and 2 is not required.…”

“…Moreover, these methods consider a single homotopy parameter compared to 1 ; 2 in OCP (18)(19)(20)(21). This problem can be taken into account, for instance, by using 1 and 2 in a cascade of two single homotopies or by reducing them to a single parameter 1 2 .…”

“…The first run Finally, the optimal solution x t; u t; t; ; T of the original OCP in Eqs. (1) and (2) is obtained for (18) and the boundary conditions (20), some freedom exists concerning how 1 and 2 are increased. In general, the number of steps N depends on the sensitivity of the problem and the length of the homotopy path.…”

“…The Eqs. (19)(20)(21)(22)(23)(24)(25) define a two-point BVP for the states xt, t, the multipliers , the optimal control ut, and the end time T in dependence of the continuation parameters 1 ; 2 .…”