Optimal switching conditions for minimum fuel fixed time transfer between non coplanar elliptical orbits

Abstract

The ordinary theory of maxima and minima is applied to develop the optimal switching conditions for the problem of two-impulse transfer with time constraint between non-coplanar elliptical orbits. It is shown that all the elements of the transfer orbit, the total characteristic velocity for the transfer and the time of flight between impulses can be expressed explicitly in terms of three variables namely the semi-latus rectum of the transfer orbit, and the true anomalies defining the locations of the impulses on the initial and the final orbit, respectively. In terms of these variables, three necessary optimal conditions are derived for solving either the problem of minimum-fuel, fixed-time transfer or the problem of minimum-time transfer for a prescribed fuel consumption. In the special cases of fixed-time coplanar transfer and free-time non-coplanar transfer, the general optimal conditions are reduced to the classical optimal conditions in the published literature. In particular, it is shown that for the minimum-fuel, fixed-time transfer between coplanar circular orbits, the optimal conditions can be expressed in terms of Lambert's invariant parameters.