Minimum-fuel, power-limited transfers between coplanar elliptical orbits

Abstract

Two methods of computing coplanar, minimum-fuel, power-limited transfers are developed, based on approximate solutions obtained by the averaging method. In the first method, the average solution provides estimates of the initial adjoint variables; in the second, it provides approximations of the optimal controls in feedback form. Both trajectory variables and orbit elements are used in developing these methods. Canonical transformations are derived to convert between these sets of coordinates. The accuracy of the methods for computing coplanar, minimum-fuel, power-limited transfers is assessed for a variety of initial and final orbits. Some initial steps are taken toward the characterization of coplanar, minimum-fuel, power-limited transfers for a wide range of thrust to weight ratios. Circle to ellipse and ellipse to ellipse transfers are considered. Details of the trajectories and thrust profiles for a few illustrative cases are presented. These trajectories and thrust profiles are compared to analytical results obtained using the averaging method and to the analytic solution for infinitesimal transfer. The secular behavior of minimum-fuel transfer is predicted by the averaging results. The shape and orientation of the osculating orbits are predicted quantitatively, while the size is predicted qualitatively. The analytic solution for infinitesimal transfer predicts the qualitative behavior of the thrust during each revolution. Some general principles of minimum-fuel, power-limited transfer are revealed.