Second-Order Averaging of Low-Thrust Transfers in Fixed Time

Abstract

An algorithm for the second-order approximation of fixed-time, low-thrust orbital transfers and generation of Bacon plots is proposed. After averaging the extremal flow of the optimal-control Hamiltonian, a one-parameter family of solutions of a reduced-order, two-point boundary value problems is computed by means of a differential-continuation scheme. Sensitivities of the shooting function are then used in conjunction with an ad hoc near-identity transformation between averaged and osculating variables to achieve an accurate solution for all longitudes of the departure and arrival orbits. Hence, a single simplified shooting problem has to be solved to approximate the solution for any combination of departure and arrival dates and to draw a Bacon plot for an arbitrary launch window. Both the averaged flow and the near-identity transformation are efficiently evaluated via the fast-Fourier transform algorithm, yielding a fully numerical procedure.