Introduction

The analysis of optimal low-thrust orbit transfer using averaging methods has benefited from the contributions in. In particular the minimum-time transfer between inclined circular orbits was solved in by rotating the orbit plane around the relative line of nodes of initial and final orbits using a piecewise-constant out-of-plane thrust angle switching signs at the relative antinodes while simultaneously changing the orbit size from its initial to its final required value. The original analysis in was further reformulated in in order to arrive at expressions that depict the evolutions of the transfer parameters in a uniform manner valid for all transfers. Further contributions using both analytic and numerical approaches were also carried out in and to solve increasingly more difficult problems, such as by considering realistic constraints and orbit precession due to the important second zonal harmonic J2, while also considering various averaging schemes for rapid computation of the solutions.

The first part of this chapter presents the reformulated Edelbaum problem of the minimum-time low-thrust transfer between inclined circular orbits by further extending it in order to constrain the intermediate orbits during the transfer to remain below a given altitude. The minimum-time problem involving an inequality constraint on the orbital velocity is shown to be equivalent to one involving an equality constraint in terms of the thrust yaw angle representing the control variable that is optimized resulting in the minimum-time solution.