Existence of Infinitely Many Optimal Iso-Impulse Trajectories in Two-Body Dynamics

Abstract

The question of how many impulses are necessary to optimize a transfer-type maneuver has remained open for decades. Recently, the introduction of optimal switching surfaces revealed the existence of iso-impulse trajectories with differing numbers of impulses for fixed-time rendezvous maneuvers. In this paper, the multiplicity of minimum- impulsive trajectories is studied for long-time-horizon maneuvers. One feature of these trajectories is that many of the impulses are applied at specific positions, which we coin as impulse anchor positions. Leveraging a novel analytic method, it is demonstrated that, under the inverse-square gravity model, multiple-impulse minimum- solutions can be generated from a two-impulse solution. In addition to recovering all impulsive solutions of a multirevolution benchmark problem, all solutions are classified, and it is shown that there are infinitely many optimal iso-impulse solutions. Two planet-centric problems are also solved to demonstrate that the proposed method is applicable to different transfer problems in addition to the benchmark problem. The proposed method provides analytic bounds on the lower (required) and upper (allowable) number of impulses for three classes of maneuvers: fixed-terminal-time rendezvous, free-terminal-time rendezvous, and phase-free transfer. A new interpretation of the primer vector for impulsive extremals with phasing orbits is proposed.