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

Abstract

Rendezvous maneuvers are designed to match the position and velocity vectors of a target body (e.g., planet, comet, satellite) whereas transfer maneuvers are designed to match the orbital elements of the target body, except for the true anomaly under a two-body dynamical model. The question of how many impulsive maneuvers are necessary to minimize the total delta-v, ∆v, for a transfer-type maneuver, has remained open for decades. In addition, efficient maneuver placement is an important step to generate impulsive trajectories. Recently, the introduction of optimal switching surfaces revealed the existence of iso-impulse trajectories with different numbers of impulses for fixed-time rendezvous maneuvers. The multiplicity of minimum-∆v impulsive trajectories for long-time-horizon maneuvers is studied in this work. One notable feature of these extremal impulsive trajectories is that many of the impulses are applied at a specific position, highlighting the significance of what we coin as “impulse anchor positions.” The impulse anchor position is chosen to break up the total impulse into multiple impulsive maneuvers while respecting the primer vector theory. It is demonstrated that under the inverse-square gravity model, multiple-impulse, minimum-∆v solutions can be generated using a fundamental two-impulse solution, which provides the impulse anchor positions as well as the impulse direction and magnitude. Leveraging the two-impulse base impulsive solution, an analytic method is developed to generate multiple-impulse minimum-∆v trajectories by forming algebraic ∆v−allocation problems. In addition to recovering all impulsive solutions for a multi-revolution benchmark problem from the Earth to asteroid Dionysus, the minimum-∆v solutions are classified and it is shown that there are infinitely many optimal iso-∆v solutions (i.e., requiring the same total ∆v). The proposed method allows providing analytic bounds on the lower (required) and upper (allowable) number of impulses for three important 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.