Towards Globally Optimal Impulsive Spacecraft Rendezvous under a Hovering Constraint: a Free Number of Impulses, Free Firing Anomalies, and Three Thrust Regimes

This article addresses minimum-fuel impulsive rendezvous about an elliptic reference orbit, in which a chaser must be brought from a given relative state onto a periodic relative trajectory that remains, for every subsequent anomaly, inside a rectangular hovering zone around a passive target. The number of impulses and their firing anomalies are left free, and three thrust models are treated: unbounded impulses, a cap on the thrust acceleration, and a cap together with a minimum firing level (dead band).Posed in the space of vector-valued measures on the manoeuvre window, the problem with free impulses is convex. Its dual bounds the primer vector componentwise, an optimal plan needs at most six impulses, and the problem splits exactly into an in-plane and a cross-track part. The hovering constraint is imposed exactly, never sampled: each face of the zone is a low-degree trigonometric polynomial whose minimum is bounded rigorously, and a violated face yields a linear cut. All problems are then solved by a single dense simplex method, written in plain C with static memory and no external library, driven by column generation at the peaks of the primer vector and row generation of exact facets. With a thrust cap the problem remains a linear program on a grid of cells, with bang-off-bang solutions; with a dead band it becomes nonconvex and is solved to proven optimality on the grid by an exact branch-and-bound on the same simplex. The prescribed evenly-spaced-impulse formulation of earlier work is kept as a baseline.Optimality is global over all impulse plans for unbounded impulses; with a thrust cap the continuous optimum is bracketed between computed bounds;with a dead band optimality is proved on the grid, and the extra cost the dead band shows there is found to vanish as the cells shrink. Every plan is checked by an independent program for periodicity, hovering, thrust limits and its optimality certificate. On one core of a 2.10 GHz virtualised server processor the median solve takes 6.9 ms, 1.11 ms and 32 ms for the three models, in 339 kB of static memory; allowing one to two orders of magnitude for a flight processor, the first two fit an onboard guidance cycle, while the third has no a priori bound on its running time.