Analytical Derivation of Single-Impulse Maneuvers Guaranteeing Bounded Relative Motion Under J2

Keeping a cluster of satellites within bounded relative distances requires active control, because natural perturbations (the most significant of which is the J2 term in the geopotential) tend to move the satellites apart. Whereas there is abundant literature on controlling the relative drift using multiple impulsive maneuvers and approximate astrodynamical models involving mean orbital elements, the works that attempt to minimize the number of impulses while using the inertial position and velocity vectors of the satellites are scarce. In this paper, single-impulse distance-keeping maneuvers are derived, without approximating the J2-perturbed dynamics. An analytical derivation of minimum-fuel impulsive maneuvers between equatorial orbits is provided, while relying on radial period and orbital angle matching conditions. Then, a continuation procedure is used to obtain single-impulse relative distance control between inclined orbits. The development of the impulsive maneuvers relies on the inertial position and velocity vectors of the satellites and does not involve mean elements. In each step, necessary and sufficient conditions for the existence of a single-impulse maneuver are provided. The results are illustrated using a number of realistic scenarios, such as a simulation that includes a 21×21 gravitational model, drag, and lunisolar attraction.