Minimum control for spacecraft formations in a J2 perturbed environment

Large ΔV amounts are often required to maintain the relative geometry which is needed to implement a formation flying concept. A wise use of the orbital environment makes the orbit keeping phase easier, reducing the overall propellant consumption. A first important step in this direction is the selection of formation configurations and orbits which, while still satisfying the mission requirements, are less subject to perturbations resulting naturally in closed relative motion. Within this frame, a number of studies have been recently carried out in order to identify possible sets of invariant relative orbits under the effects of the Earth oblateness, a conservative force commonly referred to as J2 which is also the most important perturbation for Low Earth Orbit. These efforts clearly marked the difficulties connected with achieving genuine periodic relative motion under J2 effect, but they also showed the existence of a set of conditions on the orbital parameters which allow for quasi-periodic J2 trajectories. This paper presents these particular trajectories, by means of deeper theoretical explanations, showing the dependency of the shape of the relative configurations on the orbital inclination. Since the quasi-periodic trajectories cannot be written analytically, and moreover, they are very sensitive with respect to the initial conditions, difficulties arise when trying to exploit these paths as reference for the control of a formation. This paper proposes a novel approach to find, from the actual quasi periodic natural trajectories, minimal control periodic reference trajectories. Next, it evaluates quantitatively the amount of propellant which is needed to control a spacecraft formation along these paths. The choice of Hill’s classical circular projected configuration as a nominal trajectory is considered as a comparison, showing the clear advantages of the proposed guidance design, which assumes low-perturbed periodic reference orbits as nominal trajectories.