Relative dynamics & control of spacecraft formations in eccentric orbits

Formation flying is a key technology for deep space and orbital applications that involve multiple spacecraft operations. Many future space applications will benefit from using formation flying technologies to perform distributed observations, including enhanced synthetic apertures for earth mapping interferometry, and improved coverage for communication and surveillance. Previous work has focused on designing passive apertures for these formation flying missions using a circular reference orbit. This paper extends this design approach to eccentric orbits. For closely spaced vehicles, the relative dynamics can be linearized to obtain a set of decoupled linear time varying equations for in-plane (orbit plane) and out-of-plane (perpendicular to orbit plane) motion. Using the homogenous solutions to these equations, we prove that there exists non-zero (i.e. nontrivial) initial states that will produce T-periodic solutions. T-periodicity results in relative motion where the vehicles return to the initial relative states at the end of every orbit (i.e. [v(t_o) = v(t₀ + T)]). Specifically, we show in-plane and out-of-plane aperture forming modes and analytically derive the necessary conditions to obtain them (differential energy matching condition). These equations extend and generalize solutions for passive aperture forming in circular orbits. We also compare the approximate fuel cost associated with the modeling error of ignoring the eccentricity in the reference orbit.