Linearized dynamics of formation flying spacecraft on a J2-perturbed elliptical orbit

A linearized set of equations of relative motion about a J(2)-perturbed elliptical reference orbit is developed. This model uses analytical relations that are well suited for on-board applications. The inclusion of the J(2) perturbation in a simple analytical model can lead to formation flying guidance and control algorithms that make use of the natural J(2)-induced relative motion to perform maneuvers instead of constantly compensating for this perturbation. The model uses the linearized differential drift rate of mean orbit elements to predict the impact of the J(2) perturbation on relative osculating spacecraft motion. It analytically provides the relative motion in Hill coordinates at any given true anomaly using only the initial osculating relative orbit elements and the initial orbit elements of the reference trajectory. A linear time-varying state-space form of the model is also presented. Simulation results show that relative motion prediction remains accurate over several orbits.