Future spacecraft formation flying missions will require accurate autonomous guidance systems to calculate a reference, or desired, relative motion during reconfiguration maneuvers. However, the efficiency in terms of propellant used for such maneuvers depends on the fidelity of the dynamics model used for calculating the reference relative motion. Therefore, an efficient method for calculating relative motion should have an analytical solution, be applicable to an eccentric orbit, and should take into account the J2 perturbation. This paper accomplishes this through an exact analytical solution of the relative motion between two spacecraft based on the orbital elements of each spacecraft. Specifically, by propagating the J2-perturbed osculating orbital elements forward in time and solving the exact solution at each time step, an accurate representation of the true spacecraft relative motion is obtained. When compared to a numerical simulator, the proposed analytical solution is shown to accurately model the relative motion, with bounded errors on the order of meters over a wide range of eccentricity values.

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Conference 26th AAS/AIAA Space Flight Mechanics Meeting, 2016
Kuiack, B. (Bradley), & Ulrich, S. (2016). Nonlinear analytical equations of relative motion on J2-perturbed eccentric orbits. In Advances in the Astronautical Sciences (pp. 4495–4508).