Nonlinear analytic solution for perturbed relative motion using differential equinoctial elements

This paper develops a nonlinear analytic solution for satellite relative motion in J₂-perturbed elliptic orbits by using the geometric method that can avoid directly solving the complex differential equations. The differential equinoctial elements (DEEs) are used to remove any singularities for zero-eccentricity or zero-inclination orbits. Based on the relationship between the relative states and the DEEs, state transition tensors (STTs) for transforming the osculating DEEs and propagating the mean DEEs have been derived. The formulation of these STTs has been split into a set of vector and matrix operations, which avoids directly expanding the complex second-order terms, and thus, the obtained STTs could be easy-to-understand and easy-to-code. Numerical results show that the proposed nonlinear solution is valid for zero-eccentricity and zero-inclination reference orbit and is more accurate than the previous linear or nonlinear methods for the long-term prediction of satellite relative motion.