Analytical Framework for Precise Relative Motion in Low Earth Orbits

This work presents a practical and efficient analytical framework for the precise modeling of the relative motion in low Earth orbits. Developed to support the design and verification of relative guidance navigation and control algorithms devoted to spacecraft rendezvous for active debris removal applications, only the orbital perturbation due to nonspherically symmetric mass distribution is considered. The relative motion is modeled in mean relative orbital elements, revisiting the available formulations to include the first-order expansion of the effects due to any even zonal harmonics and the second-order expansion of the unperturbed and J2 terms. Mean/osculating orbital element conversions are obtained by merging a second-order Hamiltonian approach applied to the J2 problem with the Kaula linear perturbation method for the remaining terms of the geopotential. The paper describes the main building blocks of the framework as well as their interfaces because the key aspect to achieve precision is to set up a fully consistent environment. The results show the achievable accuracy under realistic operational conditions for possible guidance and navigation applications.