Nonlinear high-fidelity modeling of spacecraft relative motion via orbit element differences

The ultimate need to design and develop high-fidelity dynamical models for future space missions is necessitated by the continuous and enormous interest in spacecraft formation flying, rendezvous and spacecraft proximity operations. In this paper, to obtain high-fidelity dynamics, higher-order relative motion model is developed via nonlinear mapping of orbit element differences and Hill coordinates. First, second-order variation of parameter technique of calculus of variations is applied to the direction cosine matrix (DCM), which maps vector components in inertial frame to vector components in Deputy Hill frame, and deputy spacecraft inertial position and velocity vectors in Deputy Hill frame. Second, after series of transformations and elimination of higher-order terms greater than quadratic terms, new, nonlinearly mapped radial, along-track and cross-track relative motion position and velocity equations are obtained. Using the new equation of motion, nonlinear state space model is developed. The new equations, validated via numerical simulations, are amenable for the analysis of spacecraft relative motion, formation flying, rendezvous and proximity operations in both circular and elliptical orbits.