Modeling Perturbed Relative Motion Using Orbital Elements

Abstract

This chapter discusses analytical methods for propagating the perturbed relative motion variables via differential orbital elements and differential Euler parameters. These methods rely on the mean element propagation scheme and constitute nonlinear theories. The unit-sphere approach is presented to the analytical propagation of relative motion. This approach projects the motion of the satellites in a formation onto a sphere of unit radius, allowing for application of the rules of spherical trigonometry. The chapter discusses the effects of J2 perturbation on the orbital elements via the Brouwer theory. It also explains linear theories including the State Transition Matrix (STM) of Gim and Alfriend (GA) and linear differential equation models for J2-perturbed relative motion about mean circular orbits. The concept of averaged elements is introduced by accounting for the orbit-averaged short-periodic corrections to the mean elements. A procedure for averaging the short-periodic corrections to the mean elements is discussed, leading to the development of averaged orbital elements and averaged relative motion. A more accurate, second-order state transition tensor is developed by incorporating quadratic nonlinearities of the two-body relative motion and the GA STM, which models the linearized J2 effects.