State-Transition Matrix for Satellite Relative Motion in the Presence of Gravitational Perturbations

Abstract

A state-transition matrix (STM) for the relative motion of artificial satellites is presented in this work that is valid in the presence of the gravitational spherical harmonic perturbations. The perturbed relative motion is modeled as a first-order variation of the equinoctial orbital elements describing the reference orbit. A satellite theory is derived for the analytical propagation of the reference orbit that includes the secular and periodic effects due to the zonal, sectorial, and tesseral harmonic perturbations. In case of the relative motion STM, the two main goals achieved are as follows. The analytic expressions for the elements of the STM are derived in a generalized form valid for an arbitrary spherical harmonic, and these expressions are in closed-form without using any series expansion in either the eccentricity or the ratio of the mean motion to the angular velocity of the gravitational body. The STM includes the short-period effects due to all the tesseral harmonics in addition to the secular, long-period, and short-period effects due to all the zonal harmonics. Avoidance of the series expansions in the eccentricity ensures that the accuracy of the STM does not degrade for satellite formations in medium to highly eccentric orbits. Additionally, the STM is nonsingular in case of the reference orbits that are near the resonant region; however, the long-period effects due to the resonant tesseral harmonics are ignored in this work. The proposed relative motion STM is implemented in MATLAB, and its accuracy is validated against numerical propagation with a Earth gravity field in NASA’s General Mission Analysis Tool.