Spacecraft Formation Flight: Analysis Of The Perturbed J2-modified Hill-Clohessy-Wiltshire Equations

Abstract

With the recent technological gains made in satellite design and manufacturing, there has been great interest in utilizing a cluster of satellites to perform the same tasks performed by larger satellites. Historically, this has been done utilizing the Clohessy-Wiltshire equations, or the Hill's equations, as a benchmark for understanding the physics of the relative motion between two spacecraft.

A variety of different models have been derived and implemented to study the relative motion dynamics since the Hill-Clohessy-Wiltshire (HCW) equations. Some of these models, like the HCW equations, are linearized and have analytical solutions. These models make the underlying physics due to higher order perturbations more easily understood, as well as more computationally efficient in solving. Perturbations to the relative dynamics are of necessary importance in order to obtain a model of high fidelity.

The research presented in this paper will explore an approximation of the relative motion under the disturbance of relative second-order differential gravity and the effects of the linearized second-order zonal harmonics (J2), which may be described analytically. This problem is the compilation of two previous models which each focused on solutions to one of these perturbations. The new solutions will be compared to those obtained by these models, as well as the HCW equations. Finally, as the true relative motion dynamics under the full effects of differential gravity and perturbations may only be described by the instantaneous osculating states, the linearized models will be compared to the true relative orbit. This will determine the overall fidelity of each model in describing the true relative dynamics between two spacecraft.