A Formulation of the Clohessy-Wiltshire Equations to Include Dynamic Atmospheric Drag

Abstract

The widely known Clohessy-Wiltshire equations are used to represent the relative motion of a chaser satellite about a target satellite about a point mass. These equations can often be limiting because they are derived assuming small deviations from a circular reference orbit and no orbital perturbations. At lower altitudes, atmospheric perturbations become more predominant, speciflcally for vehicles of difierent ballistic coe‐cients. Therefore, incorporating these efiects into the equations of relative motion can lead to more accurate mission planning, trajectory analysis, and controller/estimator performance. In this study the equations of relative motion are derived to include the in-plane efiects of atmospheric drag of spacecraft in a moving atmosphere. However, this higher fldelity formulation still assumes a circular reference orbit, small deviations from the reference state, and no other sources of perturbations in order to produce a set of linear difierential equations. The resulting equations of perturbed relative motion are validated, under varied ballistic coe‐cients and altitudes, against the higher-fldelity non-linear equations of motion. The results of this study demonstrate a signiflcant increase in the precision of these equations, speciflcally under the in∞uence of larger ballistic coe‐cient difierences at lower altitudes.