Using a Curvilinear Coordinate System for Satellite Relative Motion

The number of dynamics needed to model the motion between a Chief and Deputy satellites has grown greatly since the introduction of the Hill, Clohessy-Wilshire (HCW) equations of motion were introduced. The models have grown to include various things like perturbations, specifically drag, J2, and solar radiation pressure. Dynamics models have also been developed that use True Anomaly as the independent variable instead of time. A lot of work has been put forth to also include cases where the Chief is in an eccentric orbit. While these models have increased the fidelity of relative dynamics these models become very complicated to implement. That is why the HCW equations remain extremely popular after all these developments. However, their simplicity causes issues when there is In-Track separation between the Chief and Deputy satellites. The error in the dynamics increases as this separation increases which leads to a typical constraint that the separation between the Chief and Deputy needs to be much smaller than the radius of the Chief's orbit. That is where this works starts, by examining into ways to increase the accuracy in the HCW equations as the In-Track separation between the Chief and Deputy grows. In which, this will be done by using a curvilinear coordinate system. Furthermore, a technique of using a Virtual Chief satellite will by employed to allow for the HCW equations to be valid for cases where the Chief is in an eccentric orbit.