Curvilinear coordinate transformations for relative motion

Relative motion improvements have traditionally focused on inserting additional force models into existing formulations to achieve greater fidelity, or complex expansions to admit eccentric orbits for propagation. A simpler approach may be numerically integrating the two satellite positions and then converting to a modified equidistant cylindrical frame for comparison in a Hill's-like frame. Recent works have introduced some approaches for this transformation within the Hill's construct, and examined the accuracy of the transformation. Still others have introduced transformations as they apply to covariance operations. Each of these has some orbital or force model limitations and defines an approximate circular reference dimension. We develop a precise transformation between the Cartesian and curvilinear frame along the actual satellite orbit and test the results for various orbital classes. The transformation has wide applicability.