Relative Orbital Elements Formulation Based upon the Clohessy-Wiltshire Equations

Abstract

For on-orbit applications where two or more spacecraft are flying in close proximity, it is often convenient to apply the Clohessy-Wiltshire differential relative motion equations in order to calculate the relative motion of a deputy spacecraft about a chief spacecraft that is assumed to be in a circular orbit. Under these assumptions, the solutions to the Clohessy-Wiltshire equations can be re-parameterized as a set of relative orbital elements that fully characterize the relative motion of the deputy about the chief. In contrast to the Cartesian relative position and velocity states, relative orbital elements provide a clear geometric interpretation of the relative motion and yield an intuitive understanding of how the unforced relative motion will evolve with time. In this paper, the derivation of relative orbital elements is given, and the transformation between relative orbital elements and Cartesian state elements expressed in the local-vertical, local-horizontal frame is provided. The evolution of relative orbital elements with time is evaluated, and characteristics of the unforced motion in terms of relative orbital elements are described.