Approximate solutions of non-linear circular orbit relative motion in curvilinear coordinates

A compact, time-explicit, approximate solution of the highly non-linear relative motion in curvilinear coordinates is provided under the assumption of circular orbit for the chief spacecraft. The rather compact, three-dimensional solution is obtained by algebraic manipulation of the individual Keplerian motions in curvilinear, rather than Cartesian coordinates, and provides analytical expressions for the secular, constant and periodic terms of each coordinate as a function of the initial relative motion conditions or relative orbital elements. Numerical test cases are conducted to show that the approximate solution can be effectively employed to extend the classical linear Clohessy–Wiltshire solution to include non-linear relative motion without significant loss of accuracy up to a limit of 0.4–0.45 in eccentricity and 40–45\(^\circ \) in relative inclination for the follower. A very simple, quadratic extension of the classical Clohessy–Wiltshire solution in curvilinear coordinates is also presented.