Compact solution of circular orbit relative motion in curvilinear coordinates

A compact approximate solution of the highly non-linear relative motion in curvilinear coordinates is provided under the assumption of circular orbit for the chief spacecraft and moderately small inclination and eccentricity for the follower. The rather compact three-dimensional solution, which employs time as independent variable, is obtained by algebraic manipulation of the individual Keplerian motions in curvilinear coordinates and Taylor expansion for small eccentricity of the follower orbit. 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 0.4-0.5 in eccentricity and a few degrees in inclination.