Linearized relative motion equations through orbital element differences for general Keplerian orbits

Abstract

A new formulation of the orbital element-based relative motion equations is developed for general Keplerian orbits. This new solution is derived by performing a Taylor expansion on the Cartesian coordinates in the rotating frame with respect to the orbital elements. The resulted solution is expressed in terms of two different sets of orbital elements. The first one is the classical orbital elements and the second one is the nonsingular orbital elements. Among of them, however, the semi-latus rectum and true anomaly are used due to their generality, rather than the semi-major axis and mean anomaly that are used in most references. This specific selection for orbital elements yields a new solution that is universally applicable to elliptic, parabolic and hyperbolic orbits. It is shown that the new orbital element-based relative motion equations are equivalent to the Tschauner-Hempel equations. A linear map between the initial orbital element differences and the integration constants associated with the solution of the Tschauner-Hempel equations is constructed. Finally, the presented solution is validated through comparison with a high-fidelity numerical orbit propagator. The numerical results demonstrate that the new solution is computationally effective; and the result is able to match the accuracy that is required for linear propagation of spacecraft relative motion over a broad range of Keplerian orbits.