Article Title

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


relative motion, Keplerian orbits, orbital element, linearized equations


A new formulation of the orbital element-based relative motion equations is developed forgeneral Keplerian orbits. This new solution is derived by performing a Taylor expansion onthe Cartesian coordinates in the rotating frame with respect to the orbital elements. Theresulted solution is expressed in terms of two different sets of orbital elements. The first one isthe classical orbital elements and the second one is the nonsingular orbital elements. Amongof 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. Thisspecific selection for orbital elements yields a new solution that is universally applicable toelliptic, parabolic and hyperbolic orbits. It is shown that the new orbital element-basedrelative motion equations are equivalent to the Tschauner–Hempel equations. A linearmap between the initial orbital element differences and the integration constants associatedwith the solution of the Tschauner–Hempel equations is constructed. Finally, the presentedsolution is validated through comparison with a high-fidelity numerical orbit propagator.The numerical results demonstrate that the new solution is computationally effective; andthe result is able to match the accuracy that is required for linear propagation of spacecraftrelative motion over a broad range of Keplerian orbits.


Tsinghua University Press