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.
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This work was supported by the National Natural Science Foundation of China (Grant No. 61403416) and the “The Hundred Talents Program” of Chinese Academy of Science.
Zhaohui Dang received his B.S. and Ph.D. degrees in aerospace engineering from National University of Defense Technology, China, in 2009 and 2015, respectively. Currently, he is a cooperated research fellow at Northwestern Polytechnical University. His area of expertise is in spacecraft relative motion modeling, controller design for spacecraft formation flying, and distributed space system simulation.
Hao Zhang received his B.S. and Ph.D. degrees in aerospace engineering from Beihang University, China, in 2006 and 2012, respectively. From 2012 to 2017, he was with Asher Space Research Institute in the Technion (Israel Institute of Technology), where he was a postdoctoral fellow. He is currently an associate research fellow in the Technology and Engineering Center for Space Utilization, Chinese Academy of Sciences. His research interests include distributed space systems, astrodynamics, and GN&C.
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Dang, Z., Zhang, H. Linearized relative motion equations through orbital element differences for general Keplerian orbits. Astrodyn 2, 201–215 (2018). https://doi.org/10.1007/s42064-018-0021-1
- relative motion
- Keplerian orbits
- orbital element
- linearized equations