, Volume 2, Issue 3, pp 201–215 | Cite as

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

  • Zhaohui Dang
  • Hao Zhang


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.


relative motion Keplerian orbits orbital element linearized equations 



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.


  1. [1]
    Sullivan, J., Grimberg, S., D′Amico, S.Comprehensive survey and assessment of spacecraft relative motion dynamics models. Journal of Guidance, Control, and Dynamics, 2017, 40(8): 1837–1859.CrossRefGoogle Scholar
  2. [2]
    Hill, G. W. Researches in the lunar theory, American Journal of Mathematics, 1878, 1(1): 5–26.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Clohessy, W. H., Wiltshire, R. S. Terminal guidance system for satellite rendezvous, Journal of the Aerospace Sciences, 1960, 27(9): 653–658.CrossRefzbMATHGoogle Scholar
  4. [4]
    Tschauner, J., Hempel, P. OptimaleBeschleunigungspro-gramme fur das Rendezvous-Manover. Astronautica Acta, 1964, 10(5-6): 296–307.Google Scholar
  5. [5]
    Lawden, D. F. Fundamentals of space navigation. Journal of the British Interplanetary Society, 1954, 13(2): 87–101.Google Scholar
  6. [6]
    Carter, T., Humi, M. Fuel-optimal rendezvous near apoint in general Keplerianorbit. Journal of Guidance, Control, and Dynamics, 1987, 10(6): 567–573.CrossRefzbMATHGoogle Scholar
  7. [7]
    Carter, T. New form for the optimal rendezvous equations near a Keplerian orbit. Journal of Guidance, Control, and Dynamics, 1990, 13(1): 183–186.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Carter, T. State transition matrices for terminal rendezvous studies: brief survey and new example. Journal of Guidance, Control, and Dynamics, 1998, 21(1): 148–155.CrossRefzbMATHGoogle Scholar
  9. [9]
    Yamanaka, K., Ankersen, F. New state transition matrix for relative motion on an arbitrary elliptical orbit. Journal of Guidance, Control, and Dynamics, 2002, 25(1): 60–66.CrossRefGoogle Scholar
  10. [10]
    Dang, Z. Solutions of Tschauner-Hempel Equations. Journal of Guidance Control and Dynamics, 2017, 40(11): 2953–2957.Google Scholar
  11. [11]
    Sengupta, P., Vadali, S. Relative motion and the geometry of formations in Keplerianelliptic orbits with arbitrary eccentricity. Journal of Guidance, Control, and Dynamics, 2007, 30(4): 953–964.CrossRefGoogle Scholar
  12. [12]
    Garrison, J. L., Gardner, T. G., Axelrad, P. Relative motion in highly elliptical orbits. Space ight Mechanics 1995, 1995: 1359–1376.Google Scholar
  13. [13]
    Alfriend, K. T., Gim, D. W., Schaub, H. Gravitational perturbations, nonlinearity and circular orbit assumption effects on formation flying control strategies. Guidance and Control 2000, 2000: 139–158.Google Scholar
  14. [14]
    Alfriend, K. T., Yan, H. Orbital elements approach to the nonlinear formation flying problem. International Formation Flying Symposium, Toulouse, France. 2002.Google Scholar
  15. [15]
    Gim, D.-W., Alfriend, K. State Transition matrix of relative motion for the perturbed noncircular reference orbit. Journal of Guidance, Control, and Dynamics, 2003, 26(6): 956–971.CrossRefGoogle Scholar
  16. [16]
    Schaub, H., Alfriend, K. Hybrid Cartesian and orbit element feedback law for formation flying spacecraft. Journal of Guidance, Control, and Dynamics, 2002, 25(2): 387–393.CrossRefGoogle Scholar
  17. [17]
    Roscoe, C. T., Westphal, J. J., Griesbach, J. D., Hanspeter, S. Formation establishment and recon-flguration using differential elements in J2-perturbed orbits. Journal of Guidance, Control, and Dynamics, 2015, 38(9): 1725–1740.CrossRefGoogle Scholar
  18. [18]
    Schaub, H. Relative orbit geometry through classical orbit element differences. Journal of Guidance, Control, and Dynamics, 2004, 27(5): 839–848.CrossRefGoogle Scholar
  19. [19]
    Lane, C., Axelrad, P. Formation design in eccentric orbits using linearized equations of relative motion. Journal of Guidance, Control, and Dynamics, 2006, 29(1): 146–160.CrossRefGoogle Scholar
  20. [20]
    Broucke, R. Solution of the elliptic rendezvous problem with the time as independent variable. Journal of Guidance, Control, and Dynamics, 2003, 26(4): 615–621.CrossRefGoogle Scholar
  21. [21]
    Sinclair, A., Sherrill, R., Alan Lovell, T. Calibration of linearized solutions for satellite relative motion. Journal of Guidance, Control, and Dynamics, 2014, 37(4): 1362–1367.CrossRefGoogle Scholar
  22. [22]
    Jiang, F., Li, J., Baoyin, H. Approximate analysis for relative motion of satellite formation flying in elliptical orbits. Celestial Mechanics and Dynamical Astronomy, 2007, 98(1): 31–66.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    Vadali, S., Yan, H., Alfriend, K. Formation maintenance and fuel balancing for satellites with impulsive control. AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Guidance, Navigation, and Control and Co-located Conference, 2008, AIAA 2008–7359.Google Scholar
  24. [24]
    Alfriend, K., Vadali, S. R., Gurfil, P. How, J., Breger, L. Spacecraft formation flying: dynamics, control and navigation. Elsevier, 2009.Google Scholar
  25. [25]
    Dang, Z. New state transition matrix for relative motion on an arbitrary Keplerianorbit. Journal of Guidance, Control, and Dynamics, 2017, 40(11): 2917–2927.CrossRefGoogle Scholar

Copyright information

© Tsinghua University Press 2018

Authors and Affiliations

  1. 1.Northwestern Polytechnical UniversityXi′anChina
  2. 2.Technology and Engineering Center for Space UtilizationBeijingChina

Personalised recommendations