Skip to main content
SpringerLink
Log in

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

  • Published:
Astrodynamics Aims and scope Submit manuscript

A Correction to this article was published on 11 February 2022

This article has been updated

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Change history

References

  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.

    Article  Google Scholar 

  2. Hill, G. W. Researches in the lunar theory, American Journal of Mathematics, 1878, 1(1): 5–26.

    Article  MathSciNet  MATH  Google Scholar 

  3. Clohessy, W. H., Wiltshire, R. S. Terminal guidance system for satellite rendezvous, Journal of the Aerospace Sciences, 1960, 27(9): 653–658.

    Article  MATH  Google Scholar 

  4. Tschauner, J., Hempel, P. OptimaleBeschleunigungspro-gramme fur das Rendezvous-Manover. Astronautica Acta, 1964, 10(5-6): 296–307.

    Google Scholar 

  5. Lawden, D. F. Fundamentals of space navigation. Journal of the British Interplanetary Society, 1954, 13(2): 87–101.

    Google Scholar 

  6. Carter, T., Humi, M. Fuel-optimal rendezvous near apoint in general Keplerianorbit. Journal of Guidance, Control, and Dynamics, 1987, 10(6): 567–573.

    Article  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MATH  Google Scholar 

  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.

    Article  Google Scholar 

  10. Dang, Z. Solutions of Tschauner-Hempel Equations. Journal of Guidance Control and Dynamics, 2017, 40(11): 2953–2957.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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. 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. Alfriend, K. T., Yan, H. Orbital elements approach to the nonlinear formation flying problem. International Formation Flying Symposium, Toulouse, France. 2002.

    Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  18. Schaub, H. Relative orbit geometry through classical orbit element differences. Journal of Guidance, Control, and Dynamics, 2004, 27(5): 839–848.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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. Alfriend, K., Vadali, S. R., Gurfil, P. How, J., Breger, L. Spacecraft formation flying: dynamics, control and navigation. Elsevier, 2009.

    Google Scholar 

  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.

    Article  Google Scholar 

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

  1. Northwestern Polytechnical University, Xi′an, 710072, China

    Zhaohui Dang

  2. Technology and Engineering Center for Space Utilization, Beijing, 100194, China

    Hao Zhang

Authors
  1. Zhaohui Dang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hao Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hao Zhang.

Additional information

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.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s42064-018-0021-1

Keywords

Search

Navigation