Celestial Mechanics and Dynamical Astronomy

, Volume 99, Issue 2, pp 105–127 | Cite as

Modelling natural formations of LEO satellites

Original Article


In this paper we consider the relative motion between satellites moving along near circular orbits in LEO. We are focussing upon the natural dynamics in order that we may then develop a design tool for choosing initial conditions for satellite orbits where the satellites will follow close to a chosen configuration with little control. The approach presented here is different to much of the other published work on this topic in that we start with analytic solutions of the absolute equations of motion rather than attempt to develop linearised equations of relative motion which we then solve. This geometric approach leads to a natural decomposition of the orbital motion into three components: the motion of a guiding centre that incorporates the secular evolution of the formation due to the even zonal harmonics; the periodic motion of the formation as an approximately solid body about this guiding centre; and the periodic motion of individual satellites within the formation. By separating the motion into these three components we are able to give a full description of the motion, but in a simple form avoiding much of the complexity in other formulations. We then use this model to find expressions for the motion of one satellite with respect to another within our formation. We present propagations of satellite orbits to fully evaluate the accuracy of our expressions, noting additions that may be made to the model to further increase accuracy.


Formation flying Relative motion LEO 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bond V.R. (1999). A new solution for the rendezvous problem. Adv. Astronaut. Sci. 102(2): 1115–1143 Google Scholar
  2. Clohessy, W.H., Wiltshire, R.S.: Terminal guidance system for satellite rendezvous. J. Aerospace Sci. 27(9), 653–658, 674 (1960)Google Scholar
  3. Gill E. and Runge H. (2004). Tight formation flying for an along-track SAR interferometer. Acta Astronaut. 55(3–9): 473–485 CrossRefADSGoogle Scholar
  4. Gim D.-W. and Alfriend K.T. (2003). State transition matrix of relative motion for the perturbed non-circular reference orbit. J. Guidance, Control Dyn. 26(6): 956–971 Google Scholar
  5. Gim D.-W. and Alfriend K.T. (2005). Satellite relative motion using differential equinoctial elements. Celest. Mech. Dyn. Astron. 92: 295–336 MATHCrossRefADSMathSciNetGoogle Scholar
  6. Halsall, M., Palmer, P.L.: An analytic relative orbit model incorporating J 3. AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Keystone, Colorado, 21–24 August 2006Google Scholar
  7. Hametz, M.E., Conway, D.J., Richon, K.: Design of a formation of earth orbiting satellites: The Auroral Lites Mission. In: Proceedings of the 1999 NASA/GSFC Flight Mechanics Symposium, Greenbelt, Maryland, May 1999Google Scholar
  8. Hashida, Y., Palmer, P.: Epicyclic motion of satellites about an oblate planet. J. Guidance, Control Dyn. 24(3), 586–596, 674 (2001)Google Scholar
  9. Hashida Y. and Palmer P. (2002). Epicyclic motion of satellites under rotating potential. J. Guidance, Control Dyn. 25(3): 571–581 Google Scholar
  10. Inalhan, G., How, J.P.: Relative dynamics and control of spacecraft formations in eccentric orbits. AIAA Guidance, Navigation and Control Exhibit, Denver, Colorado, August 2000Google Scholar
  11. Karlgaard C.D. and Lutze F.H. (2004). Second-order equations for rendezvous in a circular orbit. J. Guidance, Control Dyn. 27(3): 499–501 Google Scholar
  12. Karlgaard C.D. and Lutze F.H. (2003). Second-order relative motion equations. J. Guidance, Control Dyn. 26(1): 41–49 Google Scholar
  13. Kormos, T.: Dynamics and Control of Satellite Constellations and Formations in Low Earth Orbit. Ph.D. thesis, University of Surrey (2004)Google Scholar
  14. Melton R.G. (2000). Time-explicit representation of relative motion between elliptical orbits. J. Guidance, Control Dyn. 23(4): 604–610 Google Scholar
  15. Mishne D. (2004). Formation control of satellites subject to drag variations and J2 perturbations. J. Guidance, Control Dyn. 27(4): 685–692 Google Scholar
  16. Moccia A., Vetrella S. and Bertoni R. (2000). Mission analysis and design of a bistatic synthetic aperture radar on board a small satellite. Acta Astronautica 47(11): 819–829 CrossRefADSGoogle Scholar
  17. Schaub H. (2004). Relative orbit geometry through classical orbit element differences. J. Guidance, Control, Dyn. 27(5): 839–848 CrossRefGoogle Scholar
  18. Schaub H. and Alfriend K.T. (2001). J 2 invariant relative orbits for spacecraft formations. Celest. Mech. Dyn. Astron. 79: 77–95 MATHCrossRefADSGoogle Scholar
  19. Schweighart S.A. and Sedwick R.J. (2002). High-fidelity linearized J 2 model for satellite formation flight. J. Guidance, Control Dyn. 25(6): 1073–1080 Google Scholar
  20. Vadali, S.R.: An Analytical Solution for Relative Motion of Satellites. 5th Dynamics and Control of Systems and Structures in Space Conference, Cranfield University, Cranfield, UK (2002)Google Scholar
  21. Vaddi S.S., Vadali S.R. and Alfriend K.T. (2003). Formation flying: accommodating nonlinearity and eccentricity perturbations. J. Guidance, Control Dyn. 26(2): 214–223 Google Scholar
  22. Wnuk E. and Golebiewska J. (2005). The relative motion of earth orbiting satellites. Celest. Mech. Dyn. Astron. 91: 373–389 MATHCrossRefADSGoogle Scholar
  23. Wnuk, E., Golebiewska, J.: Differential perturbations and semimajor axis estimation for satellite formation orbits. AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Keystone, Colorado, 21–24 August 2006Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.Surrey Space CentreUniversity of SurreyGuildfordUK

Personalised recommendations