Minimum sliding mode error feedback control for inner-formation satellite system with J 2 and small eccentricity

  • Lu CaoEmail author
  • Xiaoqian Chen
Research Paper


With the recent flurry of the research on Inner-Formation satellite system, there has been apparent need for a set of linearized equations to describe the relative motion of satellites under the effect of the J 2 geopotential disturbance, which is the important perturbed-factor for low-orbit Inner-Formation system. Hence, on the assumption of small eccentricity, a new set of linearized equations of motion is proposed that accounts for J 2 perturbations in an elliptical orbit. To avoid the collision between the inner satellite and the outer satellite, the Minimum Sliding Mode Error Feedback Control (MSMEFC) is developed to perform a real-time control on the outer satellite with the uncertain perturbations from the space. The highlight of MSMEFC is to introduce the concept of equivalent control error, which is the key utilization of MSMEFC. It is shown that the proposed MSMEFC can compensate any kinds of uncertain perturbations. Besides, in this paper, the relationship between the equivalent control error and uncertain perturbations is discussed. The robustness and steady-state error of MSMEFC are also analyzed to show its theoretical advantages compared with traditional SMC. Numerical simulations are employed to check the fidelity of the linearized equations. In addition, the efficacy of MSMEFC is verified by the utilization of Inner-Formation system with high control precision.


inner-formation gravity measurement minimum sliding mode error sliding mode control J2 


  1. 1.
    Neeck S P, Magner T J, Paules G E. NASA’s small satellite missions for earth observation. Acta Astronaut, 2005, 56: 187–192CrossRefGoogle Scholar
  2. 2.
    Folta D, Bristow J, Hawkins A, et al. NASA’s autonomous formation flying technology demonstration, earth observing- 1. In: Proceedings of the 1st International Symposium on Formation Flying Missions and Technologies, Toulouse, 2002Google Scholar
  3. 3.
    Cabeza I, Maetinez A. Smart-2: an ESA mission to demonstrate the Key technologies for formation flying and dragfree control. In: Proceedings of the 1st International Symposium on Formation Flying Missions and Technologies, Toulouse, 2002Google Scholar
  4. 4.
    Kristiansen R, Nicklasson P J. Spacecraft formation flying: a review and new results on state feedback control. Acta Astronaut, 2009, 65: 1537–1552CrossRefGoogle Scholar
  5. 5.
    Bauer F, Bristow J, Carpenter J, et al. Enabling spacecraft formation flying in any earth orbit through spaceborne GPS and enhanced autonomy technologies. Space Technol, 2001, 20: 175–185Google Scholar
  6. 6.
    Wang Z K, Zhang Y L. A novel concept of satellite gravity field measurement system using precision formation flying technology. In: Proceedings of the 3rd CSA-IAA Conference on Advanced Space Systems and Applications-Satellite Applications and Applied Satellites, Shanghai, 2008. 1235–1240Google Scholar
  7. 7.
    Wang Z K, Zhang Y L. Acquirement of pure gravity orbit using precision formation flying technology. In: Proceedings of the 6th Internet Workshop on Satellite Constellation and Formation Flying, Taipei, 2010. 1135–1140Google Scholar
  8. 8.
    Dang Z H, Zhang Y L. Formation control using µ-synthesis for inner-formation gravity measurement satellite system. Adv Space Res, 2012, 49: 1487–1505CrossRefGoogle Scholar
  9. 9.
    Seeber G. Satellite Geodesy. 2nd completely revised and extended edition. Berlin/New York: Walter de Gruyter, 2003CrossRefGoogle Scholar
  10. 10.
    Hill G. Researches in the lunar theory. Amer J Math, 1878, 1: 5–26MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Clohessy W H, Wiltshire R S. Terminal guidance system for satellite rendezvous. J Aerosp Sci, 1960, 27: 653–658CrossRefzbMATHGoogle Scholar
  12. 12.
    Kechichian J A. Motion in general elliptic orbit with respect to a dragging and processing coordinate frame. J Astronaut Sci, 1998, 46: 25–45MathSciNetGoogle Scholar
  13. 13.
    Sedwick R J, Miller D W, Kong E M C. Mitigation of differential perturbations in clusters of formation flying satellites. AAS/AIAA Space Flight Mechanics Meeting, 1999. AAS Paper: 99–124Google Scholar
  14. 14.
    Alfriend K, Schaub H, Gim D. Gravitational perturbations, nonlinearity and circular orbit assumption effects on formation flying control strategies. Rocky Mountain Conference, 2000. AAS Paper: 00–12Google Scholar
  15. 15.
    Gim D, Alfriend K. The state transition matrix of relative motion for the perturbed non-circular reference orbit. AAS/AIAA Space Flight Mechanics Meeting, 2001. AAS Paper: 01–222Google Scholar
  16. 16.
    Schaub H, Alfriend K T. J2 invariant relative orbits for formation flying. Celest Mech Dynam Astron, 2001, 79: 77–95CrossRefzbMATHGoogle Scholar
  17. 17.
    Schweighart S A, Sedwick R J. High fidelity linearized J2 model for satellite formation flight. J Guid Control Dynam, 2002, 25: 1073–1080CrossRefGoogle Scholar
  18. 18.
    Utkin V I. Variable structure system with sliding modes. IEEE Trans Automat Contr, 1977, 22: 212–222MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Cao L, Chen X Q, Misra A K. Minimum sliding mode error feedback control for fault tolerant reconfigurable satellite formations with J2 perturbations. Acta Astronaut, 2014, 96: 201–216CrossRefGoogle Scholar
  20. 20.
    Massey T, Shtessel Y. Continuous traditional and high-order sliding modes for satellite formation on control. J Guid Control Dynam, 2005, 28: 826–831CrossRefGoogle Scholar
  21. 21.
    Cao L, Chen X Q, Sheng T. Fault tolerant small satellite attitude control using adaptive non-singular terminal sliding mode. Adv Space Res, 2013, 51: 2374–2393CrossRefGoogle Scholar
  22. 22.
    Schaub H, Junkins L. Analytical Mechanics of Space Systems. Reston: American Institute of Aeronautics and Astronautics, 2003CrossRefzbMATHGoogle Scholar
  23. 23.
    Francis B. Satellite formation maintenance using differential atmospheric drag. Thesis. McGill University, Montreal. 2011Google Scholar
  24. 24.
    Sabatini M, Palmerini G B. Linearized formation-flying dynamics in a perturbed orbital environment. In: 2008 IEEE Aerospace Conference, Big Sky, 2008. 1310–1321Google Scholar
  25. 25.
    Godard Kumar K D. Fault tolerant reconfigurable satellite formations using adaptive variable structure techniques. J Guid Control Dynam, 2010, 33: 969–984CrossRefGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.The State Key Laboratory of Astronautic DynamicsChina Xi’an Satellite Control CenterXi’anChina
  2. 2.College of Aerospace Science and EngineeringNational University of Defense TechnologyChangshaChina

Personalised recommendations