Nonlinear orbital uncertainty propagation with differential algebra and Gaussian mixture model

  • Zhen-Jiang Sun
  • Ya-Zhong LuoEmail author
  • Pierluigi di Lizia
  • Franco Bernelli Zazzera


Nonlinear uncertainty propagation is of critical importance in many application fields of astrodynamics. In this article, a framework combining the differential algebra technique and the Gaussian mixture model method is presented to accurately propagate the state uncertainty of a nonlinear system. A high-order Taylor expansion of the final state with respect to the initial deviations is firstly computed with the differential algebra technique. Then the initial uncertainty is split to a Gaussian mixture model. With the high-order state transition polynomial, each Gaussian mixture element is propagated to the final time, forming the final Gaussian mixture model. Through this framework, the final Gaussian mixture model can include the effects of high-order terms during propagation and capture the non-Gaussianity of the uncertainty, which enables a precise propagation of probability density. Moreover, the manual derivation and integration of the high-order variational equations is avoided, which makes the method versatile. The method can handle both the application of nonlinear analytical maps on any domain of interest and the propagation of initial uncertainties through the numerical integration of ordinary differential equation. The performance of the resulting tool is assessed on some typical orbital dynamic models, including the analytical Keplerian motion, the numerical J2 perturbed motion, and a nonlinear relative motion.


nonlinear orbit uncertainty propagation differential algebra Gaussian mixture model taylor expansion 


  1. 1.
    Y. Z. Luo, and Z. Yang, Prog. Aerospace Sci. 89, 23 (2017).ADSCrossRefGoogle Scholar
  2. 2.
    J. R. Chen, J. F. Li, X. J. Wang, J. Zhu, and D. N. Wang, Sci. China–Phys. Mech. Astron. 61, 024511 (2018).ADSCrossRefGoogle Scholar
  3. 3.
    S. G. Hesar, D. J. Scheeres, and J. W. McMahon, J. Guid. Control Dyn. 40, 81 (2017).ADSCrossRefGoogle Scholar
  4. 4.
    C. Saboi, K. Hill, K. Alfriend, and T. Sukut, Acta Astronaut. 84, 69 (2013).ADSCrossRefGoogle Scholar
  5. 5.
    Y. Z. Luo, Z. Yang, and H. N. Li, Sci. China–Phys. Mech. Astron. 57, 731 (2014).ADSCrossRefGoogle Scholar
  6. 6.
    T. H. Xu, K. F. He, and G. C. Xu, Sci. China–Phys. Mech. Astron. 55, 738 (2012).ADSCrossRefGoogle Scholar
  7. 7.
    A. T. Fuller, Int. J. Control, 9, 603 (1969).CrossRefGoogle Scholar
  8. 8.
    R. M. Weisman, M. Majji, and K. T. Alfriend, Celest. Mech. Dyn. Astron. 118, 165 (2014).ADSCrossRefGoogle Scholar
  9. 9.
    I. Park, K. Fujimoto, and D. J. Scheeres, J. Guid. Control Dyn. 38, 2287 (2015).ADSCrossRefGoogle Scholar
  10. 10.
    R. Ghrist, and D. Plakalovic, "Impact of non–Gaussian error volumes on conjunction assessment risk analysis", AIAA Paper No. 2012–4965, 2012.Google Scholar
  11. 11.
    N. Arora, V. Vittaldev, and R. P. Russell, J. Guid. Control Dyn. 38, 1345 (2015).ADSCrossRefGoogle Scholar
  12. 12.
    K. Liu, B. Jia, G. Chen, K. Pham, and Erik Blasch, “A real–time orbit satellites uncertainty propagation and visualization system using graphics computing unit and multi–threading processing”, in 2015 IEEE/AIAA 34th Digital Avionics Systems Conference (DASC), IEEE No. 8A2–1, 2015.CrossRefGoogle Scholar
  13. 13.
    S. Lee, H. Lyu, and I. Hwang, J. Guid. Control Dyn. 39, 1593 (2016).ADSCrossRefGoogle Scholar
  14. 14.
    J. L. Junkins, M. R. Akella, and K. T. Alfriend, J. Astronaut. Sci. 44, 541 (1996).Google Scholar
  15. 15.
    D. K. Geller, J. Guid. Control Dyn. 29, 1404 (2006).ADSCrossRefGoogle Scholar
  16. 16.
    F. L. Markley, and J. R. Carpenter, J. Astronaut. Sci. 57, 233 (2009).ADSCrossRefGoogle Scholar
  17. 17.
    D. A. Vallado, "Covariance transformations for satellite flight dynamics operations", in AIAA/AAS Astrodynamics Specialist Conference, Big Sky, Montana, AAS–03–526, 2003.Google Scholar
  18. 18.
    M. Valli, R. Armellin, P. di Lizia, and M. R. Lavagna, J. Guid. Control Dyn. 36, 48 (2013).ADSCrossRefGoogle Scholar
  19. 19.
    A. Wittig, P. di Lizia, R. Armellin, K. Makino, F. Bernelli–Zazzera, and M. Berz, Celest. Mech. Dyn. Astron. 122, 239 (2015).ADSCrossRefGoogle Scholar
  20. 20.
    R. Armellin, and P. Di Lizia, J. Guid. Control Dyn. 41, 101 (2018).ADSCrossRefGoogle Scholar
  21. 21.
    S. Julier, J. Uhlmann, and H. F. Durrant–Whyte, IEEE Trans. Automat. Control 45, 477 (2000).CrossRefGoogle Scholar
  22. 22.
    K. Fujimoto, D. J. Scheeres, and K. T. Alfriend, J. Guid. Control Dyn. 35, 497 (2012).ADSCrossRefGoogle Scholar
  23. 23.
    E. Pellegrini, and R. P. Russell, J. Guid. Control Dyn. 39, 2485 (2016).ADSCrossRefGoogle Scholar
  24. 24.
    Z. Yang, Y. Z. Luo, J. Zhang, and G. J. Tang, J. Guid. Control Dyn. 39, 2170 (2016).ADSCrossRefGoogle Scholar
  25. 25.
    I. Park, and D. J. Scheeres, J. Guid. Control Dyn. 41, 240 (2018).ADSCrossRefGoogle Scholar
  26. 26.
    B. A. Jones, A. Doostan, and G. H. Born, J. Guid. Control Dyn. 36, 430 (2013).ADSCrossRefGoogle Scholar
  27. 27.
    S. Oladyshkin, and W. Nowak, Reliab. Eng. Syst. Saf. 106, 179 (2012).CrossRefGoogle Scholar
  28. 28.
    D. M. Luchtenburg, S. L. Brunton, and C. W. Rowley, J. Comput. Phys. 274, 783 (2014).ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    J. T. Horwood, and A. B. Poore, IEEE Trans. Automat. Contr. 56, 1777 (2011).CrossRefGoogle Scholar
  30. 30.
    K. J. DeMars, R. H. Bishop, and M. K. Jah, J. Guid. Control Dyn. 36, 1047 (2013).ADSCrossRefGoogle Scholar
  31. 31.
    M. L. Psiaki, J. R. Schoenberg, and I. T. Miller, J. Guid. Control Dyn. 38, 292 (2015).ADSCrossRefGoogle Scholar
  32. 32.
    M. Berz, Modern Map Methods in Particle Beam Physics (Academic Press, London, 1999).Google Scholar
  33. 33.
    M. Gunay, U. Orguner, and M. Demirekler, IEEE Trans. Aerosp. Electron. Syst. 52, 2732 (2016).ADSCrossRefGoogle Scholar
  34. 34.
    K. Fujimoto, and D. J. Scheeres, J. Guid. Control Dyn. 38, 1146 (2015).ADSCrossRefGoogle Scholar
  35. 35.
    V. Vittaldev, R. P. Russell, and R. Linares, J. Guid. Control Dyn. 39, 2615 (2016).ADSCrossRefGoogle Scholar
  36. 36.
    M. Massari, P. Di Lizia, F. Cavenago, and A. Wittig, “Differential Algebra software library with automatic code generation for space embedded applications”, AIAA Paper No. 2018–0398, 2018.CrossRefGoogle Scholar
  37. 37.
    P. Di Lizia, R. Armellin, and M. Lavagna, Celest. Mech. Dyn. Astr. 102, 355 (2008).ADSCrossRefGoogle Scholar
  38. 38.
    L. Isserlis, Biometrika, 12, 134 (1918).CrossRefGoogle Scholar
  39. 39.
    R. Kan, J. Multiv. Anal. 99, 542 (2008).CrossRefGoogle Scholar
  40. 40.
    P. Gurfil, and P. K. Seidelmann, Celestial Mechanics and Astrodynamics: Theory and Practice, volume 436 of Astrophysics and Space Science Library (Springer, Berlin, 2016).CrossRefzbMATHGoogle Scholar
  41. 41.
    K. T. Alfriend, Spacecraft Formation Flying: Dynamics, Control, and Navigation (Elsevier astrodynamics series, Butterworth–Heinemann/Elsevier, Oxford, 2010).Google Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Zhen-Jiang Sun
    • 1
  • Ya-Zhong Luo
    • 1
    Email author
  • Pierluigi di Lizia
    • 2
  • Franco Bernelli Zazzera
    • 2
  1. 1.College of Aerospace Science and EngineeringNational University of Defense TechnologyChangshaChina
  2. 2.Department of Aerospace Science and TechnologyPolitecnico di MilanoMilanoItaly

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