Advertisement

An investigation on cubature Kalman filter performance for orbit determination application

  • Paula Cristiane Pinto Mesquita PardalEmail author
  • Roberta Veloso Garcia
  • Helio Koiti Kuga
  • William Reis Silva
Technical Paper
  • 44 Downloads

Abstract

This work aims at discussing the cubature Kalman filter (CKF) performance when applied to a highly nonlinear problem: the artificial satellites orbit determination problem, using real global positioning system (GPS) data. The CKF is a nonlinear filter based on a third-degree spherical–radial cubature rule, which allows to numerically compute multivariate moment integrals in the Bayesian filter and also provides a set of cubature points scaling linearly with the state vector dimension. Therefore, CKF yields a systematic solution for high-dimensional nonlinear filtering problems, such as the orbit determination application addressed here. This application consists of determining the orbit of Jason-2 satellite, using real GPS data from its onboard receivers, which is a highly nonlinear problem, with respect to the dynamics and the observations equations. The standard differential equations that characterize the orbital motion and the GPS measurements are modified to accommodate the nonlinear filter, and the CKF algorithm is also used for estimating the state of the orbit. The assessment to be presented will be based on the robustness of the filter, concerning convergence speed when the measurements are scattered. The results from CKF will be compared with the unscented Kalman filter results for the same problem, in computational terms such as convergence and accuracy. According to the analysis of such criteria, the conclusions will be presented.

Keywords

Orbit determination Estimation theory Cubature Kalman filter Unscented Kalman filter GPS measurements 

Notes

Acknowledgements

The authors wish to express their appreciation to Lorena School of Engineering/University of São Paulo (EEL/USP) that kindly provided everything necessary for this paper to be developed. The authors are also grateful to the Brazilian National Council for Scientific and Technological Development (CNPq), for the funding support under contracts # 407296/2016-6, 405468/2016-4, and 307255/2018-2.

References

  1. 1.
    Arasaratnam I (2009) Cubature Kalman filtering: theory & applications. Thesis, Doctor of Philosophy (2009), p. 12, 137. McMaster University, CanadaGoogle Scholar
  2. 2.
    Gordon NJ, Salmond DJ, Smith AFM (1993) Novel approach to nonlinear/non-Gaussian Bayesian state estimation. In: IEE Proceedings-F, vol 140, no. 2, pp 107–113Google Scholar
  3. 3.
    Gelb A (1974) Applied optimal estimation. The MIT Press, CambridgeGoogle Scholar
  4. 4.
    Julier SJ, Uhlmann JK, Durrant-Whyte HF (2000) A new method for nonlinear transformation of means and covariances in filters and estimators. IEEE Trans Autom Control 45(3):477–482MathSciNetCrossRefGoogle Scholar
  5. 5.
    Arasaratnam I, Haykin S (2009) Cubature Kalman filters. IEEE Trans Autom Control 54(6):1254–1269MathSciNetCrossRefGoogle Scholar
  6. 6.
    Daum F (2005) Nonlinear filters: beyond the Kalman filter. IEEE Aerosp Electron Syst Mag 20(8):57–69MathSciNetCrossRefGoogle Scholar
  7. 7.
    Simon D (2006) Optimal state estimation. Wiley-Interscience, HobokenCrossRefGoogle Scholar
  8. 8.
    Li C, Ge Q (2011) SCKF for MAV attitude estimation. In: 2011 international conference on machine learning and cybernetics. ICMLC. Guilin, China, pp 1313–1318Google Scholar
  9. 9.
    Zhang L, Yang H, Lu H, Zhang S, Cai H, Qian S (2014) Cubature Kalman filtering for relative spacecraft attitude and position estimation. Acta Astronaut 105(1):254–264.  https://doi.org/10.1016/j.actaastro.2014.09.007 CrossRefGoogle Scholar
  10. 10.
    Pesonen H, Piché R (2010) Cubature-based Kalman filters for positioning. In: 7th workshop on positioning navigation and communication. WPNC, pp 45–49Google Scholar
  11. 11.
    Fernandez-Prades C, Vila-Valls J (2010) Bayesian nonlinear filtering using quadrature and cubature rules applied to sensor data fusion for positioning. In: 2010 IEEE international conference on communications. ICC. Cape Town, South Africa, pp 1–5Google Scholar
  12. 12.
    Li Z, Yang W, Ding D, Liao Y (2017) A Novel fifth-degree cubature Kalman filter for real-time orbit determination by radar. Math Probl Eng 2017:8526804.  https://doi.org/10.1155/2017/8526804 MathSciNetCrossRefGoogle Scholar
  13. 13.
    Jia B, Xin M, Cheng Y (2013) High-degree cubature Kaman filter. Automatica 49(2):510–518MathSciNetCrossRefGoogle Scholar
  14. 14.
    OSTM/Jason-2 Products Handbook. 67p. 03 Aug. 2009. [online JPL/NASA database]. ftp://podaac.jpl.nasa.gov/allData/ostm/preview/L2/GPS-OGDR/docs/userhandbook.pdf
  15. 15.
    Pardal PCPM, Kuga HK, Vilhena de Moraes R (2013) Analyzing the unscented Kalman filter robustness for orbit determination through global positioning system signals. J Aerosp Technol Manag 5(4):395–408.  https://doi.org/10.5028/jatm.v5i4.252 CrossRefGoogle Scholar
  16. 16.
    McGee L, Schmidt S (1985) Discovery of the Kalman filter as a practical tool for aerospace and industry. NASA Technical Memo 86847, November 1985Google Scholar
  17. 17.
    Julier SJ, Uhlmann JK (1997) A new extension of the Kalman filter for nonlinear systems. In: International symposium on aerospace/defense sensing, simulation and controls. SPIE, 1997Google Scholar
  18. 18.
    Julier SJ, Uhlmann JK, Durrant-Whyte HF (1995) A new approach for filtering nonlinear systems. In: American control conference (AACC), Seattle, Washington, pp 1628–1632Google Scholar
  19. 19.
    Pardal PCPM, Kuga HK, Vilhena de Moraes R (2009) A discussion related to orbit determination using nonlinear sigma point Kalman filter. Math Probl Eng 2009:140963.  https://doi.org/10.1155/2009/140963 CrossRefzbMATHGoogle Scholar
  20. 20.
    Brown RG, Hwang PYC (1984) A Kalman filter approach to precision GPS geodesy. Navig Glob Position Syst 2:155–166Google Scholar
  21. 21.
    Maybeck PS (1982) Stochastic models, estimation, and control, vol 2. Academic Press, New YorkzbMATHGoogle Scholar
  22. 22.
    Montenbruck O, Gill E (2001) Satellite orbits: models, methods, and applications. Springer, BerlinzbMATHGoogle Scholar
  23. 23.
    Pardal PCPM, Kuga HK, Vilhena de Moraes R (2015) The particle filter sample impoverishment problem in the orbit determination application. Math Probl Eng 2015:168045.  https://doi.org/10.1155/2015/168045 CrossRefGoogle Scholar
  24. 24.
    Pardal PCPM, Kuga HK, Vilhena de Moraes R (2010) Non linear sigma point Kalman filter applied to orbit determination using GPS measurements. In: 22nd international meeting of the satellite division of the institute of navigation (ION GNSS), Savannah, USA, 2010Google Scholar
  25. 25.
    Kaula WM (1966) Theory of satellite geodesy. Blasdell Publishing Co., WalthamzbMATHGoogle Scholar
  26. 26.
    Marshall JA, Luthcke SB (1994) Modeling radiation forces on topex/poseidon for precise orbit determination. J Spacecr Rocket 31:1CrossRefGoogle Scholar
  27. 27.
    Guan T Special cases of the three-body problem. [online Colorado University database]. http://inside.mines.edu/fs_home/tohno/teaching/PH505_2011/Paper_TianyuanGuan.pdf. Accessed 12 Sept 2018
  28. 28.
    Chiaradia APM, Kuga HK, Prado AFBA (2003) Single frequency GPS measurements in real-time artificial satellite orbit determination. Acta Astronaut 53(2):123–133CrossRefGoogle Scholar
  29. 29.
    Pardal PCPM Determinação de órbita em tempo real através de filtro não linear de Kalman sigma-ponto. Thesis (Doctorate Degree in Aerospace Engineering and Technology/Space Mechanics and Control). São José dos Campos: INPE, 2011. xxvi + 120 p. (sid.inpe.br/mtc-m19/2011/05.04.12.08.10-TDI)Google Scholar

Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  1. 1.USP/EEL (University of São Paulo)LorenaBrazil
  2. 2.ITA/DCTA (Technological Institute of Aeronautics)São José dos CamposBrazil
  3. 3.UnB/FGA (University of Brasília)GamaBrazil

Personalised recommendations