Skip to main content
SpringerLink
Log in

Design of High-Degree Student’s t-Based Cubature Filters

  • Short Paper
  • Published:
Circuits, Systems, and Signal Processing Aims and scope Submit manuscript

Abstract

The Student’s t-based nonlinear filter (STNF) can cope with the filtering problem of nonlinear systems with heavy-tailed process and measurement noises. The key problem in the design of a STNF is how to calculate the Student’s t weighted integral, and the performance of the STNF depends heavily on the used numerical integration technique. In this paper, new high-degree Student’s t spherical-radial cubature rules (STSRCRs) for the Student’s t weighted integral are proposed based on spherical-radial transformation and moment matching methods, from which new high-degree Student’s t-based cubature filters (STCFs) are developed. The proposed high-degree STSRCRs can achieve better approximation to the Student’s t weighted integral as compared with the existing third-degree Student’s t integral rules. As a result, the proposed high-degree STCFs have higher estimation accuracy than the existing STNFs. Simulation results illustrate that the proposed filters can achieve higher estimation accuracy than the existing Gaussian approximate filters, Huber-based nonlinear Kalman filters and STNFs with slightly increased computational complexities, and are computationally much more efficient than the existing Gaussian sum filter and particle filter.

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.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. B.D.O. Anderson, J.B. Moore, Optimal Filtering (Prentice Hall, Englewood Cliffs, NJ, 1979)

    MATH  Google Scholar 

  2. I. Arasaratnam, S. Haykin, Cubature Kalman filter. IEEE Trans. Autom. Control 54(6), 1254–1269 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. I. Arasaratnam, S. Haykin, R. Elliott, Discrete-time nonlinear filtering algorithms using Gauss-Hermite quadrature. Proc. IEEE 95(5), 953–977 (2007)

    Article  Google Scholar 

  4. S. Arulampalam, S. Maskell, N. Gordon, T. Clapp, A tutorial on particle filters for on-line non-linear/non-Gaussian Bayesian tracking. IEEE Trans. Signal Process. 50(2), 174–189 (2002)

    Article  Google Scholar 

  5. R. Cools, P. Rabinowitz, Monomial cubature rules since Stroud: a compilation. J. Comput. Appl. Math. 48(3), 309–326 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  6. J. Duník, O. Straka, M. Šimandl, Stochastic integration filter. IEEE Trans. Autom. Control 58(6), 1561–1566 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. F. El-Hawary, Y. Jing, Robust regression-based EKF for tracking underwater targets. IEEE J. Ocean. Eng. 20(1), 31–41 (1995)

    Article  Google Scholar 

  8. E.L. Elte, The Semiregular Polytopes of the Hyperspaces (Hoitsema, Groningen, 1912)

    MATH  Google Scholar 

  9. M.A. Gandhi, L. Mili, Robust Kalman filter based on a generalized maximum-likelihood-type estimator. IEEE Trans. Signal Process. 58(5), 2509–2520 (2010)

    Article  MathSciNet  Google Scholar 

  10. A. Genz, J. Monahan, Stochastic integration rules for infinite regions. SIAM J. Sci. Comput. 19(2), 426–439 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. A. Genz, Fully symmetric interpolatory rules for multiple integrals over hyper-spherical surfaces. J. Comput. Appl. Math. 157(1), 187–195 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. N. Gordon, A. Smith, Approximate non-Gaussian Bayesian estimation and modal consistency. J. R. Stat. Soc. B. 55(4), 913–918 (1993)

    MathSciNet  MATH  Google Scholar 

  13. Y.L. Huang, Y.G. Zhang, X.X. Wang, L. Zhao, Gaussian filter for nonlinear systems with correlated noises at the same epoch. Automatica 60(10), 122–126 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Y.L. Huang, Y.G. Zhang, N. Li, L. Zhao, Design of sigma-point Kalman filter with recursive updated measurement. Circuits Syst. Signal Process. 35(5), 1767–1782 (2016)

    Article  MATH  Google Scholar 

  15. Y.L. Huang, Y.G. Zhang, N. Li, Z. Shi, Design of Gaussian approximate filter and smoother for nonlinear systems with correlated noises at one epoch apart. Circuits Syst. Signal Process. 35(11), 3981–4008 (2016)

    Article  MATH  Google Scholar 

  16. Y.L. Huang, Y.G. Zhang, N. Li, J. Chambers, A robust Gaussian approximate fixed-interval smoother for nonlinear systems with heavy-tailed process and measurement noises. IEEE Signal Proc. Lett. 23(4), 468–472 (2016)

    Article  Google Scholar 

  17. Y.L. Huang, Y.G. Zhang, Z.M. Wu, N. Li, J. Chambers, A novel robust Student’s t based Kalman filter. IEEE Trans. Aerosp. Electron. Syst. 53(3), 1545–1554 (2017)

    Article  Google Scholar 

  18. Y.L. Huang, Y.G. Zhang, Robust Student’s t based stochastic cubature filter for nonlinear systems with heavy-tailed process and measurement noises. IEEE Access 5(5), 7964–7974 (2017)

    Article  Google Scholar 

  19. Y.L. Huang, Y.G. Zhang, A new process uncertainty robust Student’s t based Kalman filter for SINS/GPS integration. IEEE Access 5(7), 14391–14404 (2017)

    Article  Google Scholar 

  20. Y.L. Huang, Y.G. Zhang, B. Xu, Z.M. Wu, J. Chambers, A new outlier-robust Student’s t based Gaussian approximate filter for cooperative localization. IEEE/AMSE Trans. Mech. (2017). doi:10.1109/TMECH.2017.2744651

    Google Scholar 

  21. Y.L. Huang, Y.G. Zhang, N. Li, J. Chambers, Robust Student’s t based nonlinear filter and smoother. IEEE Trans. Aerosp. Electron. Syst. 52(5), 2586–2596 (2016)

    Article  Google Scholar 

  22. Y.L. Huang, Y.G. Zhang, N. Li, S.M. Naqvi, J. Chambers, A robust Student’s t based cubature filter. in 19th International Conference on Information Fusion (FUSION) (IEEE, Heidelberg, 2016), pp. 9–16

  23. K. Ito, K. Xiong, Gaussian filters for nonlinear filtering problems. IEEE Trans. Autom. Control 45(5), 910–927 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  24. R. Izanloo, S.A. Fakoorian, H.S. Yazdi, D. Simon, Kalman filtering based on the maximum correntropy criterion in the presence of non-Gaussian noise. in Proceedings of 50th Annual Conference on Information Science and Systems (IEEE, Princeton, NJ, 2016), pp. 1–6

  25. B. Jia, M. Xin, Y. Cheng, High-degree cubature Kalman filter. Automatica 49(2), 510–518 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. S.J. Julier, J.K. Uhlman, Unscented filtering and nonlinear estimation. Proc. IEEE 92(3), 401–422 (2004)

    Article  Google Scholar 

  27. C.D. Karlgaard, H. Schaub, Huber-based divided difference filtering. J. Guid. Control Dyn. 30(3), 885–891 (2007)

    Article  Google Scholar 

  28. C.D. Karlgaard, Nonlinear regression Huber-Kalman filtering and fixed-interval smoothing. J. Guid. Control Dyn. 38(2), 322–330 (2015)

    Article  Google Scholar 

  29. B. Kovacevic, Z. Durovic, S. Glavaski, On robust Kalman filtering. Int. J. Control 56(3), 547–562 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  30. S.J. Li, H. Wang, T.Y. Chai, A t-distribution based particle filter for target tracking. in American Control Conference (IEEE, Minneapolis, MN, 2006), pp. 2191–2196

  31. J. Loxam, T. Drummond, Student mixture filter for robust, realtime visual tracking. in Proceedings of 10th European Conference on Computer Vision: Part III (Springer, Berlin, 2008), pp. 372–385

  32. J. Lu, D.L. Darmofal, Higher-dimensional integration with Gaussian weight for applications in probabilistic design. SIAM J. Sci. Comput. 26(2), 613–624 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  33. I.P. Mysovskikh, The approximation of multiple integrals by using interpolatory cubature formulae, in Quantitative Approximation, ed. by R.A. DeVore, K. Scherer (Academic Press, New York, 1980)

  34. M. Roth, E. Özkan, F. Gustafsson, A Student’s t filter for heavy-tailed process and measurement noise, 2013 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (IEEE, Vancouver, BC, 2013), pp. 5770–5774

  35. A.H. Stroud, Approximate Calculation of Multiple Integrals (Prentice Hall, Englewood Cliffs, NJ, 1971)

    MATH  Google Scholar 

  36. S.Y. Wang, J.C. Feng, C.K. Tse, Spherical simplex-radial cubature Kalman filter. IEEE Signal Process. Lett. 21(1), 43–46 (2014)

    Article  Google Scholar 

  37. X.G. Wang, N.G. Cui, J.F. Guo, Huber-based unscented fltering and its application to vision-based relative navigation. IET Radar Sonar Navig. 4(1), 134–141 (2010)

    Article  Google Scholar 

  38. Y.D. Wang, S.M. Sun, L. Li, Adaptively robust unscented Kalman filter for tracking a maneuvering vehicle. J. Guid. Control Dyn. 37(5), 1696–1701 (2014)

    Article  Google Scholar 

  39. Y.G. Zhang, Y.L. Huang, N. Li, L. Zhao, Interpolatory cubature Kalman filters. IET Control Theory Appl. 9(11), 1731–1739 (2015)

    Article  MathSciNet  Google Scholar 

  40. Y.G. Zhang, Y.L. Huang, N. Li, L. Zhao, Embedded cubature Kalman filter with adaptive setting of free parameter. Signal Process. 114(9), 112–116 (2015)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Automation, Harbin Engineering University, No. 145, Nantong Street, Nangang District, Harbin, 150001, People’s Republic of China

    Yulong Huang & Yonggang Zhang

Authors
  1. Yulong Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yonggang Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yonggang Zhang.

Additional information

This work was supported by the National Natural Science Foundation of China under Grant Nos. 61773133 and 61633008 and the Natural Science Foundation of Heilongjiang Province Grant No. F2016008 and the Fundamental Research Founds for the Central University of Harbin Engineering University under Grant Nos. HEUCFP201705 and HEUCF041702 and the Ph.D. Student Research and Innovation Fund of the Fundamental Research Founds for the Central Universities under Grant No. HEUGIP201706 and China Scholarship Council Foundation.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huang, Y., Zhang, Y. Design of High-Degree Student’s t-Based Cubature Filters. Circuits Syst Signal Process 37, 2206–2225 (2018). https://doi.org/10.1007/s00034-017-0662-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-017-0662-y

Keywords

Search

Navigation