Skip to main content
SpringerLink
Log in

An Optimal Adaptive Kalman Filter

  • Original Article
  • Published:
Journal of Geodesy Aims and scope Submit manuscript

Abstract

In a robustly adaptive Kalman filter, the key problem is to construct an adaptive factor to balance the contributions of the kinematic model information and the measurements on the state vector estimates, and the corresponding learning statistic for identifying the kinematic model biases. What we pursue in this paper are some optimal adaptive factors under the particular conditions that the state vector can or cannot be estimated by measurements. Two optimal adaptive factors are derived, one of which is deduced by requiring that the estimated covariance matrix of the predicted residual vector equals the corresponding theoretical one. The other is obtained by requiring that the estimated covariance matrix of the predicted state vector equals its theoretical one. The two related optimal adaptive factors are given. These are analyzed and compared in theory and in an actual example. This shows, through the actual computations, that the filtering results obtained by optimal adaptive factors are superior to those obtained by adaptive factors based on experience.

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.

Institutional subscriptions

Similar content being viewed by others

References

  • Deng Z (2003) Self-tuning filtering theory with applications – modern time series analysis method. Press of Harbin Institute of Technology, Harbin, China

    Google Scholar 

  • Hekimoğlu S, Berber M (2003) Effectiveness of robust methods in heterogeneous linear models. J Geod 76(11–12):706-713 DOI: 10.1007/s00190-002-0289-y

    Article  Google Scholar 

  • Hide C, Moore T, Smith M (2004) Multiple model Kalman filtering for GPS and low-cost INS integration. In: Proceedings of ION-GNSS-2004, Long Beach, September 21–24, http://www.nottingham.ac.uk/iessg/research/res_publication/rp 2004

  • Koch KR, Yang Y (1998) Robust Kalman filter for rank deficient observation models. J Geod 72(7–8):436–441 DOI: 10.1007/s001900050183

    Article  Google Scholar 

  • Masreliez CJ, Martin RD (1977) Robust Bayesian estimation for the linear model and robustifying the Kalman filter. IEEE Trans Automat Control AC-22:361–371

    Article  Google Scholar 

  • Mohamed AH, Schwarz K-P (1999) Adaptive Kalman filtering for INS/GPS. J Geod 73(4):193–203 DOI: 10.1007/s001900050236

    Article  Google Scholar 

  • Niehen W (2004) Adaptive Kalman filtering based on matched filtering of the innovation sequence. In: Proceedings of the 7th International Conference on information fusion. Stockholm, Sweeden, June 28–July 04, 2004, pp. 362–369

  • Panozzo T, Born GH, Bernelli-Zazzera F (2004) Kalman filtering of accelerometric data for aerobraking navigation. Presented at the 6th conference of dynamic and control of systems and structures in space, Riomoggiore

  • Schaffrin B (1991) Generating robustified Kalman filters for the integration of GPS and INS. Technical Report 15, Institute of Geodesy, University of Stuttgart

  • Schwarz KP, Cannon ME, Wong RVC (1989) A comparison of GPS kinematic models for determination of position and velocity along a trajectory. Manuscripta geodaetica 14:345–353

    Google Scholar 

  • Teunissen PJG (1990) Quality control in navigation systems. IEEE Aerosp Electron Syst Mag 5(7):35–41

    Article  Google Scholar 

  • Tsai C, Kurz L (1983) An adaptive robustifying approach to Kalman filtering. Automatica 19:279–288

    Article  Google Scholar 

  • Wang J, Stewart MP, Tsakiri M (2000) Adaptive Kalman filtering for integration of GPS with GLONASS and INS. In: Schwarz K-P (eds) Geodesy Beyond 2000. Springer, Berlin Heidelberg New York, pp 325–330

    Google Scholar 

  • Xu T, Yang Y (2000) Improved sage adaptive filtering. Sci Surv Mapp 25(3):22–25

    Google Scholar 

  • Yang Y (1991) Robust Bayesian estimation. Bull Géodésique 65(2):145–150

    Google Scholar 

  • Yang Y (1997) Robust Kalman filter for dynamic systems. J Zhengzhou Inst Surv Mapp 14:79–84

    Google Scholar 

  • Yang Y, Gao W (2005a) Influence comparison of adaptive factors on navigation results. J Navigation 58(3):471–478

    Article  Google Scholar 

  • Yang Y, Xu T (2003) An adaptive Kalman filter based on Sage windowing weights and variance components. J Navigation 56(2):231–240

    Article  Google Scholar 

  • Yang Y, He H, Xu G. (2001a) Adaptively robust filtering for kinematic geodetic positioning. J Geod 75(2/3):109–116 DOI: 10.1007/s001900000157

    Article  Google Scholar 

  • Yang Y, Xu T, He H (2001b) On adaptively kinematic filtering. Selected papers for English edition, Acta Geodetica et Cartographica Sinica, p 25–32

  • Zhou J, Huang Y, Yang Y, Ou J (1997) Robust least squares method. Publishing House of Huazhong University of Science and Technology, Wuhan

Download references

Author information

Authors and Affiliations

  1. Xi’an Research Institute of Surveying and Mapping, No.1 Mid-Yanta Road, Xi’an, 710054, Peoples Republic of China

    Yuanxi Yang & Weiguang Gao

Authors
  1. Yuanxi Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Weiguang Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuanxi Yang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yang, Y., Gao, W. An Optimal Adaptive Kalman Filter. J Geodesy 80, 177–183 (2006). https://doi.org/10.1007/s00190-006-0041-0

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00190-006-0041-0

Keywords

Search

Navigation