Abstract
When certain constraints in the kinematic state parameters of a multi-sensor navigation system exist, they should be taken into account for the improvement of the positioning accuracy and reliability. In this paper, two types of robust estimators for integrated and two stages of Kalman filtering with state parameter constraints are derived based on the generalized maximum likelihood Lagrangian condition, respectively. The properties of the two estimators are discussed. The changes of the state estimates and their covariance matrices as well as the residual vector caused by the constraints are derived and analyzed. It is shown by a simulated example that the precision of the state estimates provided by the Kalman filtering with constraints is better than that provided by the Kalman filtering without considering the state parameter constraints; and the robust Kalman filtering with constraints further improves the reliability and robustness of the state estimates.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alouani AT, Blair WD (1993) Use of a kinematic constraint in tracking constant speed, maneuvering targets. IEEE Trans Automat Control 38(7): 1107–1111
Chia T, Chow P, Chizeck HJ (1991) Recursive parameter identification of constrained systems: an application to electrically stimulated muscle. IEEE Trans Biomed Eng 38(5): 429–442
Doran HE (1992) Constraining Kalman filter and smoothing estimates to satisfy time-varying restrictions. Rev Econ Stat 74(3): 568–572
Hampel FR, Ronchetti EM, Rousseeuw PJ, Stahel WA (1986) Robust statistics. The approach based on influence functions. Wiley, New York
Hekimoglu S (2005) Increasing reliability of the test for outliers whose magnitudes is small. Surv Rev 38: 274–285
Huber PJ (1964) Robust estimation of a location parameter. Ann Math Stat 35: 73–101
Huber PJ (1981) Robust statistics. Wiley, New York
Julier SJ, LaViola J (2007) On Kalman On Kalman filtering with nonlinear equality constraints. IEEE Trans Signal Process 55(6): 2774–2784
Koch KR (1987) Parameter estimation and hypothesis testing in linear model. Springer, New York
Koch KR (1999) Parameter estimation and hypothesis testing in linear models. 2. Springer, Berlin
Koch KR, Yang Y (1998) Robust Kalman filter for rank deficient observation model. J Geod 72(8): 436–441
Ou J (1999) Quasi-accurate detection of gross errors (QUAD). Acta Geod Cartogr Sin 28(1): 15–20
Ou J, Chai Y, Yuan Y (2004) Adaptive filtering for kinematic positioning by selection of the parameter weights. In: Zhu Y, Sun H (eds) Progress in geodesy and geodynamics. Hubei Science & Technology Press, pp 816–823
Petovello MG, O’Keefe K, Lachapelle G, Cannon ME (2009) Consideration of time-correlated errors in a Kalman filter applicable to GNSS. J Geod 83: 51–56
Pope A (1976) The statistics of residuals and outlier detection of outliers. NOAA Technical Report, Rockville
Ren C, Ou J, Yuan Y (2005) Application of adaptive filtering by selecting the parameter weight factor in precise kinematic GPS positioning. Prog Nat Sci 15(1): 41–46
Schaffrin B (1995) On some alternatives to Kalman filtering. In: Sanso F (eds) Geodetic theory today. Springer, Berlin, pp 235–245
Schwarz KP, Cannon ME, Wong RVC (1989) A comparison of GPS kinematic models for determination of position and velocity along a trajectory. Manuscr Geod 14(6): 345–353
Simon D, Chia TL (2002) Kalman filtering with state equality constraints. IEEE Trans Aerosp Electron Syst 39(1): 128–136
Simon D, Simon DL (2004) Aircraft turbofan engine health estimation using constrained Kalman Filtering. J Eng Gas Turbines Power 126(1): 1–6
Tahk M, Speyer JL (1990) Target tracking problems subject to kinematic constraints. IEEE Trans Automat Control 35(3): 324–326
Wen W, Durrant-Whyte HF (1992) Model-based multi-sensor data fusion. In: Proceedings of the 1992 IEEE international conference on robotics and automation, IEEE. IEEE Press, New York, pp 1720–1726
Wiśniewski Z (1999) Concept of robust estimation of variance coefficient (VR-estimation). Boll Geod Sci Affin 3: 291–310
Wiśniewski Z (2008) Estimation of parameters in a split functional model of geodetic observations (M-split estimation). J Geod. doi:10.1007/s00190-008-0241-x
**e L, Soh YC (1994) Robust Kalman filtering for uncertain system. Syst Control Lett 22: 122–129
Xu PL (1989) Statistical criteria for robust methods. ITC JNo 1: 37–40
Xu P (1989) On robust estimation with correlated observations. Bull Geod 63: 237–252
Xu PL (1993) Consequences of constant parameters and confidence intervals of robust estimation. Boll Geod Sci Affin 52: 231–249
Xu P (2005) Sign-constrained robust least squares, subjective breakdown point and the effect of weights of observations on robustness. J Geod 79: 146–159
Yang Y (1991) Robust Bayesian estimation. Bull Geod 65: 145–150
Yang Y (1992) Robustifying collocation. Manuscr Geod 17: 21–28
Yang Y (1993) Robust estimation and applications. Bayi Publishing House, Bei**g
Yang Y (1994) Robust estimation for dependent observations. Manuscr Geod 19: 10–17
Yang Y (1997) Robust Kalman filter for dynamical system. J Zhengzhou Inst Surv Mapp 14(2): 79–84
Yang Y (2003) Properties of the adaptive filtering for kinematic positioning. Acta Geod Cartogr Sin 32(3): 189–192
Yang Y (2006) Adaptive navigation and kinematic positioning. Press of Surveying and Map**, Bei**g
Yang Y, Cui X (2008) Adaptively robust filter with multi adaptive factors. Surv Rev 40(309): 260–270
Yang Y, He H, Xu G (2001) Adaptively robust filtering for kinematic geodetic positioning. J Geod 75(2/3): 109–116
Yang Y, Song L, Xu T (2002) Robust parameter estimation for correlated geodetic observations. Selected Papers for english edition. Acta Geod Cartogr Sin 31(2): 18–24
Zhou J, Huang Y, Yang Y, Ou J (1997) Robust least squares methods. Publishing House of Huazhong University of Science and Technology, Wuhan
Zhu J (1996) Robustness and the robust estimate. J Geod 70: 586–590
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Yang, Y., Gao, W. & Zhang, X. Robust Kalman filtering with constraints: a case study for integrated navigation. J Geod 84, 373–381 (2010). https://doi.org/10.1007/s00190-010-0374-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00190-010-0374-6