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Generalized total Kalman filter algorithm of nonlinear dynamic errors-in-variables model with application on indoor mobile robot positioning

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Abstract

In this paper, a nonlinear dynamic errors-in-variables (DEIV) model which considers all of the random errors in both system equations and observation equations is presented. The nonlinear DEIV model is more general in the structure, which is an extension of the existing DEIV model. A generalized total Kalman filter (GTKF) algorithm that is capable of handling all of random errors in the respective equations of the nonlinear DEIV model is proposed based on the Gauss–Newton method. In addition, an approximate precision estimator of the posteriori state vector is derived. A two dimensional simulation experiment of indoor mobile robot positioning shows that the GTKF algorithm is statistically superior to the extended Kalman filter algorithm and the iterative Kalman filter (IKF) algorithm in terms of state estimation. Under the experimental conditions, the improvement rates of state variables of positions x, y and azimuth ψ of the GTKF algorithm are about 14, 29, and 66%, respectively, compared with the IKF algorithm.

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References

  • Aftatah M, Lahrech A, Abounada A, Soulhi A (2016) GPS/INS/Odometer data fusion for land vehicle localization in GPS denied environment. Mod Appl Sci 11(1):62–75

    Article  Google Scholar 

  • Amiri-simkooei AR (2013) Application of least squares variance component estimation to errors-in-variables models. J Geod 87:935–944

    Article  Google Scholar 

  • Amiri-Simkooei AR (2016) Non-negative least-squares variance component estimation with application to GPS time series. J Geod 90:451–466

    Article  Google Scholar 

  • Amiri-simkooei AR, Jazaeri S (2012) Weighted total least squares formulated by standard least squares theory. J Geod Sci 2:113–124

    Google Scholar 

  • Amiri-Simkooei AR, Jazaeri S (2013) Data-snooping procedure applied to errors-in-variables models. Stud Geophys Geod 57:426–441

    Article  Google Scholar 

  • Amiri-Simkooei AR, Zangeneh-Nejad F, Asgari J (2016) On the covariance matrix of weighted total least-squares estimates. J Surv Eng 142:04015014

    Article  Google Scholar 

  • Baheti RS (1986) Efficient approximation of Kalman filter for target tracking. IEEE Trans Aerosp Electron Syst. doi:10.1109/TAES.1986.310687

    Google Scholar 

  • Bell BM, Cathey FW (1993) The iterated Kalman filter update as a Gauss–Newton method. IEEE Trans Autom Control 38:294–297. doi:10.1109/9.250476

    Article  Google Scholar 

  • Brown RG, Hwang PYC (2012) Introduction to random signals and applied Kalman filtering: with MATLAB exercises and solutions, 4th edn. Wiley, New York

    Google Scholar 

  • Chang GB (2015) On least-squares solution to 3D similarity transformation problem under Gauss–Helmert model. J Geod 89:573–576

    Article  Google Scholar 

  • Chang GB, Xu TH, Wang QX, Zhang SB, Chen GL (2017) A generalization of the analytical least-squares solution to the 3D symmetric Helmert coordinate transformation problem with an approximate error analysis. Adv Space Res 59:2600–2610

    Article  Google Scholar 

  • Crassidis JL, Junkins JL (2011) Optimal estimation of dynamic systems, 2nd edn. Applied mathematics and nonlinear science series. Chapman & Hall/CRC, New York

    Google Scholar 

  • Fan QG, Sun BW, Sun Y, Wu YH, Zhuang XP (2017) Data fusion for indoor mobile robot positioning based on tightly coupled INS/UWB. J Navig. doi:10.1017/S0373463317000194

    Google Scholar 

  • Fang X (2013) Weighted total least squares: necessary and sufficient conditions, fixed and random parameters. J Geod 87:733–749

    Article  Google Scholar 

  • Fang X (2014) On non-combinatorial weighted total least squares with inequality constraints. J Geod 88:805–816

    Article  Google Scholar 

  • Fang X (2015) Weighted total least-squares with constraints: a universal formula for geodetic symmetrical transformations. J Geod 89:459–469

    Article  Google Scholar 

  • Fang X, Wu Y (2016) On the errors-in-variables model with equality and inequality constraints for selected numerical examples. Acta Geod Geophys 51:515–525

    Article  Google Scholar 

  • Fang X, Wang J, Li BF, Zeng WX, Yao YB (2015) On total least squares for quadratic form estimation. Stud Geophys Geod 59(3):366–379

    Article  Google Scholar 

  • Fang X, Li BF, Alkhatib H, Zeng WX, Yao YB (2017) Bayesian inference for the errors-in-variables model. Stud Geophys Geod 61(1):35–52

    Article  Google Scholar 

  • Farrell J (2008) Aided navigation: GPS with high rate sensors. McGraw-Hill Inc, New York

    Google Scholar 

  • Gelb A (1974) Applied optimal estimation. MIT Press, Cambridge

    Google Scholar 

  • Golub G, van Loan C (1980) An analysis of the total least-squares problem. SIAM J Numer Anal 17:883–893

    Article  Google Scholar 

  • Han HZ, Wang J, Wang JL, Tan XL (2015) Performance analysis on carrier phase-based tightly-coupled GPS/BDS/INS integration in GNSS degraded and denied environments. Sensors 15:8685–8711

    Article  Google Scholar 

  • Han HZ, Wang J, Wang JL, Hernandez A (2017) Reliable partial ambiguity resolution for single-frequency GPS/BDS and INS integration. GPS Solut 21:251–264

    Article  Google Scholar 

  • Henderson HV, Searle SR (1981) On deriving the inverse of a sum of matrices. SIAM Rev 23:53–60

    Article  Google Scholar 

  • Julier SJ, Uhlmann JK, Durrant-Whyte HF (1995) A new approach for filtering nonlinear systems. In: Proceedings of the ACC’95, pp 1628–1632. doi: 10.1109/ACC.1995.529783

  • Li BF, Shen YZ, Li WX (2012) The seamless model for three-dimensional datum transformation. Sci China 55:2099–2108

    Article  Google Scholar 

  • Li ZK, Chang GB, Gao JX, Wang J, Hernandez A (2016) GPS/UWB/MEMS-IMU tightly coupled navigation with improved robust Kalman filter. Adv Space Res 58:2424–2434

    Article  Google Scholar 

  • Liu JN (1983) The equivalence of coordinate transformation models for the combination of satellite and terrestrial networks. J Wuhan Techn Univ Surv Mapp 8:37–50 (in Chinese with English abstract)

    Google Scholar 

  • Liu J, Chen R (1998) Sequential Monte Carlo methods for dynamic systems. J Am Stat Assoc 93:1032–1044

    Article  Google Scholar 

  • Liu JN, Liu DJ (1985) The influence of the accuracy in geodetic and geocentric coordinates on combined adjustment. Acta Geod Cartogr Sin 14:133–144 (in Chinese with English abstract)

    Google Scholar 

  • Liu JN, Liu DJ, Cui XZ (1987) Theory and applications of combined adjustment of satellite and terrestrial networks. J Wuhan Techn Univ Surv Mapp 12(4):1–9 (in Chinese with English abstract)

    Google Scholar 

  • Lu J, Chen Y, Li BF, Fang X (2014) Robust total least squares with reweighting iteration for three-dimensional similarity transformation. Surv Rev 46:28–36

    Article  Google Scholar 

  • Mahboub V (2012) On weighted total least-squares for geodetic transformation. J Geod 86:359–367

    Article  Google Scholar 

  • Mahboub V, Sharifi MA (2013) On weighted total least-squares with linear and quadratic constraints. J Geod 87:279–286

    Article  Google Scholar 

  • Mahboub V, Ardalan AA, Ebrahimzadeh S (2015) Adjustment of non-typical errors-invariables models. Acta Geod Geophys 50:207–218

    Article  Google Scholar 

  • Mahboub V, Saadatseresht M, Ardalan AA (2016) A general weighted total Kalman filter algorithm with numerical evaluation. Stud Geophys Geod 61:19–34

    Article  Google Scholar 

  • Mahboub V, Saadatseresht M, Ardalan AA (2017a) A solution to dynamic errors-invariables within system equations. Acta Geod Geophys. doi:10.1007/s40328-017-0201-0

    Google Scholar 

  • Mahboub V, Saadatseresht M, Ardalan AA (2017b) On constrained integrated total Kalman filter for integrated direct geo-referencing. Surv Rev. doi:10.1080/00396265.2017.1341736

    Google Scholar 

  • Merwe R, Wan E (2001) The square-root unscented Kalman filter for state and parameter-estimation. Proc IEEE Int Conf Acoust Speech Signal Process (ICASSP) 6:3461–3464

    Google Scholar 

  • Neitzel F (2010) Generalization of total least-squares on example of unweighted and weighted 2D similarity transformation. J Geod 84:751–762

    Article  Google Scholar 

  • Sage AP, Husa GW (1969) Adaptive filtering with unknown prior statistics. Joint Am Control Conf. doi:10.1109/JACC.1969.4169325

    Google Scholar 

  • Schaffrin B (2009) TLS collocation: the total least squares approach to EIV-models with stochastic prior information. In: 18th international, workshop on matrices and statistics, Solvakia

  • Schaffrin B, Iz HB (2008) Towards total Kalman filtering for mobile mapping. Intl Arch Photogr Rem Sens Spat Inf Sci 36:270–275

    Google Scholar 

  • Schaffrin B, Uzun S (2011) Errors-In-Variables for mobile mapping algorithms. In: The presence of OutlIERS, archives of photogrammetry, cartography and remote sensing, vol 22. pp 377–387

  • Schaffrin B, Wieser A (2008) On weighted total least squares adjustment for linear regression. J Geod 82:415–421

    Article  Google Scholar 

  • Shen YZ, Li BF, Chen Y (2011) An iterative solution of weighted total least-squares adjustment. J Geod 85:229–238

    Article  Google Scholar 

  • Shi Y, Xu PL, Liu JN, Shi C (2015) Alternative formulae for parameter estimation in partial errors-in-variables models. J Geod 89:13–16

    Article  Google Scholar 

  • Teunissen PJG (1988) The nonlinear 2D symmetric Helmert transformation: an exact nonlinear least-squares solution. J Geod 62:1–15

    Google Scholar 

  • Wang LY, Xu GY (2016) Variance component estimation for partial errors-in-variables models. Stud Geophys Geod 60:35–55

    Article  Google Scholar 

  • Wang LY, Zhao YW (2017) Unscented transformation with scaled symmetric sampling strategy for precision estimation of total least squares. Stud Geophys Geod. doi:10.1007/s11200-11016-11113-11200

    Google Scholar 

  • Wang B, Li JC, Liu C (2016) A robust weighted total least squares algorithm and its geodetic applications. Stud Geophys Geod 60:177–194

    Article  Google Scholar 

  • Wang B, Li JC, Liu C, Yu J (2017) Generalized total least squares prediction algorithm for universal 3D similarity transformation. Adv Space Res 59(3):815–823

    Article  Google Scholar 

  • Xu GC (2003) GPS-theory, algorithms and applications, 2nd edn. Springer, Berlin

    Google Scholar 

  • Xu PL (2016) The effect of errors-in-variables on variance component estimation. J Geod 90(8):681–701

    Article  Google Scholar 

  • Xu PL, Liu JN (2013) Variance components in errors-in-variables models: estimability, stability and bias analysis. In: Invited talk, VIII Hotine-Marussi symposium on mathematical geodesy, Rome, 17–21 June 2013

  • Xu PL, Liu JN (2014) Variance components in errors-in-variables models: estimability, stability and bias analysis. J Geod 88:719–734

    Article  Google Scholar 

  • Xu PL, Liu JN, Shi C (2012) Total least squares adjustment in partial errors-in-variables models: algorithm and statistical analysis. J Geod 86:661–675

    Article  Google Scholar 

  • Zhou YJ, Kou XJ, Zhu JJ, LI J (2014) A newton algorithm for weighted total least-squares solution to a specific errors-in-variables model with correlated measurements. Stud Geophys Geod 58:349–375

    Article  Google Scholar 

Download references

Acknowledgements

We are grateful to the editor László Bányai and two anonymous reviewers for their valuable comments. This research is supported by the National Key Research and Development Program of China (2016YFC0803103).

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Authors and Affiliations

  1. School of Environment Science and Spatial Informatics, China University of Mining and Technology, Xuzhou, 221116, China

    Hang Yu & Guobin Chang

  2. School of Geomatics and Urban Spatial Information, Beijing University of Civil Engineering and Architecture, Beijing, 100044, China

    Jian Wang & Houzeng Han

  3. College of Geomatics Science and Technology, Nanjing University of Technology, Nanjing, 211800, China

    Bin Wang

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  1. Hang Yu
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  2. Jian Wang
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  3. Bin Wang
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  4. Houzeng Han
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  5. Guobin Chang
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Correspondence to Jian Wang.

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Yu, H., Wang, J., Wang, B. et al. Generalized total Kalman filter algorithm of nonlinear dynamic errors-in-variables model with application on indoor mobile robot positioning. Acta Geod Geophys 53, 107–123 (2018). https://doi.org/10.1007/s40328-017-0207-7

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  • DOI: https://doi.org/10.1007/s40328-017-0207-7

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