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Huber-based high-degree cubature Kalman tracking algorithm

Zhang Wen-Jie Wang Shi-Yuan Feng Ya-Li Feng Jiu-Chao

Huber-based high-degree cubature Kalman tracking algorithm

Zhang Wen-Jie, Wang Shi-Yuan, Feng Ya-Li, Feng Jiu-Chao
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  • In recent decades, nonlinear Kalman filtering based on Bayesian theory has been intensively studied to solve the problem of state estimation in nonlinear dynamical system. Under the Gaussian assumption, Bayesian filtering can provide a unified recursive solution to the estimation problem that is described as the calculation of Gaussian weighted integrals. However it is typically intractable to directly calculate these integrals. The numerical integration methods are required from a practical perspective. Therefore, nonlinear Kalman filters are generated by different numerical integrations. As a representative of nonlinear Kalman filter, cubature Kalman filter (CKF) utilizes a numerical rule based on the third-degree spherical-radial cubature rule to obtain better numerical stability, which is widely used in many fields, e.g., positioning, attitude estimation, and communication. Target tracking can be generalized as the estimations of the target position, the target velocity and other parameters. Hence, nonlinear Kalman filters can also be used to perform target tracking, effectively. Since the CKF based on the third-degree cubature rule has a limited accuracy of estimation, it is necessary to find a CKF based a cubature rule with higher accuracy in the case of target tracking system with a large uncertainty. High-degree cubature Kalman filter is therefore proposed to implement state estimation due to its higher numerical accuracy, which is preferred to solve the estimation problem existing in target tracking. To improve the filtering accuracy and robustness of high-degree cubature Kalman filter, in this paper we present a new filtering algorithm named Huber-based high-degree cubature Kalman filter (HHCKF) algorithm. After approximating nonlinear measurements by using the statistical linear regression model, the measurement update is implemented by the Huber M estimation. As a mixed estimation technique based on the minimum of l1-norm and l2-norm, the Huber estimator has high robustness and numerical accuracy under the assumption of Gaussian measurement noises. Therefore, the Huber-based high-degree cubature Kalman tracking algorithm is generated by combining the state prediction based on the fifth-degree spherical radial rule. In this paper, the influence of tuning parameter on the tracking performance is discussed by simulations. Simulations in the context of bearings only tracking and reentry vehicle tracking demonstrate that the new HHCKF can improve the tracking performance significantly.
      Corresponding author: Wang Shi-Yuan, wsy@swu.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 61101232), the Fundamental and Frontier Research Project of Chongqing, China (Grant No. cstc2014jcyjA40020), and the Fundamental Research Funds for Central Universities, China (Grant No. XDJK2014B001).
    [1]

    Pakki K, Chandra B, Gu D W 2011 Proc. of the American Control Conference USA, June 29-July 1, 2011 p3609

    [2]

    Gustafsson F, Hendeby G 2012 IEEE Trans. Signal Process. 60 545

    [3]

    Arasaratnam I, Haykin S, Elliott R J 2007 IEEE Proc. 95 953

    [4]

    Sheng Z, Chen J Q, Xu R H 2012 Acta Phys. Sin. 61 069301 (in Chinese) [盛峥, 陈加清, 徐如海 2012 物理学报 61 069301]

    [5]

    Leung H, Zhu Z, Ding Z 2000 IEEE Trans. Signal Process. 48 1807

    [6]

    Julier S Y, Uhlmann J K 2004 IEEE Proc. 92 401

    [7]

    Hu G G, Gao S S, Zhong Y M, Gao B B 2015 Chin. Phys. B 24 070202

    [8]

    Arasaratnam I, Haykin S 2009 IEEE Trans. Autom. Control 54 1254

    [9]

    Zhang X C, Guo C J 2013 Chin. Phys. B 22 128401

    [10]

    Wang S Y, Feng J C, Tse C K 2014 IEEE Signal Process. Lett. 21 43

    [11]

    Jia B, Xin M, cheng Y 2013 Automatica 49 510

    [12]

    Zhang X C, Teng Y L 2015 Asian J. Control 17 1

    [13]

    Zhang X C 2014 Circ. Syst. Signal Process. 65 469

    [14]

    Huber P J 1964 Ann. Math. Stat. 35 73

    [15]

    Huber P J, Ronchetti E M 2009 Robust Statistics (Hoboken, New Jersey: John Wiley Sons, Inc.)

    [16]

    Petrus P 1999 IEEE Trans. Signal Process. 47 1129

    [17]

    Chang L, Hu B, Chang G, Li A 2012 IET Sci. Measur. Technol. 6 502

    [18]

    Karlgaard C D, Schaub H 2007 J. Guidance, Control, Dyn. 30 885

    [19]

    Wang X G, Cui N G, Guo J 2010 IET Radar, Sonar Navigation 4 134

    [20]

    Chang G B, Xu J N, Chang L B 2011 J. Nanjing Univ. Aeronaut. Astron. 43 754 (in Chinese) [常国宾, 许江宁, 常路宾 2011 南京航空航天大学学报 43 754]

    [21]

    Zhang Q, Qiao Y K, Kong X Y, Si X S 2014 Acta Phys. Sin. 63 110505 (in Chinese) [张琪, 乔玉坤, 孔祥玉, 司小胜 2014 物理学报 63 110505]

    [22]

    Lu Z Y, Wang D M, Wang J H, Wang Y 2015 Acta Phys. Sin. 64 150502 (in Chinese) [逯志宇, 王大鸣, 王建辉, 王跃 2015 物理学报 64 150502]

    [23]

    Karlgaard C D, Schaub H 2006 American Institute of Aeronautics and Astronautics, AIAA Paper 2006

    [24]

    Dunk J, Straka O, imandl M 2013 IEEE Trans. Autom. Cont. 58 1561

    [25]

    Bar-Shalom Y, Li X R, Kirubarajan T 2002 Estimation with Applications to Tracking and Navigation (New York: Williey Inter Science Press)

  • [1]

    Pakki K, Chandra B, Gu D W 2011 Proc. of the American Control Conference USA, June 29-July 1, 2011 p3609

    [2]

    Gustafsson F, Hendeby G 2012 IEEE Trans. Signal Process. 60 545

    [3]

    Arasaratnam I, Haykin S, Elliott R J 2007 IEEE Proc. 95 953

    [4]

    Sheng Z, Chen J Q, Xu R H 2012 Acta Phys. Sin. 61 069301 (in Chinese) [盛峥, 陈加清, 徐如海 2012 物理学报 61 069301]

    [5]

    Leung H, Zhu Z, Ding Z 2000 IEEE Trans. Signal Process. 48 1807

    [6]

    Julier S Y, Uhlmann J K 2004 IEEE Proc. 92 401

    [7]

    Hu G G, Gao S S, Zhong Y M, Gao B B 2015 Chin. Phys. B 24 070202

    [8]

    Arasaratnam I, Haykin S 2009 IEEE Trans. Autom. Control 54 1254

    [9]

    Zhang X C, Guo C J 2013 Chin. Phys. B 22 128401

    [10]

    Wang S Y, Feng J C, Tse C K 2014 IEEE Signal Process. Lett. 21 43

    [11]

    Jia B, Xin M, cheng Y 2013 Automatica 49 510

    [12]

    Zhang X C, Teng Y L 2015 Asian J. Control 17 1

    [13]

    Zhang X C 2014 Circ. Syst. Signal Process. 65 469

    [14]

    Huber P J 1964 Ann. Math. Stat. 35 73

    [15]

    Huber P J, Ronchetti E M 2009 Robust Statistics (Hoboken, New Jersey: John Wiley Sons, Inc.)

    [16]

    Petrus P 1999 IEEE Trans. Signal Process. 47 1129

    [17]

    Chang L, Hu B, Chang G, Li A 2012 IET Sci. Measur. Technol. 6 502

    [18]

    Karlgaard C D, Schaub H 2007 J. Guidance, Control, Dyn. 30 885

    [19]

    Wang X G, Cui N G, Guo J 2010 IET Radar, Sonar Navigation 4 134

    [20]

    Chang G B, Xu J N, Chang L B 2011 J. Nanjing Univ. Aeronaut. Astron. 43 754 (in Chinese) [常国宾, 许江宁, 常路宾 2011 南京航空航天大学学报 43 754]

    [21]

    Zhang Q, Qiao Y K, Kong X Y, Si X S 2014 Acta Phys. Sin. 63 110505 (in Chinese) [张琪, 乔玉坤, 孔祥玉, 司小胜 2014 物理学报 63 110505]

    [22]

    Lu Z Y, Wang D M, Wang J H, Wang Y 2015 Acta Phys. Sin. 64 150502 (in Chinese) [逯志宇, 王大鸣, 王建辉, 王跃 2015 物理学报 64 150502]

    [23]

    Karlgaard C D, Schaub H 2006 American Institute of Aeronautics and Astronautics, AIAA Paper 2006

    [24]

    Dunk J, Straka O, imandl M 2013 IEEE Trans. Autom. Cont. 58 1561

    [25]

    Bar-Shalom Y, Li X R, Kirubarajan T 2002 Estimation with Applications to Tracking and Navigation (New York: Williey Inter Science Press)

  • [1] Hu Xiaoliang, Liang Hong, Wang Huili. Lattice Boltzmann method simulations of the immiscible Rayleigh-Taylor instability with high Reynolds numbers. Acta Physica Sinica, 2020, 69(4): 1-10. doi: 10.7498/aps.69.20191504
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  • Received Date:  16 September 2015
  • Accepted Date:  05 January 2016
  • Published Online:  20 April 2016

Huber-based high-degree cubature Kalman tracking algorithm

    Corresponding author: Wang Shi-Yuan, wsy@swu.edu.cn
  • 1. School of Electronic and Information Engineering, Southwest University, Chongqing 400715, China;
  • 2. School of Electronic and Information Engineering, South China University of Technology, Guangzhou 510641, China
Fund Project:  Project supported by the National Natural Science Foundation of China (Grant No. 61101232), the Fundamental and Frontier Research Project of Chongqing, China (Grant No. cstc2014jcyjA40020), and the Fundamental Research Funds for Central Universities, China (Grant No. XDJK2014B001).

Abstract: In recent decades, nonlinear Kalman filtering based on Bayesian theory has been intensively studied to solve the problem of state estimation in nonlinear dynamical system. Under the Gaussian assumption, Bayesian filtering can provide a unified recursive solution to the estimation problem that is described as the calculation of Gaussian weighted integrals. However it is typically intractable to directly calculate these integrals. The numerical integration methods are required from a practical perspective. Therefore, nonlinear Kalman filters are generated by different numerical integrations. As a representative of nonlinear Kalman filter, cubature Kalman filter (CKF) utilizes a numerical rule based on the third-degree spherical-radial cubature rule to obtain better numerical stability, which is widely used in many fields, e.g., positioning, attitude estimation, and communication. Target tracking can be generalized as the estimations of the target position, the target velocity and other parameters. Hence, nonlinear Kalman filters can also be used to perform target tracking, effectively. Since the CKF based on the third-degree cubature rule has a limited accuracy of estimation, it is necessary to find a CKF based a cubature rule with higher accuracy in the case of target tracking system with a large uncertainty. High-degree cubature Kalman filter is therefore proposed to implement state estimation due to its higher numerical accuracy, which is preferred to solve the estimation problem existing in target tracking. To improve the filtering accuracy and robustness of high-degree cubature Kalman filter, in this paper we present a new filtering algorithm named Huber-based high-degree cubature Kalman filter (HHCKF) algorithm. After approximating nonlinear measurements by using the statistical linear regression model, the measurement update is implemented by the Huber M estimation. As a mixed estimation technique based on the minimum of l1-norm and l2-norm, the Huber estimator has high robustness and numerical accuracy under the assumption of Gaussian measurement noises. Therefore, the Huber-based high-degree cubature Kalman tracking algorithm is generated by combining the state prediction based on the fifth-degree spherical radial rule. In this paper, the influence of tuning parameter on the tracking performance is discussed by simulations. Simulations in the context of bearings only tracking and reentry vehicle tracking demonstrate that the new HHCKF can improve the tracking performance significantly.

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