access icon free Modification of unscented Kalman filter using a set of scaling parameters

This work, based on the standard unscented Kalman filter (UKF), proposes a modified UKF for highly non-linear stochastic systems, assuming that the associated probability distributions are normal. In the standard UKF with 2n + 1 sample points, the estimated mean and covariance match the true mean and covariance up to the third order, besides, there exists a scaling parameter that plays a crucial role in minimising the fourth-order errors. The proposed method consists of a computationally efficient formulation of the unscented transform that incorporates , , constant parameters to scale sample points such that the order errors are minimised. The scaling parameters are obtained by solving a set of algebraic equations. Through rigorous analytical processes and numerical simulations, it is demonstrated that the new filter provides consistent estimates and the estimation error of the modified UKF is smaller than that of the standard UKF. With the help of a well-studied case, univariate non-stationary growth model, the authors evaluate the estimation performance of the new technique using 4n + 1 sample points over 100 independent runs.

Inspec keywords: Kalman filters; algebra; probability

Other keywords: probability distributions; unscented Kalman filter; modified UKF; scaling parameters; univariate nonstationary growth model; estimated mean; covariance match; algebraic equations; nonlinear stochastic systems; fourth-order errors

Subjects: Filtering methods in signal processing; Algebra; Other topics in statistics

References

    1. 1)
      • 17. Straka, O., Duník, J., Šimandl, M.: ‘Unscented Kalman filter with advanced adaptation of scaling parameter’, Automatica, 2014, 50, (10), pp. 26572664.
    2. 2)
      • 30. Arulampalam, M.S., Maskell, S., Gordon, N., et al: ‘A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking’, IEEE Trans. Signal Process., 2002, 50, (2), pp. 174188.
    3. 3)
      • 14. Julier, S.J., Uhlmann, J.K.: ‘A general method for approximating nonlinear transformations of probability distributions’ (Technical report, Robotics Research Group, Department of Engineering Science, University of Oxford., 1996), pp. 127.
    4. 4)
      • 5. Zhang, Y., Huang, Y., Li, N., et al: ‘Interpolatory cubature Kalman filters’, IET Control Theory Applic., 2015, 9, (11), pp. 17311739.
    5. 5)
      • 28. Gordon, N.J., Salmond, D.J., Smith, A.F.: ‘Novel approach to nonlinear/non-Gaussian Bayesian state estimation’, IEE Proc. F (Radar Signal Process.), 1993, 140, (2), pp. 107113.
    6. 6)
      • 19. Turner, R., Rasmussen, C.E.: ‘Model based learning of sigma points in unscented Kalman filtering’, Neurocomputing, 2012, 80, pp. 4753.
    7. 7)
      • 21. Lerner, U.N.: ‘Hybrid Bayesian networks for reasoning about complex systems’. PhD thesis, Stanford University, 2002.
    8. 8)
      • 23. Van, Z., James, R.: ‘A more robust unscented transform’. Proc. Int. Symp. Optical Science and Technology (SPIE), 2001, pp. 371380.
    9. 9)
      • 22. Tenne, D., Singh, T.: ‘The higher order unscented filter’. Proc. American Control Conf., Denver, Colorado, June 2003, pp. 24412446.
    10. 10)
      • 3. Roy, A., Mitra, D.: ‘Multi-target trackers using cubature Kalman filter for Doppler radar tracking in clutter’, IET Signal Process., 2016, 10, (8), pp. 888901.
    11. 11)
      • 31. Drummond, O.E., Li, X.R., He, C.: ‘Comparison of various static multiple-model estimation algorithms’. Proc. Int. Conf. Aerospace/Defense Sensing and Controls, Orlando, Florida, April 1998, pp. 510527.
    12. 12)
      • 18. Sakai, A., Kuroda, Y.: ‘Discriminatively trained unscented Kalman filter for mobile robot localization’, J. Adv. Res. Mech. Eng., 2010, 1, (3), pp. 153161.
    13. 13)
      • 33. Li, X.R., Zhao, Z., Jilkov, V.P.: ‘Practical measures and test for credibility of an estimator’. Proc. Int. Workshop on Estimation, Tracking, and Fusion—A Tribute to Yaakov Bar-Shalom, Monterey, California, May 2001, pp. 481495.
    14. 14)
      • 4. Aghili, F.: ‘A prediction and motion-planning scheme for visually guided robotic capturing of free-floating tumbling objects with uncertain dynamics’, IEEE Trans. Robot., 2012, 28, (3), pp. 634649.
    15. 15)
      • 24. Isserlis, L.: ‘On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables’, Biometrika, 1918, 12, (1/2), pp. 134139.
    16. 16)
      • 7. Jazwinski, A.H.: ‘Stochastic processes and filtering theory’ (Academic Press, New York, 1970, 1st edn.).
    17. 17)
      • 9. Julier, S., Uhlmann, J., Durrant-Whyte, H.F.: ‘A new method for the nonlinear transformation of means and covariances in filters and estimators’, IEEE Trans. Autom. Control, 2000, 45, (3), pp. 477482.
    18. 18)
      • 29. Liu, J., Wang, Y., Zhang, J.: ‘A linear extension of unscented Kalman filter to higher-order moment-matching’. 53rd IEEE Conf. Decision and Control, Los Angeles, California, December 2014, pp. 50215026.
    19. 19)
      • 16. Scardua, L.A., Da Cruz, J.J.: ‘Automatic tuning of the unscented Kalman filter and the blind tricyclist problem: an optimization problem’, IEEE Control Syst., 2016, 36, (3), pp. 7085.
    20. 20)
      • 15. Julier, S.J.: ‘The scaled unscented transformation’. Proc. Int. Conf. American Control, Anchorage, Alaska, May 2002, pp. 45554559.
    21. 21)
      • 26. Rhudy, M., Gu, Y., Gross, J., et al: ‘Evaluation of matrix square root operations for UKF within a UAV GPS/INS sensor fusion application’, Int. J. Navig. Obs., 2011, 2011, pp. 111.
    22. 22)
      • 6. Kalman, R.E.: ‘A new approach to linear filtering and prediction problems’, J. Basic Eng., 1960, 82, (1), pp. 3545.
    23. 23)
      • 8. Julier, S.J., Uhlmann, J.K.: ‘New extension of the Kalman filter to nonlinear systems’. Proc. Int. Conf. AeroSense'97, Orlando, Florida, April 1997, pp. 182193.
    24. 24)
      • 2. Zarei-Jalalabadi, M., Malaek, S.M.: ‘Practical method to predict an upper bound for minimum variance track-to-track fusion’, IET Signal Process., 2017, 11, (8), pp. 961968.
    25. 25)
      • 11. Skoglund, M.A., Hendeby, G., Axehill, D.: ‘Extended Kalman filter modifications based on an optimization view point’. Proc. Int. Conf. Information Fusion, Grand Hyatt Washington, Washington DC, July 2015, pp. 18561861.
    26. 26)
      • 20. Jia, B., Xin, M., Cheng, Y.: ‘High-degree cubature Kalman filter’, Neurocomputing, 2013, 49, (2), pp. 510518.
    27. 27)
      • 27. Kitagawa, G.: ‘Monte Carlo filter and smoother for non-Gaussian nonlinear state space models’, J. Comput. Graph. Stat., 1996, 5, (1), pp. 125.
    28. 28)
      • 12. Bass, R.W., Norum, V.D., Schwartz, L.: ‘Optimal multichannel nonlinear filtering’, J. Math. Anal. Appl., 1966, 16, (1), pp. 152164.
    29. 29)
      • 25. Larsen, R.J., Marx, M.L.: ‘An introduction to mathematical statistics and its applications (Vol. 2)’ (Prentice-Hall, New Jersey, 1981, 2012, 5th edn.).
    30. 30)
      • 32. Bar-Shalom, Y., Birmiwal, K.: ‘Consistency and robustness of PDAF for target tracking in cluttered environments’, Automatica, 1983, 19, (4), pp. 431437.
    31. 31)
      • 1. Goh, Y.H., Raveendran, P., Goh, Y. L.: ‘Robust speech recognition system using bidirectional Kalman filter’, IET Signal Process., 2015, 9, (6), pp. 491497.
    32. 32)
      • 10. Simon, D.: ‘Optimal state estimation: Kalman, H infinity, and nonlinear approaches’ (John Wiley & Sons, Hoboken, New Jersey, 2006, 1st edn.).
    33. 33)
      • 13. Ienkaran, A., Haykin, S.: ‘Cubature Kalman filters’, IEEE Trans. Autom. Control, 2009, 6, (54), pp. 12541269.
http://iet.metastore.ingenta.com/content/journals/10.1049/iet-spr.2017.0300
Loading

Related content

content/journals/10.1049/iet-spr.2017.0300
pub_keyword,iet_inspecKeyword,pub_concept
6
6
Loading