UD Covariance Factorization for Unscented Kalman Filter using Sequential Measurements Update

Extended Kalman Filter (EKF) is probably the most widely used estimation algorithm for nonlinear systems. However, not only it has difficulties arising from linearization but also many times it becomes numerically unstable because of computer round off errors that occur in the process of its implementation. To overcome linearization limitations, the unscented transformation (UT) was developed as a method to propagate mean and covariance information through nonlinear transformations. Kalman filter that uses UT for calculation of the first two statistical moments is called Unscented Kalman Filter (UKF). Square-root form of UKF (SRUKF) developed by Rudolph van der Merwe and Eric Wan to achieve numerical stability and guarantee positive semi-definiteness of the Kalman filter covariances. This paper develops another implementation of SR-UKF for sequential update measurement equation, and also derives a new UD covariance factorization filter for the implementation of UKF. This filter is equivalent to UKF but is computationally more efficient. Keywords—Unscented Kalman filter, Square-root unscented Kalman filter, UD covariance factorization, Target tracking.

[1]  Rudolph van der Merwe,et al.  Sigma-point kalman filters for probabilistic inference in dynamic state-space models , 2004 .

[2]  L. Teng.  Theory and application of Kalman filtering , 1970 .

[3]  H. Kushner.  Dynamical equations for optimal nonlinear filtering , 1967 .

[4]  Jeffrey K. Uhlmann,et al.  Unscented filtering and nonlinear estimation , 2004, Proceedings of the IEEE.

[5]  N. Gordon,et al.  Novel approach to nonlinear/non-Gaussian Bayesian state estimation , 1993 .

[6]  Jeffrey K. Uhlmann,et al.  New extension of the Kalman filter to nonlinear systems , 1997, Defense, Security, and Sensing.

[7]  A. Jazwinski.  Stochastic Processes and Filtering Theory , 1970 .

[8]  Hugh F. Durrant-Whyte,et al.  A new method for the nonlinear transformation of means and covariances in filters and estimators , 2000, IEEE Trans. Autom. Control..

[9]  Rudolph van der Merwe,et al.  The Unscented Kalman Filter , 2002 .

[10]  Rudolph van der Merwe,et al.  The square-root unscented Kalman filter for state and parameter-estimation , 2001, 2001 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.01CH37221).

[11]  H.F. Durrant-Whyte,et al.  A new approach for filtering nonlinear systems , 1995, Proceedings of 1995 American Control Conference - ACC'95.

[12]  N E Manos,et al.  Stochastic Models , 1960, Encyclopedia of Social Network Analysis and Mining. 2nd Ed..

[13]  Kazufumi Ito,et al.  Gaussian filters for nonlinear filtering problems , 2000, IEEE Trans. Autom. Control..

[14]  Mohinder S. Grewal,et al.  Kalman Filtering: Theory and Practice Using MATLAB , 2001 .

[15]  S. Julier,et al.  A General Method for Approximating Nonlinear Transformations of Probability Distributions , 1996 .

[16]  H. Kushner.  Approximations to optimal nonlinear filters , 1967, IEEE Transactions on Automatic Control.

[17]  Rudolph van der Merwe,et al.  Efficient derivative-free Kalman filters for online learning , 2001, ESANN.

[18]  S. Haykin,et al.  Adaptive Filter Theory , 1986 .

[19]  Rudolph van der Merwe,et al.  Sigma-Point Kalman Filters for Integrated Navigation , 2004 .

[20]  G. Bierman.  Factorization methods for discrete sequential estimation , 1977 .

[21]  Rudolph van der Merwe,et al.  Dual Estimation and the Unscented Transformation , 1999, NIPS.

[22]  C. L. Thornton.  Triangular covariance factorizations for , 1976 .