Modeling, Estimation and Optimal Filtration in Signal Processing

Appendix K

The Unscented Kalman Filter (UKF)

The unscented Kalman filter (UKF) is based on the “unscented transformation” (UT). First proposed by Julier et al. [1] the UT allows for the estimation of the mean and the covariance of an arbitrary analytical transformation y = f() of a random Gaussian vector with a mean value and a covariance matrix .

If L denotes the size of the vector , the method put forth by Julier et al. runs in three steps:

1) 2L+1 particles or σ-points [1] are generated as follows:

where (M)_i is the i^ith row or column of matrix M and λ = α²(L + κ)– L is a scaling factor. Element α is a parameter which allows us to control the dispersion of the σ-points around the mean . κ is a secondary scaling factor.

2) The σ-points are transformed using function f:

3) The mean and ...