Use of Cholesky square roots amidst the UD-factorization implementation of Kalman filters in real-time airborne tracking systems

Abstract

In airborne Track-While-Scan systems, target parameters are updated via individual Kalman Filters. To insure numerical stability and accuracy, UD-Factorization techniques are used. A preset number of individual tracks is typically maintained in dense target environments. In real-time application the processing load is of concern. The UD-Factorization algorithm for measurement updates operates on each measurement individually when the error covariance matrix of the measurements is diagonal. In the inertial X-Y-Z TWS Kalman Filter for each track, this matrix is inherently non-diagonal and consequently needs to be operated upon. The proposed algorithm utilizes the lower triangular Cholesky square root technique to determine the normalized measurement vector and observation matrix, and yields an identity measurement error covariance matrix. To perform all the computations necessary requires considerable effort, and this paper delineates what is involved. The computationally cost-effective way to operate reduces to only a few subsidiary calculations above what would be necessary had the measurement error covariance matrix been diagonal to begin with. This algorithm is invoked prior to performing the Kalman Measurement Update equations.