Critical remarks on the linearised and extended Kalman filters with geodetic navigation examples

Abstract

The Kalman filter has been used in a wide variety of engineering applications. There are two typical forms in implementing the nonlinear Kalman filter in conjunction with linearisation trajectory for which calculation of Jacobian matrices is involved: the linearised Kalman filter (LKF) and the extended Kalman filter (EKF). This paper aims to serve as a tutorial to the readers for providing better understanding and correctly implementing the two forms of nonlinear filters. Some critical remarks useful in designing a nonlinear Kalman filter are pointed out. Linearisation of the trajectory for the LKF and EKF is discussed. Performance degradation due to linearisation error is illustrated. Divergence problem for the LKF is presented. Implementation practice for the LKF and the EKF via total state estimate in conjunction with the error state estimate (in which the state variables are incremental quantities) are provided. Discussion on increase of dynamic process noise to the estimation precision in LKF and EKF is involved. Clear description of the implementation algorithms is provided. The step-by-step procedure for performing the filters is provided accompanied with two geodetic navigation examples. The materials prepared in this paper can be modified for further development in various applications.