Bayesian Estimation and Tracking: A Practical Guide

7.3 An Alternate Derivation of the Multidimensional Covariance Prediction Equations

For future reference, an alternate set of EKF state covariance prediction equations will be derived here. In the next chapter, these will be turned into finite difference solutions that are used to show the relationship between the finite difference method and the unscented Kalman filter. For numerical integration methods other than the finite difference methods, this is the preferred method for evaluating covariance integrals.

Define the matrix

(7.67)

Now (7.34) can be rewritten as

(7.68)

This has the exact same form as the state prediction equation (5.51), resulting in

(7.69)

where has been used to distinguish this form from (7.41).

Similarly, the observation covariance prediction becomes

(7.70)

where

(7.71)

The cross-covariance matrix prediction equation (5.41) can be transformed into

(7.72)

where ...