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) equation

Now (7.34) can be rewritten as

(7.68) equation

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

(7.69) equation

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

Similarly, the observation covariance prediction becomes

(7.70) equation

where

(7.71) equation

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

(7.72)

where ...

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