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Bayesian Estimation and Tracking: A Practical Guide by Anton J. Haug

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8.1 One-Dimensional Finite Difference Kalman Filter

8.1.1 One-Dimensional Finite Difference State Prediction

The scalar derivation for the EKF state prediction equation in terms of img was given in (7.10)), repeated here for clarity

(8.1) equation

As noted previously, this is accurate to third order in c.

Replacing the second-order derivative with its finite difference approximation from (2.47), (8.1) becomes

(8.2) equation

Gathering terms, this can be rewritten as

(8.3) equation

Let

(8.4) equation

so that

(8.5) equation

and define the finite difference evaluation points as

(8.6) equation

where

(8.7) equation

See Section 2.2 for an explanation of the notation . Using (8.4) and (8.6), (8.3) becomes

(8.8)

Replacing the continuous c defined in (7.3) with ...

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