Equation 40 could be used as the dynamics update in an extended Kalman filter (EKF), where is the prior state and is the dynamic model.
Note that Equation 42 is actually the EKF measurement update for the covariance and Equation 45 is the measurement update for the mean, assuming a trivial measurement Jacobian (identity matrix). The tangent vector is the innovation.
In this case of trivial measurement Jacobian, the Kalman gain is defined
(46) |
so that the Kalman update can be written in its standard form:
Labelling the above in the standard EKF framework, the state covariance is given by and the measurement noise is given by . Note that Eq. 48 is mathematically identical to Eq. 45, and Eq. 49 is identical to Eq. 42.
The case of non-trivial measurement or dynamics Jacobians is a simple modification of the equations given here.
Ethan Eade 2012-02-16