The big idea: linearise, then filter

A Kalman filter needs \(F\) and \(H\) to be matrices. But a rover’s motion is

The \(\cos\theta\) and \(\sin\theta\) make it nonlinear. A sensor that measures the distance to a landmark, \(r = \sqrt{(x_L-x)^2 + (y_L-y)^2}\), is nonlinear too.

The EKF’s trick is simple: zoom in. Any smooth curve looks like a straight line if you zoom in far enough. At each step the EKF:

1. pushes the estimate through the true nonlinear model \(f\) to get the prediction, and
2. uses the slope of \(f\) (its Jacobian) to push the covariance.

The same goes for the measurement: predicted measurement from the nonlinear \(h\), covariance from its Jacobian \(H\).

When the uncertainty is small compared with how curved the model is, this works very well. When the uncertainty is large (for example a robot with no idea of its heading), the straight-line approximation fails and the filter can diverge. That limitation motivates the UKF and particle filter later in this track.

Think of the EKF as “a Kalman filter that re-draws its straight lines every step”.