Prediction model

Source: SLAM for Dummies

Used in the prediction step .

How to compute an expected position of the robot given the old position and the control input (so basically based on odometry .

Control terms are $\Delta x, \Delta y, \Delta \theta$

$$ \begin{align} f &= \left[ \begin{array}{c} x + \Delta t \cos \theta + q \Delta t \cos \theta \
y + \Delta t \sin \theta + q \Delta t \sin \theta \
\theta + \Delta \theta + q \Delta\theta \end{array} \right]\
&= \left[ \begin{array}{c} x + \Delta x + q \Delta x \
y + \Delta y + q \Delta y \
\theta + \Delta \theta + q \Delta\theta \end{array} \right] \end{align} $$

Jacobian (assuming linearised version)

$$ \left[ \begin{array}{ccc} 1 & 0 & -\Delta t \sin\theta\
0 & 1 & \Delta\cos\theta\
0 & 0 & 1 \end{array} \right] $$

Not extended for landmarks because only used for robot position prediction