To update the belief when the robot moves, we have to specify the action model . Based on the assumption of normally distributed errors in translation and rotation, we use a mixture of two independent, zero-centered Gaussian distributions whose tails are cut off [Burgard et al. 1996]. The variances of these distributions are proportional to the length of the measured motion.
Figure 3 illustrates the resulting densities for two example paths if the robot's belief starts with a Dirac distribution. Both distributions are three-dimensional (in -space) and Figure 3 shows their 2D projections into -space.