Code analysis in the gmapping of velocity motion model of probabilistic robot

The configuration of a rigid mobile robot is usually described in 6 variables: his three-dimensional Cartesian coordinate system, and the relative external coordinate system of three Euler angles (RPY roll, pitch, yaw), so in the plane environment generally with three variables can be described, called posture.
So in general, the posture information of a robot is a two-dimensional planar coordinate (x, y) of a robot and its azimuth Θ\theta, which is represented by this vector:
⎛⎝⎜xyθ⎞⎠⎟\begin{pmatrix} x \ y \ \theta \end{pmatrix}
A posture without a direction is called a position (location). Then we know that the conditional density in probability kinematics is: P (xt|ut,xt−1) p (x_t|u_t,x_{t-1})
We all know that the XT x_t and xt−1 x_{t-1} are all robot postures, UT u_t is motion control, so this model describes the posterior distribution of the kinematic state obtained by the robot cable after the motion control UT xt−1 for X_{t-1 u_t}.

Velocity Motion Model is controlled by two speed: translational speed and rotational speed.
So control ut= (VTWT) u_t = \begin{pmatrix} v_t \ w_t \ \end{pmatrix}
The counter-clockwise rotation angular velocity is positive, the forward movement line velocity is positive. The following gives a direct
motion_model_velocity Pseudo-code

The second line is the most difficult to understand, although I am self-understanding but do not know right or wrong, so still do not write, the specific meaning of the pseudo-code: initial Posture xt−1= (x yθ) T x_{t-1}= (x\ y\ \theta) ^t control ut= (v W) t u_t= (v \ W) ^t and imaginary successor Posture Xt= (x′y′θ′) T