## 3 Model of the Robot  
### Kinematics of the Robot  
The kinematic structure of the robot and the distribution of the velocity vectors are shown in Figure 1. It is possible to distinguish the following main components of the robot: 0--mobile platform; and 1--6--wheels. The following designations for the \(i\)th wheel have been introduced in the robot model: \(A_{i}\)--geometric center; \(r_{i}\) (m)--geometric (unloaded) radius; \(L_{f}\) (m), \(L_{b}\) (m)--distances of the front and back axes from the middle axle, respectively; \(W\) (m)--track width; and \(\theta_{i}\) (rad)--spin angle. It is assumed that the robot motion is performed in the \(Oxy\) plane of the fixed coordinate system \(\{O\}\). In turn, a moving co-ordinate system \(\{R\}\) is associated with the robot and its origin lies halfway between the centers of wheels 3 and 4. The robot's actual pose (or posture) in the \(\{O\}\) co-ordinate system, including its position and course, can be presented in the form of the vector of generalized co-ordinates:  
\[{}^{O}\boldsymbol{q}= \begin{bmatrix}{}^{O}x_{R},{}^{O}y_{R},{}^{O}\varphi_{ \Omega}\end{bmatrix} ^{T}, \tag{1}\]  
where \({}^{O}x_{R}\) (m), \({}^{O}y_{R}\) (m) are co-ordinates defining the position of point \(R\) of the robot, and \({}^{O}\varphi_{\Omega}\) (rad) is its course angle, that is, the rotation angle of the mobile platform about the \({}^{O}z\) axis. In turn, the vectors of generalized velocities of the robot, respectively, in the \(\{O\}\) and \(\{R\}\) co-ordinate systems can be written as:  
\[{}^{O}\dot{\boldsymbol{q}}= \begin{bmatrix}{}^{O}v_{Rx},{}^{O}v_{Ry},{}^{O} \omega_{0z}\end{bmatrix} ^{T},\quad\quad{}^{R}\dot{\boldsymbol{q}}= \begin{bmatrix} {}^{R}v_{Rx},{}^{R}v_{Ry},{}^{R}\omega_{0z}\end{bmatrix} ^{T}, \tag{2}\]  
Figure 1: Kinematic structure of the robot and distribution of the velocity vectors during turning. where \({}^{O}\upsilon_{Rx}={}^{O}\dot{x}_{R}\) (m/s), \({}^{O}\upsilon_{Ry}={}^{O}\dot{y}_{R}\) (m/s) are co-ordinates of the linear velocity vector of point \(R\) of the robot and \({}^{O}\omega_{0z}={}^{O}\dot{q}_{0z}\) (rad/s) is the co-ordinate of the angular velocity vector of the mobile platform, both in the fixed co-ordinate system \(\{O\}\), and \({}^{R}\upsilon_{Rx}\) (m/s), \({}^{R}\upsilon_{Ry}\) (m/s) and \({}^{R}\omega_{0z}\) (rad/s) are similar co-ordinates of velocity vectors, but described in the robot co-ordinate system \(\{R\}\). Those two vectors of generalized velocities satisfy the following relationship:  
\[{}^{O}\dot{q}={}^{O}_{R}\mathbf{I}^{R}\dot{q}, \tag{3}\]  
where matrix \({}^{Q}_{R}\mathbf{J}\) has the following form:  
\[{}^{O}_{R}\mathbf{J}= \begin{bmatrix}\cos({}^{O}\varphi_{0z})&-\sin({}^{O}\varphi_ {0z})&0\\ \sin({}^{O}\varphi_{0z})&\cos({}^{O}\varphi_{0z})&0\\ 0&0&1\end{bmatrix} . \tag{4}\]  
The subject of the research is a skid-steered robot in which the slip of the wheels is an inherent feature of its motion when turning and rotating in place. This means that the location of the instantaneous center of rotation of the robot's mobile platform results from dynamics, i.e., from the interaction of the wheels with the ground. An example location of the instantaneous center of rotation of the robot's mobile platform, i.e., point C, is illustrated in Figure 1. In turn, the projection of this point on the \(Rx\) axis is indicated with point \(B\). The velocity vectors of characteristic points \(A_{i}\) of the geometric centers of the robot's wheels can be determined from the distribution of velocity vectors, an example of which is shown in Figure 1. They can be calculated based on the knowledge of the linear velocity vector \({}^{R}\upsilon_{Rx}\), and the angular velocity vector \({}^{R}\omega_{0}\) of the mobile platform from the following equation:  
\[{}^{R}\upsilon_{Ai}={}^{R}\upsilon_{R}+{}^{R}\omega_{0}\times{}^{R}\mathbf{r}_{ Ai}, \tag{5}\]  
where \({}^{R}\mathbf{r}_{Ai}= \begin{bmatrix}{}^{R}\chi_{Ai}&{}^{R}\upsilon_{Ai},&{}^{R} \chi_{Ai}\end{bmatrix}