## 4 Motion Control System  
### Desired Motion of the Robot  
The desired motion of the robot can be described by the vector of the robot's desired pose, which has the form of the vector of generalized co-ordinates:  
\[{}^{O}\boldsymbol{q}_{d}=\begin{bmatrix}{}^{O}x_{Rd},\ {}^{O}y_{Rd},\ {}^{O} \varphi_{0 zd}\end{bmatrix}^{T}, \tag{24}\]  
where \({}^{O}x_{Rd}\) (m), \({}^{O}y_{Rd}\) (m) are the desired co-ordinates of the position of point \(R\) of the robot, and \({}^{O}\varphi_{0 zd}\) (rad) is a desired course of the robot's mobile platform with respect to the \({}^{O}z\) axis.  
It should be noted that the analyzed robot is a non-holonomic system and is characterized by the fact that it cannot move in the lateral direction, except in the case of movement under the condition of the slip of the wheels. Therefore, it is assumed that \({}^{R}v_{Ryd}=0\).  
Thus, the desired motion of the trajectory of the robot is represented in the form of the vector of desired generalized velocities:  
\[\boldsymbol{v}_{d}=\begin{bmatrix}{}^{R}v_{Rd},{}^{R}\omega_{0 zd}\end{bmatrix} ^{T}=\begin{bmatrix}v_{Rd},\omega_{0 zd}\end{bmatrix}^{T}, \tag{25}\]  
where \({}^{R}v_{Rxd}=v_{Rd}\) (m/s) is a value of the desired linear velocity of the characteristic point \(R\) of the robot and \({}^{R}\omega_{0 zd}\) (rad/s) is a value of the desired angular velocity of its mobile platform, both in the robot co-ordinate system \(\{R\}\).  
The relationship between the generalized desired velocities of the robot in the fixed co-ordinate system \(\{O\}\) and the one associated with the robot \(\{R\}\) can be presented in the following form:  
\[{}^{O}\dot{\mathbf{q}}_{d}=\begin{bmatrix}{}^{O}v_{Rxd}\\ {}^{O}v_{Ryd}\\ {}^{O}\omega_{0 zd}\end{bmatrix}=\begin{bmatrix}{}^{O}\dot{x}_{Rd}\\ {}^{O}\dot{y}_{Rd}\\ {}^{O}\dot{y}_{0 zd}\end{bmatrix}=\begin{bmatrix}{}^{O}\cos({}^{O}\varphi_{0 zd} )&0\\ {}^{O}\dot{y}_{0 zd}\end{bmatrix}\begin{bmatrix}{}^{R}v_{Rd}\\ {}^{R}\omega_{0 zd}\end{bmatrix}. \tag{26}\]  
It can be noticed that, if the robot moves on horizontal ground, then, even though the co-ordinates of the velocity vectors \({}^{R}v_{Rd}\) and \({}^{O}v_{Rd}\) have different values, their modules are equal, that is, \({}^{R}v_{Rd}={}^{O}v_{Rd}=v_{Rd}\) (m/s). From the fact that the axes of both reference systems are parallel to each other, it also follows that \({}^{R}\omega_{0 zd}={}^{O}\omega_{0 zd}=\omega_{0 zd}\) (rad/s). The same applies to accelerations, i.e., \({}^{R}a_{Rd}={}^{O}a_{Rd}\) (m/s2) and \({}^{R}e_{0 zd}={}^{O}e_{0 zd}=e_{0 zd}\) (rad/s2). Therefore, in the analyzed case, the desired values of the robot's generalized velocities and accelerations do not depend on whether they are expressed in the fixed co-ordinate system \(\{O\}\) or in the moving co-ordinate system associated with the robot \(\{R\}\).  
In the case of the plane motion of the robot's mobile platform, the desired velocity of the point \(R\), that is, \({}^{R}v_{Rxd}=v_{Rd}\) (m/s), depends on the desired angular velocity \({}^{R}\omega_{0 zd}=\omega_{0 zd}\) (rad/s) and the radius of curvature of the path \(R_{z}\) (m) according to the following formula:  
\[v_{Rd}=\omega_{0 zd}R_{z}. \tag{27}\]