## 4 Motion Control System
### Wheels' Controller  
As mentioned earlier, various control system structures and techniques can be used for the trajectory tracking control of wheeled robots. However, this problem ultimately involves controlling the movement of the robot's wheels based on the desired motion parameters, which, in particular, may come from a higher layer of the control system. Let us assume that the wheels' controller is the only layer of the control system, and the desired robot movement is based on the robot's generalized velocity vector \(\mathbf{u}_{s}\). Then this vector is equal to the vector of desired generalized velocities \(\mathbf{v}_{d}\), that is:  
\[\mathbf{u}_{s}=\mathbf{v}_{d}= \begin{bmatrix}v_{Rd},\omega_{0 zd}\end{bmatrix} ^{ \mathsf{T}}. \tag{28}\]  
In order to control the motion of the wheels, the vector \(\mathbf{u}_{s}\) must be transformed into a vector of the desired angular velocities of the spin of wheels \(\mathbf{\omega}_{d}\). Therefore, in this case, the trajectory tracking control problem is to determine the control vector \(\mathbf{u}(t)\) such that \(\mathbf{\omega}(t)\rightarrow\mathbf{\omega}_{d}(t)\) for \(t\rightarrow\infty\). Assuming that the same desired angular velocity of spin \(\omega_{ld}\) is used for all left wheels of the robot, and the same \(\omega_{nd}\) for all right wheels, the following relationship can be used to determine these velocities:  
\[ \begin{bmatrix}\omega_{ld}\\ \omega_{rd}\end{bmatrix} =\frac{1}{r} \begin{bmatrix}1&-W/2\\ 1&W/2\end{bmatrix} \mathbf{u}_{s}. \tag{29}\]  
The relationship (29) results from the kinematics of the robot for the case of motion without slip of the wheels and the use of differential steering, and can be applied to wheeled robots with various kinematic structures. In the case of the analyzed six-wheeled robot with non-steered wheels, the desired angular velocities \(\omega_{id}\) (rad/s) for individual wheels are equal to the following:  
\[\omega_{id}= \begin{cases}\omega_{ld}&\text{for }i\in\mathbb{L},\\ \omega_{rd}&\text{for }i\in\mathbb{R},\end{cases}  \tag{30}\]  
where \(\mathbb{L}_{w}=\{1,\,3,\,5\}\), \(\mathbb{R}_{w}=\{2,\,4,\,6\}\) are the indices for the left and right wheels, respectively. In order to execute the trajectory tracking control of the robot, let us introduce the following error vector of the angular velocities of the spin of the wheels \(\mathbf{e}_{\omega}=\left[\mathbf{e}_{\omega 1},\dots,\mathbf{e}_{\omega n}\right]^{T}\) and the error vector of the rotation angles of these wheels \(\mathbf{e}_{\theta}=\left[\mathbf{e}_{\theta 1},\dots,\mathbf{e}_{\theta n}\right]^{T}\):  
\[\mathbf{e}_{\omega}=\mathbf{\omega}_{d}-\mathbf{\omega}=\dot{\mathbf{\theta}}_{d}-\dot{\mathbf{ \theta}},\quad\mathbf{e}_{\theta}=\mathbf{\theta}_{d}-\theta. \tag{31}\]  
Let us also assume that a linear controller is used for the motion control of the robot's wheels. Then, the control vector \(\mathbf{u}=\left[U_{1},\dots,U_{n}\right]^{T}\) can be determined based on the following relationship:  
\[\mathbf{u}=(k_{\omega}\mathbf{e}_{\omega}+k_{\theta}\mathbf{e}_{\theta})|\text{sgn}(\mathbf{ \omega}_{d})|, \tag{32}\]  
where \(k_{\omega}\) (Vs/rad) and \(k_{\theta}\) (V/rad) are the controller gains, and \(U_{i}\) (V) is the \(i\)th motor voltage input, \(i=1,\dots n\). The control vector \(\mathbf{u}\) is further limited using the saturation function so that individual control signals do not exceed the maximum motor voltage input \(U_{max}\) (V). The expression \(|\text{sgn}(\mathbf{\omega}_{d})|\) was introduced to force the robot to stop when the desired angular velocities \(\mathbf{\omega}_{d}\) are equal to zero and the trajectory tracking errors \(\mathbf{e}_{\omega}\) or \(\mathbf{e}_{\theta}\) are non-zero. The schematic diagram of the analyzed control system is illustrated in Figure 4.  
It should be noted that such a control system does not take into account the actual pose of the robot or its real velocities.