## 3 Model of the Robot
### Dynamics of the Robot
It is assumed that the tangential co-ordinates of the ground reaction forces, that is, \({}^{R}F_{Aix}\) (N) and \({}^{R}F_{Aiy}\) (N), are proportional to the normal co-ordinates of the ground reaction forces \({}^{R}F_{Aix}\), and the actual values of the tire adhesion coefficient in the longitudinal and lateral directions, respectively. Thus, the following dependencies are satisfied:  
\[{}^{R}F_{Aix}=\mu_{ix}{}^{R}F_{Aiz},\quad{}^{R}F_{Aiy}=\mu_{iy}{}^{R}F_{Aiz}. \tag{16}\]  
Strictly speaking, in the case of combined slip conditions, including both longitudinal and lateral slip, the value of the resultant tangential component of the ground reaction force should lie inside the friction ellipse. This would require a reduction in the value of lateral force \({}^{R}F_{Aiy}\). However, in the developed robot model, for simplicity, this aspect is omitted because it does not significantly affect the robot's movement. Figure 3 shows the \({}^{R}F_{Aix}(\lambda_{i})\) characteristic resulting from the adopted modified version of the Kiencke tire model. As a result of the forces acting on the robot, it moves with linear acceleration \({}^{R}a_{CM}\), whose co-ordinates result from the following formulae:  
\[{}^{R}a_{CMx}=\sum_{i=1}^{n}{}^{R}F_{Aix}/m_{R},\quad{}^{R}a_{CMy}=\sum_{i=1}^ {n}{}^{R}F_{Aiy}/m_{R},\quad{}^{R}a_{CMz}=0. \tag{17}\]  
If the robot turns or rotates in place and all its wheels are driven, then the longitudinal forces \({}^{R}F_{Aix}\) (N) cause movement, while the lateral forces \({}^{R}F_{Aiy}\) (N) counteract it. It is,  
Figure 3: Illustration of longitudinal force vs. longitudinal slip ratio dependency. therefore, intuitive that, in such a case, the wider the wheel track is in relation to the axle distance, the more advantageous it will be. It can also be assumed that the values of the so-called self-aligning torques \({}^{R}T_{Aiz}\) (Nm) are negligibly small in comparison to the values of the remaining moments of force acting about the \({}^{R}z\) axis; therefore, they can be omitted in the robot's dynamics model. Therefore, the \({}^{R}\varepsilon_{0z}\) (rad/s\({}^{2}\)) co-ordinate of the robot's angular acceleration vector results from the equation:  
\[{}^{R}\varepsilon_{0z}=(\sum_{r\in\mathbb{R}}{}^{R}F_{Arx}w_{r}-\sum_{l\in \mathbb{L}}{}^{R}F_{Alx}w_{l}+\sum_{f\in\mathbb{F}}{}^{R}F_{Afy}l_{f}-\sum_{b \in\mathbb{B}}{}^{R}F_{Aby}l_{b})/I_{Rz}, \tag{18}\]  
where the sets \(\mathbb{L}_{w}=\{1,\ 3,\ 5\}\), \(\mathbb{R}_{w}=\{2,\ 4,\ 6\}\), \(\mathbb{F}_{w}=\{1,\ 2\}\), and \(\mathbb{B}_{w}=\{5,\ 6\}\) define the indices for the left, right, front, and back wheels, respectively. The individual distances between the centers of these wheels and the center of mass of the robot, measured along the \({}^{R}x\) and \({}^{R}y\) axes, are, respectively, equal to \(l_{f}=\left(L_{f}-{}^{R}x_{CM}\right)\), \(l_{b}=\left(L_{b}+{}^{R}x_{CM}\right)\), \(w_{r}=\left(W/2+{}^{R}y_{CM}\right)\), and \(w_{l}=\left(W/2-{}^{R}y_{CM}\right)\). Moreover, \(I_{Rz}\) (kg m\({}^{2}\)) is the mass moment of inertia of the robot about an axis parallel to \({}^{R}z\) and passing through its center of mass. Knowing the forces acting on the robot's wheels, one can determine the angular acceleration of their spin based on the following equation:  
\[\varepsilon_{i}=\left(\tau_{i}-{}^{R}F_{Aix}r_{i}-{}^{R}F_{Aix}r_{i}f_{r}{\rm sgn }(\omega_{i})\right)/I_{Wy}, \tag{19}\]  
where \(\tau_{i}\) (Nm) is the value of the driving torque, the expression \(-{}^{R}F_{Aix}rf_{r}{\rm sgn}(\omega_{i})\) reflects the value of the rolling resistance torque acting on the \(i\)th wheel, and \(f_{r}\) (-) is the rolling resistance coefficient. Equation (19) omits the moments of motion resistance occurring in kinematic pairs, in particular, those related to the phenomenon of friction in the bearings. This is due to the fact that they are negligible in relation to the remaining moments of force acting on wheels. They can only have a noticeable effect at a very low velocity of robot movement, when they can be comparable to driving torques.