## 5 Simulation Studies
### Results of Simulation Studies  
As part of the simulation studies, various sets of desired motion parameters were analyzed, taking into account motion with different maximum values of velocities and accelerations, as well as turning with different radii. In the first stage, simulation studies were carried out for the nominal geometric parameters of the robot, including the track width \(W\) (m). The characteristic motion parameters of the robot for the selected cases presented in this article are summarized in Table 1. In turn, Figure 7 shows the exemplary time histories of the desired motion parameters corresponding to extreme cases, i.e., 1 and 3. They are desired linear velocity and acceleration of the robot's point \(R\) (Figures 6(a) and 6(b), respectively) as well as desired angular velocity and acceleration of the robot's mobile platform (Figures 6(c) and 6(d), respectively). In the simulation studies, zero initial conditions were assumed in terms of the robot's pose and velocities. Moreover, in addition to the cases of desired motion, the research considered two cases regarding the structure of the control system, including the following:  
1. The control system with the wheels' controller only;
2. The control system with the wheels' controller and pose controller. Figure 8 shows the results obtained from the simulation of the robot's motion using the wheels' controller for the desired motion cases 1 and 3. Based on the analysis of the motion paths of the robot's \(R\) point (Figure 7(a)), it can be seen that, after making a turn, the deviation of the actual motion path from the desired one increases, which is related to the occurrence of wheel slip during turning. This effect is \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Case No.** & \(\begin{array}{c}v_{R\alpha}\\ \text{(m/s)}\end{array}\) & \(\begin{array}{c}a_{Rmax}\\ \text{(m/s^{2})}\end{array}\) & \(\begin{array}{c}L_{r}\\ \text{(m)}\end{array}\) & \(\begin{array}{c}L_{h}\\ \text{(m)}\end{array}\) & \(\begin{array}{c}I\\ \text{(m)}\end{array}\) & \(\begin{array}{c}R_{z}\\ \text{(m)}\end{array}\) & \(\begin{array}{c}\omega_{02\pi}\\ \text{(m)}\end{array}\) & \(\begin{array}{c}\epsilon_{02\pi max}\\ \text{(rad)}\end{array}\) \\ \hline
1 & 0.3 & 0.675 & 0.5 & 0.75 & 0.1 & \(-\)0.6 & \(-\)0.5 & 2.60 & \(2/3\,\pi\) \\
2 & 0.6 & 2.700 & 0.5 & 0.75 & 0.1 & \(-\)0.6 & \(-\)1.0 & 10.38 & \(2/3\,\pi\) \\
3 & 0.9 & 6.075 & 0.5 & 0.75 & 0.1 & \(-\)0.6 & \(-\)1.5 & 23.37 & \(2/3\,\pi\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Characteristic motion parameters of the robot for the selected cases. more visible when the robot moves at a higher velocity, but it is expected for each case that this error increases as the robot moves. In addition to wheel slip, other factors influencing the tracking errors are the wheels' controller itself and the dynamic properties of the drive units, which are reflected in the time histories of the wheel spin angular velocities (Figure 8b). The largest errors in the angular velocities of wheel spin (Figure 8c) occur at the moments of the highest linear and angular accelerations, i.e., during acceleration and braking. Additionally, in the case of higher velocities, the process of reducing tracking errors takes much longer. It should be noted that a simple linear controller is used as a wheels' controller, which is not able to compensate for the nonlinearities occurring in the robot dynamics model. This functionality can be achieved by using a controller based on the robot dynamics model, e.g., an adaptive or robust controller. Moreover, due to the fact that the actual robot pose is not transmitted to the wheels' controller, it is unable to minimize the pose error, which is confirmed by the time histories of the pose errors (Figure 8d). It can be seen that the highest values of driving torques occur when accelerating the robot and while turning (Figure 8e).