## 5 Simulations and Comparison
### Vehicle Models and Test Conditions
#### 5.1.1 SIRENE Test Using Proposed Algorithm  
The vehicle data were taken from [59; 62]. Conditions of the investigation were as follows: the forces and torques values were limited here, based on the references, i.e., \(|\tau_{u}|\leq 500\) N, \(|\tau_{r}|\leq 500\) Nm, and the velocities \(u_{max}=2\) m/s, \(v_{max}=0.8\) m/s. The operating conditions of the vehicle are in accordance with the technical data provided in the cited literature. For the controller used, the inaccuracy of the model parameters was assumed as W = 0.2 (20%) and the following set of control parameters:  
\[k_{x}=20,\ \ k_{\psi}=2,\ \ \lambda_{1}=\lambda_{2}=0.1,\ \ \Gamma_{1}=\Gamma_{2}=0.1,\ \ \beta_{1}=\beta_{2}=0.1. \tag{69}\]  
The set of coefficients was selected to ensure that the tracking task could be carried out under the specified constraints. The results for the desired linear trajectory (62) and constant disturbances (64) are given in Figure 2. From Figure 2a-c it can be seen that the trajectory is tracked correctly. Likewise, the velocities \(u,r\) tend towards the desired ones (large values of \(u_{d}\) and \(r_{d}\) at the beginning of the movement are due to the conditions of the method and the vehicle dynamics) as it arises from Figure 2d,e. All velocities \(u,v,r\) have acceptable values (Figure 2f) and force \(\tau_{u}\) and torque \(\tau_{r}\) only have large values at the beginning, as results from Figure 2g. Very similar results were obtained for the same trajectory but using variable disturbances (65)-(67). They are shown in Figure 3a-g.  
Simulation results for the cycloid trajectory (63) and constant disturbances (64) are presented in Figure 4a-g. As can be seen, also in the case of a curvilinear trajectory, the tracking task is carried out correctly. Compared to the previous graphs, some fluctuations of the variables can be observed (position, velocity, control signals). Figure 2: Simulation results for SIRENE-IQV controller and linear trajectory (constant disturbances): (**a**) desired and realized trajectory; (**b**) position errors (E-frame); (**c**) position errors (B-frame) and angular error; (**d**) desired and realized velocity \((u_{d},u)\); (**e**) desired and realized velocity \((r_{d},r)\); (**f**) realized velocities; (**g**) applied force and torque. When the variable disturbances described by (65)-(67) were applied, one can found that the performance was very close to that obtained for the constant disturbances as can be observed from Figure 5a-g.  
For a more accurate comparison, the results obtained using criteria (68) are summarized in Table 2 (C. dist.--constant disturbances, V. dist.--variable disturbances). It was found that for the SIRENE vehicle, the mean position and velocity errors were smaller when a cycloid trajectory was used instead of the linear trajectory. The forces and the torque also had smaller values while the mean kinetic energy had a significantly larger value. The use of constant and variable disturbances (with the given parameters) resulted in only minor changes in the performance. \begin{table}
\begin{tabular}{c c c c c c} \hline  & & **Linear** & **Trajectory** & **Cycloid** & **Trajectory** \\ \hline
**Index** & & **C. Dist.** & **V. Dist.** & **C. Dist.** & **V. Dist.** \\ \hline MIA & \(x_{e}^{E}\) & 0.2807 & 0.2821 & 0.0752 & 0.0757 \\  & \(y_{e}^{E}\) & 0.0972 & 0.1047 & 0.0771 & 0.0756 \\  & \(y_{e}\) & 0.0287 & 0.0262 & 0.0155 & 0.0128 \\  & \(x_{e}\) & 0.2807 & 0.2820 & 0.0753 & 0.0759 \\  & \(y_{e}\) & 0.0981 & 0.1054 & 0.0769 & 0.0754 \\  & \(u_{e}\) & 5.3909 & 9.4245 & 1.3491 & 1.3603 \\  & \(r_{e}\) & 0.0360 & 0.0353 & 0.0361 & 0.0362 \\ MIAC & \(\tau_{u}\) & 22.817 & 21.654 & 7.9840 & 15.953 \\  & \(r_{e}\) & 6.2126 & 5.2778 & 4.5685 & 4.2121 \\ RMS & \(||e||\) & 1.0112 & 1.0176 & 0.4257 & 0.4297 \\ KE & \(K_{m}\) & 253.92 & 253.96 & 429.61 & 429.61 \\ \hline \end{tabular}
\end{table} 
Table 2: Performance for SIRENE.