## 5 Tests and Performance Comparison
### Tests and Comparison of the nomRL-PFC and the \(J\)-randRL-PFC  
To examine the robustness of the nomRL-PFC and the \(J\)-randRL-PFC against variations in the yaw inertia, both agents are executed on the training path several times, where each time a different value for \(J_{z,\text{md}}^{\text{C}}\) is chosen according to the inequalities (13). The returns of both agents are shown in Figure 10, which are evaluated at \(80\%,85\%,\dots,\ 120\%\) of the nominal inertia value \(J_{z,\text{nom}}^{\text{C}}\) of the ROboMObil. The blue line represents the returns of the nomRL-PFC, whereas the orange line depicts the returns of the \(J\)-randRL-PFC.  
In Figure 10, it can be observed that the returns of the nomRL-PFC and the \(J\)-randRL-PFC both stay at a relatively constant level for all considered inertia values. The reason for this can be explained with the help of Table 2. It shows the RMS errors of the nomRL-PFC and the \(J\)-randRL-PFC for the inertia values \(80\%\cdot J_{z,\text{nom}}^{\text{C}}\), \(J_{z,\text{nom}}^{\text{C}}\) and \(120\%\cdot J_{z,\text{nom}}^{\text{C}}\). The nomRL-PFC and the \(J\)-randRL-PFC each provide constant RMS errors for all three considered inertia values. Therefore, the returns of the agents do not vary notably. A possible explanation for this might be that the inertia does not have an overall major influence on the dynamics of the system for the considered motion control task, which is why the agents are able to perform equally well for all considered inertia values. Furthermore, both agents achieve similar RMS values for the position and orientation errors, with the nomRL-PFC providing slightly higher ones for both errors. For these errors, both agents receive similar overall rewards. Nevertheless, the nomRL-PFC is able to achieve higher overall returns in Figure 10 mainly due to the smaller RMS for the velocity error, which is rewarded higher due to the choice of \(\theta_{v,2}\) in Equation (15).  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{1}{c}{**Agent**} & \multicolumn{2}{c}{**nomRL-PFC**} & \multicolumn{3}{c}{**J-randRL-PFC**} \\ \hline \multicolumn{1}{c}{**\% of Inertia \(J_{z,\text{nom}}^{\text{C}}\)**} & **80\%** & **100\%** & **120\%** & **80\%** & **100\%** & **120\%** \\ \hline \(c_{y}^{\text{r}}\) [m] (RMS) & 0.013 & 0.013 & 0.013 & 0.010 & 0.010 & 0.010 \\ \(c_{y_{y}}^{\text{r}}\) [m/s] (RMS) & 0.106 & 0.106 & 0.106 & 0.144 & 0.144 & 0.144 \\ \(c_{y}^{\text{r}}\) [rad] (RMS) & 0.020 & 0.020 & 0.020 & 0.013 & 0.014 & 0.014 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The root mean square (RMS) errors of the nomRL-PFC and the \(J\)-randRL-PFC after executing the agents on the path for different inertia values. The best metric for each inertia value is marked green.  
Figure 10: Return of the nomRL-PFC (blue line) and the \(J\)-randRL-PFC (orange line) for the inertia values considered during the training of the agent with dynamics randomization.  
With these observations, it can be stated that alternating the values of the yaw inertia does not affect the control performances of the agents. The RMS errors of both agents remain at a constant level, which indicates their robustness against different inertia values. More specifically, this shows that the nominal agent can still perform well even under uncertain inertia values and that the randomization of the yaw inertia during training does not provide any advantages.