## 5 Tests and Performance Comparison
### Tests and Comparison of the nomRL-PFC and the \(m\)-randRL-PFC  
To evaluate the robustness and compare the performance of the nomRL-PFC and the \(m\)-randRL-PFC, both agents are executed several times on the path in Figure 6, where each time a different external mass \(m_{\text{ext}}\) is chosen within the interval \(m_{\text{ext}}\in[0\text{ kg},\ 300\text{ kg}]\), i.e., the interval on which the agent with dynamics randomization was trained. The returns both agents obtained during these executions are shown in Figure 8. Here, the dark blue dot represents the return of the nomRL-PFC for the external mass \(m_{\text{ext}}=0\text{ kg}\). The light-blue dashed line with the crosses represents the nominal agent's returns for executions on the path that were aborted due to early termination (cf. Inequalities (17)). The orange line shows the return of the \(m\)-randRL-PFC for all external masses \(m_{\text{ext}}\). In Figure 8, it can be seen that the nomRL-PFC achieves a 9% higher return than the \(m\)-randRL-PFC for \(m_{\text{ext}}=0\text{ kg}\), which is the value for which the nomRL-PFC was trained. However, the \(m\)-randRL-PFC outperforms the nominal agent for cases in which \(m_{\text{ext}}\geq 25\text{ kg}\). In all cases with an additional mass, the execution of the nominal agent on the path is aborted early because it triggers one or more of the safety-critical termination conditions introduced in Equation (17). In the case of \(m_{\text{ext}}=300\text{ kg}\), for example, the execution of the nomRL-PFC is aborted because the position error exceeds the respective pre-defined threshold, i.e., \(\left|\epsilon_{y}^{p}\right|>2\text{ m}\) (cf. Inequalities (17)). This is illustrated in Figure 9. Here, the pathway of the nominal agent is depicted in blue, while the pathway of the \(m\)-randRL-PFC is depicted in orange. The reference path is presented by the dashed black line. The solid black lines represent the path boundaries. It can be observed that the nomRL-PFC starts to slightly deviate from the reference path until the agent eventually leaves the road. However, the \(m\)-randRL-PFC continues to successfully follow the path closely. Furthermore, for the complete interval of the values considered for \(m_{\text{ext}}\), the returns of the \(m\)-randRL-PFC remain at a relatively high level compared with the nomRL-PFC's  
Figure 8: Return of the normRL-PFC (blue line) and \(m\)-randRL-PFC (orange line) after executions on the path for different external mass values \(m_{\text{ext}}\). returns, which decrease continually. This underlines the robustness of the \(m\)-randRL-PFC against varying mass values. Table 1 shows the root mean square error (RMS) of the lateral position, velocity, and orientation errors \(e_{y}^{\mathrm{P}}\), \(e_{v_{z}}^{\mathrm{P}}\), and \(e_{\mathrm{p}}^{\mathrm{P}}\) of the nomRL-PFC and the \(m\)-randRL-PFC during their execution along the entire path with \(\Delta k=0.05\) s. More specifically, we consider the cases \(m_{\mathrm{ext}}=0\) kg and \(m_{\mathrm{ext}}=300\) kg, which represent both ends of the randomization interval. The errors of the nomRL-PFC are not provided for \(m_{\mathrm{ext}}=300\) kg since the agent on the path failed to completely execute due to the early termination criteria described above. For \(m_{\mathrm{ext}}=0\) kg, the lateral position error \(e_{y}^{\mathrm{P}}\)-RMS of the nomRL-PFC is higher than that of the \(m\)-randRL-PFC. However, the nominal agent achieves a smaller RMS for both the velocity error \(e_{v_{z}}^{\mathrm{P}}\) and the orientation error \(e_{\mathrm{p}}^{\mathrm{P}}\). In summary, the nomRL-PFC is able to achieve a higher overall return for \(m_{\mathrm{ext}}=0\) kg, as shown in Figure 8, resulting from the lower RMS of the velocity and orientation errors throughout the entire execution on the path. For \(m_{\mathrm{ext}}=300\) kg, the \(m\)-randRL-PFC's returns decrease slightly compared with the case with \(m_{\mathrm{ext}}=0\) kg because the RMS of all errors increases.