## 5 Tests and Performance Comparison
### Tests and Comparison of the nomRL-PFC and the \(\mu\)-randRL-PFC  
To evaluate and compare the performance of nomRL-PFC and \(\mu\)-randRL-PFC for varying friction values, the agents both control the ROboMObil over the training path from start to finish multiple times, where each time a different tire-road friction coefficient \(\mu\) is chosen. The performance of the agents is analyzed for friction values from the interval on which the \(\mu\)-randRL-PFC was trained; see Equation (14). The returns of both the nomRL-PFC and the \(\mu\)-randRL-PFC are shown in Figure 11 for several different friction values. The blue line represents the return of the nomRL-PFC, whereas the orange line depicts the return of the \(\mu\)-randRL-PFC. It can be seen that the nomRLPFC is able to obtain a higher return for friction values close to \(\mu=1.0\), which is the friction value for which it was trained. However, for \(\mu\leq 0.925\), the nominal agent is outperformed by the \(\mu\)-randRL-PFC. Furthermore, the return of the nominal agent decreases significantly for smaller friction values, whereas the return of the agent trained with friction randomization is able to keep the return at a high level for all considered values of \(\mu\), showing robust behavior on varying road conditions. Table 3 summarizes the RMS errors of the nomRL-PFC and the \(\mu\)-randRL-PFC on the path for the friction values \(\mu=0.6\) and \(\mu=1.0\). It can be stated that both agents achieve a similar RMS for the lateral position error \(e_{y}^{\mathrm{p}}\) on a dry road surface, i.e., \(\mu=1.0\), with the nomRL-PFC providing lower errors for both velocity and orientation tracking. In the case of a snowy road with \(\mu=0.6\), however, the path-following performance of the nomRL-PFC declines, which increases its RMS for \(e_{y}^{\mathrm{p}}\) throughout the path. This is illustrated in Figure 12, which shows the road section around the arc length \(s=493\) m for \(\mu=0.6\), with the blue line depicting the pathway of the nomRL-PFC and the orange one illustrating the pathway of the \(\mu\)-randRL-PFC. It can be observed that both agents are able to successfully follow the reference path, while the \(\mu\)-randRL-PFC is able to achieve  
Figure 11: Return of the nomRL-PFC (blue line) and \(\mu\)-randRL-PFC (orange line) for the friction values considered during the training of the agent with dynamics randomization. smaller lateral position errors to the reference path. This is the main reason why the overall return of the nomRL-PFC also decreases in Figure 11 for small friction values. The lateral position error \(e_{y}^{\text{P}}\) determines the value of the Gaussian-like function \(g_{y}\left(e_{y}^{\text{P}}\right)\) in Equation (5), which is multiplied with the remainder of the reward function as part of the hierarchical design of the reward function. With increasing lateral position errors \(e_{y}^{\text{P}}\), the value of the function \(g_{y}\left(e_{y}^{\text{P}}\right)\) decreases. Consequently, this leads to a smaller overall return during the execution on the path. The \(\mu\)-randRL-PFC, on the other hand, is able to obtain a similar \(e_{y}^{\text{P}}\)-RMS for both road conditions (cf. Table 3), which further underlines its robustness against different friction values. The performance comparison of the nomRL-PFC and the \(\mu\)-randRL-PFC at rather challenging road sections further demonstrates the robustness of the latter agent. The tight road turns between the arc lengths \(s=300\) m and \(s=600\) m of the path shown in Figure 6 represent such sections. The lateral position errors \(e_{y}^{\text{P}}\) induced by executing both agents in this particular part of the path are shown in Figure 13 for the friction value \(\mu=1.0\), with the blue line representing the nomRL-PFC and the orange line representing the \(\mu\)-randRL-PFC.