## 5 Simulation Examples
### Simulation Results and Discussion  
The simulation results of the PFA with RRR in a seaway are shown in Figures 8-10, which indicate that the proposed scheme can achieve good performances of path following, course changing, course keeping and roll reduction. Figure 8 shows the test routes in X-\(Y\) plane. It illustrates that the PFA with RRR control is well established despite the existence of model perturbation and sea state disturbances. Figure 9 indicates the contrast experimental results of LOS-CT and LOS between \(WP_{1}\) and \(WP_{2}\). Figure 10 demonstrates the position errors, course angles, roll angles and the rudder efforts. For the motions before and after the RRR is activated, the comparative analysis is carried out according to  
Figure 7: The wave graphs of sea state 5.  
the two stages of the track maintenance and the turning near the waypoints, as shown in Table 1. During the path following stage, the LOS-CT scheme and course controller performs well with an energy-efficient steering gear input. As shown in Figure 9, the LOS-CT guidance reduces the rudder movements 48.7% under the guarantee of path following accuracy. When the RRR is activated, the roll amplitude is reduced by 48.5% with about 10 times rudder movement, and both the track error and the course error have increased by nearly fivefold, but they are all within the rational range of navigation practice. During the ship's turning near the waypoints, the system responses quickly with a reasonable steering mode, the track and course show certain overshoots. Although the roll reduction is negligible because of the outward heeling in turning, the activation of RRR attenuates the overshoots of track and course greatly by 47.6% and 94.9%, respectively. Furthermore, it can be seen from the variation chart of ship speed that, when the ship turns, there is obvious speed drop, and when the RRR is turned on, the turning speed drop is reduced, while the speed drop of the path following stage appears.  
As shown in Table 2, the control performances of PFA-RRR in different sea states prove the robustness and effectiveness. When the wave-induced disturbances get bigger, the accuracy of track tracking is worse, the rudder roll reduction is better in less than 30\({}^{\circ}\), and the rudder movements increases gradually. It is noted that the reduction rate is significant but decreases when the maximum roll angle reaches 35 degrees.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Autopilot**} & \multicolumn{4}{c}{**Path Following**} & \multicolumn{4}{c}{**Waypoint Turning**} \\ \cline{2-9}  & _d (m)_ & \(\epsilon_{c}\) (\({}^{\circ}\)) & \(\phi\) (\({}^{\circ}\)) & \(\delta\) (\({}^{\circ}\)) & _d (m)_ & \(\psi\) (\({}^{\circ}\)) & \(\phi\) (\({}^{\circ}\)) & \(\delta\) (\({}^{\circ}\)) \\ \hline RRR-OFF & 0.47 & 0.22 & 5.69 & 0.65 & 86.32 & 3.13 & 4.64 & 14.03 \\ RRR-ON & 2.12 \(\uparrow\) & 0.90\(\uparrow\) & 2.93\(\downarrow\) & 7.61\(\uparrow\) & 45.24\(\downarrow\) & 0.16\(\downarrow\) & 4.61\(\downarrow\) & 13.81\(\downarrow\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: The standard deviation (STD) values of the control performances with the RRR on/off.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{Sea State} & \(\phi_{\text{max}}\) (\({}^{\circ}\)) & 10.0 & 15.0 & 20.0 & 25.0 & 30.0 & 35.0 \\  & \(\phi_{\text{std}}\) (\({}^{\circ}\)) & 3.9 & 5.2 & 6.8 & 7.7 & 9.8 & 10.3 \\  & 1.4 & 1.9 & 2.6 & 2.9 & 3.3 & 5.9 \\  & 40.9 & 44.7 & 49.2 & 50.5 & 57.0 & 43.8 \\  & 6.0 & 7.3 & 8.5 & 9.3 & 10.3 & 13.8 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Control performances of PFA-RRR in different sea states.  
Figure 8: The path following results in _X-Y_ plane.