## 5 Results and Discussion
### Autopilot Simulation  
In order to test turning and diving abilities of the SFRM with the PD controller, a 3D map with numerous waypoints was established. The coordinates of waypoints in the 3D map are described in Table 7. An autopilot system consists of a 6-DOF manoeuvring mathematical model and a course-keeping PD controller for an LOS-based guidance system. Simulation results of tuning PD control on the rudder planes and the stern planes would be presented by comparing with two types of controlling modes, i.e., A-type and B-type. It is worth mentioning that the PD control gains of the A-type controller are tuned by the horizontal zigzag test and the vertical zigzag test, whereas the ones of the B-type controller are tuned by the horizontal zigzag test and the meander test, respectively.  
Figure 25a-c shows the trajectories of the SFRM with three-view angles based on \(U=1.5\) m/s in the 3D map by comparing the A-type with the B-type controllers. Comprehensively, the total sailing distance of the A-type controller is 382 m with the sailing time of 297 s, whereas the total sailing distance of the B-type controller is 387 m with the sailing time of 301 s. Compared with the B-type controller, the A-type controller is capable of saving 1.3% sailing distance and 0.7% sailing time. Although both simulated trajectories are very similar to each other, the A-type controller appears to have better performances of turning and diving abilities than those of the B-type.  
Figure 26a,b presents variations of the rudder angle and the stern plane angle in the time domain. It is clearly found that both the rudder angle and the stern plane angle of the A-type controller are more sensitive to changes in the course angles during the passage.  
On the other hand, time records of 6-DOF velocity components of the SFRM with the A-type and B-type controllers are illustrated in Figure 27a-f, respectively. There are large differences of heave rates (Figure 27c) and pitch rates (Figure 27e) between the A-type and the B-type controllers. Generally speaking, the A-type controller appears to have faster motion responses, especially for the heave rate and the pitch rate, than the B-type. The average power spent by the A-type controller is 5.02 W, which is better than 5.01 W spent by the B-type controller. In addition, the entire work done by using the A-type controller is 1490.94 J, whereas the result by using the B-type controller is 1508.01 J. In the case of efficiency, it would be more appropriate to adopt the A-type controller than the B-type in the autopilot simulation of the SFRM.  
\begin{table}
\begin{tabular}{c c} \hline
**Waypoint Number** & **Coordinate (X, Y, Z)** \\ \hline
1 & (20, 0, 0) \\ \hline
2 & (20, –35, –5) \\ \hline
3 & (55, –35, –10) \\ \hline
4 & (55, 0, –15) \\ \hline
5 & (90, 0, –20) \\ \hline
6 & (90, –35, –25) \\ \hline
7 & (125, –35, –30) \\ \hline
8 & (125, 0, –35) \\ \hline
9 & (145, 0, –40) \\ \hline \end{tabular}
\end{table}
Table 7: The coordinates of waypoints in the 3D map. Unit: [m].  
Figure 25: Sailing trajectories of the SFRM by comparing the A-type controller with the B-type controller in (**a**) the stereo view; (**b**) the top view; and (**c**) the side view, respectively.  
Figure 27: _Conts._  
Figure 26: Time series of (**a**) rudder angle, and (**b**) stern plane angles for the A-type and B-type controllers, respectively.