## 5 Results and Discussion
### Manoeuvring Tests
#### 5.2.4. Vertical Zigzag Test  
In the vertical zigzag test, the stern planes were pulled to a fixed angle (\(\delta_{H}\)), and its position was maintained until the desired pitch angle (\(\theta_{d}\)) was reached before the stern planes were pulled to the counter angle (\(-\delta_{H}\)) and maintained until the desired pitch angle of the SFRM reached \(-\theta_{d}\). Different stern plane angles (\(\delta\)) and pitch angles (\(\theta\)) were experimented on in the test. The overshooting angle of each pitch movement, the width of depth overshoot, and time to the maximum depth excursion of each stern plane angle change were measured to determine the SFRM's manoeuvrability in depth change. Vertical zigzag manoeuvre is widely used to evaluate the manoeuvrability of submarines, so this can be definitely used to evaluate the manoeuvrability of the SFRM. In the simulation, the SFRM first sailed straight for \(100\) s; then, the stern planes were turned to \(20^{\circ}\), and its position was maintained for a certain period. When the pitch angle reached \(20^{\circ}\), the stern plane was immediately turned to \(-20^{\circ}\), and its position was maintained for a certain period. When the pitch angle reached  
Figure 17: Relationships between the yaw rates and the rudder plane angles at different sailing speeds for the spiral test. \(-20^{\circ}\), the procedure was repeated. However, the pitch overshoot was reached only in case of \(U=1.5\) m/s. For comparison, the stern plane operations of \(U=0.5\) and \(1.0\) m/s were set to be the same as the ones of \(U=1.5\) m/s. In the vertical zigzag test, the exact turning time of the stern plane was set to be \(1\) s. In addition, the maintaining time of the stern plane angle at \(\pm 20^{\circ}\) was all given as \(6\) s for \(U=0.5\), \(1\), and \(1.5\) m/s. Figure 18 compares the SFRM's trajectories in the vertical zigzag test at sailing speeds of \(U=0.5\), \(1\), and \(1.5\) m/s. When \(U=1.5\) m/s, the trajectory was significantly longer than those of \(U=1\) m/s and \(U=0.5\) m/s. When \(U=1.5\) m/s, the trajectory was significantly longer than those of \(U=1\) m/s and \(U=0.5\) m/s. The pitching amplitude (width of the depth overshoot) was found to be larger for the case of the faster sailing speed and varied according to different periods of the stern plane operations. For \(U=0.5\), \(1\), and \(1.5\) m/s, the pitching amplitudes are \(0.44\), \(0.48\), and \(1.14\) m. The SFRM moved significantly farther at higher speeds, and a slight pitch motion occurred due to the movement of the critical point [35]. In addition, the simulated trajectories and time between stern plane position changes varied at different sailing speeds. Figure 19a-c illustrates the variations of time series in stern plane and pitch angles at three sailing speeds. A full cycle of turning the pitch angle from \(0^{\circ}\) to \(-20^{\circ}\) and then \(20^{\circ}\) required \(10\) s. Specifically, time to the maximum depth excursion of the first overshoot from the initial operation of the stern planes required \(4\) s. On the other hand, time to the maximum depth excursion of the second overshoot from the initial operation of the stern planes required \(10\) s.  
In terms of overshooting angles, Figure 19a indicates that when \(U=0.5\) m/s, the maximum overshooting angle was \(-14.15^{\circ}\). In Figure 19b, the maximum overshooting angle was \(-6.94^{\circ}\) when \(U=1\) m/s. Finally, Figure 19c demonstrates that when \(U=1.5\) m/s, the maximum overshooting angle was \(2.1^{\circ}\). Therefore, at high sailing speeds, the maximum overshooting angle of the SFRM increased significantly compared with that at lower sailing speeds. Given that the stern-plane operation period is the same, the vertical zigzag movement is verified to be highly correlated with sailing speed.