## 5 Results and Discussion
### Manoeuvring Tests
#### 5.2.2 Horizontal Zigzag Test  
In the horizontal zigzag test, the rudder planes were immediately pulled to a fixed angle (\(\delta_{V}\)) and remained in that position until the desired course angle (\(\Psi_{d}\)) of the SFRM is reached; then, the rudder was pulled to the counter rudder angle (\(-\delta_{V}\)) immediately and remained in that position until the desired course angle turned to \(-\Psi_{d}\). Specifically, different rudder angles (\(\delta\)) and course (yaw) angles (\(\Psi\)) were experimented on in the test. The course overshooting angle and the time required to reach maximum overshooting within the duration of each rudder angle change were measured to determine the SFRM's manoeuvrability in course change. In the simulation, the SFRM sailed straight for 100 s. Then, the rudder plane was immediately turned to \(20^{\circ}\), and the position of the rudder plane was maintained for a certain period. When the yaw angle reached \(20^{\circ}\), the rudder plane was immediately turned to \(-20^{\circ}\), and the position of the rudder plane was maintained for a certain period. After the yaw angle changed to \(-20^{\circ}\), the previous procedure was repeated. In the horizontal zigzag test, the exact turning time of the rudder was set to be 1 s. Meanwhile, the maintaining time of the rudder angle at \(\pm 20^{\circ}\) was given as 15, 8, and 6 s for \(U\) = 0.5, 1, and 1.5 m/s, respectively. Figure 14 compares the SFRM's trajectories (top view) at three different sailing speeds: \(U\) = 0.5, 1, and 1.5 m/s, respectively. When \(U\) = 1.5 m/s, the trajectory was significantly longer than those of \(U\) = 1 m/s and \(U\) = 0.5 m/s. Additionally, the yawing amplitude (width of the path overshoot) was found to be larger for the case of the faster sailing speed and varied according to different periods of the rudder operations. For \(U\) = 0.5, 1, and 1.5 m/s, the yawing amplitudes are 0.89, 2.07, and 3.36 m. The trajectory of the SFRM at the highest sailing speed is not straight due to the slight variation of yaw overshoot in the subsequent horizontal zigzag manoeuvres. Figure 15a-c presents the time series of rudder and yaw angles at different sailing speeds. Figure 15a indicates that when \(U\) = 0.5 m/s, a full cycle of turning the yaw angle from \(0^{\circ}\) to \(-20^{\circ}\) and then \(20^{\circ}\) required 21 s. Figure 15b illustrates that when \(U\) = 1 m/s, a full turning cycle was 11 s long. Finally, Figure 15c indicates that when \(U\) = 1.5 m/s, a full turning cycle was 9 s long. Specifically, the time to the maximum width excursion of the first overshoot from the initial operation of the rudder planes required 7, 4, and 3 s for \(U\) = 0.5, 1.0, and 1.5 m/s. On the other hand, time to the maximum width excursion of the second overshoot from the initial operation of the rudder planes required 21, 11, and 9 s for \(U\) = 0.5, 1.0, and 1.5 m/s. Figure 14: The SFRM’s trajectories in the horizontal zigzag test at different sailing speeds. The SFRM's reaction speed increased considerably at higher sailing speeds. In terms of overshooting angles, Figure 15a indicates that when \(U=0.5\) m/s, the maximum overshooting angle was \(9.92^{\circ}\). In Figure 15b, when \(U=1\) m/s, the maximum overshooting angle was \(12.62^{\circ}\). Finally, Figure 15c indicates that when \(U=1.5\) m/s, the maximum overshooting angle was \(18.46^{\circ}\). The overshooting angle increased considerably in accordance with higher speeds. Given that the time at which the deflection of rudder angle is initiated is the same, i.e., \(t=100\) s, the time to the maximum width excursion of the yaw angle is shorter for the higher speed than that for the lower speed. Additionally, more turning cycles were completed at higher speeds than at lower speeds. Therefore, sailing speed was found to be a critical parameter influencing the relationship between yaw angle, rudder angle, and overshooting angle.