## 5 Results and Discussion
### Optimal PD Control Gains  
After each manoeuvring test was conducted to determine the operation limit for rudder performance, the optimal PD control gains for each test would be calculated by using the PID tuner. Figure 24a-e show the step responses, i.e., the output of the control system for a step input, of different manoeuvring tests with three sailing speeds. Since the characteristics parameters, including rise time, settling time, and overshoot, change with the variation of the control gains \((K_{P},K_{D})\), a set of optimal control gains can be obtained by examining the values of the characteristics parameters. To determine the efficacy of the proposed 6-DOF manoeuvring mathematical model combined with the PD controller, the PD control gains and the characteristics parameters of manoeuvring tests with the PID tuner are introduced in Table 6.  
Figure 24: _Conts._For the manoeuvring tests in the horizontal plane, i.e., turning circle, horizontal zigzag and spiral tests, the turning circle test has the largest percentage of overshoot on the basis of different sailing speeds. By contrast, the horizontal zigzag test and the spiral test have less amounts of overshoot than the turning circle test. Furthermore, the horizontal zigzag test has the minimum amount of overshoot, rise time, and settling time among all of the horizontal plane tests. In terms of the optimum action of the PD controller, the horizontal zigzag test has a better performance than the other two horizontal-plane tests.  
For the manoeuvring tests in the vertical plane, i.e., vertical zigzag and meander tests, the amounts of overshoot for the meander test appear to be less than those for the vertical zigzag test mostly. On the other hand, the vertical zigzag test has a better performance of settling time than the meander test. In general, the values of rise time for the meander test are much smaller than those for the horizontal zigzag test.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Test** & \(\mathbf{U}\) **[m/s]** & \(\mathbf{K_{P}}\) & \(\mathbf{K_{D}}\) & **Rise Time [s]** & **Settling Time [s]** & **Overshoot [\%]** & **Peak** \\ \hline \multirow{3}{*}{Turning Circle} & 0.5 & 0.63 & 0.32 & 1.11 & 7.19 & 29.1 & 1.29 \\ \cline{2-8}  & 1 & 0.82 & 0.46 & 0.918 & 8.06 & 34.4 & 1.32 \\ \cline{2-8}  & 1.5 & 1.03 & 0.43 & 0.74 & 6.79 & 37.8 & 1.38 \\ \hline \multirow{3}{*}{Horizontal ZigZag} & 0.5 & 0.98 & 0.32 & 1.09 & 6.66 & 17.8 & 1.19 \\ \cline{2-8}  & 1 & 1.23 & 0.52 & 0.817 & 6.48 & 24.9 & 1.25 \\ \cline{2-8}  & 1.5 & 1.47 & 0.61 & 0.764 & 6.42 & 31 & 1.31 \\ \hline \multirow{3}{*}{Spiral} & 0.5 & 1.12 & 0.62 & 1.08 & 6.19 & 28.1 & 1.28 \\ \cline{2-8}  & 1 & 1.33 & 0.67 & 0.93 & 6.83 & 32.4 & 1.31 \\ \cline{2-8}  & 1.5 & 1.48 & 0.73 & 0.84 & 6.79 & 36.8 & 1.33 \\ \hline \multirow{3}{*}{Vertical ZigZag} & 0.5 & 0.49 & 0.29 & 0.856 & 5.42 & 12.8 & 1.15 \\ \cline{2-8}  & 1 & 0.78 & 0.75 & 0.981 & 6.71 & 22.8 & 1.25 \\ \cline{2-8}  & 1.5 & 1.21 & 0.35 & 0.869 & 6.04 & 29.7 & 1.3 \\ \hline \multirow{3}{*}{Meander} & 0.5 & 0.56 & 0.52 & 1.28 & 6.69 & 17.5 & 1.18 \\ \cline{2-8}  & 1 & 0.92 & 0.82 & 0.866 & 6.03 & 19.5 & 1.22 \\ \cline{2-8}  & 1.5 & 1.13 & 0.85 & 0.809 & 6.68 & 28.4 & 1.28 \\ \hline \hline \end{tabular}
\end{table}
Table 6: The characteristics parameters and the gains of the PD controller with the PID tuner for different manoeuvring tests.  
Figure 24: Step responses of the PD controller with respect to three sailing speeds, \(U=0.5\), \(1\), and \(1.5\,\mathrm{m/s}\), for (**a**) the turning circle test, (**b**) the horizontal zigzag test, (**c**) the spiral test, (**d**) the vertical zigzag test, and (**e**) the meander test.  
are smaller than those for the vertical zigzag test. Consequently, it is difficult to judge the efficacy of the PID tuner by comparing the vertical zigzag test with the meander test.