## 6 Simulation Results
In the proposed guidance law, the lateral position suffers the smallest overshoot with the fastest convergence speed, which demonstrates the superiority of the proposed guidance law. When the water depth varies, the vessel-bank interaction changes due to the varying hydrodynamic coefficients, and without proper control, it will make the vessel shift away from the desired path. To maintain the path, the proposed path following control design should be robust to the varying water depths. It can be seen from Figure 4 that the vessel using all the guidance laws can return to the desired path when water depth changes, while the vessel using the proposed guidance law can return to the desired path faster with smaller overshoot. The quantitative measure of the overshoot and settling time under different guidance laws are compared in Table 2, it can be seen that the overshoot is the least and the settling time is the shortest using the proposed guidance law when the water depth changes. Figure 4: Comparison of lateral position with different guidance laws at \(y_{d}=0.1L\). The time histories of the rudder angle are presented in Figure 5. It can be seen that the rudder operations with different guidance laws meet the rudder saturation and rate limits. A larger rudder angle is needed when the water depth decreases. The vessel-bank interaction becomes aggravated as the water depth decreases and makes control more challenging. The desired heading angle and the true heading angle in the proposed path following control scheme are presented in Figure 6. It can be seen that the true heading angle can asymptotically converge to the desired heading angle without steady-state errors, which illustrates that the designed yaw tracking control system is robust to model uncertainties, unknown parameters, and external disturbances. \begin{table}
\begin{tabular}{c c c c c c c} \hline  & \multicolumn{3}{c}{**2.03\(\rightarrow\)1.72**} & \multicolumn{3}{c}{**1.72\(\rightarrow\)1.37**} \\ \cline{2-7}  & \multicolumn{2}{c}{**Overshoot**} & \multicolumn{2}{c}{**Settling Time (s)**} & \multicolumn{2}{c}{**Overshoot**} & \multicolumn{2}{c}{**Settling Time (s)**} \\ \hline ILOS & 8.43 m & 61.67\% & 1937 & 0.71 m & 5.19\% & 2108+ \\ ALOS & 7.68 m & 56.18\% & 2674 & 0.91 m & 6.67\% & 2108+ \\ ELOS & 6.77 m & 49.52\% & 3140 & 0.33 m & 2.41\% & 2108+ \\ HGESO & 3.43 m & 25.09\% & 1937 & 0.04 m & 0.29\% & 1208 \\ \hline \end{tabular}
\end{table} 
Table 2: Comparison of overshoot and settling time at \(y_{d}=0.1L\). Figure 5: Comparison of rudder angle with different guidance laws at \(y_{d}=0.1L\). **Case 2. ** The desired lateral position is set as \(y_{d}=0.05L\); other conditions are the same as those in Case 1. The time histories of the lateral position with different guidance laws are presented in Figure 7. It can be seen that the vessel under all the guidance laws can converge to the desired path asymptotically, and it converges to the desired path with the smallest overshoot and the fastest speed in the proposed guidance law, which demonstrates the superiority of the proposed guidance law. From the comparisons of Figures 4 and 7, it can be seen that the dynamic process of the lateral position greatly changes when \(y_{d}\) turns from \(0.1L\) to \(0.05L\) using the ILOS, ALOS and ELOS guidance laws. However, it only slightly changes in the proposed guidance law, which demonstrates that the proposed guidance law is much more robust to different desired paths than the other guidance laws. In Table 3, the quantitative measure of the overshoot and settling time under different guidance laws are compared, while the benefits of the proposed guidance law are objectively evaluated. Figure 6: The desired heading angle vs. the true heading angle in the proposed guidance law at \(y_{d}=0.1L\). Figure 7: Comparison of lateral position with different guidance laws at \(y_{d}=0.05L\). **Case 3.