## 6 Simulation Results  
To validate the theoretical results, simulations were carried out in this part. The developed strategy was applied to an underactuated vessel in the simulations. The main parameters of the vessel are given in Table 1, the dynamics of the vessel are given by [PERSON] et al. [7]. The control objective is to make the vessel follow a straight-line path parallel to the bank with varying water depths. The surge speed of the vessel is assumed to be constant. The initial kinematics and dynamics of the vessel are \([x_{0},\,y_{0},\,\psi_{0}]^{T}=[0,\,0,\,0]^{T}\), \([u_{0},\,v_{0},\,r_{0}]^{T}=[3.075\,\mathrm{m}/\mathrm{s},0,0]\). The rudder saturation and rate limits (\(|\delta|\leq 35^{\circ},|\dot{\delta}|\leq 5^{\circ}/\mathrm{s}\)) are considered. **Case 1**. The desired lateral position is set as \(y_{d}=0.1L\) (\(L\) denotes vessel length), the water depth to draft ratio is set as 2.03 in the beginning. When the vessel sails 100 \(L\) (\(t=4446\,\mathrm{s}\)), the water depth to draft ratio begins to decrease gradually until its value reaches 1.72; then the water depth is maintained unchanged until the vessel sails 200 \(L\) (\(t=8892\,\mathrm{s}\)), the water depth to draft ratio begins to decrease gradually again until its value reaches 1.37, then the water depth is maintained unchanged till the end of the simulation. In the process of varying water depths, the slope of the varying bottom is set as 1. The initial values for the high-gain extended state observer is \([-y_{d},\,0]\), the design parameters of the proposed guidance law are \(\alpha_{1}=10\), \(\alpha_{2}=2\), \(\varepsilon=0.2\). Comparison studies with ILOS, ALOS and ELOS guidance laws are conducted to give a more vivid demonstration of the proposed guidance law. An ILOS guidance law proposed by [PERSON] et al. [19] is  
\[\left\{ \begin{array}{l}\psi_{d}=-\tan^{-1}\left(\frac{y_{e}+\sigma_{ILOS}y_{ \mathrm{int}}}{\Delta_{ILOS}}\right),\\ \dot{y}_{\mathrm{int}}=\frac{\Delta_{ILOS}y_{\mathrm{int}}}{(y_{e}+\sigma_{ILOS }y_{\mathrm{int}})^{2}+\Delta_{ILOS}}.\end{array} \right. \tag{62}\]  
An ALOS guidance law proposed by [PERSON] et al. [20] is  
\[\left\{ \begin{array}{l}\psi_{d}=-\tan^{-1}\Bigl{(}\frac{1}{\Delta_{ALOS}}y_{ e}+\dot{\beta}\Bigr{)},\\ \dot{\hat{\beta}}=\gamma_{ALOS}\frac{\
u\Delta_{ALOS}}{\sqrt{\Delta_{ALOS} ^{2}+(y_{e}+\Delta_{ALOS}\hat{\beta})^{2}}}y_{e}.\end{array} \right. \tag{63}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Length (m) & 136.7 \\ Breadth (m) & 22.0 \\ Mean draft (m) & 10.5 \\ Longitudinal position of the vessel’s center of gravity (m) & \(-0.89\) \\ Displacement volume (m\({}^{3}\)) & 15,317 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main parameters of the vessel. An ELOS guidance law proposed by [PERSON] et al. [22] is  
\[\left\{ \begin{array}{l}\psi_{d}=-\tan^{-1}\left(\frac{y_{e}}{x_{ ELOS}}+\hat{\beta}\right),\\ \hat{\beta}=\frac{\hat{\beta}}{U\cos\psi},\\ \hat{g}=p+k_{ELOS}y_{e},\\ \hat{p}=-k_{ELOS}p-k_{ELOS}{}^{2}y_{e}-k_{ELOS}U\sin\psi.\end{array} \right. \tag{64}\]  
The design parameters in these guidance laws are properly selected by a trial and error approach to guarantee that the closed-loop system, under the corresponding guidance law, performs better. The look-ahead distance in ILOS is \(\Delta_{ILOS}=4L\), the integral coefficient is \(\sigma_{ILOS}=0.08\). The look-ahead distance in ALOS is \(\Delta_{ALOS}=4L\), the adaptive coefficient is \(\gamma_{ALOS}=0.01\). The look-ahead distance in ELOS is \(\Delta_{ELOS}=5.5L\), the observer gain is \(k_{ELOS}=0.12\), the initial value of ELOS is set as \(p(t_{0})=-k_{ELOS}y_{e}(t_{0})\). In Figure 4, comparison of lateral position with different guidance laws at \(y_{d}=0.1L\) is presented. It can be seen that the vessel with all the guidance laws can track the desired straight-line path in the steady state, while the response in the dynamic process is significantly different.