## 5 Simulation  
In this section, the effectiveness and superiority of the proposed finite-time control scheme are verified by comparative simulation. The model parameters of the underactuated MSV are given as: \(m_{11}=25.8\), \(m_{22}=33.8\), \(m_{33}=2.76\), \(d_{11}=0.93\), \(d_{22}=2.89\), \(d_{33}=0.5\). The desired path is chosen as \(X_{p}=[50\sin(\vartheta/50),\vartheta]^{T}\). The initial position is chosen as \(X_{0}=[2,2]^{T}\). The initial surge velocity is chosen as \(u_{0}=0.2\), and the desired surge velocity is chosen as \(u_{d}=1\). The initial value of other states is set as zero. The time-varying external disturbances are simulated as: \(\tau_{wu}=0.1\sin(0.1t)+0.5\sin(0.05t)\), \(\tau_{wv}=0.05\sin(0.05t)\), \(\tau_{wr}=0.1\sin(0.1t)+0.1\sin(0.05t)\). The position error constraints are selected as \(k_{x}(t)=6e^{-0.15t}+0.5\), \(k_{y}(t)=6e^{-0.15t}+0.5\), and the input saturation limitations are set as \(\tau_{u\max}=4N\), \(\tau_{u\min}=-4N\), \(\tau_{r\max}=7N\) and \(\tau_{r\min}=-7N\). The control parameters are selected as \(k_{1}=0.1\), \(k_{2}=2\), \(k_{3}=70\), \(k_{4}=1\), \(\Delta=10\), \(k_{\psi}=10\), \(k_{\psi\updownarrow 1}=0.7\), \(b_{1}=2\), \(b_{2}=1\), \(k_{\chi_{r1}}=1\), \(k_{\chi_{r2}}=1\), \(k_{\chi_{r}}=0.01\), \(k_{r}=5\), \(k_{r1}=0.01\), \(b_{3}=2\), \(b_{4}=1\), \(k_{\chi_{u1}}=1\), \(k_{\chi_{u2}}=0.01\), \(k_{\chi_{u}}=0.01\), \(k_{u}=15\), \(k_{u1}=0.02\), \(a_{r}=2.5\), \(a_{r1}=0.01\), \(b_{r}=2.5\), \(c_{r}=2.5\). To better show the effectiveness and superiority of the proposed finite-time path following control scheme, the comparison is constructed with the EC-LOS guidance law [14] and backstepping controller. The EC-LOS guidance law is given as follows:  
\[ \begin{cases}\dot{\theta}=\dfrac{U\cos(\psi-\psi_{p}+\beta)+\theta_{s}}{\sigma }\\ \psi_{d}=\psi_{p}-\beta+\arctan(-\theta_{e},\Delta)\end{cases}  \tag{83}\]  
where \(\theta_{s}=\frac{k_{u}}{\varepsilon_{s}}\frac{k_{u}^{2}}{\pi}\sin\left(\frac {\pi\varepsilon_{s}^{2}}{2k_{u}^{2}}\right)\cos\left(\frac{\pi\varepsilon_{s} ^{2}}{2k_{u}^{2}}\right)+k_{\varepsilon 0}\varepsilon_{s}\) and \(\theta_{e}=\frac{k_{e}}{\varepsilon_{e}}\frac{k_{e}^{2}}{\pi}\sin\left(\frac {\pi\varepsilon_{e}^{2}}{2k_{e}^{2}}\right)\cos\left(\frac{\pi\varepsilon_{e }^{2}}{2k_{e}^{2}}\right)+k_{\varepsilon 0}\varepsilon_{e}\). In addition, the EC- LOS controller parameters are given as follows: \(k_{\varepsilon s}=2\), \(k_{\varepsilon e}=2\), \(k_{\varepsilon s0}=0.2\), \(k_{\varepsilon e0}=0.2\), \(\Delta=12\). The simulation results are described in Figures 4-10. Figure 4 shows the desired path and the actual path of the underactuated MSV under two guidance laws. It is shown that the proposed control scheme can control the vessel to follow the desired path quickly, which converges faster than the control scheme based on the EC-LOS guidance law. Figure 5 illustrates the velocities \(u\), \(v\) and \(r\) of the underactuated MSV under two guidance laws, in which the vessel can quickly reach the desired surge velocity \(u_{d}\) under the proposed control scheme. Figures 6 and 7 show the along-track error and the cross-track error of the underactuated MSV under two guidance laws. As shown in the Figures 6 and 7, both the along-track error and the cross-track error can quickly converge into a small neighborhood around zero under two guidance laws, and they will not exceed the position error constraints. However, the convergence rate of the along-track error and the cross-track error under the FT-LOS guidance law is faster than that under the EC-LOS guidance law.