## 4 Simulations  
The effectiveness and robustness of the proposed path following control method were evaluated based on the platform of MATLAB. The Cybership II [31] was taken as the control object whose parameters were as follows: \(m_{11}=25.8,m_{22}=33.8,m_{33}=2.76,d_{11}=0.9257,d_{22}=2.8909,d_{33}=0.5\). For simplicity, we ignored the off-diagonal terms of the inertia and damping. The maximum actuated force and moment were 2 N and 1.5 Nm. From the beginning of the simulations, the ocean currents with time-varying speed were given as \(V_{x}=0.08\sin\left(0.1t\right)\) m/s and \(V_{y}=0.04\sin(0.1t)\) m/s. The time-varying external disturbances were generated with the first-order Markov process \(\tau_{uu}+\zeta_{1}\tau_{uu}=w_{1},\tau_{uv}+\zeta_{2}\tau_{uv}=w_{2},\tau_{ uv}+\zeta_{3}\tau_{uv}=w_{3}\), where \(w_{i}\) and \(\zeta_{i}(i=1,2,3)\) are zero-mean Gaussian white noise and constants, respectively [32]. The parameters of the controller are listed in Table 1. The node numbers and widths of RBFNNs were chosen as: node number \(l=21\) and the widths \(b_{i}=3(i=1,2,\ldots,l)\). The neural active region was chosen as \(\left[\left|u\right|,\left|v\right|,\left|r\right|\right]\in\left[\left[0,2 \right],\left[0,1.5\right],\left[0,1.5\right]\right]^{T}\). Performance comparisons between the proposed finite-time currents observer based ILOS guidance with adaptive RBFNN (FCONN) controller and the indirect adaptive observer based [10] ILOS guidance with adaptive RBFNN (IAONN) controller are presented in the following two control scenarios. Case 1: The desired path and speed assignment were chosen as \(\mathbf{P}_{d}=\left[\theta,\theta\right]^{T}\) and \(u_{d}=0.5\) m/s. The initial states were given as \(\left[x(0),y(0),u_{r}(0),v_{r}(0),r\right]^{T}=\left[0\text{ m},2\text{ m},0.01\text{ m/s},0\text{ m/s},0\text{ rad}/\text{s}\right]^{T}\). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Notation** & **Value** & **Notation** & **Value** & **Notation** & **Value** & **Notation** & **Value** & **Notation** & **Value** \\ \hline \(k_{1}\) & 1 & \(l_{2}\) & 4 & \(\Gamma_{1}ii(i=21)\) & 500 & \(\xi_{2}\) & 20 & \(\chi_{r}\) & 0.01 \\ \(k_{2}\) & 2 & \(\rho_{1}\) & 0.01 & \(\Gamma_{2}ii(i=21)\) & 50 & \(\lambda_{1}\) & 0.01 & \(\chi_{u}\) & 1 \\ \(k_{3}\) & 4 & \(\rho_{2}\) & 0.03 & \(\iota_{1}\) & 0.05 & \(\lambda_{2}\) & 0.01 & \(\Delta\) & 2.51 \\ \(k_{4}\) & 5 & \(k_{\zeta_{1}}\) & 1.2 & \(\iota_{2}\) & 0.05 & \(\delta_{1}^{0}\) & 0.1 & & & \\ \(l_{1}\) & 100 & \(k_{\zeta_{2}}\) & 1.2 & \(\xi_{1}\) & 50 & \(\delta_{1}^{0}\) & 0.1 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the path following controller. The simulation results are shown in Figure 2(a)-g. Table 2 summarizes the performance indices based on the integrated absolute error (IAE) and the time integrated absolute error (ITAE), which were used to evaluate the transient performance and steady-state performance. The proposed FCONN control method could drive the USV following the desired path with a high-precision and better transient process (Figure 2(a)). Their detailed distinction is more clearly shown in Figure 2(b),c, specifically the smaller along- and cross-tracking errors and the smaller heading and surge velocity tracking errors. Meanwhile, the lower IAE and ITAE metrical values of the cross-tracking error revealed the better transient and steady-state performance. Figure 2(d) shows that the proposed finite-time currents observer could identify the time-varying currents accurately, whereas the IAONN control scheme had an obvious oscillation during the transient process, as well as lower accuracy in the steady state. Figure 2(d) demonstrates that the RBFNNs could capture the unknown dynamical uncertainties precisely and Figure 2(e) presents the compound bounds and their estimation, which played decisive roles in driving the dynamics state \(r\) and \(u_{r}\) to their real value.