## 4 Numerical Simulations  
The comparison with the finite-time terminal sliding mode control method in [21] is implemented to demonstrate the advantages and effectiveness of the proposed method. In simulations, the major parameters of the hovercraft can be found in [29]. The initial values of the hovercraft under the condition of Assumption 1 are selected as \(\psi(0)=30\), \(u(0)=30\), \(v(0)=0\) and \(r(0)=0\). The control parameters are designed as \(k_{x}=0.02\), \(k_{y}=0.068\), \(k_{u}=1.999\), \(k_{v}=2.5\) and \(k_{r}=0.99\). The parameters of the virtual vessel are chosen as \(x_{d}(0)=0\), \(y_{d}(0)=0\), \(\psi_{d}(0)=45\), \(u_{dset}(t)=35\) and \(v_{dset}(t)=0\). The hovercraft's initial position \(x(0)\), \(y(0)\) and the parameter \(r_{dset}(t)\) of the virtual vessel are given in two cases to show the advantages of the proposed method. **Case 1:** The hovercraft's initial position is selected as \(x(0)=-50\), \(y(0)=50\) and the parameter \(r_{dset}(t)\) of the desired trajectory is given by:  
\[r_{dset}\left(t\right)=\left\{ \begin{array}{ll}0&t<100\\ sat\left(k_{rest}\left(t-100\right),0.5,-0.5\right)&t\geq 100\end{array} \right. \tag{57}\]  
**Case 2:** The hovercraft's initial position is selected as \(x(0)=-60\), \(y(0)=60\) and the parameter \(r_{dset}(t)\) of the desired trajectory is given by:  
\[r_{dset}\left(t\right)=\left\{ \begin{array}{ll}0&t<100\\ sat\left(-k_{rest}\left(t-100\right),0.5,-0.5\right)&100\leq t<460\\ sat\left(k_{rest}\left(t-100\right),0.5,-0.5\right)&t\geq 460\end{array} \right. \tag{58}\]  
where  
\[sat(x,x\_max,x\_min)=\left\{ \begin{array}{ll}x\_max,&x>x\_max\\ x,&x\_min\leq x\leq x\_max\\ x\_min,&x<x\_min\end{array} \right. \tag{59}\]  
The parameters of the performance constraint function are designed as \(\bar{k}_{xb}=\bar{k}_{yb}=100\), \(k_{xb}=k_{yb}=10\), \(\bar{k}_{ub}=20\), \(\bar{k}_{vb}=5\), \(\bar{k}_{rb}=5\), \(k_{ub}=0.6\), \(k_{vb}=0.6\), \(k_{rb}=0.6\), \(T_{xa}=T_{ya}=20\) and \(T_{ua}=T_{va}=T_{ra}=60\). Since ARBFNN is used to approximate the model uncertainties \(f_{u}\), \(f_{v}\), and \(f_{r}\), the hidden node numbers for \(\hat{W}_{u}^{T}H_{u}\left(Z_{u}\right)\), \(\hat{W}_{v}^{T}H_{v}\left(Z_{v}\right)\) and \(\hat{W}_{r}^{T}H_{r}\left(Z_{r}\right)\) are set to \(n=13\), with centers of the radial basis functions \(\phi_{l_{1}}\left(l_{1}=1,\cdots,n\right)\), \(\phi_{l_{2}}\left(l_{2}=1,\cdots,n\right)\) and \(\phi_{l_{3}}\left(l_{3}=1,\cdots,n\right)\) evenly spaced in \([-1.5,1.5]\), \([-1,1]\) and \([-2,2]\), respectively. The widths of the radial basis functions are chosen as \(\varepsilon_{l_{1}}=30\), \(\varepsilon_{l_{2}}=20\) and \(\varepsilon_{l_{2}}=25\), respectively. The model uncertainties and external disturbances are described by the following formula:  
\[\left[ \begin{array}{c}f_{u}\\ f_{v}\\ f_{r}\end{array} \right]=\left[ \begin{array}{cc}\frac{2\sin(0.05t)}{m_{0}} \\ \frac{0.2\cos(0.03t)}{m_{0}}\\ \frac{\cos(0.02t)}{J_{z0}}\end{array} \right]\left[ \begin{array}{c}b_{1}\\ b_{2}\\ b_{3}\end{array} \right] \tag{60}\]  
where \(b=\left[ \begin{array}{cc}b_{1}&b_{2}&b_{3}\end{array} \right]^{T}\in R^{3}\), \(\dot{b}=-T^{-1}b+Aw_{n}\) is the first-order Markov process, \(w_{n}\in R^{3}\) is the vector of zero-mean Gaussian white noises, the other parameters of the first-order Markov process are set as:  
\[ \begin{array}{l}b(0)=[2\times 10^{4},2\times 10^{4},2\times 10^{4}]^{T}\\ T=diag(10^{3},10^{3},10^{3})\\ A=diag(1\times 10^{4},1\times 10^{4},1\times 10^{4})\end{array}  \tag{61}\]  
The simulation results of the two different methods with and without appointed-time performance constraints under cases 1 and 2 are shown in Figures 2-10 and labeled as _proposed method_ and _sliding  
Figure 4: Speed and turn rate tracking errors of two methods under case 1. Figure 5: Control inputs of two methods under case 1. Figure 6: Trajectory of the hovercraft under case 2.