# NESO-Based Path Following Control for
## 4 Case Study  
In this section, the designed safe control strategy given by 0 for path following is applied on an underactuated hovercraft to validate the theoretical results. The main characteristics of the underactuated hovercraft are length \(L=29m\), width \(W=13.6m\), mass \(m=164000 Kg\), and inertia \(I_{z}=1.696\times 10^{7}\text{Kg}\cdot m^{2}\). The curved path following of the underactuated hovercraft with nonlinear time-varying ocean disturbance (the wind) is considered herein. The initial attitudes of the underactuated hovercraft are \((x,y,\psi)=(0m,200m,-30^{\circ})\) and initial velocities are \((u,v,r)=(0m/s,0m/s,0 rad/s)\). As the external environmental disturbance, the relative wind speed can be described as  
\[\left\{ \begin{array}{l}x_{wind}=u+10\cos(30^{\circ}-\psi)\\ y_{wind}=v+10\sin(30^{\circ}-\psi)\end{array} \right. \tag{48}\]  
Therefore, the disturbances are nonlinear time-varying for the hovercrafts according to Equation (48). In this study, the cubic algorithm in [35] is used to a generate a desired curved path, and the waypoint data base is shown in Figure 3a. The parameterized curved reference path with the starting pose \(wpt0=(0m,0m)\) and the end pose \(wpt6=(5100m,5000m)\) is shown in Figure 3a using a black solid line. And the waypoints data are shown in the Table 1.  
To better illustrate the effectiveness of the designed NESO, a comparative study of LESO is also carried out. The LESO proposed in [12] is given as  
\[\left\{ \begin{array}{l}\dot{\mathbf{\eta}}=\mathbf{R}(\psi)\hat{\mathbf{o}} -(\beta_{01}/\varepsilon)(\hat{\mathbf{\eta}}-\mathbf{\eta})\\ \dot{\mathbf{\phi}}=\mathbf{\hat{d}}-(\beta_{02}/\varepsilon^{2})\mathbf{R}^{ T}(\psi)(\hat{\mathbf{\eta}}-\mathbf{\eta})+\mathbf{M}^{-1}\tau\\ \dot{\mathbf{\hat{d}}}=-\Big{(}\beta_{03}/\varepsilon^{3}\Big{)}\mathbf{R}^{ T}(\psi)(\hat{\mathbf{\eta}}-\mathbf{\eta})\end{array} \right. \tag{49}\]  
where the designed parameters of LESO are selected as \(\beta_{01}\)\(=1.5\), \(\beta_{02}\)\(=0.75\), \(\beta_{03}\)\(=0.125\) such that the matrix  
\[\left[ \begin{array}{ccc}-\beta_{01}&1&0\\ -\beta_{02}&0&1\\ -\beta_{03}&0&0\end{array} \right] \tag{50}\]  
is Hurwitz, and \(\varepsilon=0.9\). The parameters of the proposed NESO are selected as \(\alpha=0.78\), \(\beta_{1}\)\(=0.5\), \(\beta_{2}\)\(=0.6\), \(\beta_{3}=0.15\). The parameters of ILOS (14) are chosen as \(\gamma_{\Delta}=2\times 10^{-3}\), \(\varepsilon=0.5\), \(\Delta_{\max}=240\) \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Waypoints** & **Wpt0** & **Wpt1** & **Wpt2** & **Wpt3** & **Wpt4** & **Wpt5** & **Wpt6 1**  \\ \hline \(x(m)\) & 0 & 1300 & 2500 & 2700 & 3000 & 3800 & 5100 \\ \(y(m)\) & 0 & 700 & 1500 & 2700 & 3700 & 4300 & 5000 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The waypoints data base for the curved path of the underactuated hovercraft. Figure 3: The curved path following results. (**a**) The path following results of the proposed safe control strategy; (**b**) the path following tracking errors \(x_{e}\), \(y_{e}\), and \(\psi_{e}\) of the proposed control scheme. **Remark 8**. : _The criteria for selecting parameters: (1) For the NESO, the designed process indicates that \(2/3<\alpha<1\), and \(\beta_{i}\) such that the matrix like (50) is Hurwitz; (2) for the ILOS, the integral parameter \(\varsigma\) is generally selected around 1, the changing rate is small such that the lookahead distance is changing slowly, and \(\Delta\) is about 3-10 times the length of the vehicle; (3) for the control law, the control parameters (except the error constraint \(\ell\) that is the smaller the better) are almost exponentially convergent, and the parameters are generally selected near 1._  
The curved path following simulation results of the underactuated hovercraft are plotted in Figure 3a and the along-tracking, cross-tracking, and heading-tracking errors are shown in Figure 3b, which indicates that the proposed control scheme can achieve the path following objective and ensure all tracking errors converge to zero.