# NESO-Based Path Following Control for
## 2 Preliminaries and Problem Formulation
### Problem Formulation
#### 2.2.1 Nonlinear Hovercraft Dynamics with Uncertainties and SLC  
To describe the motion of an underactuated hovercraft in horizontal plan, we define the inertial north-east-down (NED) reference frame and the body-fixed (BF) reference frame as shown in Figure 1b. Then, a mathematical motion model of three-degree-of-freedom (including surge, sway, and yaw) for underactuated hovercrafts is established as follows. The three-degree-of-freedom kinematics equations of an underactuated hovercraft can be written as [30]  
\[\left\{ \begin{array}{l}\dot{x}=u\cos\psi-v\sin\psi\\ \dot{y}=u\sin\psi+v\cos\psi\\ \dot{\psi}=r\end{array} \right. \tag{5}\]  
where \(x\), \(y\), and \(\psi\) denote the north, east positions and heading attitude of an underactuated hovercraft in the inertial NED frame, respectively. \(u\), \(v\), and \(r\) denote the surge linear velocity, sway linear velocity, and yaw angular velocity in the BF frame for an underactuated hovercraft, respectively. The three-degree-of-freedom kinetic equations of an underactuated hovercraft with turning SLC can be expressed as [31]  
\[\left\{ \begin{array}{l}m\Big{(}\dot{u}-vr\Big{)}=F_{xu}+F_{xh}+F_{xm}+F_{xP }\\ m\Big{(}\dot{v}+ur\Big{)}=F_{yu}+F_{yh}+F_{ym}+F_{yc}\\ I_{z}\dot{r}=M_{zu}+M_{zh}+M_{zm}+M_{zc}+M_{zR}\end{array} \right. \tag{6}\]  
subject to  
\[r_{\rm{Imin}}(t)\leq r\leq r_{\rm{Imax}}(t) \tag{7}\]in which \(m\) represent the mass and \(I_{z}\) is the rotational inertia of an underactuated hovercraft. \(F_{xa}\), \(F_{yu}\), and \(M_{za}\) represent the aerodynamic forces and moment. \(F_{xh}\), \(F_{yh}\), and \(M_{2h}\) are the hydrodynamic forces and moment. \(F_{xm}\), \(F_{ym}\), and \(M_{zm}\) denote the air momentum forces and moment. \(F_{yc}\) and \(M_{zc}\) denote the cushion force and moment. \(F_{xP}\) is the propulsion force of the aerial propeller that should be designed to achieve the yaw control, and \(M_{zR}\) is the force of the rudder to be designed later for controlling the surge motion of an underactuated hovercraft. All above forces and moments have their own complex nonlinear mathematical models, and because the focus of this paper is the design of the control strategy and considering the limitations of the length of the paper, the details of the model are not expanded. The details can be referred to the specific modeling description in the literature [4]. \(r_{\rm{Imax}}(t)\) and \(r_{\rm{Imin}}(t)\) represent the maximum and minimum of the turning safety limit (control area) parameters shown in Figure 2a, respectively. **Remark 2**. : _The safety limit, including tuning velocity, roll angle, and sideslip angle three parameters, is a series of most essential constraint indicators to ensure the safety of an underactuated hovercraft. The change of turning velocity directly affects the change of sideslip angle. Thus, the indirect control of the sideslip angle can be realized by controlling the turning velocity. Additionally, the roll angle is affected by the cushion system, which is not included here. Therefore, this paper addresses a more safe control strategy of underactuated hovercrafts subject to the turning SLC._  
The turning safety limit of an underactuated hovercraft is shown in Figure 2a, which is divided into three parts: safety area, alert area, and control area. When the turning velocity belongs to the alert area, the underactuated hovercraft will alert the sailor. The limit of the control area is the allowable maximum turning rate. Note that the turning safety limit of an underactuated hovercraft is a nonlinear function with respect to velocity, thus it is time-varying. To facilitate the NESO design, define the total uncertainties as  
\[\left\{ \begin{array}{l}d_{u}(t){=:}f_{u}(t,u,v,r,\tau_{wu})=\left(F_{xa}+F_{ xh}+F_{xm}\right)/m+cr\\ d_{v}(t){=:}f_{v}(t,u,v,r,\tau_{uv})=\left(F_{yu}+F_{yh}+F_{ym}+F_{yc}\right) /m-ur\\ d_{r}(t){=:}f_{r}(t,u,v,r,\tau_{vr})=\left(M_{2a}+M_{2h}+M_{zc}+M_{zm}\right) /I_{z}\end{array} \right.