## 2 Hovercraft Model  
The model of a hovercraft moving horizontally that is considered in this work is shown in Figure 1.  
Using the North-East-Down (NED) frame (the inertial frame), the position and orientation of the vehicle is \(\eta=[x,y,\phi]^{T}\). Based on [22; 29], the equations of motion of the hovercraft moving horizontally can be written as follows (symbols are explained in Appendix A):  
\[\dot{\eta}=R(\phi)\
u, \tag{1}\] \[M\dot{\
u}=-C(\
u)-D(\
u)+\tau, \tag{2}\]  
where \(\
u=[u,\
u,r]^{T}\) denotes the vector of velocities in the body frame (containing the surge velocity, the sway velocity and the yaw velocity). If the hovercraft is equipped with thrusters and a rudder, then the \(\tau\) vector can be written as follows [22]:  
\(\tau=[b_{T}T\cos\psi,\ p_{T}T\sin\psi,\ ab_{T}T\sin\psi]^{T}\). The remaining matrices and vectors have the following forms:  
\[ \begin{array}{c}M=\left[\begin{array}{ccc}m&0&-my_{g}\\ 0&m&mx_{g}\\ -my_{g}&mx_{g}&J\end{array}\right]\qquad\qquad R(\phi)=\left[\begin{array}{ cccc}\cos\phi&-\sin\phi&0\\ \sin\phi&\cos\phi&0\\ 0&0&1\end{array}\right],\\ C(\
u)=\left[\begin{array}{c}-mvr-mx_{g}r^{2}\\ mur+my_{g}r^{2}\\ mx_{g}ur-my_{g}vr\end{array}\right],\quad D(\
u)=\left[\begin{array}{c}d_{u 0}\mathrm{sign}\,u+d_{u}u+d_{ur}r\\ d_{v0}\mathrm{sign}\,v+d_{v}v+d_{vr}r\\ d_{r0}\mathrm{sign}\,r+d_{ur}u+d_{vr}v+d_{rr}r\end{array}\right],\end{array}  \tag{3}\]  
in which \(m\), \(J\) are the vehicle mass and inertia, respectively; \(y_{g}\), \(x_{g}\) are the mass displacements; \(d_{0w},d_{0w},d_{0r}\) mean the friction coefficients; and coefficients \(d_{u},d_{v},d_{r},d_{ur},d_{vr}\) represent linear drag. Figure 1: Underactuated hovercraft model sketch. _Comment._ In Equations (1)-(3), there are mass components, coefficients of friction linear drag and displacements. In the proposed method, it is not necessary to know the exact displacements (after transformation their influence is small but rather the mass components \(mx_{g}\), \(my_{g}\) are important). Although these are difficult to determine for a real vehicle, there are methods by which such information can be obtained. For example, one can use the parameter identification methods described in [22; 30] for a hovercraft (a more detailed description of the general method is also known from [31]). After velocity transformation, e.g., [32], the new equations instead of (2) are of the form  
\[N\dot{\zeta}=-\mathrm{Y}^{T}C(\
u)-\mathrm{Y}^{T}D(\
u)+\mathrm{ Y}^{T}\tau, \tag{4}\] \[\
u=\mathrm{Y}\zeta,\] (5) \[\mathrm{Y}=\left[ \begin{array}{ccc}1&0&\mathrm{Y}_{13}\\ 0&1&\mathrm{Y}_{23}\\ 0&0&1\end{array} \right],\ \ \ \ N=\mathrm{diag}\{N_{1},N_{2},N_{3}\},\]  
where \(\zeta=[\zeta_{1},\zeta_{2},\zeta_{3}]^{T}\) is the vector of inertial quasi-velocities. Other symbols mean \(N_{1}=m\), \(N_{2}=m\), \(N_{3}=J-m(x_{g}^{2}+y_{g}^{2})\), \(Y_{13}=y_{g}\), \(Y_{23}=-x_{g}\). Equations of motion instead of (4) and (5) can be written as follows:  
\[\zeta_{1}=u-\mathrm{Y}_{13}r, \tag{6}\] \[\zeta_{2}=v-\mathrm{Y}_{23}r,\] (7) \[\zeta_{3}=r,\] (8) \[N_{1}\dot{\zeta}_{1}=mvr+mx_{g}r^{2}-d_{u0}\mathrm{sign}\,u-d_{u} u-d_{ur}r+b_{T}T\cos\psi,\] (9) \[N_{2}\dot{\zeta}_{2}=-mur-my_{g}r^{2}-d_{v0}\mathrm{sign}\,v-d_{ v}v-d_{vv}r+b_{T}T\sin\psi,\] (10) \[N_{3}\dot{\zeta}_{3}=-mx_{g}ur+my_{g}vr-d_{ur}u-d_{vv}v-d_{vv} \mathrm{sign}\,r-d_{r}r\] \[-ab_{T}T\sin\psi+\mathrm{Y}_{13}b_{T}T\cos\psi+\mathrm{Y}_{23}b_{ T}T\sin\psi. \tag{11}\]  
Finally, after dividing into two sub-systems, the set of Equations (6)-(11) can be given in the compact vector matrix form:  
\[\dot{p}=R_{1}\
u_{tr}, \tag{12}\] \[\dot{R}_{1}=R_{1}S\,\zeta_{3},\] (13) \[m\dot{\zeta}_{tr}=-mSv_{tr}+f;\] (14) \[\zeta_{2}=v-\mathrm{Y}_{23}r,\]0  
where \(v_{tr}=[u\ \ \ v]^{T}\) is the linear velocity vector, \(\dot{\zeta}_{tr}=[\zeta_{1}\ \ \ \zeta_{2}]\) is the corresponding quasi-velocity vector, \(R_{1}=\left[ \begin{array}{cc}\cos\phi&-\sin\phi\\ \sin\phi&\cos\phi\end{array} \right]\), \(S=\left[ \begin{array}{cc}0&-1\\ 1&0\end{array} \right]\) and  
\[f=\left[