## 3 Control Scheme Using Quasi-Velocities  
The proposed controller is a modification of the one coming from [22] and is an extension to vehicles with a model that takes into account the asymmetry caused by the shift of the center of mass. Thus, the goal is to also track the desired trajectory \(p_{d}\), which has a fixed point defined in the body frame \(\delta\in R^{2}\). The disturbances are not taken into account. This means that the algorithm only considers the friction model as presented in Section 2, while it does not handle other types of external disturbances (deterministic or stochastic). The tracking error \(z_{1}\) is introduced as a measure of accuracy. Calculating its time derivative, one has  
\[z_{1}=p-p_{d}+R_{1}\,\delta,\;\;\;z_{2}=\dot{z}_{1}=R_{1}v_{tr}-\dot{p}_{d}+R_ {1}S\,\delta\,\zeta_{3},\;\;\;\delta=[\delta_{x}\;\;0]^{T}. \tag{18}\]  
Recalling now (6) and (7), denoting \(\mathrm{Y}_{3}=[\mathrm{Y}_{13}\;\;\mathrm{Y}_{23}]^{T}\), one obtains \(v_{tr}=\zeta_{tr}+\mathrm{Y}_{3}\zeta_{3}\) and its time derivative \(\dot{v}_{tr}=\dot{\zeta}_{tr}+\mathrm{Y}_{3}\dot{\zeta}_{3}\). Thus, the time derivative of \(z_{2}\) has the following form:  
\[\dot{z}_{2}=R_{1}Sv_{tr}\zeta_{3}+m^{-1}R_{1}(-mSv_{tr}\zeta_{3}+f+m\mathrm{Y} _{3}N_{3}^{-1}\tau_{r})-\dot{p}_{d}+R_{1}S^{2}\,\delta\,\zeta_{3}^{2}+R_{1}S\, \delta\,N_{3}^{-1}\tau_{r}. \tag{19}\]  
The following LFC is proposed:  
\[V=\frac{1}{2}k_{1}z_{1}^{T}z_{1}+\frac{1}{2}z_{2}^{T}z_{2}+\mu z_{1}^{T}z_{2}, \tag{20}\]  
where \(k_{1}>0\), \(\mu>0\) are some coefficients. Determining analogously as in [22] the time derivative of \(V\), one obtains  
\[\dot{V}=-A(z_{1},z_{2})+(\mu z_{1}+z_{2})^{T}(\dot{z}_{2}+k_{1}z_{1}+k_{2}z_{2 }), \tag{21}\]  
with  
\[A(z_{1},z_{2})=k_{1}\mu z_{1}^{T}z_{1}+k_{2}\mu z_{1}^{T}z_{2}+(k_{2}-\mu)z_{2 }^{T}z_{2}, \tag{22}\]  
and \(k_{2}>0\). Moreover, in order to guarantee that \(A(z_{1},z_{2})\) and \(V(z_{1},z_{2})\) are positively definite, it is assumed that [22]  
\[\mu<\sqrt{k_{1}},\;\;\;\;\mu<\frac{4k_{1}k_{2}}{4k_{1}+k_{2}^{2}}. \tag{23}\]  
Using \(\dot{z}_{2}\) and grouping the terms, Equation (21) can be written in the form:  
\[\dot{V}=-A(z_{1},z_{2})+(\mu z_{1}+z_{2})^{T}R_{1}\cdot\bigg{(}B \,T\bigg{[} \begin{array}{c}\cos\psi\\ \sin\psi\end{array} \bigg{]}+m^{-1}f_{0}\] \[-R_{1}^{T}\dot{p}_{d}-\bigg{[} \begin{array}{c}\delta_{x}\\ 0\end{array} \bigg{]}\zeta_{3}^{2}+\bigg{[} \begin{array}{c}\mathrm{Y}_{13} \\ \mathrm{Y}_{23}+\delta_{x}\end{array} \bigg{]}N_{3}^{-1}\tau_{r0}+R_{1}^{T}(k_{1} z_{1}+k_{2}z_{2})\bigg{)}, \tag{24}\]  
where\[B=b_{T}\bigg{[} \begin{array}{cc}\beta_{11}&\beta_{12}\\ \beta_{21}&\beta_{22}\end{array} \bigg{]}=b_{T}\bigg{[} \begin{array}{cc}m^{-1} +N_{3}^{-1}Y_{13}^{2}&-N_{3}^{-1}Y_{13}(a-Y_{23})\\ (Y_{23}+\delta_{x})N_{3}^{-1}Y_{13}&m^{-1}-(Y_{23}+\delta_{x})N_{3}^{-1}(a-Y_{ 23})\end{array} \bigg{]}, \tag{25}\] \[f_{0}=\bigg{[} \begin{array}{cc}mx_{g}r^{2}-d_{u0}\text{sign}(u )-d_{u}u-d_{ur}r\\ -my_{g}r^{2}-d_{v0}\text{sign}(v)-d_{v}v-d_{vr}r\end{array} \bigg{]},\] (26) \[\tau_{r0}=-mx_{g}ur+my_{g}vr-d_{ur}u-d_{ur}v-d_{r0}\text{sign}(r) -d_{r}r. \tag{27}\]  
Under such conditions, a stabilizing controller should be designed in order to track the system error. Theorem 1: _It is assumed that the equations of motion of a hovercraft expressed in terms of quasi-velocities can be written as (6)-(11), with the external force (16) and (17) taking into consideration the states (18) and the bounded desired trajectory \(p_{d}\) and bounded its time derivative. If an invertible matrix \(B\) (25) is selected, this means that_  
\[\beta_{11}\beta_{22}-\beta_{12}\beta_{21}\
eq 0, \tag{28}\]  
_and the control thrust \(T\) as well as the rudder angle \(\psi\) are used in the form_  
\[T\bigg{[} \begin{array}{c}\cos\psi\\ \sin\psi\end{array} \bigg{]} =-B^{-1}\bigg{(}m^{-1}f_{0}-R_{1}^{T}\bar{p}_{d}-\bigg{[} \begin{array}{c}\delta_{x}\\ 0\end{array} \bigg{]}\zeta_{3}^{2}+\bigg{[} \begin{array}{c}Y_{13}\\ Y_{23}+\delta_{x}\end{array}