## 3 Vehicle Model in Terms of Quasi-Velocities  
For a symmetric inertia matrix \(M\), it is possible to use some decomposition method, e.g., [61], which was successfully applied for marine vehicles for example in [62; 63] (Equation (1) is valid). In the method \(M=\mathrm{Y}^{-T}N\mathrm{Y}^{-1}\), which leads to a diagonal matrix \(N=\mathrm{Y}^{T}M\mathrm{Y}\). Consequently, instead of (2), one gets:\[\dot{\eta}=R(\psi)v_{r}+V,\quad v_{r}=\Upsilon\zeta, \tag{18}\] \[\dot{\zeta}+N^{-1}\mathrm{Y}^{T}C(v_{r})v_{r}+N^{-1}\mathrm{Y}^{T}Dv _{r}=N^{-1}\mathrm{Y}^{T}\tau_{ua}, \tag{19}\]  
where \(\zeta=[\zeta_{1},\zeta_{2},\zeta_{3}]^{T}\), with  
\[\mathrm{Y}=\left[ \begin{array}{ccc}1&0&0\\ 0&1&\mathrm{Y}_{23}\\ 0&0&1\end{array} \right],\quad\quad N=\mathrm{diag}\{N_{1},N_{2},N_{3}\}. \tag{20}\]  
The introduced quantities are defined as: \(N_{1}=m_{11}\), \(N_{2}=m_{11}\)\(N_{3}=m_{33}-(m_{23}^{2}/m_{22})\), \(\mathrm{Y}_{23}=-(m_{23}/m_{22})\), \(\zeta_{1}=u_{r}\), \(\zeta_{2}=v_{r}-\mathrm{Y}_{23}r\), \(\zeta_{3}=r\), which means that only \(v_{r}\
eq\zeta_{2}\) because \(v_{r}=\zeta_{2}+\mathrm{Y}_{23}\zeta_{3}\). Instead of \(M^{-1}Bf=[\tau_{u},0,\tau_{r}]^{T}\), it is \(N^{-1}Bf=[\tau_{u},0,\tau_{r}]^{T}\). Equations replacing (1) and (2) are written as follows:  
\[\dot{x}=\zeta_{1}\cos\psi-(\zeta_{2}+\mathrm{Y}_{23}\zeta_{3}) \sin\psi+V_{x}, \tag{21}\] \[\dot{y}=\zeta_{1}\sin\psi+(\zeta_{2}+\mathrm{Y}_{23}\zeta_{3}) \cos\psi+V_{y},\] (22) \[\dot{\psi}=\zeta_{3},\] (23) \[\dot{\zeta}_{1}=F_{1}(\zeta)+\tau_{u}^{*},\] (24) \[\dot{\zeta}_{2}=F_{2}(\zeta)\] (25) \[\dot{\zeta}_{3}=F_{3}(\zeta)+\tau_{r}^{*}, \tag{26}\]  
where:  
\[F_{1}(\zeta)=-N_{1}^{-1}(c_{13}^{*}\zeta_{3}+d_{11}\zeta_{1}), \tag{27}\] \[F_{2}(\zeta)=-N_{2}^{-1}(c_{23}^{*}\zeta_{3}+d_{22}(\zeta_{2}+ \mathrm{Y}_{23}\zeta_{3})+d_{23}\zeta_{3}),\] (28) \[F_{3}(\zeta)=N_{3}^{-1}(c_{13}^{*}\zeta_{3}+(c_{23}^{*}-(d_{32} +\mathrm{Y}_{23}d_{22}))(\zeta_{2}+\mathrm{Y}_{23}\zeta_{3})-(\mathrm{Y}_{23} (c_{23}^{*}+d_{23})+d_{33})\zeta_{3}), \tag{29}\]  
and \(c_{13}^{*}=-(m_{22}(\zeta_{2}+\mathrm{Y}_{23}\zeta_{3})+m_{23}\zeta_{3})\), \(c_{23}^{*}=m_{11}\zeta_{1}\). The change of coordinates is the same as in [56] because only the velocity transformation was applied and the system dynamics remains unchanged. Therefore, in terms of the QV, one has:  
\[z_{1}=\psi, \tag{30}\] \[\dot{y}=\zeta_{1}\sin\psi+(\zeta_{2}+\mathrm{Y}_{23}\zeta_{3}) \cos\psi+V_{y},\]0 (31) \[\dot{y}=\zeta_{1}\sin\psi+(\zeta_{2}+\mathrm{Y}_{23}\zeta_{3}) \cos\psi+V_{y},\]1 (32) \[\dot{y}=\zeta_{1}\sin\psi+(\zeta_{2}+\mathrm{Y}_{23}\zeta_{3}) \cos\psi+V_{y},\]2 (33) \[\dot{y}=\zeta_{1}\sin\psi+(\zeta_{2}+\mathrm{Y}_{23}\zeta_{3}) \cos\psi+V_{y},\]3 (34) \[\dot{y}=\zeta_{1}\sin\psi+(\zeta_{2}+\mathrm{Y}_{23}\zeta_{3}) \cos\psi+V_{y},\]4  
and also:  
\[\dot{y}=\zeta_{1}\sin\psi+(\zeta_{2}+\mathrm{Y}_{23}\zeta_{3}) \cos\psi+V_{y},\]5 \[\dot{y}=\zeta_{1}\sin\psi+(\zeta_{2}+\mathrm{Y}_{23}\zeta_{3}) \cos\psi+V_{y},\]6  
\[\left[ \begin{array}{c}\dot{\xi}_{1}\\ \dot{\xi}_{2}\end{array} \right]=\left[ \begin{array}{c}\xi_{1}\\ \dot{\xi}_{2}\end{array} \right]+\left[ \begin{array}{c}V_{x}\\ V_{y}\end{array} \right], \tag{38}\] \[\left[ \begin{array}{c}\dot{\xi}_{3}\\ \dot{\xi}_{4}\end{array} \right]=\left[ \begin{array}{c}F_{\xi_{3}}^{*}(z_{1}, \xi_{3},\xi_{4})\\ F_{\xi_{4}}^{*}(z_{1},\xi_{3},\xi_{4})\end{array} \right]+\left[ \begin{array}{ c}\cos z_{1}&-(l+\mathrm{Y}_{23})\sin z_{1}\\ \sin z_{1}&(l+\mathrm{Y}_{23})\cos z_{1}\end{array} \right]\left[ \begin{array}{ c}\tau_{u}^{*}\\ \tau_{r}^{*}\end{array} \right], \tag{39}\]where:  
\[\left[ \begin{array}{c}F_{\xi_{3}}^{*}(\cdot)\\ F_{\xi_{4}}^{*}(\cdot)\end{array} \right]=\left[ \begin{array}{cc}\cos\psi&-\sin \psi\\ \sin\psi&\cos\psi\end{array} \right]\left[ \begin{array}{c}F_{1}(\zeta)-\zeta_{ 2}\zeta_{3}-(l+Y_{23})\zeta_{3}^{2}\\ F_{2}(\zeta)+\zeta_{1}\xi_{3}+(l+Y_{23})F_{3}(\zeta)\end{array} \right].