# Modification of Nonlinear Controller for Asymmetric Marine Vehicles Moving in Horizontal Plane
## 2 Marine Vehicle Model
### Transformed Equations of Motion  
If the inertia matrix \(M\) is symmetric, then it is possible to decompose it, e.g., using the method given in [49]. This approach was successfully applied for marine vehicles, for example, in [50; 51]. The decomposition leads to \(M=\mathrm{Y}^{-T}N\mathrm{Y}^{-1}\) and a diagonal matrix \(N=\mathrm{Y}^{T}M\mathrm{Y}\) is obtained. As a result, instead of (2), one has:  
\[\dot{\eta}=R(\psi)\
u_{r}+V,\ \ \ \
u_{r}=\mathrm{Y}\zeta, \tag{5}\] \[\dot{\zeta}+N^{-1}\mathrm{Y}^{T}C(\
u_{r})\
u_{r}+N^{-1}\mathrm{Y }^{T}D\
u_{r}=N^{-1}\mathrm{Y}^{T}\tau, \tag{6}\]  
where \(\zeta=[\zeta_{1},\zeta_{2},\zeta_{3}]^{T}\), with:  
\[\mathrm{Y}=\left[ \begin{array}{ccc}1&0&\mathrm{Y}_{13}\\ 0&1&\mathrm{Y}_{23}\\ 0&0&1\end{array} \right],\ \ \ \ \ N=\mathrm{diag}\{N_{1},N_{2},N_{3}\}, \tag{7}\]  
where \(N_{1}=m_{11}\), \(N_{2}=m_{22}\)\(N_{3}=m_{33}-(m_{13}^{2}/m_{11})-(m_{23}^{2}/m_{22})\), \(\mathrm{Y}_{13}=-(m_{13}/m_{11})\), \(\mathrm{Y}_{23}=-(m_{23}/m_{22})\), \(\zeta_{1}=u_{r}-\mathrm{Y}_{13}r\), \(\zeta_{2}=v_{r}-\mathrm{Y}_{23}r\), \(\zeta_{3}=r\) which means that \(v_{r}\
eq\zeta_{2}\) because \(u_{r}=\zeta_{1}+\mathrm{Y}_{13}\zeta_{3}\) and \(v_{r}=\zeta_{2}+\mathrm{Y}_{23}\zeta_{3}\). Moreover, \(N^{-1}Bf=[\tau_{u},0,\tau_{r}+\mathrm{Y}_{13}\tau_{u}]^{T}=[\tau_{\zeta u},0, \tau_{\zeta r}]^{T}\). **Remark 1**. : _The \(Q\mathrm{V}\)\(\zeta_{1}\), \(\zeta_{2}\), and \(\zeta_{3}\) have a physical sense. They can be understood as a speed disturbance due to dynamic couplings in the vehicle, namely, \(\zeta_{1}=u_{r}+\Delta u_{r}\), \(\zeta_{2}=v_{r}+\Delta v_{r}\), \(\zeta_{3}=r\)._The equations replacing (1) and (2) are as follows:  
\[\dot{x} =(\zeta_{1}+\mathrm{Y}_{13}\zeta_{3})\cos\psi-(\zeta_{2}+\mathrm{Y} _{23}\zeta_{3})\sin\psi+V_{x}, \tag{8}\] \[\dot{y} =(\zeta_{1}+\mathrm{Y}_{13}\zeta_{3})\sin\psi+(\zeta_{2}+\mathrm{Y }_{23}\zeta_{3})\cos\psi+V_{y},\] (9) \[\dot{\psi} =\zeta_{3},\] (10) \[\dot{\zeta}_{1} =F_{1}(\zeta)+\tau_{\psi},\] (11) \[\dot{\zeta}_{2} =F_{2}(\zeta)\] (12) \[\dot{\zeta}_{3} =F_{3}(\zeta)+\tau_{\psi}, \tag{13}\]  
where:  
\[F_{1}(\zeta) =-\mathrm{N}_{1}^{-1}\Big{(}c_{13}(\zeta)\,\zeta_{3}+d_{11}( \zeta_{1}+\mathrm{Y}_{13}\zeta_{3})+d_{13}\zeta_{3}\Big{)}, \tag{14}\] \[F_{2}(\zeta) =-\mathrm{N}_{2}^{-1}\Big{(}c_{23}(\zeta)\zeta_{3}+d_{22}( \zeta_{2}+\mathrm{Y}_{23}\zeta_{3})+d_{23}\zeta_{3}\Big{)},\] (15) \[F_{3}(\zeta) =\mathrm{N}_{3}^{-1}\Big{(}c_{13}(\zeta)(\zeta_{1}+\mathrm{Y}_{1 3}\zeta_{3})+c_{23}(\zeta)(\zeta_{2}+\mathrm{Y}_{23}\zeta_{3})-c_{33}(\zeta) \zeta_{3}\] \[\quad-(\mathrm{Y}_{13}d_{11}+d_{31})(\zeta_{1}+\mathrm{Y}_{13} \zeta_{3})-(\mathrm{Y}_{23}d_{22}+d_{32})(\zeta_{2}+\mathrm{Y}_{23}\zeta_{3})\] \[\dot{y} =(\zeta_{1}+\mathrm{Y}_{13}\zeta_{3})\sin\psi+(\zeta_{2}+\mathrm{Y }_{23}\zeta_{3})\cos\psi+V_{y},\]0  
and \(c_{13}(\zeta)=-(m_{22}(\zeta_{2}+\mathrm{Y}_{23}\zeta_{3})+m_{23}\zeta_{3})\), \(c_{23}(\zeta)=m_{11}(\zeta_{1}+\mathrm{Y}_{13}\zeta_{3})-m_{13}\zeta_{3}\), \(c_{33}(\zeta)=\mathrm{Y}_{13}c_{13}(\zeta)+\mathrm{Y}_{23}c_{23}(\zeta)\). The change in coordinates after using the velocity transformation is applied.