## 3 Equations of Motion in Terms of Quasi-Velocities  
To use the decomposition method, one must assume that the matrix \(M\) is symmetric. It is possible to meet this assumption as long as one takes into account all model inaccuracies and external disturbances in the disturbance function with limitations on its values, as will be shown in the next section. The \(M\) symmetric matrix can be decomposed using one of the known methods, such as [45] which was successfully applied for marine vehicles, for example in [49] (Equation (1) is valid). This method was developed for mechanical systems [45]. It is easy to implement [49] and has low computational complexity which reduces simulation time. In addition, it allows physical interpretation of IQV, and the physical units are the same as in classical equations of motion. The inertia matrix is factorized as \(M=\mathrm{Y}^{-T}N_{diag}\mathrm{Y}^{-1}\) which leads to a diagonal matrix \(N_{diag}=\mathrm{Y}^{T}\mathrm{M}\mathrm{Y}\). However, since the control algorithm must use known quantities, the decomposed matrix is of the form \(N=\hat{\mathrm{Y}}^{T}M\hat{\mathrm{Y}}\) (the matrix \(\hat{\mathrm{Y}}\) contains nominal parameters), so the resulting errors \(\Delta\mathrm{Y}\) are included in the new disturbance vector which is defined as \(f=f_{E}+\Delta\mathrm{Y}\). Then, the kinematic Equation (1) remains unchanged, but instead of (2) one obtains:  
\[N_{\xi}^{T}+\hat{\mathrm{Y}}^{T}C(v)v+\hat{\mathrm{Y}}^{T}D(v)v =\hat{\mathrm{Y}}^{T}\tau+\hat{\mathrm{Y}}^{T}f, \tag{4}\] \[v=\hat{\mathrm{Y}}_{\xi},\] (5) \[\hat{\mathrm{Y}}=\left[ \begin{array}{ccc}1&0&0\\ 0&1&\hat{\mathrm{Y}}_{23}\\ 0&0&1\end{array} \right],\hskip 14.226378 ptN=\mathrm{diag}\{N_{1},N_{2},N_{3}\}, \tag{6}\]  
where the new vector of quasi-velocities \(\zeta=[\zeta_{1},\zeta_{2},\zeta_{3}]^{T}\). The introduced quantities are defined as: \(N_{1}=m_{11}\), \(N_{2}=m_{22}\), \(N_{3}=m_{33}-(m_{23}^{2}/m_{22})\), \(\hat{\mathrm{Y}}_{23}=-(\hat{m}_{23}/\hat{m}_{22})\). Equations of motion replacing (4) and (5) may be written as:  
\[\zeta_  
\(\{E\}\) must be smooth and continuous. The desired attitude angle trajectory is determined from the desired trajectory from the Equation (such as in [15]):  
\[\psi_{d}=\arctan\left(\frac{\dot{y}_{d}}{\dot{x}_{d}}\right). \tag{16}\]  
Moreover, the coordinate transformation is applied:  
\[x_{e} =\cos\psi\ x_{e}^{E}+\sin\psi\ y_{e}^{E} \tag{17}\] \[y_{e} =-\sin\psi\ x_{e}^{E}+\cos\psi\ y_{e}^{E}, \tag{18}\]  
because \(x_{e}\), \(y_{e}\) are defined in the body frame \(\{B\}\). Calculating the time derivative of the kinematic tracking errors and making use of (1), the following can be written:  
\[\dot{x}_{e} =u-u_{d}\cos\psi_{e}+ry_{e} \tag{19}\] \[\dot{y}_{e} =v+u_{d}\sin\psi_{e}-rx_{e},\] (20) \[\dot{\psi}_{e} =r-r_{d}, \tag{21}\]  
where \(u_{d}=\sqrt{x_{d}+\dot{y}_{d}}\) and \(r_{d}=\dot{\psi}_{d}\) mean the reference velocities. **Assumption 1**. : _The model parameter perturbations are bounded, i.e., \(|m_{ij}-\dot{m}_{ij}|\leq\widetilde{m}_{ij}\), \(|d_{ij}-\hat{d}_{ij}|\leq\widetilde{d}_{ij}\) where \(i,j=1,2,3\), \(|N_{3}-\widetilde{N}_{3}|\leq\widetilde{N}_{3}\), and \(\hat{\cdot}\) means the nominal value of the real parameter. This means that there exists the upper bound of the parameter perturbation. Moreover, all the vehicle states are measurable and can be applied in the controller._  
**Assumption 2**. : _The external disturbances \(f_{u_{e}}f_{v_{e}}f_{r}\) are unknown but bounded (as, e.g., in [52; 53]). This is also true for their first time derivatives [53]. The disturbances may be constant or time-varying._  
**Assumption 3**. : _In order to avoid chattering during the vehicle motion under the proposed controller, the discontinuous function signum is approximated by the hyperbolic tangent function, i.e., \(sgn(S_{i})\approx tanh(\gamma_{i}S_{i})\) and \(\gamma_{i}\) is a positive scalar where \(i=1,2\)[9]._  
**Remark 1**. : _The control strategy consists of two control algorithms.