## 3 Vehicle Model in Terms of Quasi-Velocities
\tag{40}\]  
In order to keep the same idea of changing variables as in [56], it was assumed that \(u_{r}=\xi_{3}\cos z_{1}+\xi_{4}\sin z_{1}\), \(v_{r}=-\xi_{3}\sin z_{1}+\xi_{4}\cos z_{1}-z_{2}l\), and \(r=z_{2}\). The change of input to linearize the external dynamics has the form:  
\[\left[ \begin{array}{c}\tau_{u}^{*}\\ \tau_{r}^{*}\end{array} \right]=\left[ \begin{array}{cc}\cos\psi&-(l+Y_{23}) \sin\psi\\ \sin\psi&(l+Y_{23})\cos\psi\end{array} \right]^{-1}\left[ \begin{array}{c}-F_ {\xi_{3}}^{*}(z_{1},\xi_{3},\xi_{4})+\mu_{1}\\ -F_{\xi_{4}}^{*}(z_{1},\xi_{3},\xi_{4})+\mu_{2}^{*}\end{array} \right], \tag{41}\]  
where \(\mu_{1}\) is defined by (14), whereas \(\mu_{2}^{*}\) by:  
\[\mu_{2}^{*}=-k_{xy}(l+Y_{23})(\xi_{4}-\xi_{4_{d}})-k_{xy}(\xi_{2}-\xi_{2_{d}}) -k_{xy}(\xi_{2_{d}}-\xi_{2_{d}})+\xi_{4_{d}}. \tag{42}\]  
However, from the used decomposition method, it follows that \(\tau_{u}^{*}=N_{1}^{-1}\tau_{u}\) and \(\tau_{r}^{*}=N_{3}^{-1}\tau_{r}\). This relationship can be used for the input signals normalization, and it is described by the equation:  
\[\left[ \begin{array}{c}\tau_{u}\\ \tau_{r}\end{array} \right]=s_{f}\left[ \begin{array}{cc}N_{1}&0\\ 0&N_{3}\end{array} \right]\left[ \begin{array}{c}\mu_{1}\\ \mu_{2}\end{array} \right], \tag{43}\]  
where \(s_{f}\) means a scaling factor (a constant value). In addition, the following relationships arise from this \(\tau_{u}=s_{f}N_{1}\,\mu_{1}=\beta_{1}\mu_{1}\) and \(\tau_{r}=s_{f}N_{3}\,\mu_{2}=\beta_{2}\mu_{2}\). It is necessary if the values \(N_{1}\) and \(N_{3}\) are too large or too small, which would make the control signal ineffective. The benefit of such normalization is that the control algorithm action will be directly related to the dynamics of the vehicle. Note that in reference [56], the control gains are selected without reference to vehicle dynamics. **Information accessible using QV**. Some benefits of decomposition of the inertia matrix can be pointed out:  
1. Diagonalization of the inertia matrix instead of computing the inverse inertia matrix;
2. An insight (additional information) into the dynamics of the vehicle;
3. A heuristic algorithm in matching the QV and classical regulator gains;
4. Control signal related to the vehicle's dynamics. In order to give an insight into the vehicle dynamics, consider some quantities known from the literature that were applied for fully actuated marine vehicles [62; 63]. We can use the following of them:  
1. Norm of the matrix \(\left\|Y^{-1}\right\|\) which determine couplings in the dynamic model of the vehicle. The matrix \(Y\) depend on elements of the symmetric matrix \(M\). 2. The kinetic energy \(K\) expressed in terms of the vector \(\zeta\) for 3-DOF vehicle: \[K{=}\frac{1}{2}v_{r}^{T}Mv_{r}{=}\frac{1}{2}\xi^{T}N\xi{=}\frac{1}{2}\sum_{i=1 }^{3}N_{i}\xi_{i}^{2}{=}\sum_{i=1}^{3}K_{i}.\] (44) Because each variable \(\zeta_{i}\) takes into account a component of the kinetic energy arising from all other velocities, which are coupled with the \(i\)-th variable, then we can examine the contribution of kinetic energy associated with each variable and the total kinetic energy of the vehicle. It is possible to determine \(K_{i}\) reduced by each variable. 3. The average kinetic energy associated with each variable and the total kinetic energy: \[K_{m}=\text{mean}\left(K\right)=\sum_{i=1}^{3}\text{mean}\left(K_{i}\right).\] (45)4. Deformation of each velocity. The variables \(\zeta_{i}\) allow one to evaluate the impact of other velocities at the \(i\)-th velocity, i.e., \[\zeta_{i}=\text{Y}_{ii}^{-1}\
u_{ri}+\sum_{j=i+1}^{3}\text{Y}_{ij}^{-1}\
u_{rj}.\] (46) This result means that each \(\zeta_{i}\) takes into account a coupling between itself and the other velocities. Thus, using the controller, one can observe the effects of couplings for dynamics of the vehicle. Effect of the dynamical couplings is obtained by \(\text{Y}_{ij}\).