## 3 Equations of Motion in Terms of Quasi-Velocities
### Kinematic Control Algorithm  
In order to design the kinematic controller the following Lyapunov function candidate is proposed, e.g., [54]:  
\[V_{k}=\frac{1}{2}x_{e}^{2}+\frac{1}{2}y_{e}^{2}+(1-\cos\psi_{e}). \tag{22}\]  
Its time derivative, after using (19)-(21), has the form:  
\[\dot{V}_{k}=x_{e}\dot{x}_{e}+y_{e}\dot{y}_{e}+\dot{\psi}_{e}\sin\psi_{e}=(u-u_{ d}\cos\psi_{e})x_{e}+(r-r_{d}+u_{d}y_{e})\sin\psi_{e}+vy_{e}. \tag{23}\]  
Next, because the velocities \(u\), \(r\) are understood as the virtual control variables to ensure that \(\dot{V}_{k}<0\), then the desired velocities are proposed as [8; 55]:  
\[v_{du} =u_{d}\cos\psi_{e}-k_{x}x_{e}, \tag{24}\] \[v_{dr} =r_{d}-u_{d}y_{e}-k_{\psi}\sin\psi_{e}, \tag{25}\]where \(k_{x}>0\), \(k_{\psi}>0\) are some gain coefficients. If the virtual variables are equal to the desired variables, then substituting the Equations (24) and (25) into the expression (23), the following can be written:  
\[\dot{V}_{k}=-k_{x}x_{e}^{2}-k_{\psi}\sin^{2}\psi_{e}+vy_{e}\leq-k_{x}x_{e}^{2}-k _{\psi}\sin^{2}\psi_{e}+|vy_{e}|. \tag{26}\]  
The following lemma is proposed because the direct control in sway direction cannot be ensured. **Lemma 1**. : _The velocity \(v\) is bounded if the disturbances \(f_{v}\) are bounded and consequently also the quasi-velocity \(\zeta_{2}\) is limited._  
Proof. : The Lyapunov function candidate is as follows:  
\[V_{\zeta_{2}}=\frac{1}{2}\zeta_{2}^{2}. \tag{27}\]  
Since the drag coefficients are dependent on the velocity \(v\) and the equation of motion (11) in IQV, to make the analysis easier it can be transformed and expressed by the formula:  
\[\dot{\zeta}_{2}=N_{2}^{-1}(-m_{11}ur-d_{22}v-d_{23}r+f_{v}). \tag{28}\]  
The time derivative of (27) can be given in the form:  
\[\dot{V}_{\zeta_{2}}=\zeta_{2}\dot{\zeta_{2}}=\zeta_{2}N_{2}^{-1}(-m_{11}ur-d_{ 22}v-d_{23}r+f_{v}). \tag{29}\]  
Moreover, the drag coefficients can be given in the form \(d_{22}=Y_{22}(r)+Y_{|v|j}|v|\), \(d_{23}=Y_{23}(r)+Y_{|v|r}|v|\) where \(Y_{22}(r)=Y_{v}+Y_{|v|p}|r|\) and \(Y_{23}(r)=Y_{r}+Y_{|r|r}|r|\). Therefore, using the above and (8) it can be written that:  
\[\dot{V}_{\zeta_{2}} = \zeta_{2}N_{2}^{-1}(-m_{11}ur-d_{22}v-d_{23}r+f_{v}) \tag{30}\] \[= N_{2}^{-1}(v-\hat{Y}_{23}r)\big{(}-m_{11}ur-(Y_{22}(r)+Y_{|v|v|} |v|)v-(Y_{23}(r)+Y_{|v|r}|v|)r+f_{v}\big{)}.\]  
Let \(A=-m_{11}ur-Y_{23}(r)r-Y_{|v|r}|v|+f_{v}\). The function \(\dot{V}_{\zeta_{2}}<0\) if the following conditions are fulfilled:  
1. \(v<\hat{Y}_{23}r<0\) and \(\big{(}A-Y_{22}(r)v-Y_{|v|p}|v|v\big{)}>0\), and \((v-\hat{Y}_{23}r)<0\), (31) 2. \(\hat{Y}_{23}r<v<0\) and \(\big{(}A-Y_{22}(r)v-Y_{|v|v|}v\big{)}<0\), and \((v-\hat{Y}_{23}r)>0\), (32) 3. \(v>\hat{Y}_{23}r>0\) and \(\big{(}A-Y_{22}(r)v-Y_{|v|v|}v\big{)}<0\), and \((v-\hat{Y}_{23}r)>0\), (33) 4. \(0<v<\hat{Y}_{23}r\) and \(\big{(}A-Y_{22}(r)v-Y_{|v|v|}v\big{)}>0\), and \((v-\hat{Y}_{23}r)<0\). (34)  
Taking into account boundedness of \(u\) and \(r\) (they are controlled and bounded) and the disturbance function \(f_{v}\) bounded by condition (31)-(34) one obtains:  
\[\dot{V}_{k}=x_{e}\dot{x}_{e}+y_{e}\dot{y}_{e}+\dot{\psi}_{e}\sin\psi_{e}=(u-u_{ d}\cos\psi_{e})x_{e}+(r-r_{d}+u_{d}y_{e})\sin\psi_{e}+vy_{e}. \tag{23}\]0 \[\dot{V}_{k}=x_{e}\dot{x}_{e}+y_{e}\dot{y}_{e}+\dot{\psi}_{e}\sin\psi_{e}=(u-u_{ d}\cos\psi_{e})x_{e}+(r-r_{d}+u_{d}y_{e})\sin\psi_{e}+vy_{e}. \tag{23}\]1  
where \(\Delta_{1}=Y_{22}^{2}(r)-4Y_{|v|p}A\) and \(\Delta_{2}=Y_{22}^{2}(r)+4Y_{|v|p}A\). It follows from this condition that \(v\) is decreasing and therefore bounded. Considering that \(\zeta_{2}=v-\hat{Y}_{23}r\), it can be concluded that also \(\zeta_{2}\) is limited. Because the desired trajectory \(u_{d}\) arises from trajectory planning and values of the velocity \(u\) are limited by thrusters, then the desired trajectory is bounded. Moreover, the error \(y_{e}\) is bounded. From the analysis it follows that \(v\) and \(y_{e}\) are both bounded.