## 3 Equations of Motion in Terms of Quasi-Velocities
### Dynamic Control Algorithm  
The task of the dynamic controller is to move the vehicle to the desired velocities using \(\tau_{u}\) and \(\tau_{r}\), while the velocity errors are assumed as:  
\[[u_{e},\mathbf{r}_{e}]^{T}=[u-u_{d},r-r_{d}]^{T}. \tag{39}\]  
Because, the algorithm is expressed using IQV therefore \(\zeta_{1e}=\zeta_{1}-\zeta_{1d}\) and \(\zeta_{3e}=\zeta_{3}-\zeta_{3d}\) are applied. From (7)-(9) it is noticeable that it is also \(u_{e}=u-u_{d}\) and \(r_{e}=r-r_{d}\). For the surge force \(\tau_{u}\) the following integral sliding surface is defined:  
\[S_{1}=\zeta_{1e}+\lambda_{1}\int_{0}^{t}\zeta_{1e}(\sigma)d\sigma, \tag{40}\]  
where \(\lambda_{1}>0\) is a control coefficient. The time derivative of \(S_{1}\) taking into account (10) is \(\dot{S}_{1}=\dot{\zeta}_{1e}+\lambda_{1}\zeta_{1e}\) (where \(\dot{\zeta}_{1e}=\dot{\zeta}_{1}-\dot{\zeta}_{1d}\)):  
\[\dot{S}_{1}=N_{1}^{-1}\big{(}F_{1}(\zeta)-N_{1}\dot{\zeta}_{1d}+N_{1}\lambda_{ 1}\zeta_{1e}+\tau_{u}+f_{u}\big{)}. \tag{41}\]  
To ensure the convergence of the error to zero along the sliding surface \(S_{1}\) the following control algorithm is proposed:  
\[\tau_{u}=-\hat{F}_{1}(\zeta)+\hat{N}_{1}\dot{\zeta}_{1d}-\hat{N}_{1}\lambda_{ 1}\zeta_{1e}-\hat{f}_{u}-\Gamma_{1}\text{sgn}(S_{1}), \tag{42}\]  
where the last term, designed according to Assumption 3, is related to the parameters' perturbation reducing, and \(\hat{f}_{u}\) means the estimated value of the external disturbance force \(f_{u}\). In order to prove the stability of the vehicle under the parameters' perturbation and external disturbances, the following Lyapunov function candidate is assumed:  
\[V_{1}=\frac{1}{2}N_{1}S_{1}^{2}+\frac{1}{2}\beta_{1}\vec{f}_{u}^{2}, \tag{43}\]where \(\tilde{f}_{u}=f_{u}-\hat{f}_{u}\) denotes the estimated error of the disturbance force and \(\beta_{1}>0\) a constant coefficient (control parameter). The time derivative of \(V_{1}\) calculated using (41) and (42) is:  
\[\hat{V}_{1}=N_{1}S_{1}\dot{S}_{1}+\beta_{1}\tilde{f}_{u}\dot{\tilde {f}}_{u}=S_{1}\big{(}F_{1}(\zeta)-\hat{F}_{1}(\zeta)+(\hat{N}_{1}-N_{1})\dot{ \zeta}_{1d}+\lambda_{1}(N_{1}-\hat{N}_{1})\zeta_{1e}\] \[-\Gamma_{1}\text{sgn}(S_{1})\big{)}+\tilde{f}_{u}S_{1}+\beta_{1} \tilde{f}_{u}\dot{\tilde{f}}_{u}, \tag{44}\]  
where:  
\[F_{1}(\zeta)-\hat{F}_{1}(\zeta)=(m_{22}-\hat{m}_{22})vr+(m_{23}-\hat{m}_{23})r^ {2}+(\hat{d}_{11}-d_{11})u. \tag{45}\]  
Next, making use of Assumption 1 the gain term \(\Gamma_{1}\) is applied:  
\[\Gamma_{1}=\widetilde{m}_{22}|vr|+\widetilde{m}_{23}|r^{2}|+\widetilde{d}_{1 1}|u|+\widetilde{N}_{1}|\dot{\zeta}_{1d}|+\lambda_{1}\widetilde{N}_{1}|\zeta_ {1e}|+\delta_{1}. \tag{46}\]  
Then under Assumption 2 (if \(f_{u}=0\), i.e., for constant disturbances), Equation (44) becomes:  
\[\hat{V}_{1}=N_{1}S_{1}\dot{S}_{1}+\beta_{1}\tilde{f}_{u}\dot{ \tilde{f}}_{u}\leq-\delta_{1}|S_{1}|+\tilde{f}_{u}S_{1}+\beta_{1}\tilde{f}_{u }\dot{\tilde{f}}_{u}=-\delta_{1}|S_{1}|+\tilde{f}_{u}S_{1}+\beta_{1}\tilde{f}_ {u}(\dot{f}_{u}-\dot{\tilde{f}}_{u})\] \[S_{1}=\zeta_{1e}+\lambda_{1}\int_{0}^{t}\zeta_{1e}(\sigma)d\sigma, \tag{40}\]0  
Designing the adaptive term \(\dot{\tilde{f}}_{u}=\beta_{1}^{-1}S_{1}\) one obtains:  
\[S_{1}=\zeta_{1e}+\lambda_{1}\int_{0}^{t}\zeta_{1e}(\sigma)d\sigma, \tag{40}\]1  
This result means that \(\hat{V}_{1}=0\) provided \(\zeta_{1e}\) converges to zero along the sliding surface \(S_{1}\). However, \(\zeta_{1e}=u_{e}\), thus also \(u_{e}\) tends to zero. Therefore, the surge velocity error \(u_{e}\) converges asymptotically to zero if the signal \(\tau_{u}\) (42) is used. For the yaw moment (torque) the integral sliding surface is given in the form:  
\[S_{1}=\zeta_{1e}+\lambda_{1}\int_{0}^{t}\zeta_{1e}(\sigma)d\sigma, \tag{40}\]2  
where \(\lambda_{2}>0\) is a control coefficient.