## 3 Equations of Motion in Terms of Quasi-Velocities
### Dynamic Control Algorithm
The time derivative of \(S_{2}\) taking into account (12) is \(\dot{S}_{2}=\dot{\zeta}_{3e}+\lambda_{2}\zeta_{3e}\) (where \(\dot{\zeta}_{3e}=\dot{\zeta}_{3}-\dot{\zeta}_{3d}\)):  
\[S_{1}=\zeta_{1e}+\lambda_{1}\int_{0}^{t}\zeta_{1e}(\sigma)d\sigma, \tag{40}\]3  
The proposed input control signal \(\tau_{r}\) has the form:  
\[S_{1}=\zeta_{1e}+\lambda_{1}\int_{0}^{t}\zeta_{1e}(\sigma)d\sigma, \tag{40}\]4  
where the last term meets Assumption 3 and is related to the parameters' perturbation function, whereas \(\hat{f}_{\zeta_{3}}\) is the estimated value of the external disturbance force \(f_{\zeta_{3}}\). The Lyapunov function candidate is assumed as follows:  
\[S_{1}=\zeta_{1e}+\lambda_{1}\int_{0}^{t}\zeta_{1e}(\sigma)d\sigma, \tag{40}\]5  
where \(\hat{f}_{\zeta_{3}}=f_{\zeta_{3}}-\hat{f}_{\zeta_{3}}\) is the estimated error of the disturbance force and \(\beta_{2}>0\) is a constant control parameter. Calculating the time derivative of \(V_{2}\) and applying (50) and (51) it can be written:  
\[S_{1}=\zeta_{1e}+\lambda_{1}\int_{0}^{t}\zeta_{1e}(\sigma)d\sigma, \tag{40}\]6 \[S_{1}=\zeta_{1e}+\lambda_{1}\int_{0}^{t}\zeta_{1e}(\sigma)d\sigma, \tag{40}\]7  
Now using Assumption 1, the following gain function is adopted to guarantee the stability of the control system against parameter perturbations:  
\[S_{1}=\zeta_{1e}+\lambda_{1}\int_{0}^{t}\zeta_{1e}(\sigma)d\sigma, \tag{40}\]8 \[S_{1}=\zeta_{1e}+\lambda_{1}\int_{0}^{t}\zeta_{1e}(\sigma)d\sigma, \tag{40}\]9  
Then, given Assumption 2 and the first case, i.e., \(\hat{f}_{\zeta_{3}}=0\) (which implies from (12) that both \(\hat{f}_{v}=0\) and \(\hat{f}_{r}=0\), i.e., for constant disturbances), the function \(\hat{V}_{2}\) (53) can be expressed as:  
\[\dot{S}_{1}=N_{1}^{-1}\big{(}F_{1}(\zeta)-N_{1}\dot{\zeta}_{1d}+N_{1}\lambda_{ 1}\zeta_{1e}+\tau_{u}+f_{u}\big{)}. \tag{41}\]0 \[\dot{S}_{1}=N_{1}^{-1}\big{(}F_{1}(\zeta)-N_{1}\dot{\zeta}_{1d}+N_{1}\lambda_{ 1}\zeta_{1e}+\tau_{u}+f_{u}\big{)}. \tag{41}\]1  
Designing the adaptive term \(\dot{\tilde{f}}_{\zeta_{3}}=\beta_{2}^{-1}S_{2}\) one obtains:  
\[\dot{S}_{1}=N_{1}^{-1}\big{(}F_{1}(\zeta)-N_{1}\dot{\zeta}_{1d}+N_{1}\lambda_{ 1}\zeta_{1e}+\tau_{u}+f_{u}\big{)}. \tag{41}\]2  
This result means that \(\dot{V}_{2}=0\) provided \(\zeta_{3x}\) converges to zero along the sliding surface \(S_{2}\). However, \(\zeta_{3x}=r_{e}\), thus also \(r_{e}\) tends to zero. Therefore the yaw velocity error \(r_{e}\) converges asymptotically to zero if the signal \(\tau_{r}\) (51) is used. Finally, for the entire control system, the following inequality is obtained:  
\[\dot{S}_{1}=N_{1}^{-1}\big{(}F_{1}(\zeta)-N_{1}\dot{\zeta}_{1d}+N_{1}\lambda_{ 1}\zeta_{1e}+\tau_{u}+f_{u}\big{)}. \tag{41}\]3  
which means that the proposed dynamic control algorithm can stabilize velocity errors \(u_{e}\) and \(r_{e}\). **Remark 2**. : _The discontinuous sign function can be replaced by using another smooth function which allows us to avoid the well-known chattering problem. For example, in [18], a definition of a smooth function was given. The properties of such a function satisfies, e.g., \(f_{s}(\chi)=\text{tanh}(\chi)\). Replacing the discontinuous function \(\text{sgn}\) by the \(\text{tanh}(a\chi)\) function, which, however, is only an approximation \(\text{sgn}(\chi)\approx\text{tanh}(a\chi)\) where \(a\) is a positive scalar can be found, e.g., in [9]. In [56] it was shown that if instead of using \(\text{sgn}(\chi)\), the approximation by the use of \(\text{sat}(\chi/\epsilon)\) is carried out, then the sliding mode control algorithm guarantees ultimate boundedness with an bound that can be controlled by the design of parameters \(\epsilon\) (globally uniformly asymptotically stabilization can be proven). Taking the above into account and knowing that both functions, i.e., sat and tanh, are smooth saturation functions, one may conclude that also for the tanh function only globally uniformly asymptotically stabilization is ensured._  
**Remark 3**. : _A more realistic situation than before will now be considered.