## 3 Control Design and Stability Analysis
### Yaw Control  
The yaw control law will be designed to guarantee that the yaw angular velocity finishes tracking the desired trajectory within a user-appointed time while ensuring that the yaw angular velocity tracking error remains within the predefined appointed-time performance constraint boundary. Consider the following integral barrier Lyapunov function combining an appointed-time performance constraint function:  
\[V_{r}=\int_{0}^{r_{e}}\frac{k_{ra}^{2}\left(t\right)\xi}{k_{ra}^{2}\left(t \right)-\xi^{2}}d\xi+\frac{1}{2}\hat{W}_{r}^{T}\Gamma_{r}^{-1}\hat{W}_{r} \tag{41}\]  
where \(\hat{W}_{r}=W_{r}^{*}-\hat{W}_{r}\) represents the estimation error with \(\hat{W}_{r}\) representing estimated value of \(W_{r}^{*}\) and \(\Gamma_{r}\) is a positive definite diagonal matrix. \(k_{ra}\left(t\right)\) is given by:  
\[k_{ra}\left(t\right)=\left\{ \begin{array}{l}\frac{\tilde{k}_{rb}-k_{rb}}{T_{ ra}^{2}}t^{2}-2\frac{k_{rb}-k_{rb}}{T_{ra}}t+\bar{k}_{rb},t\in\left[0,T_{ra} \right)\\ k_{rb},t\in\left[T_{ra},+\infty\right)\end{array} \right. \tag{42}\]We know that the candidate Lyapunov function \(V_{r}\) is positive definite, continuously differentiable, and radially unbounded in the set \(\Omega_{r}=\left\{r_{e}:\left|r_{e}\right|<k_{ra}\left(t\right)\right\}\); therefore, \(V_{r}\) is a valid candidate Lyapunov function. Calculating the time derivative of \(V_{r}\) yields:  
\[ \begin{split}\dot{V}_{r}=&\frac{\partial V_{r}}{ \partial r_{e}}\dot{r}_{e}+\frac{\partial V_{r}}{\partial k_{ra}\left(t\right)} k_{ra}\left(t\right)-\ddot{W}_{r}^{T}\Gamma_{r}^{-1}\dot{W}_{r}\\ =&\frac{k_{ra}^{2}\left(t\right)r_{e}}{k_{ra}^{2} \left(t\right)-r_{e}^{2}}\dot{r}_{e}+r_{e}\left(\frac{k_{ra}\left(t\right)}{r_ {e}}\log\frac{k_{ra}^{2}\left(t\right)}{k_{ra}^{2}\left(t\right)-r_{e}^{2}}- \frac{k_{ra}\left(t\right)r_{e}}{k_{ra}^{2}\left(t\right)-r_{e}^{2}}\right)k _{ua}\left(t\right)\\ &-\ddot{W}_{r}^{T}\Gamma_{r}^{-1}\dot{W}_{r}\end{split}  \tag{43}\]  
Substituting \(\dot{r}_{e}\) into (43), we have:  
\[ \begin{split}\dot{V}_{r}=&\frac{k_{ra}^{2}\left(t \right)r_{e}}{k_{ra}^{2}\left(t\right)-r_{e}^{2}}\left(\frac{M_{zD0}}{J_{z0}}+f _{r}+\frac{\tau_{r}}{J_{z0}}-\dot{x}_{r}\right)\\ &+r_{e}\left(\frac{k_{ra}\left(t\right)}{r_{e}}\log\frac{k_{ra}^ {2}\left(t\right)}{k_{ra}^{2}\left(t\right)-r_{e}^{2}}-\frac{k_{ra}\left(t \right)r_{e}}{k_{ra}^{2}\left(t\right)-r_{e}^{2}}\right)\dot{k}_{ua}\left(t \right)\\ &-\ddot{W}_{r}^{T}\Gamma_{r}^{-1}\dot{W}_{r}\end{split}  \tag{44}\]  
According to [PERSON]'s direct method, the yaw control law and the adaptive updating law are designed as:  
\[ \begin{split}\tau_{r}=& J_{z0}\left(-k_{r}r_{e}- \frac{M_{zD0}}{J_{z0}}-\dot{W}_{r}^{T}H_{r}\left(Z_{r}\right)-\delta_{r}sgn \left(r_{e}\right)+\dot{x}_{r}\right.\\ &-\frac{k_{ra}^{2}\left(t\right)-r_{e}^{2}}{k_{ra}^{2}\left(t \right)}\left(\frac{k_{ra}\left(t\right)}{r_{e}}\log\frac{k_{ra}^{2}\left(t \right)}{k_{ra}^{2}\left(t\right)-r_{e}^{2}}-\frac{k_{ra}\left(t\right)r_{e}}{ k_{ra}^{2}\left(t\right)-r_{e}^{2}}\right)\dot{k}_{ua}\left(t\right)\\ &\left.+\frac{k_{ra}^{2}\left(t\right)-r_{e}^{2}}{k_{ra}^{2}\left( t\right)}\frac{k_{ua}^{2}\left(t\right)v_{e}}{k_{ua}^{2}\left(t\right)-v_{e}^{2}}u \right)\\ &\dot{W}_{r}=\Gamma_{r}\left(\frac{k_{ra}^{2}\left(t\right)r_{e}}{ k_{ra}^{2}\left(t\right)-r_{e}^{2}}H_{r}\left(Z_{r}\right)-\sigma_{r}\dot{W}_{r} \right)\end{split}  \tag{45}\]  
where \(k_{r}\), \(\delta_{r}\), and \(\sigma_{r}\) are positive constants. **Remark 8**. : _Using L'Hopital's rule, we have:_  
\[\lim_{r_{e}\to 0}\frac{k_{ra}\left(t\right)}{r_{e}}\log\frac{k_{ra}^{2}\left(t \right)}{k_{ra}^{2}\left(t\right)-r_{e}^{2}}=\frac{2k_{ra}\left(t\right)r_{e}}{ k_{ra}^{2}\left(t\right)-r_{e}^{2}} \tag{47}\]  
_Therefore, the yaw control law \(\tau_{r}\) is well defined._  
Substituting (45) and (46) into (44) yields:  
\[