## 3 Control Design and Stability Analysis  
### Position Control  
To guarantee that the position tracking errors always satisfy the appointed-time performance constraint condition (1), the new integral barrier Lyapunov function combined with appointed-time performance constraint function is constructed as:  
\[V_{p}=\int_{0}^{x_{e}}\frac{k_{xa}^{2}\left(t\right)\xi}{k_{xa}^{2}\left(t \right)-\varsigma^{2}}d\varsigma+\int_{0}^{y_{e}}\frac{k_{ya}^{2}\left(t \right)\xi}{k_{ya}^{2}\left(t\right)-\varsigma^{2}}d\varsigma \tag{19}\]  
where  
\[k_{ia}\left(t\right)=\left\{\begin{array}{l}\frac{\bar{k}_{ib}-k_{b}}{T_{ia}^ {2}}t^{2}-2\frac{\bar{k}_{ib}-k_{b}}{T_{ia}}t+\bar{k}_{ib},t\in\left[0,T_{ia} \right)\\ k_{ib},t\in\left[T_{ia},+\infty\right)\end{array}\right. \tag{20}\]  
with \(i=x,y\) and the definition of the parameters is similar to (1). We know that the candidate Lyapunov function \(V_{p}\) is positive definite, continuously differentiable, and radially unbounded in the sets \(\left|x_{e}\right|<k_{xa}\left(t\right)\) and \(\left|y_{e}\right|<k_{yu}\left(t\right)\); therefore, \(V_{p}\) is a valid candidate Lyapunov function.  
Taking the time derivative of \(V_{p}\) along (15) yields:  
\[\dot{V}_{p}= \frac{k_{xa}^{2}\left(t\right)x_{e}}{k_{xa}^{2}\left(t\right)-x_{ e}^{2}}\dot{x}_{e}+\left(k_{xa}\left(t\right)\log\frac{k_{xa}^{2}\left(t \right)}{k_{xa}^{2}\left(t\right)-x_{e}^{2}}-\frac{k_{xa}\left(t\right)x_{e}^{ 2}}{k_{xa}^{2}\left(t\right)-x_{e}^{2}}\right)k_{xa}\left(t\right) \tag{21}\] \[+\frac{k_{ya}^{2}\left(t\right)y_{e}}{k_{ya}^{2}\left(t\right)-y_ {e}^{2}}\dot{y}_{e}+\left(k_{ya}\left(t\right)\log\frac{k_{ya}^{2}\left(t \right)}{k_{ya}^{2}\left(t\right)-y_{e}^{2}}-\frac{k_{yu}\left(t\right)y_{e}^{ 2}}{k_{ya}^{2}\left(t\right)-y_{e}^{2}}\right)\dot{k}_{ya}\left(t\right)\] \[= \left[\begin{array}{c}\frac{k_{xa}^{2}\left(t\right)x_{e}}{k_{xa }^{2}\left(t\right)-x_{e}^{2}}\\ \frac{k_{xa}^{2}\left(t\right)y_{e}}{k_{xa}^{2}\left(t\right)-y_{e}^{2}}\\ \end{array}\right]^{T}\left[\begin{array}{c}\dot{x}_{e}\\ \dot{y}_{e}\\ \end{array}\right]+\left[\begin{array}{c}k_{xa}\left(t\right)\log\frac{k_{xa }^{2}\left(t\right)}{k_{xa}^{2}\left(t\right)-x_{e}^{2}}-\frac{k_{xa}\left(t \right)x_{e}^{2}}{k_{xa}^{2}\left(t\right)-x_{e}^{2}}\\ k_{yu}\left(t\right)\log\frac{k_{yu}^{2}\left(t\right)}{k_{yu}^{2}\left(t\right) -y_{e}^{2}}-\frac{k_{yu}\left(t\right)y_{e}^{2}}{k_{yu}^{2}\left(t\right)-y_{e }^{2}}\\ \end{array}\right]^{T}\left[\begin{array}{c}k_{xa}\left(t\right)\\ k_{yu}\left(t\right)\\ \end{array}\right]\]  
According to [PERSON]'s direct method, the virtual position control laws are designed as follows:  
\[\left[\begin{array}{c}\alpha_{u}\\ \alpha_{v}\end{array}\right] = \left[\begin{array}{cc}\cos\psi&\sin\psi\\ -\sin\psi/\cos\phi&\cos\psi/\cos\phi\end{array}\right] \tag{22}\] \[\times \left[\begin{array}{c}\dot{x}_{d}-k_{x}x_{e}-\frac{k_{xa}^{2} \left(t\right)-x_{e}^{2}}{k_{xa}^{2}\left(t\right)}\left(\frac{k_{xa}}{x_{e}} \log\frac{k_{xa}^{2}\left(t\right)}{k_{xa}^{2}\left(t\right)-x_{e}^{2}}-\frac{k _{xa}\left(t\right)x_{e}}{k_{xa}^{2}\left(t\right)-x_{e}^{2}}\right)k_{xa} \left(t\right)\\ \dot{y}_{d}-k_{y}y_{e}-\frac{k_{yu}^{2}\left(t\right)-y_{e}^{2}}{k_{yu}^{2} \left(t\right)}\left(\frac{k_{yu}\left(t\right)}{y_{e}}\log\frac{k_{yu}^{2} \left(t\right)}{k_{yu}^{2}\left(t\right)-y_{e}^{2}}-\frac{k_{yu}\left(t\right)x _{e}}{k_{yu}^{2}\left(t\right)-y_{e}^{2}}\right)k_{yu}\left(t\right)\end{array}\right]\]  
where \(k_{x}\) and \(k_{y}\) are positive constants.  
**Remark 4**: _In the control process of the hovercraft, \(\left|\phi\right|<90^{\circ}\) is always satisfied due to the effect of roll restoring moment. In addition, using L'Hopital's rule, we have:_  
\[\lim_{i_{e}\to 0}\frac{k_{ia}\left(t\right)}{i_{e}}\log\frac{k_{ia}^{2}\left(t \right)}{k_{ia}^{2}\left(t\right)-i_{e}^{2}}=\frac{2k_{ia}\left(t\right)i_{e}}{ k_{ia}^{2}\left(t\right)-i_{e}^{2}},\quad i=x,y \tag{23}\]_Therefore, the virtual position control laws are well defined._  
Substituting (22) into (21) yields:  
(24)