## 2 Preliminaries and Problem Formulation  
**Lemma 1**. : _For \(\forall g_{i}\in R\), \(i=1,2,\ldots,n\), two positive real numbers \(0<v_{1}\leq 1\) and \(v_{2}>1\), the inequalities that hold are as follows [41]:_  
\[\left(\sum_{i=1}^{n}\lvert g_{i}\rvert\right)^{v_{1}}\leq\sum_{i=1 }^{n}\lvert g_{i}\rvert^{v_{1}}\leq n^{1-v_{1}}\biggl{(}\sum_{i=1}^{n}\lvert g _{i}\rvert\biggr{)}^{v_{1}} \tag{1}\] \[\sum_{i=1}^{n}\lvert g_{i}\rvert^{v_{2}}\leq\biggl{(}\sum_{i=1}^{ n}\lvert g_{i}\rvert\biggr{)}^{v_{2}}\leq n^{v_{2}-1}\sum_{i=1}^{n}\lvert g _{i}\rvert^{v_{2}} \tag{2}\]  
**Lemma 2**. : _For \(\forall(a,b)\in R^{2}\), one can obtain the following [42]:_  
\[\text{ab}\leq\frac{\xi^{\widetilde{p}}}{\widetilde{p}}\lvert a\rvert^{ \widetilde{p}}+\frac{1}{\widetilde{q}\xi^{\widetilde{q}}}\lvert b\rvert^{ \widetilde{q}} \tag{3}\]  
_where \(\xi>0\), \(\widetilde{q}>1\), \(\widetilde{p}>1\), and \((\widetilde{p}-1)(\widetilde{q}-1)=1\)._  
**Lemma 3**. : _For real variables \(\phi\) and \(\gamma\) and any of the positive constants \(\zeta_{1}\), \(\zeta_{2}\), and \(\zeta_{3}\), the satisfied inequality is as follows [43]:_  
\[\lvert\phi\rvert^{\zeta_{1}}\lvert\gamma\rvert^{\zeta_{2}}\leq\frac{\zeta_{1 }}{\zeta_{1}+\zeta_{2}}\zeta_{3}\lvert\phi\rvert^{\zeta_{1}+\zeta_{2}}+\frac {\zeta_{2}}{\zeta_{1}+\zeta_{2}}\zeta_{3}^{-\frac{\zeta_{1}}{\zeta_{2}}}\lvert \gamma\rvert^{\zeta_{1}+\zeta_{2}} \tag{4}\]  
**Lemma 4**. : _Consider a nonlinear system \(\dot{\zeta}=f(\zeta)\) with \(\zeta\in R^{n\times 1}\). On one hand, if there exists a Lyapunov function \(V(\zeta)\) satisfying the following [44]:_  
\[\dot{V}+h_{1}V^{v_{3}}+h_{2}V^{v_{4}}\leq\Delta_{v} \tag{5}\]  
_for some \(h_{1}\in R^{+}\), \(h_{2}\in R^{+}\), \(0<v_{3}<1\), \(v_{4}\geq 1\), and \(\Delta_{v}\in R^{+}\), then, the system is finite-time stable._  
**Lemma 5**. : _Consider a nonlinear system \(\dot{\zeta}=f(\zeta)\) with \(\zeta\in R^{n\times 1}\). If there exists a Lyapunov function \(V(\zeta)\) satisfying the following [45]:_  
\[\dot{V}\leq-\gamma V^{P} \tag{6}\]  
_where \(\gamma>0\), \(0<p<1\), then, the system will converge to zero in a finite time \(T\leq\frac{1}{\gamma(1-p)}V^{1-p}\)._  
**Assumption 1**. : _Assuming that the rate of change of the total disturbance to which the system is subjected is bounded, \(\left|\dot{d}_{j}\right|\leq\theta_{j}\) where \(j=p,q,r\)._  
**Remark 1**. : _Although the wind disturbances acting on the quadrotor UAV are readily variable, the energy of the disturbances is also finite; at the same time, although we are unable to measure each of the basic physical parameters of the UAV with any great degree of accuracy, the resulting modeling uncertainty is nearly constant, and the total disturbances to the system can therefore be regarded as signals with a bounded rate of change. In summary, Assumption 1 is reasonable._  
**Definition 1**. : _Define \(|\mathbf{a}|^{m/n}sign(\mathbf{b})=\left[|a_{1}|^{m/n}sign(b_{1})\ \ |a_{2}|^{m/n}sign(b_{2})\ \ \ldots\ \ |a_{3}|^{m/n}sign(b_{3})\right]\), and the vector \(\mathbf{a}\in R^{n\times 1}\) and \(\mathbf{b}\in R^{n\times 1}\)._  
The attitude model of the quadrotor UAV includes an attitude kinematics model and an attitude dynamics model, which is modeled based on the Newton-Euler method with the following results:  
\[\dot{\mathbf{x}}_{1}=\mathbf{A}(\mathbf{x}_{1})\mathbf{x}_{2} \tag{7}\]  
\[\dot{\mathbf{x}}_{2}=\mathbf{B}(\mathbf{x}_{2})+\mathbf{G}\mathbf{u}_{1}+ \mathbf{d} \tag{8}\]  
\[\mathbf{A}(x_{1})=\left[ \begin{array}{ccc}1&\sin\phi\tan\theta&\cos\phi\tan \theta\\ 0&\cos\phi&-\sin\phi\\ 0&\sin\phi/\cos\theta&\cos\phi/\cos\theta\end{array} \right],\mathbf{G}=\left[ \begin{array}{ccc}J_{x}&0&0\\ 0&J_{y}&0\\ 0&0&J_{z}\end{array} \right]\]  
\[\mathbf{B}(x_{2})=\left[ \begin{array}{ccc}\frac{J_{y}-I_{z}}{J_{x}}qr&\frac {I_{z}-I_{x}}{J_{y}}pr&\frac{I_{x}-I_{y}}{I_{z}}pq\end{array} \right]^{T}\]  
where \(\mathbf{x}_{1}=\left[ \begin{array}{ccc}\phi&\theta&\varphi\end{array} \right] ^{T}\) and \(\mathbf{x}_{2}=\left[ \begin{array}{ccc}p&q&r\end{array}