## 2 Preliminaries
### Tracking Differentiator  
In this section, a tracking differentiator is considered to avoid the differentiation of the virtual controls in the BSC scheme. **Theorem 1** ([26]). : _If any solution of the following system:_  
\[ \begin{array}{c}\dot{z}_{1}=z_{2},\\ z_{2}=f(z_{1},z_{2})\end{array}  \tag{13}\]  
_satisfies \(z_{1}\to 0\), \(z_{2}\to 0\) as \(t\to\infty\), then for an arbitrary bounded and integrable input function \(\alpha\) and a constant \(T>0\), the solution of the following system_  
\[ \begin{array}{c}\dot{\chi}_{1}=\chi_{2},\\ \dot{\chi}_{2}=R^{2}f(\chi_{1}-\alpha,\chi_{2}/R)\end{array}  \tag{14}\]  
_satisfies \(\lim\limits_{R\to\infty}\int_{0}^{T}\lvert\chi_{1}-\alpha\rvert dt=0\) if the solutions of (14) satisfy \(\chi_{1}\to 0\) and \(\chi_{2}\to 0\) as \(t\to\infty\)._  
**Theorem 2** ([26]). : _Consider the following system_  
\[ \begin{array}{c}\dot{\chi}_{1}=\chi_{2},\\ \dot{\chi}_{2}=-r_{1}\mathrm{sinh}(a_{1}\chi_{1})-r_{2}\mathrm{sinh}(a_{2}\chi _{2}).\end{array}  \tag{15}\]  
_The system (15) is global uniformly asymptotically stable if \(r_{i}>0\) and \(a_{i}>0\), \(i=1,2\), are satisfied._  
**Theorem 3**. : _Consider the following system_  
\[ \begin{array}{c}\dot{\chi}_{1}=\dot{\chi}_{2},\\ \dot{\chi}_{2}=-R^{2}[r_{1}\mathrm{sinh}(a_{1}\varphi_{1}(\chi_{1}-\alpha))+r_{ 2}\mathrm{sinh}(a_{2}\varphi_{2}(\chi_{2})/R)],\end{array}  \tag{16}\]  
_where \(\varphi_{1}(\chi_{1}-\alpha)=\kappa sig^{1/2}(\chi_{1}-\alpha)+(1-\kappa)sig^{ (2+\gamma)/2}(\chi_{1}-\alpha)\), \(\varphi_{2}(\chi_{2})=sig^{1/2}(\chi_{2})\), \(sig^{1/2}(\cdot)=\lvert\cdot\rvert^{1/2}sign(\cdot)\), \(0<\kappa<1\) and \(0<\gamma<1\) are constants. For an arbitrary bounded and integrable function \(\alpha\) and a constant \(T>0\), \(\lim\limits_{R\to\infty}\int_{0}^{T}\lvert\chi_{1}-\alpha\rvert dt=0\) can be obtained if the positive conditions of all parameters are satisfied._  
Figure 1: Description of the hexacopter UAV. **Proof. ** Transforming \(\omega_{1}=\chi_{1}-\alpha\) and \(\omega_{2}=\chi_{2}/R\) in (16), it follows that \(\omega_{1}r_{1}\mathrm{sinh}(a_{1}\varphi(\omega_{1}))\geq 0\) because \(\mathrm{sinh}(\cdot)\) is an odd function. Then, \(\int_{0}^{\omega_{1}}r_{1}\mathrm{sinh}(a_{1}\varphi(\tau))d\tau\geq 0\), where \(\tau\) means an introduced integral variable. Thus, we can consider the following Lyapunov function:  
\[V(\omega_{1},\omega_{2})=R^{2}\int_{0}^{z_{1}}r_{1}\mathrm{sinh}(a_{1}\varphi( \tau))d\tau+\frac{1}{2}\omega_{2}^{2}. \tag{17}\]  
Taking the time derivative of (17) and using (16) result in  
\[ \begin{split}\dot{V}(\omega_{1},\omega_{2})=R_{2}^{2}r_{1} \mathrm{sinh}(a_{1}\varphi(\omega_{1}))\omega_{2}+\omega_{2}\dot{\omega_{2}} \\ =R_{2}^{2}r_{1}\mathrm{sinh}(a_{1}\varphi(\omega_{1}))\omega_{2}- R^{2}\omega_{2}[r_{1}\mathrm{sinh}(a_{1}\varphi_{1}(\omega_{1}))+r_{2}\mathrm{ sinh}(a_{2}\varphi_{2}(\omega_{2}))]\\ =-R^{2}\omega_{2}r_{2}\mathrm{sinh}(a_{2}\varphi_{2}(\omega_{2}) )\leq 0\end{split}  \tag{18}\]  
Thus, it is sure that \(\dot{V}(\omega_{1},\omega_{2})=0\) only when \(\omega_{2}=0\). Based on (16), when \(\omega_{2}=0\), \(\omega_{1}\) will always be equivalent to zero. If \(\omega_{1}\rightarrow\infty\) and \(\omega_{2}\rightarrow\infty\), we obtain that \(V(\omega_{1},\omega_{2})\rightarrow\infty\). Hence, the solutions of \(\dot{V}(\omega_{1},\omega_{2})\) do not include any other whole trajectory, except the origin (0,0). Therefore, the system (16) is globally asymptotically stable. This concludes the proof. \(\square\)  
For comparison of the above differentiators, three tracking differentiators were selected. The first one, which is proposed by [PERSON] et al. [27], is the following hyperbolic tangent tracking differentiator:  
\[ \begin{split}\dot{\chi}_{1}=\chi_{2}\\ \dot{\chi}_{2}=-R^{2}[r_{1}\mathit{tansig}(a_{1}(\chi_{1}-\alpha ))+r_{2}\mathit{tansig}(a_{1}\chi_{2}/R)],\end{split}  \tag{19}\]  
where \(\mathit{tansig}(a_{1}x)=\frac{2}{\mathit{c}^{-2a_{1}x}+1}-1\). The second one, which was proposed by [PERSON] et al.