## 4 Method
### Fasteso
#### 4.1.1 Construction of Traditional Super-Twisting Extended State Observer (STESO)  
In this part, according to our previous work [30], a traditional SMESO named STESO for quadrotor UAV manipulator is constructed. In this work, the situation, in which the robotic arm keeps constant while the UAV faces external disturbances during the flight, is mainly considered: (\(\dot{\eta}=\ddot{\eta}=\mathbf{0}\)). That is a general scenario that the quadrotor UAV manipulator often encounters when conducting aerial tasks. Nevertheless, usually there is unwanted vibration of the equipped robotic arm caused by the high-speed motors and propellers. To consider all mentioned disturbances, the model (25) could be reconstructed as  
\[(M_{\mathbf{21}}+\Delta M_{\mathbf{21}})\ddot{\mathbf{p}}^{\mathbf{I}}+(M_{\mathbf{22}}+\Delta M_{ \mathbf{22}})\ddot{\mathbf{\Psi}}+C_{\mathbf{21}}\dot{\mathbf{p}}^{\mathbf{I}}+(M_{\mathbf{23}}+\Delta M _{\mathbf{23}})\Delta\ddot{\mathbf{\eta}}+C_{\mathbf{22}}\dot{\mathbf{\Psi}}+C_{\mathbf{23}}\Delta \dot{\eta}+G_{\mathbf{2}}=u_{\mathbf{r}}+d_{\mathbf{r}} \tag{26}\]  
where \(\Delta M_{\mathbf{21}}\), \(\Delta M_{\mathbf{22}}\), \(\Delta M_{\mathbf{23}}\) represent model uncertainties. \(\Delta\dot{\eta}\) and \(\Delta\ddot{\eta}\), considered as a portion of model uncertainty, denote the residuals from the temporary variation of the robotic arm parameters \(\ddot{\mathbf{\eta}},\ddot{\mathbf{\eta}}\). In order to build the observer, reconstruct the attitude dynamic Equation (26) as  
\[M_{\mathbf{22}}\ddot{\mathbf{\Psi}}=u_{\mathbf{r}}+d_{\mathbf{r}}-M_{\mathbf{21}}\ddot{\mathbf{p}}^{ \mathbf{I}}-\Delta M_{\mathbf{21}}\ddot{\mathbf{p}}^{\mathbf{I}}-\Delta M_{\mathbf{22}}\ddot{\mathbf{ \Psi}}-(M_{\mathbf{23}}+\Delta M_{\mathbf{23}})\Delta\ddot{\mathbf{\eta}}-C_{\mathbf{23}}\Delta \dot{\eta}-C_{\mathbf{21}}\dot{\mathbf{p}}^{\mathbf{I}}-C_{\mathbf{22}}\dot{\mathbf{\Psi}}-G_{\mathbf{ 2}} \tag{27}\]  
Figure 2: Whole control structure of the proposed method. Introduce the lumped disturbance as  
\[{{d_{r}}^{*}}={{d_{r}}}-\Delta{{M_{21}}}{{\tilde{p}}^{I}}-\Delta{{M_{22}}}{{ \tilde{\Psi}}}-({{M_{23}}}+\Delta{{M_{23}}})\Delta{{\tilde{\eta}}}-{{C_{23}}} \Delta{{\tilde{\eta}}} \tag{28}\]  
Combining Equations (27) and (28), we get  
\[{{M_{22}}}{{\tilde{\Psi}}}={{u_{r}}}+{{d_{r}}^{*}}-{{M_{21}}}{{\tilde{p}}^{I}}-{ {C_{21}}}{{\tilde{p}}^{I}}-{{C_{22}}}{{\tilde{\Psi}}}-{{G_{2}}} \tag{29}\]  
where the variable \({{\tilde{p}}^{I}}\) is measured directly from the sensors or velocity differential is too noisy to use, so TD is introduced here to estimate the system acceleration for noise alleviation. Additionally, \({{\dot{p}}^{I}}\) and \({{\tilde{\Psi}}}\) measured from the sensors are also a little noisy, in that case, TD would be used to reduce noise, too. Moreover, the items \({{M_{21}}}\), \({{C_{21}}}\) and \({{G_{2}}}\) could be obtained in advance according to the presented UAV model. As for dynamics model (29), considering the feedback linearization method, we could reformulate the control input as  
\[{{u_{r}}}={{u_{r}^{*}}}+{{M_{21}}}{{\tilde{p}}^{I}}+{{C_{21}}}{{\dot{p}}^{I}}+ {{C_{22}}}{{\tilde{\Psi}}}+{{G_{2}}} \tag{30}\]  
Combining Equations (29) and (30), we get  
\[{{M_{22}}}{{\tilde{\Psi}}}={{u_{r}^{*}}}+{{d_{r}}^{*}} \tag{31}\]  
When building the STESO, it is supposed that every channel is independent from each other. Therefore, only one channel will be presented here and the other two are completely identical. As for model Equation (31), the one-dimensional model for STESO design is provided as  
\[{{J_{i}}}{{\dot{\Psi}}_{i}}={{u_{ri}^{*}}}+{{d_{ri}}^{*}} \tag{32}\]  
Introduce a new extended state vector \({\boldsymbol{\zeta}}=[{{\zeta_{1}}}\;\;\;\;\;{{\zeta_{2}}}\;]^{T}\), where \({{\zeta_{1}}}={{J_{i}}}{{\dot{\Psi}}_{i}}\) and \({{\zeta_{2}}}={{d_{ri}}^{*}}\), \((i=\varphi,\theta,\psi)\). Reconstruct the model (32) as  
\[\left\{ \begin{array}{l}{{\dot{\zeta}}_{1}}={{u_{ri}^{*}}}+{{\zeta_{2}}}\\ {{\dot{\zeta}}_{2}}=\chi\end{array} \right.