## 3 Design of Controller and Nonlinear Disturbance Observer
### Design of Longitudinal and Latitudinal Controllers  
Let us define the state variables for x-, y-axes and roll, pitch axes from (6), (7), (9), and (10) as \(x_{3}=[x\;\;y]^{T}\), \(x_{4}=[\dot{x}\;\;\dot{y}]^{T}\), \(x_{5}=[\dot{\phi}\;\;\theta]^{T}\), and \(x_{6}=[\dot{\phi}\;\;\dot{\theta}]^{T}\). The state-space model is represented as follows:  
\[\dot{x}_{3}=x_{4},\dot{x}_{4}=R_{4}(\psi,u_{1})T_{4}(x_{5})-\rho_{4}, \tag{41}\]  
\[\dot{x}_{5}=x_{6},\dot{x}_{6}=f_{6}+g_{6}u_{\phi\theta}-\rho_{6}, \tag{42}\]  
where \(R_{4}(\psi,u_{1})=\frac{1}{m}\left[ \begin{array}{cc}\cos\psi&\sin\psi\\ \sin\psi&-\cos\psi\end{array} \right]u_{1}\), \(T_{4}(x_{5})=[\cos\phi\sin\theta\sin\phi]^{T}\), \(\rho_{4}=[\rho_{x}\;\rho_{y}]^{T}\), \(g_{6}=diag(1/\;l_{x}\;l/y)\), \(u_{\phi\theta}=[\tau_{\phi}\;\tau_{\phi}]^{T}\), and \(f_{6}=[\dot{\phi}\dot{\psi}(l_{y}-l_{z})/\;l_{x}+\dot{l}\dot{\theta}\dot{ \Omega}_{r}/\;l_{x}\;\dot{\psi}\dot{\phi}(l_{z}-l_{x})/\;l_{y}-l_{y}\dot{ \phi}\dot{\Omega}_{r}/\;l_{y}]^{T}\), \(\rho_{6}=[\rho_{\phi}\;\rho_{\theta}]^{T}\). We define the sliding mode vectors including the tracking command, errors, and state variables as  
\[s_{3}=x_{3}-x_{3d}-\xi_{31}, \tag{43}\]\[s_{4} =x_{4}-\chi_{31}-\xi_{32}, \tag{44}\] \[s_{5} =x_{5}-x_{5d}-\xi_{51},\] (45) \[s_{6} =x_{6}-\chi_{51}-\xi_{52}. \tag{46}\]  
In the sliding mode surfaces, the dynamics of the compensating signals \(\xi_{3j}\), \(j=1,2\), is described as  
\[ \begin{array}{l}\dot{\xi}_{i1}=-k_{1}\xi_{i1}+\xi_{12}+\chi_{i1}-\alpha_{i}- \beta_{\zeta,i1}sig^{1/2}(\xi_{i1}),\\ \dot{\xi}_{i2}=-k_{2}\xi_{i2}-\dot{\xi}_{i1}-\beta_{\zeta,j2}sig^{1/2}(\xi_{i2 }),\,i=3,5,\end{array}  \tag{47}\]  
where \(\beta_{\zeta,ij}>0,\ i=3,5,\ j=1,2\), are constants. Furthermore, \(\chi_{i1}\) is obtained from the tracking differentiator described as follows:  
\[ \begin{array}{l}\dot{\chi}_{i1}=\chi_{i2},\\ \dot{\chi}_{i2}=-R_{i}^{2}[r_{i1}\sinh(a_{i1}\varphi_{i1}(\chi_{i1},\alpha_{i 1}))+r_{i2}\sinh(a_{i2}R_{i}^{-1}\varphi_{i2}(\chi_{i2}))],i=3,5,\end{array}  \tag{48}\]  
where \(\varphi_{i1}(\chi_{i1},\alpha_{i})=\kappa_{i}sig^{1/2}(\chi_{i1}-\alpha_{i})+( 1-\kappa_{i})sig^{(2+\gamma_{i1})/2}(\chi_{i1}-\alpha_{i})\), \(\varphi_{i2}(\chi_{i2})=sig^{1/2}(\chi_{i2})\), \(\kappa_{i}>1\) and \(0<\gamma_{i}<1\) are constants. We define the following super-twisting state variable vectors to design the controller using the variables defined previously  
\[\xi_{3}=\left[ \begin{array}{cc}\zeta_{31}&\zeta_{32}\end{array} \right]^{T}= \left[ \begin{array}{cc}k_{31}s_{3}+sig^{1/2}(s_{3})\\ -k_{32}\int_{0}^{t}\left[k_{31}^{2}s_{3}+\frac{3k_{32}}{2}sig^{1/2}(s_{3})+ \frac{1}{2}sign(s_{3})\right]d\tau\end{array} \right], \tag{49}\]  
\[\xi_{4}=\left[ \begin{array}{cc}\zeta_{41}&\zeta_{42}\end{array} \right]^{T}= \left[ \begin{array}{cc}k_{41}s_{4}+sig^{1/2}(s_{4})\\ -k_{42}\int_{0}^{t}\left[k_{41}^{2}s_{4}+\frac{3k_{42}}{2}sig^{1/2}(s_{4})+ \frac{1}{2}sign(s_{4})\right]d\tau\end{array} \right], \tag{50}\]  
\[\xi_{5}=\left[ \begin{array}{cc}\zeta_{51}&\zeta_{52}\end{array} \right]^{T}= \left[ \begin{array}{cc}k_{51}s_{5}+sig^{1/2}(s_{5})\\ -k_{52}\int_{0}^{t}\left[k_{51}^{2}s_{5}+\frac{3k_{51}}{2}sig^{1/2}(s_{5})+ \frac{1}{2}sig^{1/2}(s_{5})\right]d\tau\end{array} \right], \tag{51}\]  
\[\xi_{6}=\left[ \begin{array}{cc}\zeta_{61}&\zeta_{62}\end{array} \right]^{T}= \left[ \begin{array}{cc}k_{61}s_{6}+sig^{1/2}(s_{6})\\ -k_{62}\int_{0}^{t}\left[k_{61}^{2}s_{6}+\frac{3k_{41}}{2}sig^{1/2}(s_{6})+ \frac{1}{2}sig^{1/2}(s_{6})\right]d\tau\end{array} \right]. \tag{52}\]  
The differentiation of the above state vectors is as follows:  
\[\dot{\xi}_{3}=\left[ \begin{array}{cc}\dot{\xi}_{31}&\dot{\xi}_{32}\end{array}  \right]^{T}=\left[ \begin{array}{cc}\mu_{3}(s_{4}+\xi_{31}+\xi_{41}-\dot{x}_ {3d}-\dot{\xi}_{31})\\ -\mu_{3}k_{31}\dot{\xi}_{31}\end{array} \right], \tag{53}\]  
\[\dot{\xi}_{4}=\left[ \begin{array}{cc}\dot{\xi}_{41}&\dot{\xi}_{42}\end{array}  \right]^{T}=\left[