## 3 Aggressive Control of Quadrotors with Extended Finite-Time Observer
### Quadrotor Model
\begin{bmatrix}\cos\psi_{i}\cos\theta_{i}&\cos\psi_{i}\sin\theta_{i}\sin \phi_{i}-\cos\phi_{i}\sin\psi_{i}&\sin\psi_{i}\sin\phi_{i}+\cos\psi_{i}\cos \phi_{i}\sin\theta_{i}\\ \cos\theta_{i}\sin\psi_{i}&\cos\psi_{i}\cos\phi_{i}+\sin\psi_{i}\sin\theta_{i} \sin\phi_{i}&\cos\phi_{i}\sin\psi_{i}\sin\theta_{i}-\cos\psi\sin\phi_{i}\\ -\sin\theta_{i}&\cos\theta_{i}\sin\phi_{i}&\cos\theta_{i}\cos\phi_{i}\end{bmatrix} \]  
Thus, according to Equation (3), we obtain the translational dynamics as follows  
\[ \begin{split}\tilde{X}_{i}&=(\sin\psi_{i}\sin\phi_{i}+ \cos\psi_{i}\cos\phi_{i}\sin\theta_{i})\frac{F_{T_{i}}}{m}\\ \tilde{Y}_{i}&=(\cos\phi_{i}\sin\psi_{i}\sin\theta_{i}-\cos\psi_{i} \sin\phi_{i})\frac{F_{T_{i}}}{m}\\ \tilde{Z}_{i}&=-g+(\cos\theta_{i}\cos\phi_{i})\frac{F_{T_{i}}}{m} \end{split}  \tag{5}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Description** \\ \hline \(x_{e}y_{e}z_{e}\) & Fixed inertial frame \\ \(\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3}\) & Basis vector of inertial frame \(\mathbf{e}_{1}=[1,0,0]^{T}\), \(\mathbf{e}_{2}=[0,1,0]^{T}\), \(\mathbf{e}_{3}=[0,0,1]^{T}\) \\ \(\mathcal{X}_{i}=[X_{i},Y_{i},Z_{i}]^{T}\) & Coordinates of the center of mass of a quadrotor in \(x_{e}y_{e}z_{e}\) \\ \(\Theta_{i}=[\phi_{i},\theta_{i},\psi_{i}]^{T}\) & Euler angles (pitch, roll and yaw) \\ \(R_{z}(\psi_{i}),R_{y}(\theta_{i}),R_{x}(\phi_{i})\) & Rotation matrices in yaw, pitch and roll \\ \(R_{i}\) & Rotation matrix \(R_{i}=R_{z}(\psi_{i})R_{y}(\theta_{i})R_{x}(\phi_{i})\) \\ \(J=\text{diag}\{I_{x_{p}},I_{y_{p}},I_{z_{b}}\}\) & Inertia matrix represented in the body-fixed frame \(x_{p}y_{p}z_{b}\) \\ \(\Omega_{i}=[p_{i},q_{i},r_{i}]^{T}\) & Angular velocity of the quadrotor \(i\) in the body-fixed frame \\ \(\mathbf{S}(\cdot):\mathbb{R}^{3}\to\mathbb{R}^{3\times 3}\) & Operation from vector in \(\mathbb{R}^{3}\) to skew-symmetric matrix \(\mathbb{R}^{3\times 3}\) \\ \(\tau_{i}=[\tau_{\phi_{i}},\tau_{\theta_{i}},\tau_{\psi_{i}}]^{T}\in\mathbb{R}^ {3}\) & The moments of roll, pitch and yaw \\ \(F_{T_{i}}\) & Total thrust force of quadrotor \(i\) \\ \(\mathcal{X}_{i}=[\mathcal{X}_{i}^{T},\tilde{X}_{i}^{T}]\) & State of quadrotor \(i\) \\ \(u_{i}=[F_{T_{i}},\tau_{i}^{T}]^{T}\in\mathbb{R}^{4}\) & Control input of the quadrotor \(i\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The denotations of the variables in the dynamic model of quadrotor. According to the second equation in (3), we can obtain  
\[\Omega_{i}= \begin{bmatrix}p_{i}\\ q_{i}\\ r_{i}\end{bmatrix} =R_{i}^{T}\psi_{i}\mathbf{e}_{3}+\dot{\theta}_{i}R_{x}^{T}(\phi_ {i})R_{y}^{T}(\theta_{i})\mathbf{e}_{2}+\dot{\phi}_{i}R_{x}^{T}(\phi_{i}) \mathbf{e}_{1}=T_{i}\cdot \begin{bmatrix}\dot{\phi}_{i}\\ \dot{\theta}_{i}\\ \dot{\psi}_{i}\end{bmatrix}  \tag{6}\]  
where  
\[T_{i}= \begin{bmatrix}R_{x}^{T}(\phi_{i})\mathbf{e}_{1}&R_{x}^{T}(\phi_{i})R_{ y}^{T}(\theta_{i})\mathbf{e}_{2}&R_{i}^{T}\mathbf{e}_{3}\end{bmatrix} = \begin{bmatrix}1&0&-\sin\theta_{i}\\ 0&\cos\phi_{i}&\cos\theta_{i}\sin\phi_{i}\\ 0&-\sin\phi_{i}&\cos\theta_{i}\cos\phi_{i}\end{bmatrix} \]  
Let us denote  
\[\tilde{T}_{i}= \begin{bmatrix}\tan\theta_{i}&0&\cos\theta_{i}+\tan\theta_{i} \sin\theta_{i}\\ 0&-\cos\theta_{i}&0\\ \sec\theta_{i}&0&\tan\theta_{i}\end{bmatrix} \]  
According to (6) and using \(\tilde{T}_{i}\), we can obtain the rotational dynamics as follows  
\[ \begin{bmatrix}\dot{\phi}_{i}\\ \dot{\theta}_{i}\\ \dot{\psi}_{i}\end{bmatrix} =(JT_{i})^{-1}\tau_{i}+\tilde{T}_{i}\cdot \begin{bmatrix} \dot{\phi}_{i}\dot{\theta}_{i}\\ \dot{\phi}_{i}\dot{\psi}_{i}\\ \dot{\theta}_{i}\dot{\psi}_{i}\end{bmatrix} +(JT_{i})^{-1}\mathbf{S}^{T} \left(T_{i}\cdot \begin{bmatrix}\dot{\phi}_{i}\\ \dot{\theta}_{i}\\ \dot{\psi}_{i}\end{bmatrix} \right)JT_{i}\cdot \begin{bmatrix}\dot{\phi}_{i}\\ \dot{\theta}_{i}\\ \dot{\psi}_{i}\end{bmatrix}  \tag{7}\]  
The quadrotor is known as an under-actuated system. When the quadrotor tilts, a component of the total thrust is directed sideways and the aircraft translates horizontally.