## 3 Aggressive Control of Quadrotors with Extended Finite-Time Observer
### Nonlinear-Flatness Based Decoupling Control  
We firstly define the aggressive formation of quadrotors as follows. [Aggressive formation assumption] The attitude angles pitch \(\theta_{i}\) and roll \(\phi_{i}\) of each quadrotor can be greater than \(15^{\circ}\) (but smaller than \(30^{\circ}\)) in aggressive formation modes. According to Proposition 3, we conclude that the desired motion of a quadrotor can be uniquely specified by the 3D coordinates \(\mathcal{X}_{i}\) and the yaw angle \(\psi_{i}\). Therefore, the decentralized formation tracking problem is to find the _guidance vector_\([X_{i}^{d},Y_{i}^{d},Z_{i}^{d},\psi_{i}^{d}]^{T}\) for each quadrotor. The guidance vector is composed by the neighbors states, which will be designed in Section 4. The control input \(u_{i}=[F_{T_{i}},\varphi_{i},\tau_{\theta_{i}},\tau_{\theta_{i}}]^{T}\) is designed in order to minimize the tracking errors of \(\mathcal{X}_{i}-\mathcal{X}_{i}^{d}\) and \(\psi_{i}-\psi_{i}^{d}\), where \(\mathcal{X}_{i}^{d}\) and \(\psi_{i}^{d}\) represent the desired 3D position and the heading direction (yaw angle). According to Equation (A1), the controller for the altitude is designed as follows  
\[F_{T_{i}}=\frac{(\dot{Z}_{i}^{d}-k_{2Z}(\dot{Z}_{i}-\dot{Z}_{i}^{d})-k_{1Z}(Z_{ i}-Z_{i}^{d})+g)m}{\cos\theta_{i}\cos\phi_{i}} \tag{10}\]  
where \(Z_{i}^{d}\) represents the desired value of the altitude. Let us denote by \(e_{Z_{i}}=Z_{i}-Z_{i}^{d}\) the tracking error of the altitude and substitute (10) into the third equation in (5). Then, we have  
\[\ddot{e}_{Z_{i}}=-k_{2Z}e_{Z_{i}}-k_{1Z}e_{Z_{i}}\]  
Obviously, the altitude can exponentially track the given desired altitude trajectory \(Z_{i}^{d}\) for some selected positive scalars \(k_{1z}\) and \(k_{2z}\). The desired altitude trajectory should satisfy \(Z_{i}^{d}\in C^{3}(\mathbb{R},\mathbb{R})\). Then, the altitude dynamics are decoupled with the other states of the quadrotor. We rewrite the dynamic of the attitude angles in (7) as follows  
\[\dot{\Theta}_{i}=\bar{T}_{i}\cdot \begin{bmatrix}\phi_{i}\dot{\theta}_{i}\\ \dot{\phi}_{i}\dot{\psi}_{i}\\ \dot{\theta}_{i}\dot{\psi}_{i}\end{bmatrix} +(JT_{i})^{-1}\mathbf{S}^{T}\big{(}T_{i} \cdot\dot{\Theta}_{i}\big{)}JT_{i}\cdot\dot{\Theta}_{i}+(JT_{i})^{-1}\tau_{i} \tag{11}\]  
The inputs \(\tau_{i}=[\tau_{\dot{\Theta}_{i}},\tau_{\dot{\Theta}_{i}},\tau_{\dot{\psi}_{i}} ]^{T}\) are designed as follows  
\[\tau_{i}=JT_{i}\bar{\tau}_{i}+JT_{i} \begin{bmatrix}\dot{\phi}_{i}\dot{\theta}_ {i}\\ \dot{\phi}_{i}\dot{\psi}_{i}\\ \dot{\theta}_{i}\dot{\psi}_{i}\end{bmatrix} -\mathbf{S}^{T}\big{(}T_{i}\cdot\dot{ \Theta}_{i}\big{)}JT_{i}\cdot\dot{\Theta}_{i} \tag{12}\]  
Substituting (12) into (11), we obtain  
\[\dot{\Theta}_{i}=\bar{\tau}_{i} \tag{13}\]  
where \(\bar{\tau}_{i}=\dot{\Theta}_{i}^{d}-k_{2\Theta_{i}}(\dot{\Theta}_{i}-\dot{ \Theta}_{i}^{d})-k_{1\Theta_{i}}(\Theta_{i}-\Theta_{i}^{d})\). Notations \(k_{1\Theta_{i}}\) and \(k_{2\Theta_{i}}\) represent two diagonal gain matrices. If \(\Theta_{i}^{d}\), \(\dot{\Theta}_{i}^{d}\) and \(\dot{\Theta}_{i}^{d}\) are calculated according Ito the equations in (A4), namely, replacing \(F_{i}\), \(F_{i}\), \(F_{i}\), \(\bar{F}_{i}\), and \(\bar{\bar{F}}_{i}\) by \(F_{i}^{d}\), \(F_{i}^{d}\), \(\bar{F}_{i}^{d}\), \(\bar{\bar{F}}_{i}^{d}\), and \(\bar{\bar{F}}_{i}^{d}\), it is trivial to observe that the attitude angles can exponentially track the desired attitude angles that are in terms of \(F_{i}\) and its derivatives. However, the controller (12) is open-loop on the translational dynamics, because the position feedback is not used in the controller design. If the dynamics of the quadrotor is precisely modeled and no external disturbances and sensors noise exist, this controller can perfectly perform. However, in practice, the dynamics of the system can be never precisely modeled, additionally, the noise and disturbance always exist. Therefore, the closed-loop control should be used, such that the feedback of the positions and translational velocities are required.