## 3 Aggressive Control of Quadrotors with Extended Finite-Time Observer
### Nonlinear-Flatness Based Decoupling Control
We observe from the equations in (A4) that the yaw angle is a component of the flat output \(F_{i}\), then, the desired values of the yaw angle (\(\psi_{i}^{d}\), \(\psi_{i}^{d}\) and \(\dot{\psi}_{i}^{d}\)) are trivial to obtain. However, the desired pitch and roll angles are not explicitly given by the components of the flat output \(F_{i}^{d}\) and its derivatives. We will discuss how to obtain the terms \(\phi_{i}^{d}\), \(\dot{\phi}_{i}^{d}\)\(\dot{\phi}_{i}^{d}\), \(\theta_{i}^{d}\), \(\dot{\theta}_{i}^{d}\) and \(\dot{\bar{\theta}}_{i}^{d}\) as follows. As analyzed before, the design of \(\tau_{\dot{\theta}_{i}}\) is simple. Normally, in order to represent the control input by using the flatness output, the high-order derivatives of \(F_{i}\) are calculated. Specifically, in our case, in order to represent the torques \(\tau_{\dot{\theta}_{i}}\) and \(\tau_{\dot{\theta}_{i}}\), two supplementary derivatives of \(\dot{X}_{i}\) and \(\dot{Y}_{i}\) in Equation (A2) are necessary and we obtain  
\[ \begin{bmatrix}X_{i}\\ Y_{i}\end{bmatrix} ^{(4)}=A \begin{bmatrix}\dot{\bar{\theta}}_{i}\\ \dot{\bar{\theta}}_{i}\end{bmatrix} +\zeta(\Theta_{i},\Theta_{i},\dot{\bar{ \psi}}_{i},\dot{Z}_{i},\dot{\bar{Z}}_{i},\dot{\bar{Z}}_{i})=v_{i}\]  
where matrix \(A\) satisfies  
\[A=(\dot{Z}_{i}+g)\times \begin{bmatrix}\sin\psi_{i}\text{sec}^{2}\phi_{i}\text {sec}\,\theta_{i}&\cos\psi_{i}\text{sec}^{2}\,\theta_{i}+\sin\psi_{i}\tan \phi_{i}\text{tan}\,\theta_{i}\text{sec}\,\theta_{i}\\ -\cos\psi_{i}\text{sec}^{2}\phi_{i}\text{sec}\,\theta_{i}&\sin\psi_{i}\text{ sec}^{2}\,\theta_{i}-\cos\psi_{i}\tan\phi_{i}\text{tan}\,\theta_{i}\text{sec}\, \theta_{i}\end{bmatrix} \]  
The function \(\zeta\) represents the terms of the second-order derivative of the right side of Equation (A2) except the term that contains \(\ddot{\phi}_{i}\) and \(\ddot{\theta}_{i}\). The vector \(v_{i}=[v_{i1}\quad v_{i2}]^{T}\) is composed by the new control inputs \(v_{i1}\) and \(v_{i2}\), which are decoupled on axes \(X\) and \(Y\). Then, the new trajectory tracking controllers in closed loop on the plane \(X-Y\) are given as follows  
\[ \begin{split} v_{i1}&=\dddot{X}_{i}^{d}-k_{4X}(\ddot{X}_{i}- \ddot{X}_{i}^{d})-k_{3X}(\ddot{X}_{i}-\dot{X}_{i}^{d})-k_{2X}(\dot{X}_{i}-\dot{X }_{i}^{d})-k_{1X}(X_{i}-X_{i}^{d})\\ v_{i2}&=\dddot{Y}_{i}^{d}-k_{4Y}(\ddot{Y}_{i}-\ddot{Y}_{i}^{d})-k_{3 Y}(\ddot{Y}_{i}-\ddot{Y}_{i}^{d})-k_{2Y}(\dot{Y}_{i}-\dot{Y}_{i}^{d})-k_{1Y}(Y_{i}-Y_{i}^{d}) \end{split}  \tag{14}\]Observe that \(\det A=(\dot{Z}_{i}+g)\text{sec}^{2}\,\phi_{i}\text{sec}^{3}\,\theta_{i}\). Therefore, when \(\dot{Z}_{i}\
eq-g\) and \(|\phi_{i}|<\frac{\pi}{2}\), \(|\theta_{i}|<\frac{\pi}{2}\), \(\det A\) is finite and positive such that \(A\) is invertible. In fact, if \(|\phi_{i}|<\frac{\pi}{2}\), \(|\theta_{i}|<\frac{\pi}{2}\), we have \(\dot{Z}_{i}>-g\). \(\dot{Z}_{i}\) will be never equal to \(-g\), unless that all the rotors of the quadrotor stop rotating. This case is not considered in this work. Then, according to Equation (13), we can design the torque \(\tilde{\tau}_{i}=[\tilde{\tau}_{\phi_{i}}\quad\tilde{\tau}_{\theta_{i}}\quad \tilde{\tau}_{\phi_{i}}]^{T}\) as follows  
\[ \begin{bmatrix}\tilde{\tau}_{\phi_{i}}\\ \tilde{\tau}_{\theta_{i}}\\ \tilde{\tau}_{\phi_{i}}\end{bmatrix} = \begin{bmatrix}A^{-1}(v_{i}-\zeta)\\ \tilde{\psi}_{i}^{d}-k_{2\tilde{\tau}_{i}}(\dot{\psi}_{i}-\dot{\psi}_{i}^{d})- k_{1\psi_{i}}(\psi_{i}-\psi_{i}^{d})\end{bmatrix}  \tag{15}\]  
Then, substituting (15) into (12), we obtain the three torques. In practice, the velocities \(\dot{X}_{i}\), \(\dot{Y}_{i}\) and its higher-order derivatives \(\ddot{X}_{i}\), \(\dddot{X}_{i}\), \(\dot{Y}_{i}\) and \(\dddot{Y}_{i}\) are not directly measurable (not contained in the output \(y_{i}\) (9)), they are obtained by using Equation (A2) and its derivatives. We will present in the following subsection how to obtain the these derivatives.