## 3 Aggressive Control of Quadrotors with Extended Finite-Time Observer  
### Quadrotor Model  
The dynamics of a quadrotor is modelled as the motion of a rigid body in 3-D space under a thrust force and three moments, which are generated by the thrust forces of the four rotors. The notations used in the quadrotor's model are shown in Table 1. The orientation of the quadrotor with respect to the inertial frame is represented by the rotation matrix \(R_{i}\in\text{SO}(3)\), where SO(3) represents a special orthogonal group, and whose determinant is one. Note that \(R_{i}^{T}R_{i}=I\), where \(I\) represents an identical matrix. The dynamics of a quadrotor \(i\) is shown as follows  
\[\left\{ \begin{aligned} m\tilde{\mathcal{X}}_{i}=-mg\mathbf{e}_{3}+R _{i}F_{T_{i}}\mathbf{e}_{3}\\ \tilde{R}_{i}=R_{i}S(\Omega_{i})\\ J\Omega_{i}+S(\Omega_{i}))\Omega_{i}=\tau_{i}\end{aligned} \right. \tag{3}\]  
where \(\mathcal{X}_{i}=[X_{i},Y_{i},Z_{i}]^{T}\) represents the coordinates of the center of mass of a quadrotor in the fixed inertial frame \(x_{d}y_{e}z_{e}\). The Euler angles (pitch, roll, and yaw) are represented by the vector \(\Theta_{i}=[\phi_{i},\theta_{i},\psi_{i}]^{T}\). The first equation of (3) represents the translational dynamics in the inertial frame, where \(m\) represents the mass of a quadrotor and \(g\) represents the gravity. The rotation matrix \(R_{i}\) in the second equation of (3) can transform the coordinates of a point from the body-fixed frame to the inertial frame. The third equation of (3) represents the rotational dynamics of quadrotor \(i\). The inertia matrix \(J\) of a quadrotor \(i\) is represented in the body-fixed frame, according to the assumptions on the physical structure of a quadrotor, we can obtain that \(J=\text{diag}\{I_{x_{p}},I_{y_{p}},I_{z_{b}}\}\) is diagonal, where scalars \(I_{x_{p}},I_{y_{p}}\) and \(I_{z_{b}}\) represent the moments of inertia with respect to \(x_{p},y_{p}\) and \(z_{b}\) respectively. Additionally, we assume that all the quadrotors in the flock have the same physical coefficients such as the mass and the inertia matrix. The angular velocity of the quadrotor \(i\) in the body-fixed frame is represented by \(\Omega_{i}\in\mathbb{R}^{3}\). The function \(S(\cdot):\mathbb{R}^{3}\to\mathbb{R}^{3\times 3}\) represents an operation that transforms a vector in \(\mathbb{R}^{3}\) to a skew-symmetric matrix \(\mathbb{R}^{3\times 3}\). Given two arbitrary vectors \(v_{1},v_{2}\in\mathbb{R}^{3}\), the function \(S(\cdot)\) satisfies the property \(S(v_{1})\cdot v_{2}=v_{1}\times v_{2}\). Then, according to the definition of cross product and the operation \(S(\cdot)\), we conclude that \(S(\Omega_{i})\in\mathbb{R}^{3\times 3}\) is skew-symmetric matrix in terms of the members of \(\Omega_{i}=[p_{i},q_{i},r_{i}]^{T}\) as follows  
\[S(\Omega_{i})= \begin{bmatrix}0&-r_{i}&q_{i}\\ r_{i}&0&-p_{i}\\ -q_{i}&p_{i}&0\end{bmatrix}  \tag{4}\]  
where \(p_{i},q_{i}\) and \(r_{i}\) represent the angular velocities with respect to the body-fixed frame of the quadrotor \(i\). We refer to \(\phi_{i},\theta_{i},\psi_{i}\) as the roll, pitch, and yaw angles, \(\tau_{i}=[\tau_{\phi_{i}},\tau_{\theta_{i}},\tau_{\psi_{i}}]^{T}\in\mathbb{R}^ {3}\) represents the roll, pitch, and yaw moments. The thrust force is represented by \(F_{T_{i}}\). We note that \(\mathbf{e}_{3}=[0,0,1]^{T}\) is a constant vector. The attitude dynamics of each quadrotor is generated by moments \(\tau_{\phi_{i}},\tau_{\theta_{i}}\) and \(\tau_{\psi_{i}}\) and thrust force \(F_{T_{i}}\). The rotation matrix \(R_{i}\) from the body-fixed frame to the inertial frame is the sequence of roll-pitch-yaw (namely \(\phi-\theta-\psi\)), i.e., \(R_{i}=R_{z}(\psi_{i})R_{y}(\theta_{i})R_{x}(\phi_{i})\), where \(R_{z}(\psi_{i}),R_{y}(\theta_{i})\), \(R_{x}(\phi_{i})\) represents the rotation matrices of yaw, pitch, and roll. \[R_{i}=