## 2 Model Description and Transformation  
### Dynamics Model  
The mechanical structure of a quadrotor UAV and the coordinate reference system are shown in Figure 1. \(\xi_{E}\{\mathbf{\varrho}_{E}x_{E}y_{E}z_{E}\}\) is the Earth-fixed inertial frame. \(\xi_{B}\{\mathbf{\varrho}_{B}x_{E}y_{B}z_{B}\}\) denotes the body-fixed frame. In the frame \(\xi_{E}\), \(\mathbf{\zeta}=[x,y,z]^{\mathsf{T}}\in\mathcal{R}^{3}\) presents the position state vector. \(\mathbf{\Theta}=[\phi,\theta,\psi]^{\mathsf{T}}\in\mathcal{R}^{3}\) is the Euler angle vector, in which \(\phi,\theta\) and \(\psi\) denote the roll, pitch, and yaw angles. \(\mathbf{\eta}=[u,v,w]^{\mathsf{T}}\in\mathcal{R}^{3}\) and \(\mathbf{\Omega}=[p,q,r]^{\mathsf{T}}\in\mathcal{R}^{3}\) are the linear and angular velocity vectors under the frame \(\xi_{B}\), respectively. Furthermore, the kinematic transformation relationships of the quadrotor UAV between the frame \(\xi_{E}\) and \(\xi_{B}\) are introduced as [32]:  
\[\mathbf{\zeta}=\mathbf{R}_{B}^{E}\mathbf{\eta},~{}\mathbf{\dot{\Theta}}=\mathbf{\Phi}_{B}^{E}\mathbf{ \Omega}, \tag{1}\]  
where the transformation matrices \(\mathbf{R}_{B}^{E}\) and \(\mathbf{\Phi}_{B}^{E}\) from the frame \(\xi_{B}\) to \(\xi_{E}\) can be expressed as  
\[\mathbf{R}_{B}^{E}= \begin{bmatrix}C\partial C\psi&S\phi S\partial C\psi-C\phi S\psi&C \phi S\partial C\psi+S\phi S\psi\\ C\partial S\psi&S\phi S\partial S\psi+C\phi S\psi&C\phi S\partial\psi-S\phi S \phi\\ -S\theta&S\phi C\theta&C\phi C\theta\end{bmatrix} ,\]  
\[\mathbf{\Phi}_{B}^{E}= \begin{bmatrix}1&S\phi T\theta&C\phi T\theta\\ 0&C\phi&-S\phi\\ 0&S\phi/C\theta&C\phi/C\theta\end{bmatrix} ,\]with \(S\cdot,C\cdot\) and \(T\cdot\) denoting \(\sin(\cdot),\cos(\cdot)\) and \(\tan(\cdot)\), respectively. The dynamical model [36; 37] with respect to the frame \(\xi_{B}\) can be established as  
\[ \begin{array}{l}m\dot{\eta}=-\mathbf{\Omega}_{\times}m\eta-\mathbf{F}_{f}+\mathbf{ F}_{g}+\mathbf{U}_{f}+\mathbf{D}_{f},\\ J\dot{\mathbf{\Omega}}=-\mathbf{\Omega}_{\times}J\mathbf{\Omega}-\mathbf{M}_{t}+\mathbf{G}_{a}+ \mathbf{U}_{t}+\mathbf{D}_{t},\end{array}  \tag{2}\]  
where \(m\) is the total mass of the quadrotor UAV. \(\mathbf{J}=\text{diag}\big{\{}J_{x},J_{y},J_{z}\big{\}}\in\mathcal{R}^{3\times 3}\) represents the moments of inertia. \(\mathbf{F}_{f}=[k_{1}u,k_{2}v,k_{3}w]^{\mathrm{T}}\in\mathcal{R}^{3}\) and \(\mathbf{M}_{t}=[k_{4}p,k_{5}q,k_{6}r]^{\mathrm{T}}\in\mathcal{R}^{3}\) represent aerodynamic friction vectors with \(k_{1},k_{2},\ldots,k_{6}\) being the aerodynamic friction coefficients. \(\mathbf{U}_{f}=[0,0,f]^{\mathrm{T}}\in\mathcal{R}^{3}\) is the total thrust vector, and \(\mathbf{U}_{t}=[U_{t1},U_{t2},U_{t3}]^{\mathrm{T}}\in\mathcal{R}^{3}\) denotes the control torque vector. \(\mathbf{D}_{f}=[D_{f1},D_{f2},D_{f3}]^{\mathrm{T}}\in\mathcal{R}^{3}\) and \(\mathbf{D}_{t}=[D_{t1},D_{t2},D_{t3}]^{\mathrm{T}}\in\mathcal{R}^{3}\) denote the external disturbance vectors of the quadrotor UAV. \(\mathbf{F}_{g}=(\mathbf{R}_{B}^{E})^{-1}[0,0,-mg]^{\mathrm{T}}\in\mathcal{R}^{3}\) denotes the gravity vector with \(g\) being the gravity coefficient. \(\mathbf{G}_{a}=[J_{r}q\bar{\omega},-J_{r}p\bar{\omega},0]^{\mathrm{T}}\in\mathcal{ R}^{3}\) represents the gyroscopic moment vector, where \(J_{r}\) is the inertia of each motor, and \(\bar{\omega}=\omega_{1}-\omega_{2}+\omega_{3}-\omega_{4}\) is the total residual rotor speed. Note that the inaccessibility of the rotor speed, \(\mathbf{G}_{a}\) is considered as an additional disturbance during the control design [38]. In addition, \((\cdot)_{\times}\) denotes the skew-symmetric matrix satisfying  
\[\mathbf{x}_{\times}= \begin{bmatrix}0&-x_{3}&x_{2}\\ x_{3}&0&-x_{1}\\ -x_{2}&x_{1}&0\end{bmatrix}