## 2 Dynamics Model of Quadrotor UAV  
### Modeling of Attitude Motion Characteristics  
The quadrotor's flight is sustained by the forces generated by each of its four rotors, with each rotor producing thrust that contributes to the overall lift, stability, and control of the vehicle [27; 28]. An inertial frame \(o\)\(x_{e}y_{e}z_{e}\) and a body-fixed frame \(o\)\(x_{b}y_{b}z_{b}\), with the origin at the center of mass of the quadrotor as shown in Figure 1, are considered. Based on the principles of rigid body dynamics, the rotational dynamics of a quadrotor can be expressed using the Newton-Euler method in the body fixed frame with the inertia matrix \(\mathbf{I}=\text{diag}(I_{x},I_{y},I_{z})\), assuming the neglect of gyroscopic torque and air resistance [29; 30]:  
\[\mathbf{I}\cdot\dot{\mathbf{\omega}}=-\mathbf{\omega}\times(\mathbf{I}\cdot\mathbf{\omega})+\mathbf{ \tau}, \tag{1}\]  
where \(\mathbf{\omega}=(p,q,r)^{\text{T}}\) is the angular velocity vector and \(\dot{\mathbf{\omega}}\) is its derivative with respect to time; \(\mathbf{\omega}\times(\mathbf{I}\cdot\mathbf{\omega})\) represents the gyroscopic effect due to the rotational motion of the quadrotor; and \(\mathbf{\tau}=\left(\tau_{x},\tau_{y},\tau_{z}\right)^{\text{T}}\) is the external torque vector acting on the quadrotor. To facilitate control-oriented modeling, the external torque vector \(\mathbf{\tau}\) is represented by the pitch, roll, and yaw control inputs \(U_{1}\), \(U_{2}\) and \(U_{3}\). Accordingly, the rotational dynamics of the quadrotor can be derived from (1):  
\[\left\{ \begin{array}{l}\dot{p}=\frac{I_{y}-I_{z}}{L_{z}}qr+\frac{1}{L_{y}}U_ {1}\\ \dot{q}=\frac{I_{z}-I_{y}}{L_{z}}rp+\frac{1}{L_{y}}U_{2}\\ \dot{r}=\frac{I_{z}-I_{y}}{L_{z}}pq+\frac{1}{L_{y}}U_{3}\end{array} \right., \tag{2}\]  
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Figure 1: Configuration, inertial and body fixed frame of the quadrotor. The attitude of the quadrotor is given by the three Euler angles: yaw angle \(\phi\), pitch angle \(\theta\) and roll angle \(\psi\) that together form the vector \(\mathbf{\Omega}=\left(\phi,\theta,\psi\right)^{\text{T}}\). The conversion relationship between Euler angle velocities \(\hat{\mathbf{\Omega}}\) and angular velocities in the body frame \(\omega\) is as follows:  
\[ \begin{pmatrix}p\\ q\\ r\end{pmatrix} =\mathbf{R}_{x}\mathbf{R}_{y}\mathbf{R}_{z} \begin{pmatrix}0\\ 0\\ \dot{\psi}\end{pmatrix} +\mathbf{R}_{x}\mathbf{R}_{y} \begin{pmatrix}0\\ \dot{\theta}\\ 0\end{pmatrix} +\mathbf{R}_{x} \begin{pmatrix}\dot{\phi}\\ 0\\ 0\end{pmatrix} = \begin{pmatrix}1&0&-\sin\theta\\ 0&\cos\phi&\cos\theta\sin\phi\\ 0&-\sin\phi&\cos\theta\cos\phi\end{pmatrix}\begin{pmatrix}\dot{\phi}\\ \dot{\theta}\\ \dot{\psi}\end{pmatrix} , \tag{3}\]  
where  
\[ \begin{split}\mathbf{R}_{x}=\begin{pmatrix}1&0&0\\ 0&\cos\phi&\sin\phi\\ 0&-\sin\phi&\cos\phi\end{pmatrix},\ \mathbf{R}_{y}=\begin{pmatrix}\cos \theta&0&-\sin\theta\\ 0&1&0\\ \sin\theta&0&\cos\theta\end{pmatrix},\ \mathbf{R}_{z}=\begin{pmatrix}\cos\psi&\sin \psi&0\\ -\sin\psi&\cos\psi&0\\ 0&0&1\end{pmatrix}.\end{split} \]  
The above equation can be written as:  
\[ \begin{pmatrix}\dot{\phi}\\ \dot{\theta}\\ \dot{\psi}\end{pmatrix} = \begin{pmatrix}1&\tan\theta\sin\phi&\tan\theta\cos \phi\\ 0&\cos\phi&-\sin\phi\\ 0&\sin\phi\sec\theta&\cos\phi\sec\theta\end{pmatrix}\begin{pmatrix}p\\ q\\ r\end{pmatrix} , \tag{4}\]  
In summary, from (2) and (4), the nonlinear dynamics model of the quadrotor attitude can be obtained as:  
\[ \begin{pmatrix}\dot{\phi}\\ \dot{\theta}\\ \dot{\psi}\\ \dot{p}\\ \dot{q}\\ \dot{r}\end{pmatrix} = \begin{pmatrix}p+q\tan\theta\sin\phi+r\tan\theta\cos \phi\\ q\cos\phi-r\sin\phi\\ q\sin\phi\sec\theta+r\cos\phi\sec\theta\\ \frac{l_{y}-l_{z}}{l_{z}}qr\\ \frac{l_{z}-l_{y}}{l_{y}}rp\\ \frac{l_{x}-l_{y}}{l_{z}}pq\end{pmatrix} + \begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&0\\ \frac{1}{l_{x}}&0&0\\ 0&\frac{1}{l_{y}}&0\\ 0&0&\frac{1}{l_{z}}\end{pmatrix}\begin{pmatrix}U_{1}\\ U_{2}\\ U_{3}\end{pmatrix}