## 3 Dynamic Model
### Nonlinear Dynamic Model of Quadrotor UAV  
The flight action performed is derived from the lift \(\Gamma_{i}=k_{t}\gamma_{i}^{2},i=1,2,3,4\), which are generated by the rotational speed \(\gamma\) of the rotor, and \(k_{t}\) is the lift coefficient. The four rotors are fixed vertically on the same fuselage plane, so the total lift \(u_{1}\) produced is also vertically upward. After adjusting the motor to make the speed difference between the rotors, the input torque \(u_{2}\) of the roll channel, the input torque \(u_{3}\) of the pitch channel, and the input torque \(u_{4}\) of the yaw channel will be generated, and then the corresponding rotational motion will be performed. Figure 1: Free atmospheric turbulence components. (**a**) U-direction, (**b**) V-direction, and (**c**) W-direction. Figure 2: Structure and coordinate transformation of quadrotor UAV. \[ \begin{bmatrix}u_{1}\\ u_{2}\\ u_{3}\\ u_{4}\end{bmatrix} = \begin{bmatrix}\Gamma_{1}+\Gamma_{2}+\Gamma_{3}+\Gamma_{4}\\ l(\Gamma_{4}-\Gamma_{2})\\ l(\Gamma_{3}-\Gamma_{1})\\ -T_{1}+T_{2}-T_{3}+T_{4}\end{bmatrix} = \begin{bmatrix}k_{t}(\gamma_{1}^{2}+ \gamma_{2}^{2}+\gamma_{3}^{2}+\gamma_{4}^{2})\\ k_{t}(\gamma_{4}^{2}-\gamma_{2}^{2})l\\ k_{t}(\gamma_{3}^{2}-\gamma_{1}^{2})l\\ k_{d}(-\gamma_{1}^{2}+\gamma_{2}^{2}-\gamma_{3}^{2}+\gamma_{4}^{2})\end{bmatrix} , \tag{9}\]  
where \(l\) represents the length of the UAV arm, \(T_{i}=k_{d}\gamma_{i}^{2},i=1,2,3,4\) is the anti-torque moment, and \(k_{d}\) is the anti-torque coefficient. According to the force balance equation, the following expression is established:  
\[m\overset{\rightarrow}{a}=\overset{\rightarrow}{F}_{\Gamma}-\overset{ \rightarrow}{G}-\overset{\rightarrow}{f}_{d} \tag{10}\]  
where \(\overset{\rightarrow}{a}=\left[\ddot{X}\quad\ddot{Y}\quad\ddot{Z}\right]^{T}\) is the acceleration matrix of the three axial directions, and \(m\) is the weight of the fuselage. In order to study the motion state conveniently, it is necessary to transform the force in the airframe coordinate system \(\overset{\rightarrow}{F}_{B}\) into the ground coordinate system \(\overset{\rightarrow}{F}_{E}\) for analysis through transformation matrix \(R_{E}^{B}\). \(\overset{\rightarrow}{F}_{\Gamma}\) represents the total lift matrix generated by the rotor in the ground coordinate system [21]. \(\overset{\rightarrow}{G}= \begin{bmatrix}0&0&mg\end{bmatrix} ^{T}\) represents the fuselage gravity. \(\overset{\rightarrow}{f}_{d}\) represents the atmospheric drag force, and \(k_{i}(i=1,2,3)\) is the drag coefficient of the translating air. Therefore, the linear motion equation of UAV can be obtained as:  
\[\left\{ \begin{array}{l}\ddot{X}=\frac{(\sin\vartheta\sin\vartheta+\cos \vartheta\sin\vartheta\cos\vartheta)u_{1}}{m}-\frac{k_{1}\dot{X}}{m}\\ \ddot{Y}=\frac{(-\cos\vartheta\sin\vartheta+\sin\vartheta\sin\cos\vartheta) u_{1}}{m}-\frac{k_{2}\dot{Y}}{m}\\ \ddot{Z}=\frac{(\cos\theta\cos\varphi)u_{1}-mg}{m}-\frac{k_{3}\dot{Z}}{m} \end{array} \right., \tag{11}\]  
where \(\theta,\phi,\psi\) represent the pitch angle, roll angle and yaw angle generated by the air movement of the aircraft, respectively. According to the moment balance equation, the following expression is established:  
\[M_{\tau}+M_{f}+M_{g}=J\dot{\omega}+\omega\times J\omega, \tag{12}\]  
where \(M_{\tau}\) is the torque generated by the lift of the rotor, \(M_{g}\) is the gyro torque, \(M_{f}\) is the air resistance torque, and the rotational air resistance coefficient is recorded as \(k_{j}(j=4,5,6)\). \(J\) is the inertia matrix, represented by a 3-by-3 diagonal matrix \(J=diagonal\left[I_{X}\quad I_{Y}\quad I_{Z}\right]^{T}\). \(\omega= \begin{bmatrix}p&q&r\end{bmatrix} ^{T}\) represents the angular velocity vector. \(J_{r}\) is the moment of inertia of the rotor, and \(\Omega_{r}=\gamma_{1}-\gamma_{2}+\gamma_{3}-\gamma_{4}\) is the residual error of rotor speed. Next, the angular motion equation of quadrotor UAV can be obtained as:  
\[\left\{