## 2 UAV Dynamics Model
### Aerodynamic Model Considering the Influence of Distributed Slip Flow
\right]\]  
Similarly, the pitch moment of the entire drone is expressed as follows:  
\[m_{free}=\frac{1}{2}C_{m\_free\_no\delta_{r}}\rho V_{free}^{2}S_{free}\overline {c}+\frac{1}{2}C_{m\_free\_delta_{r}}V_{free}^{2}\Big{(}S_{free}+S_{slip} \Big{)}\overline{c} \tag{7}\]  
\[ \begin{array}{l}m_{slip}=\sum\limits_{i=1}^{8}\left(\frac{1}{2}C_{m\_slip \_i}\rho V_{slip\_i}^{2}\frac{1}{8}S_{slip}\overline{c}\right)\\ +\sum\limits_{i=1}^{8}\left[\frac{1}{2}C_{m\_slip\_i}\delta_{r}V_{slip\_i}^{2} \Big{(}S_{free}+S_{slip}\Big{)}\overline{c}\right]\\ \end{array}  \tag{8}\]  
\[m_{A}=m_{free}+m_{slip} \tag{9}\]  
For the lateral roll moment and yaw moment, first calculate the lateral yaw moment coefficient of the entire aircraft without slip flow:  
\[C_{l}=C_{l}^{\beta}\beta+\sum\limits_{i=1}^{8}\left(C_{l}^{\delta_{\alpha,j}} \cdot\delta_{\alpha a,j}\right)+C_{l}^{\delta_{r}}\delta_{r}+C_{l}^{p} \overline{p}+C_{l}^{r}\overline{r} \tag{10}\]  
\[C_{n}=C_{n}^{\beta}\beta+\sum\limits_{i=1}^{8}\left(C_{n}^{\delta_{\alpha,j}} \cdot\delta_{\alpha a,j}\right)+C_{n}^{\delta_{r}}\delta_{r}+C_{n}^{p} \overline{p}+C_{n}^{r}\overline{r} \tag{11}\]  
Calculate the lateral yaw moment generated by the aerodynamic force increment in the unmanned aerial vehicle in each slip zone:  
\[ \begin{array}{l}\Delta l_{slip\_i}=\Delta L_{slip\_i}\cdot y_{slip\_i}\\ \Delta n_{slip\_i}=\Delta D_{slip\_i}\cdot y_{slip\_i}\end{array}  \tag{12}\]  
Among them, \(y_{slip\_i}\) is the x-axial distance between the aerodynamic center of each slip zone and the aerodynamic center of the whole machine. The lateral heading torque of the drone is obtained as follows:  
\[ \begin{array}{l}l_{A}=\frac{1}{2}C_{l}\rho V_{free}^{2}(S_{free}+S_{slip})b+ \sum\limits_{i=1}^{8}\Delta l_{slip\_i}\\ n_{A}=\frac{1}{2}C_{n}\rho V_{free}^{2}(S_{free}+S_{slip})b+\sum\limits_{i=1}^{8} \Delta n_{slip\_i}\end{array}  \tag{13}\]  
The aerodynamic force and torque derivation of unmanned aerial vehicles considering slip flow have been completed above. Obtain the scalar equations for the dynamics of the center of mass of the body axis system:\[\left\{ \begin{array}{l}m(\dfrac{du}{dt}+qw-rv)=\sum\limits_{i=1}^{8}(T_{i}\cos \theta_{tdi})-D\cos\alpha\cos\beta\\ -Y\cos\alpha\sin\beta+L\sin\alpha-mg\sin\theta\\ m(\dfrac{dv}{dt}+ru-pw)=-D\sin\beta+Y\cos\beta+mg\sin\phi\cos\theta\\ m(\dfrac{dw}{dt}+pv-qu)=-\sum\limits_{i=1}^{8}(T_{i}\sin\theta_{tdi})-D\sin \alpha\cos\beta\\ -Y\sin\alpha\sin\beta-L\cos\alpha+mg\cos\phi\cos\theta\end{array} \right. \tag{14}\]  
The scalar equation system for the rotational dynamics of the body axis around the center of mass is as follows:  
\[\left\{ \begin{array}{l}I_{x}\dfrac{dp}{dt}+(I_{z}-I_{y})qr-I_{zx}(pq+\dfrac{ dr}{dt})=l+\sum\limits_{i=1}^{8}Q_{xi}+\sum\limits_{i=1}^{8}I_{Tl}\\ I_{y}\dfrac{dq}{dt}+(I_{x}-I_{z})rp+I_{zx}(p^{2}-r^{2})=m+\sum\limits_{i=1}^{8 }Q_{yi}+\sum\limits_{i=1}^{8}m_{Ti}\\ I_{z}\dfrac{dr}{dt}+(I_{y}-I_{x})pq+I_{zx}(qr-\dfrac{dp}{dt})=n+\sum\limits_{i= 1}^{8}Q_{zi}+\sum\limits_{i=1}^{8}n_{Ti}\end{array} \right. \tag{15}\]  
Among them, \(Q_{xi}\), \(Q_{yi}\), and \(Q_{zi}\) are related to aerodynamic moments or torques acting on the UAV. These terms contribute to the overall rotational motion of the UAV around its center of mass.