# A Lateral-Directional Control Method for High Aspect Ratio Full-Wing UAV and Flight Tests
## 2 Mathematical Model and Flying Qualities  
### Lateral-Directional Mathematical Model  
The lateral-directional mathematical model of a high aspect ratio full-wing UAV can be expressed as:  
\[\begin{cases}\dot{\psi}=p\omega-ru+g\cos\theta\sin\phi+Y/M\\ \dot{\phi}=p+q\sin\phi\tan\theta+r\cos\phi\tan\theta\\ \dot{\psi}=q\sin\phi\sec\theta+r\cos\phi\sec\theta\\ \dot{p}=\Gamma_{1}pq-\Gamma_{2}qr+\Gamma_{3}l+\Gamma_{4}n\\ \dot{r}=\Gamma_{5}pq-\Gamma_{1}qr+\Gamma_{4}l+\Gamma_{6}n\end{cases} \tag{1}\]  
and:  
\[\begin{cases}\Gamma=I_{x}I_{z}-I_{xz}^{2}\Gamma_{1}=I_{xz}\left(I_{x}-I_{y}+I _{z}\right)/\Gamma,\Gamma_{2}=\left(I_{z}\left(I_{z}-I_{y}\right)+I_{xz}^{2} \right)/\Gamma\\ \Gamma_{3}=I_{z}/\Gamma,\Gamma_{4}=I_{xz}/\Gamma,\Gamma_{5}=\left(I_{x}\left(I _{x}-I_{y}\right)+I_{xz}^{2}\right)/\Gamma,\Gamma_{6}=I_{x}/\Gamma\end{cases} \tag{2}\]  
\[Y=qSC_{Y\beta}\beta \tag{3}\]  
Figure 1: High aspect ratio full-wing configuration.  
\[l=\bar{q}SbC_{l\beta}\beta+\frac{\bar{q}b^{2}}{2V_{a}}S\left(C_{lp}p+C_{lr}r\right) \tag{4}\]  
\[n=\bar{q}SbC_{n\beta}\beta+\frac{\bar{q}b^{2}}{2V_{a}}S\left(C_{np}p+C_{nr}r \right)+\Delta F_{p}d_{p} \tag{5}\]  
In the equations, \(I_{x}\), \(I_{y}\), \(I_{z}\), and \(I_{xz}\) are the moments of inertia and the product of inertia, respectively. \(M\) is the mass. \(u,v,w\) are velocities in the body-fixed frame. \(p,q,r\) are angular rates in the body-fixed axes. \(\beta,\theta,\phi\), and \(\psi\) are the sideslip angle, pitch angle, roll angle, and yaw angle. \(Y,l,n\) are the lateral force, roll, and yaw moments. \(C_{Y\beta},C_{l\beta},C_{n\beta}\) are the lateral force derivative, roll, and yaw static stability derivative. \(C_{np},C_{lr}\) are the dynamic cross derivatives, and \(C_{lp},C_{nr}\) are the roll and yaw damping derivatives. \(Va,S,b\) are the airspeed, planform area, and wingspan of the aircraft. \(\bar{q}\) is the dynamic pressure. \(\Delta F_{p}\) is the differential thrust of propellers, while \(d_{p}\) is the distance between the propellers and the symmetric plane of the UAV.  
The author of this paper improved the propeller thrust model proposed by [PERSON] [21] through experimental data in the article [22].  
\[F_{p}=\frac{1}{2}\rho S_{p}C_{p}\left(k_{p1}{\delta_{p}}^{2}+k_{p2}\delta_{p}- V_{p}^{2}\right) \tag{6}\]  
The testing picture of the propeller thrust and the propeller thrust fitting results is shown in Figures 2 and 3. The fitting R-squared reached 0.997.  
Define \(\Delta\delta_{p}\) as the differential throttle; the throttles of the left and right propellers are \(\delta_{pl}=\delta_{p}+\Delta\delta_{p}\) and \(\delta_{pr}=\delta_{p}-\Delta\delta_{p}\). Then, \(V_{p}\) of the left and right propellers are \(V_{pl}=V_{a}+d_{p}r\) and \(V_{pr}=V_{a}-d_{p}r\). The differential thrust of the propellers is:  
\[\Delta F_{p}=F_{p_{l}}-F_{pr}=\rho S_{p}C_{p}(2k_{p1}\delta_{p}\Delta\delta_{p }+k_{p2}\Delta\delta_{p}-2V_{a}d_{p}r) \tag{7}\]  
When the error between the course angle \(\chi\) and the yaw angle \(\psi\) is not considered, the horizontal position of the drone can be expressed as:  
\[\left\{\begin{aligned} \dot{P}_{x}&=V_{e}\sin\psi\\ \dot{P}_{y}&=V_{e}\cos\psi\end{aligned}\right. \tag{8}\]  
where \(V_{e}\) is the ground speed of the drone. \(P_{x}\) and \(P_{y}\) are the positions of the drone in the east and north directions.  
Figure 2: The testing picture of the propeller thrust.