# Design and Control of an Ultra-Low-Cost Logistic Delivery Fixed-Wing UAV
## 2 Overall UAV Design
### Fixed-Wing UAV Dynamic Modeling
\right) \tag{3}\]  
Above is the dynamic model of three angular displacements. \[ \begin{pmatrix}o\\ \dot{v}\\ \dot{a}\\ \end{pmatrix} = \begin{pmatrix}rv-qa\\ pa-ro\\ qo-pv\\ \end{pmatrix} +\frac{1}{m} \begin{pmatrix}f_{x}\\ f_{y}\\ f_{z}\\ \end{pmatrix}  \tag{4}\]  
Figure 1: Coordinate system of the drone. Above is the model of three velocity dynamics. \[ \begin{pmatrix}\dot{p}\\ \dot{q}\\ \dot{r}\end{pmatrix} = \begin{pmatrix}\Gamma_{1}pq-\Gamma_{2}qr+\Gamma_{3}l+\Gamma_{4 }n\\ \Gamma_{5}pr-\Gamma_{6}\big{(}p^{2}-r^{2}\big{)}+\frac{1}{l_{y}}m\\ \Gamma_{7}pq-\Gamma_{1}qr+\Gamma_{4}l+\Gamma_{8}n\end{pmatrix}  \tag{5}\]  
Above is the dynamic model corresponding to the three angular velocities, where m is the mass of the body, and f is the external force. \[ \begin{array}{l}\Gamma_{1}=\frac{l_{x}\left(l_{x}-l_{y}+L_{z}\right)}{ \Gamma},\Gamma_{2}=\frac{l_{x}\left(l_{x}-l_{y}\right)+l_{xz}^{2}}{\Gamma},\\ \Gamma_{3}=\frac{l_{x}}{\Gamma},\Gamma_{4}=\frac{l_{yz}}{\Gamma},\Gamma_{5}= \frac{l_{x}-l_{y}}{l_{y}},\Gamma_{6}=\frac{l_{yz}}{l_{y}},\\ \Gamma_{7}=\frac{\left(l_{x}-l_{y}\right)l_{x}+l_{xz}^{2}}{\Gamma},\Gamma_{8}= \frac{l_{x}}{\Gamma}\end{array} , \tag{6}\]  
where \(p_{n}\) is the displacement in the \(i^{i}\) direction (\(F^{i}\) inertial system), \(p_{e}\) is the displacement in the \(j^{i}\) direction (\(F^{i}\) inertial system), \(p_{d}\) is the displacement in the \(k^{i}\) direction (\(F^{i}\) inertial system), \(o\) is the displacement in the \(i^{b}\) direction (\(F^{b}\) inertial system), \(v\) is the displacement in the \(j^{b}\) direction (\(F^{b}\) inertial system), \(a\) is the displacement in the \(k^{b}\) direction (\(F^{b}\) inertial system), \(\gamma\) is the roll angle, \(\theta\) is the pitch angle, \(\lambda\) is the yaw Angle, \(p\) is the rotation rate in roll \(i^{b}(F^{b}inertial\ system)\), \(q\) is the pitch angle velocity \(j^{b}(F^{b}inertial\ system)\), and \(r\) is the yawing rate \(k^{b}F^{b}(inertial\ system)\). Of course, the above dynamic model assumes that the inertial system of the ground is on the horizontal ground, ignoring the rotation of the Earth and ignoring the external forces. Next, the forces and torques generated by the external forces of the model are established. The aerodynamic model of the gravity is as follows:  
\[f_{g}^{b}=\mathcal{R}_{v}^{b} \begin{pmatrix}0\\ 0\\ mg\end{pmatrix} = \begin{pmatrix}-mg\sin\theta\\ mg\cos\theta\sin\gamma\\ mg\cos\theta\cos\gamma\end{pmatrix}  \tag{7}\]  
The aerodynamic model of the lift is as follows:  
\[F_{lift}=\frac{1}{2}\rho V_{a}^{2}S\bigg{[}C_{lo}+C_{Ie}\alpha+C_{Iq}\frac{c} {2V_{\alpha}}q+C_{Ie}\delta_{e}\bigg{]} \tag{8}\]  
The aerodynamic model of the drag is as follows:  
\[F_{\text{drag}}\ \approx\frac{1}{2}\rho V_{a}^{2}S\bigg{[}C_{To}+C_{Te}\alpha+C_{Tq }\frac{c}{2V_{\alpha}}q+C_{Te}\delta_{e}\bigg{]} \tag{9}\]  
The aerodynamic model of the pitching force is as follows:  
\[M=\frac{1}{2}\rho V_{a}^{2}Sc\bigg{[}C_{Zo}+C_{Ze}\alpha+C_{Zq}\frac{c}{2V_{ \alpha}}q+C_{Zi_{ke}}\delta_{e}\bigg{]} \tag{10}\]  
where \(S\) is the wing area; \(c\) is the average aerodynamic chord of the wing; \(\rho\) is the density of the air; \(\alpha\) is the angle of the attack; \(q\) is the rotation rate in the pitch; \(\delta_{e}\) is the elevator; \(C_{lo},C_{Ie},C_{Iq},C_{Ie}\) are dimensionless lift coefficients; \(C_{Tq},C_{Te},C_{Tq},C_{T_{Se}}\) are dimensionless drag coefficients; and \(C_{Zo},C_{Ze}\alpha,C_{Zq},C_{Zi_{ke}}\) are dimensionless moment coefficients. The above three equations are the longitudinal aerodynamic model for the case of small angles of attack, and similarly, the transverse dynamic model can be expressed as follows:  
\[