# Design and Control of an Ultra-Low-Cost Logistic Delivery Fixed-Wing UAV
## 2 Overall UAV Design
### Fixed-Wing UAV Dynamic Modeling  
This paper initially establishes the coordinate system for fixed-wing UAVs, as shown in Figure 1. The kinematic equations for the position, ground speed, and heading angle are relative to the inertial coordinate system, while the aircraft's motion and aerodynamic interactions are described through the body coordinate system. The fixed-wing UAV \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Start Year** & **Project** & **Demonstration Place** & **Material** \\ \hline
2023 & [35] & Queensland University of Technology & 3D-printed polylactic acid (PLA) \\ \hline
2022 & [36] & Aristotle University of Thessaloniki & Carbon fiber \\ \hline
2022 & [37] & Warsaw University of Technology & Carbon fiber \\ \hline
2021 & [38] & Kempten University of Applied Sciences & Carbon fiber \\ \hline
2019 & [39] & University of Putra Malaysia (UPM) & Expanded polystyrene (EPS) \\ \hline
2018 & [40] & Yildiz Mah University & Carbon fiber \\ \hline
2012 & [41] & Autonomous University of Barcelona & Polypropylene foam \\ \hline
2012 & [42] & University of Maryland, College Park & Carbon fiber \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Material display of fixed-wing UAVs by country in the past decade. coordinate system, as illustrated in Figure 1, includes \(F_{i}(i,\,j,\,k)\) as the inertial reference frame, with its origin typically at the UAV's home point (usually the takeoff point), and the unit vectors \(i,\,j,\) and k pointing toward Earth's north and east and toward the center of the Earth, respectively. \(\text{Fb}\left(x,\,y,\,z\right)\) represents the body coordinate system. The origin is at the UAV's center of mass, with \(x\) and \(y\) pointing along the UAV's longitudinal and lateral axes, and \(z\) being the axis perpendicular to the \(\left(x,\,y\right)\) plane, pointing downward. \(\gamma\) is for the roll angle, \(\theta\) for the pitch angle, \(\lambda\) for the yaw angle, and \(\left(x,\,y,\,z\right)\) represents the position of the centroid in an inertial coordinate system. \[ \begin{split} R_{i}^{b}=R(\gamma)R(\theta)R(\lambda)\\ =\begin{pmatrix}1&0&0\\ 0&\cos\gamma&\sin\gamma\\ 0&-\sin\gamma&\cos\gamma\\ \end{pmatrix}\begin{pmatrix}\cos\theta&0&-\sin\theta\\ 0&1&0\\ \sin\theta&0&\cos\theta\\ \end{pmatrix}\begin{pmatrix}\cos\lambda&\sin\lambda&0\\ -\sin\lambda&\cos\lambda&0\\ 0&0&1\\ \end{pmatrix}\\ =\begin{pmatrix}c_{\theta}c_{\lambda}&c_{\theta}s_{\lambda}&-s_{\theta}\\ s_{\gamma}s_{\theta}c_{\lambda}-c_{\gamma}s_{\lambda}&s_{\gamma}s_{\theta}s_{ \lambda}+c_{\gamma}c_{\lambda}&s_{\gamma}c_{\theta}\\ c_{\gamma}s_{\theta}c_{\lambda}+s_{\gamma}s_{\lambda}&c_{\gamma}s_{\theta}s_{ \lambda}-s_{\gamma}c_{\lambda}&c_{\gamma}c_{\theta}\\ \end{pmatrix}\end{split}  \tag{1}\]  
Next, this paper defines 12 state quantities of fixed-wing UAVs, namely, three position states \((P^{n},P^{e},P^{d})\); the three velocities correspond to translational motion \((o,v,a)\). Three angular displacements and three angular velocities correspond to rotational motion \((p,q,r)\). Note that this is defined here as the angular velocity, not the Euler angular rate, that is, not equal to the time derivative of the Euler Angle; height \(h=-p_{d}\). \[\left( \begin{array}{c}\dot{p}_{n}\\ \dot{p}_{e}\\ \dot{p}_{d}\\ \end{array} \right)= \begin{pmatrix}c_{\theta}c_{\lambda}&c_{\theta}s_{\lambda}& -s_{\theta}\\ s_{\gamma}s_{\theta}c_{\lambda}-c_{\gamma}s_{\lambda}&s_{\gamma}s_{\theta}s_{ \lambda}+c_{\gamma}c_{\lambda}&s_{\gamma}c_{\theta}\\ c_{\gamma}s_{\theta}c_{\lambda}+s_{\gamma}s_{\lambda}&c_{\gamma}s_{\theta}s_{ \lambda}-s_{\gamma}c_{\lambda}&c_{\gamma}c_{\theta}\\ \end{pmatrix} ^{T}\left( \begin{array}{c}o\\ v\\ a\\ \end{array} \right) \tag{2}\]  
Above is the dynamic model of the position state. \[ \begin{pmatrix}\dot{\gamma}\\ \dot{\theta}\\ \dot{\lambda}\\ \end{pmatrix} = \begin{pmatrix}1&\sin\gamma\tan\theta&\cos\gamma\tan\theta\\ 0&\cos\gamma&-\sin\gamma\\ 0&\sin\gamma/\cos\theta&\cos\gamma/\cos\theta\\ \end{pmatrix} \left( \begin{array}{c}p\\ q\\ r\\ \end{array}