## 3 Design Space
### Performance Constraint Analysis  
The large-scale electric cargo UAV studied in this article belongs to the general aircraft category. In order to further complete the closed loop of the equations, meet more design constraints, and obtain relevant initial parameters such as wing area and power (refer to the related studies [10; 17] two new key design parameters of wing load and power load are used to conduct a two-dimensional design space constraint analysis, as shown in Figure 2. Considering the flight conditions, five flight constraints, including horizontal constant speed hovering, climb rate, take-off run, cruising speed, and practical ceiling, are analyzed. (1) Level Constant-Velocity Turn  
The following expression is used to determine the thrust-to-weight ratio (\(T/W\)) to maintain a specific banking load factor (\(n\)) at a specific airspeed and altitude without losing altitude. The electric aircraft studied in this article belongs to the \"normal\" category; maneuvers shall be limited to normal flying maneuvers, including straight and steady stalls and turns in which the angle of bank does not exceed \(60^{\circ}\):  
\[\frac{T}{W}=q\left[\frac{C_{D0}}{W/S}+k\left(\frac{n}{q}\right)^{2}\left( \frac{W}{S}\right)\right], \tag{6}\]  
where \(C_{D0}\) is the zero-lift drag coefficient; \(k\) is the lift-induced drag constant; \(q\) is the dynamic pressure at the selected airspeed and altitude (\(N/\text{m}^{2}\)); \(S\) is the wing area (\(m^{2}\)); \(T\) is the thrust (\(N\)); \(W\) is the aircraft weight (\(N\)); and \(n\) is the load factor, expressed as \(n{=}1/\cos\phi\). The value of \(C_{D0}\) can be matched according to the reference of typical aerodynamic characteristics of selected classes of aircraft ([PERSON], 2014 [17]). In the conceptual design stage, the relationship between the drag coefficient of the UAV, the lift drag factor, and the zero-lift drag coefficient can be expressed through the polar curve equation \(C_{D}=C_{D0}+k\cdot C_{L}^{2}\), where \(k\) is obtained by Equation (7):  
\[k=\frac{1}{\pi\cdot AR\cdot e}. \tag{7}\]  
Figure 2: Schematic plot of performance constraints. The aspect ratio (\(AR\)) is initially selected based on the designer's needs and experience, and the Oswald efficiency factor (\(e\)) is calculated by selecting an empirical formula (\(e=1.78(1-0.045 AR^{0.68})-0.64\)) suitable for the low aspect ratio ([PERSON], 2018 [18]). (2) Desired Rate of Climb  
Rate of climb is a measure of the vertical altitude an aircraft can reach per unit of time. For propeller aircrafts, assuming that the climb angle satisfies the small angle relationship and \(\cos\theta\) is approximately equal to 1, the following expression is used to determine the thrust-to-weight ratio required to achieve a given rate of climb:  
\[\frac{T}{W}=q\frac{C_{D0}}{W/S}+\frac{k}{q}\cdot\left(\frac{W}{S}\right)+\frac {V_{v}}{V}, \tag{8}\]  
where \(q\) is the dynamic pressure at the selected airspeed and altitude; \(V\) is the airspeed; \(V_{V}\) is the vertical speed. (3) Desired Take-off Run Distance  
The length of each runway is fixed. The thrust-to-weight ratio required is determined by the Equation (9) to achieve a given ground run distance during take-off:  
\[\frac{T}{W}=\frac{V_{LOF}^{2}}{2g\cdot S_{G}}+q\frac{C_{D,TO}}{W/S}+\mu(1- \frac{q\cdot C_{L,TO}}{W/S}), \tag{9}\]  
where \(C_{L,TO}\) is the lift coefficient during take-off run; \(C_{D,TO}\) is the drag coefficient during take-off run; \(V_{LOF}\) is the liftoff speed; \(q\) is the dynamic pressure at \(V_{LOF}/\sqrt{2}\) and selected altitude; \(S_{G}\) is the ground run; \(\mu\) is the ground friction constant; and \(g\) is the acceleration due to gravity. (4) Desired Cruise Airspeed  
The thrust-to-weight ratio required is determined by the following expression to achieve a given cruising speed at a desired altitude:  
\[\frac{T}{W}=qC_{D0}\left(\frac{1}{W/S}\right)+k\left(\frac{1}{q}\right)\cdot \left(\frac{W}{S}\right), \tag{10}\]  
where \(q\) is the dynamic pressure at the selected airspeed and altitude.