## 2 Materials and Methods
### Establishment of the Optimization Problem  
By parameterizing the morphology and kinematics of the wing model, a total of eight decision variables remain to be determined for this optimization: the three parameters related to the geometry of the wing model, \(r_{0}\), \(L\), \(C_{R}\), and the five associated kinematic parameters, \(f\), \(\varphi_{m}\), \(\theta_{m,r}\), \(\theta_{m,t}\), and \(C_{\theta}\). Biological scaling law was introduced additionally to limit the range of geometric variables within which the expected wing model is more like a real hummingbird wing in shape. Consequently, some parameters of the hummingbird-inspired flapping wing model are explored only in a limited sample space. Next, we will elaborate on this. According to the relation between body weights and parameters of wing performance [45], the predicted values of wing length and wing area are expressed as follows:  
\[\hat{L}=21.07m^{0.62} \tag{34}\]  
\[\text{and }\hat{S}=133.42m^{1.15}. \tag{35}\]  
where \(m\) is the body mass in grams (g), \(\hat{L}\) is the estimated value of wing length in millimeters (mm), and \(\hat{S}\) is the estimated wing area in square millimeters (mm\({}^{2}\)). With the assumption that the flat wing has a quarter elliptic shape, the predicted value of the root chord can be simplified as  
\[\hat{C}_{R}=\frac{4\hat{S}}{\pi\hat{L}}=8.06m^{0.53}. \tag{36}\]  
Given a scaling factor \(\sigma\) (\(0<\sigma<1\)) and the target mass \(m\) of the FWMAV to be designed, the wing length and root chord will be confined to a range close to the predicted value obtained in Equations (42) and (43). As a result, we wrote those as  
\[L\in\big{[}\hat{L}(1-\sigma),\hat{L}(1+\sigma)\big{]} \tag{37}\]  
\[\text{and }C_{R}\in\big{[}\hat{C}_{R}(1-\sigma),\hat{C}_{R}(1+\sigma)\big{]}. \tag{38}\]  
Under the conditions of \(\sigma=0.2\) and \(m=30\) g, the constraint boundaries for these two variables are displayed in Table 1. In addition, the root offset \(r_{0}\) is specially set aside for a length of up to 100 mm to meet the needs of the robotic arm design. We can also see from Table 1 that the kinematic parameters are explored in a relatively larger sample space than the morphological parameters. \[\text{Re}=\frac{4\rho_{f}\varphi_{m}fR_{2}\overline{c}}{\mu}\leq 13,000 \tag{39}\]  
Although Table 1 lists the boundaries of \(r_{0}\), \(L\), \(C_{R}\), \(\varphi_{m}\), and \(f\), these parameters are bounded additionally by the two conditions. One of them is the applicable condition of the used aerodynamic model. As expressed in Equation (39), the permissible Reynolds number cannot exceed 13,000 [27]. Thus, \(r_{0}\) and \(C_{R}\) are restricted further because \(R_{2}\) and \(\overline{c}\) depend on these two parameters in that formula. The other is that the vertical force generated by a couple of flapping wings must be equal to or greater than the total body weight. That means the optimized wing can have enough ability to support the actual MAV weight or go upward against gravity. Given a set of design parameters, including the eight decision variables, the wing performance can be evaluated quickly using the UBET introduced in Section 2.2. Following the earlier discussion, the forces in the \(\zeta\)-direction produced \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variables** & **Description** & **Min** & **Max** \\ \hline \(r_{0}\) & Wing root offset & 0.0 mm & 100 mm \\ \(L\) & Wing length & 78.9 mm & 188.3 mm \\ \(C_{R}\) & Root chord & 39.1 mm & 58.7 mm \\ \(f\) & Frequency & 0.0 Hz & 30.0 Hz \\ \(\varphi_{m}\) & Flapping Amplitude & 0.0 deg & 90.0 deg \\ \(\theta_{rm,r}\) & Pitching Amplitude at the root & 0.0 deg & 90.0 deg \\ \(\theta_{rm,t}\) & Pitching Amplitude at the wingtip & 0.0 deg & 90.0 deg \\ \(C_{\theta}\) & Regulating parameter of rotation angles & 0.0 deg & 5.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The decision variables and their boundary values.