# Wing Geometry and Kinematic Parameters Optimization of Flapping Wing Hovering Flight
## 2 Wing Morphological Parametrization
### Characterization of Dynamically Scaled Wing  
The posterior optimization problem involving WGP firstly needs the characterization of non-dimensional parameters and mass properties of dynamically scaled wing planform. In the current study, non-dimensional leading-edge (\(\hat{z}_{\text{le}}\left(\hat{r}\right)\)), and non-dimensional trailing-edge profiles (\(\hat{z}_{\text{tr}}\left(\hat{r}\right)\)), which are from the geometry outlines of fruit fly, are assumed to remain unchanged. Figure 1: Definition of wing shoulder frame (\(X_{\text{s}}O_{\text{s}}Z_{\text{s}}\)) and basic size for fruit fly wing planform. The variable wing effective length (\(R_{\text{eff,vari}}\)) and variable mean chord (\(C_{\text{aver,vari}}\)) are chosen to regulate the dynamically scaled wing planform outlines. Moreover, the variable \(x\)-root offset (\(x_{\text{r,vari}}\)) and variable non-dimensional pitch axis location relative to the maximum point of actual leading-edge profile (\(\hat{x}_{0,\text{vari}}\)) are adopted to regulate the aerodynamic performance of the dynamically scaled wing planform. The distance of the wing root and shoulder (\(x_{r,\text{vari}}\)) in engineering design is always arbitrarily determined only according to the assembly gap between transmission of drivetrain and overlap joint of the wing planform. This determining way neglects its effect on aerodynamic characteristics of hovering wing planform due to the fact that a constant velocity contribution from the interval of wing shoulder and root is added to linearly varying velocity of the whole wing planform [29; 30]. While the non-dimensional pitch axis location (\(\hat{x}_{0,\text{vari}}\)) of the dynamically scaled fruit fly wing model shall influence the experimental rotational coefficient [32]. So it is necessary to choose the \(x\)-root offset and non-dimensional rotational axis location as two additional variables to complete the optimized parameters' characterization of the dynamically scaled wing planform. The mass properties for the dynamically scaled wing planform are prepared for the optimization involving WGP by assuming that the wing mass was uniform isotropic distribution with very small thickness. Firstly, a three-dimensional (3D) geometry model is constructed from the original wing geometry data by CAD software (UGS NX 7.5, siemens plm software, Plano, TX, USA). The mass of wing (\(m_{\text{wing,orig}}\)), center of mass (COM) in the frame of \(X_{\text{s}}\text{O}_{\text{s}}Z_{\text{s}}\) (\(x_{\text{com,orig}}\) and \(z_{\text{com,orig}}\)), and moment of inertia relative to the COM (\(I_{xx,\text{com,orig}}\) and \(I_{zz,\text{com,orig}}\)) are calculated and listed in Table S1. Then, two ratios between dynamically scaled wing planform and original wing planform of the fruit fly are developed. They are wing length ratio (\(R_{\text{ratio}}\)) between variable wing effective length (\(R_{\text{eff,vari}}\)) and original wing effective length (\(R_{\text{eff,orig}}\)), and mean chord ratio (\(C_{\text{ratio}}\)) between variable mean chord length (\(C_{\text{aver,vari}}\)) and original mean chord length (\(C_{\text{aver,orig}}\)), respectively. It is not difficult to find the relation of COM in the frame of \(X_{\text{s}}\text{O}_{\text{s}}Z_{\text{s}}\) for the wing planform before and after it is dynamically changed (as seen in Figure 2, the variable pitch axis (\(x_{pl}\)) are also illustrated). The calculating formulation for COM of the dynamically changed wing planform with non-dimensional conformal feature in the frame of \(X_{\text{s}}\text{O}_{\text{s}}Z_{\text{s}}\) can be expressed as:  
\[x_{\text{com,vari}}=R_{\text{ratio}}(x_{\text{com,orig}}-x_{\text{r,orig}})+x _{\text{r,vari}}, \tag{1}\]  
\[z_{\text{com,vari}}=C_{\text{ratio}}z_{\text{com,orig}}. \tag{2}\]  
The dynamically changed wing planform's mass has the form of \(m_{\text{wing,vari}}=R_{\text{ratio}}C_{\text{ratio}}m_{\text{wing,orig}}\).