# Wing Geometry and Kinematic Parameters Optimization of Flapping Wing Hovering Flight
## 2 Wing Morphological Parametrization
### Description of Wing Morphology  
The complete wing planform outline, wing shoulder frame (\(X_{\mathrm{s}}O_{\mathrm{s}}Z_{\mathrm{s}}\)) and some basic sizes are illustrated in Figure 1. Here, the intersection of \(X_{\mathrm{s}}\)-axis and \(Z_{\mathrm{s}}\)-axis (\(O_{\mathrm{s}}\)) can be termed wing shoulder [1; 28], so we term the coordinate frame of \(X_{\mathrm{s}}O_{\mathrm{s}}Z_{\mathrm{s}}\) as wing shoulder frame (or wing planform fixed frame). We assume that in the initial case, \(X_{\mathrm{s}}\)-axis of \(X_{\mathrm{s}}O_{\mathrm{s}}Z_{\mathrm{s}}\) is aligned with pitch axis, where \(r\) is the radial distance along it. It is worth noting that the wing hinge always exists in the insect body, so the wing root coordinate offset distance between wing shoulder (\(O_{\mathrm{s}}\)) and wing root (\(O_{\mathrm{r}}\)), which is commonly necessary to design the airfoil of FWMAV in engineering, is termed \(x_{\mathrm{r,orig}}\)[29; 30]. The variable pitch axis (\(x_{\mathrm{p}}\)) is schemed by the red dash-dotted line. Here, \(x_{0,\mathrm{vari}}\) denotes the projected distance between pitch-axis and maximum variable leading-edge point. As a starting point, we choose the \(X_{\mathrm{s}}\)-axis of the wing shoulder frame as the original location of the pitch axis,which corresponds to the non-dimensional value of 0.36 from the maximum leading-edge point. In other words, the original leading-edge profiles (\(z_{\text{le,orig}}(r)\)) and trailing-edge profiles (\(z_{\text{tr,orig}}(r)\)) are prescribed in the wing shoulder frame firstly, then the value of 0.36 times the projected distance between maximum point on original leading-edge profiles and minimum point on original trailing-edge profiles relative to the original leading-edge profiles defines the initial pitch axis location, namely, \(z_{\text{le,orig,maxp}}/C_{\text{max,letotr,orig}}=0.36\) (Figure 1).  
For conciseness of description, a statement is made that if there is no specific suffix on the right lower quadrant of some arguments, then the notation for this argument is possessed by the original or variable wing morphological parameters collectively, such as \(R_{\text{eff}}\) is possessed by \(R_{\text{eff,orig}}\) and \(R_{\text{eff,vari}}\). In the wing shoulder frame, and chord length between leading-edge and trailing-edge profiles is \(C(r)\). Wing effective length (\(R_{\text{eff}}\)) is defined as the projected distance between the most proximal point (wing root) and the most distal point (wing tip) on the wing planform along the \(X_{\text{s}}\)-axis. Mean chord length (\(C_{\text{aver}}\)) is defined as the area of single wing (\(A_{\text{w}}\)) divided by wing effective length, \(A_{\text{w}}/R_{\text{eff}}\) (Table S1). The actual original wing outline of leading-edge (\(z_{\text{le,orig}}(r)\)) and trail-edge (\(z_{\text{tr,orig}}(r)\)) for fruit fly has been obtained through polynomial fitting the geometrical data located in the coordinate frame of \(X_{\text{s}}O_{\text{s}}Z_{\text{s}}\) (Figure 1 and Tables S2 in the Supplementary Materials). The geometrical data of the wing is transposed and translated from the original right wing geometrical data measured by [PERSON] [27].