# Wing Geometry and Kinematic Parameters Optimization of Flapping Wing Hovering Flight
## 6 Optimization Results and Analysis
### WKP optimization Results  
As for WKP optimization, the sole optimization of WKP is performed to minimize the specific power output of a hovering insect by using 2D translational aerodynamic coefficients, which is also formulated as Equation (47) but does not involve linear equation constraints for \(AR\). For this optimization problem, the WGP are unchanged and they are taken from the actual morphological parameters of fruit fly (as listed in Table S1 and S2). The optimal results are also listed in Tables 3 and 4. Figure 8: **(a–d)** Single-parameter sensitivity analyses for optimal WGP. The ratio of lift-to-weight (\(L/W\)) and power density (\(P^{\star}\)) as the function of some given WGP are illustrated by red solid lines and blue dashed lines in Figure 8, respectively. The black dot-dash vertical lines indicate the position of optimal value for the parameter in question. The black dot-dash horizontal lines denote the location where \(L/W=1\). The black dash horizontal lines indicate the corresponding value of power density when \(L/W=1\). The optimal wing kinematic pattern is characterized by round trapezoidal profile for pitch angle, and harmonic shape profile for flapping angle (Figure 9a). The round trapezoidal pitch angle signifies the pitch reversal of wing must be completed at certain faster time scale regulated by \(C_{\psi}\). Moreover, the geometry angle of attack needs to be kept constant during the large range of the midstroke. Here, the optimal harmonic shape flapping angle profile is different from those reported by [PERSON] and [PERSON] and [PERSON] et al. [12; 42], who acquired the triangular flapping angle profile. This must be attributed to the fact that the effect of rotational circulation mechanism and added-mass effect on lift and power consumption has been neglected in their power consumption estimating model. Similarly, once acquiring optimal WKP, the pitch power output and flapping power output of a single wing planform can be estimated again with given fixed WGP of fruit fly (Table 3) and illustrated in Figure 9b,c. Likewise, the pitch power consumption mainly distributes around the pitch reversal while flapping power consumption distributes during the large range of the midstroke, and peaks before the middle of the half stroke due to the peak of flapping inertial power before half of the forward stroke. Figure 9: Wingbeat motion (**a**), pitch and flapping power output of single wing (**b**,**c**) for optimal WKP. For pitch aerodynamic power (\(P_{x,\text{total}}\)) during the forward stroke (Figure 9b), the added-mass power (\(P_{x,\text{add}}\)) has a nearly equivalent peak to the rotational damping power (\(P_{x,\text{rd}}\)), which arises from the larger pitch velocity and acceleration induced by faster time scale of pitch angle during pitch reversal. The translational circulation power (\(P_{x,\text{trans}}\)) and rotational circulation power (\(P_{x,\text{rot}}\)) make minimum contribution to pitch power consumption. Their synergistic effect spans the interval of pitch reversal, and one positive peak of pitch aerodynamic power occurs at the end of the pitch reversal, which is mainly contributed by \(P_{x,\text{add}}\) (Figure 9b). The pitch inertial power (\(P_{x,\text{inert}}\)) is likewise reverse to the pitch aerodynamic power, their mutual subtraction results in a minor positive peak of total pitch mechanical power output (\(P_{x,\text{total,posi}}\)) around the pitch starting point (Figure 9b). For the backward stroke, a completely equivalent trend of pitch power distribution can be observed without any asymmetrical amplitude distribution like the situation in Figure 7b.