# Wing Geometry and Kinematic Parameters Optimization of Flapping Wing Hovering Flight
## 6 Optimization Results and Analysis
### Sensitivity Analysis for Combined Optimal WGP and WKP  
Similarly, given the combined optimal WGP and WKP listed in Table 3, the effect of perturbing single parameter on lift-to-weight ratio and power density is also analyzed. As seen in Figure 12, the distribution feature of some of optimal values in their own constraint boundary is completely similar to those for sole WGP or WKP optimization although with different peak or amplitude difference of power density, such as the effective variable wing length (\(R_{\text{eff},\text{vari}}\)), variable average chord (\(C_{\text{aver},\text{vari}}\)), wing-root offset (\(x_{\text{r},\text{vari}}\)), and non-dimensional location of pitch axis (\(\hat{x}_{0,\text{vari}}\)) for WGP (Figures 8 and 12a-d), and flapping frequency, amplitude of flapping angle (\(\psi_{m}\)), amplitude of pitch angle (\(\psi_{m}\)) and pitch phase offset (\(\zeta\)) for WKP (Figures 10a,b,d,f and 12e,f,h,j). Here, the optimal values for non-dimensional location of the pitch axis, which consistently approach the leading edge for the optimization involving WGP (Table 3), is only the numerical solution but not reflect the real situation since the pitch axis might be not a straight line. In engineering design, the location of pitch axis needs to be determined by experimentally measuring the optimal angle of attack [57]. In order to survive in a complex environment such as predation or evading maneuverability with limited energy, fruit fly might evolve to have pony-size WGP adaptive to its body size in millimeter scale but high flapping frequency and large flapping amplitude under the strategy of minimum power consumption. For the design of FWMAV, high frequency will lead to the dilemma of designing actuator and avionics system. Thus, the optimal large-sized wing length and average chord might be favorable to lower flapping frequency with minimum power consumption. Figure 12: (**a–j**) Single-parameter sensitivity analyses for combined optimal WGP and WKP. The layout is identical to Figure 8.  
Moreover, the flapping frequency and flapping amplitude are two competitive parameters to realize the production of high lift and minimum power consumption. The flapping amplitude determines the travel distance of chordwise section of wing planform, and thus the vortex dynamical evolutionary process. While high flapping frequency means high frequency evolution of highly unsteady flow field, namely, the starting vortex, LEV, trailing edge vortex, wing tip vortex, downwash vortex and wake vortex are interwoven and interact at different instants thus influence the aerodynamic performance of wing planform [62; 67]. As mentioned above, lower flapping frequency and appropriate flapping amplitude is expected for the designer to successfully solve the engineering problem of Bio-FWMAV although simultaneously compromising high maneuverability yielded by high lift from unsteady mechanism. The distinction of sensitivity trend for optimal regulating parameter of flapping angle (\(K_{\phi}\)) and pitch angle (\(C_{\psi}\)) can be observed between Figure 12g,i and Figure 10c,e for sole WKP optimization. In Figure 12g, the lift decreases monotonically with \(K_{\phi}\) during the whole variable interval, but the power density nearly maintains constant during the approximate front half of variable interval, and then monotonically increases during the second half of variable interval. The sensitivity trend of power density about \(K_{\phi}\) is different from the situation shown in Figure 10c, which might arise from the effect of the strong coupling relationship between WGP and WKP via \(AR\) and \(Re\) when acquiring the sensitivity data of power density about \(K_{\phi}\) with other optimal WGP and WKP unchanged. However, the flapping angle still manifests harmonic shape profile with an optimal value (\(K_{\phi}\) = 0.189, Table 3) such that the lift constraint is met while the power density achieves the minimum.