# Wing Geometry and Kinematic Parameters Optimization of Flapping Wing Hovering Flight
## 6 Optimization Results and Analysis
### Combined Optimization Results for WGP and WKP  
Lastly, for the combined optimization of WGP and WKP, the single objection function including nonlinear penalty for lift-to-weight constraint, penalty of boundary constraints, linear penalty of \(AR\) constraint, and nonlinear penalty of \(Re\) constraint is performed to minimize specific power output by using 3D translational aerodynamic coefficients (refer to Equation (47)). Here, both WGP and WKP are variable and constrained in particular boundary listed in Table 1. As seen from the optimal results listed in Tables 3 and 4, the pitch angle is also characterized by round trapezoidal profile but with lower flapping frequency and shorter pitch time scale controlled by \(C_{\psi}\) than for those for sole WKP optimization, and the flapping angle also varies with harmonic shape profile (Figure 11a). Here, it is worth noting that the trapezoidal pitch angle profile might not be an optimal solution in engineering. Since this profile is not easy to be realized on account of the actuator mode selection for the wingbeat motion of FWMAV [13]. Thus, the dynamics problem for this kind of pitch motion needs to be probed deeply in theory, and the feasible design for FWMAV with active or passive pitch motion may require  
Figure 10: **(a–f)** Single-parameter sensitivity analyses for optimal wing kinematics parameters (WKP). The layout is identical to Figure 8.  
a large amount of elaborate redesign and experiments with or without torsional hinge or/and actuator along the pitch axis [2; 5; 33; 65; 66]. The pitch angle amplitude determines the geometric angle of attack at the midstroke. Thus, it is always expected for flapping flight that the optimal angle of attack will bring about the most efficient translational lift and drag coefficient to minimize power consumption. In this study, the optimal pitch angle amplitude from the results of sole WKP optimization (\(\psi_{m}\) = 70.6 deg) and combined optimal WGP and WKP (\(\psi_{m}\) = 70.4 or 72.9 deg) is nearly equivalent (Table 3), both of them are approximately equal to the reported value (\(\psi_{m}\) = 72.7 deg for fruit fly) [8]. The pitch phase offset mainly influences rotational circulation force and wake capture [34]. Because the current power density model, which is based on extended quasi-steady aerodynamic model with the absence of some other unsteady lift effect (such as wake capture), is set to achieve the minimization of power consumption but not the maximum of lift, thus the symmetry pattern between  
Figure 11: Wingbeat motion (**a**), pitch and flapping power output of single wing (**b**,**c**) for combined optimal WGP and WKP. flapping angle and pitch angle is dug out when the zero pitch angle crosses to the stroke reversal point. Here, the optimal values of pitch phase offset from the results of sole WKP optimization and combined optimal WGP and WKP is approximate to \(-\pi/2\) (Table 3), and both of them are similar to the reported value [8]. Moreover, all of the WGPs approach their upper bounds except the non-dimensional location of the pitch axis, which is near to zero. The combined optimal result with the feature of larger WGP and lower frequency arises from the fact that the strong linear and nonlinear coupling relationships between WGP and WKP via \(AR\) and \(Re\) have an intense effect on the optimization procedure. Likewise, for the combined optimal WGP and WKP, the pitch power output and flapping power output of a single wing planform are estimated again and illustrated in Figure 11b,c. The time histories of those power outputs have similar trend to those from sole WKP optimization but with different amplitude distribution due to different optimal WGP and WKP adopted (Table 3, Figures 9b,c and 11b,c).