# Wing Geometry and Kinematic Parameters Optimization of Flapping Wing Hovering Flight
## 8 Conclusions  
In this paper, the separate or combined optimizations of wing geometry parameters or/and wing kinematic parameters are systematically performed to minimize the energy of flapping wing hovering flight, firstly on the basis of extended quasi-steady aerodynamic model by using hybrid-GA. Before the elaboration of the optimization problem, the parametrization description of dynamically scaled wing with non-dimensional conformal feature of insect-scale rigid wing is firstly proposed. This parametrization method provides an effective simplifying method for aerodynamic analysis based on insect-scale rigid wings' non-dimensional outline, and thus the feasibility for optimization involving WGP. Then, the extended quasi-steady aerodynamic model is derived on basis of previous quasi-steady aerodynamic model by using 2D or/and 3D translational lift and drag coefficients with consideration of the possible effect of any variation of wing shape on the aerodynamic coefficients. Further, the objective function of optimization is formed by hovering flight power density with additional penalty items for lift-to-weight ratio, boundary constraints, linear constraint of AR and nonlinear constraint of \(Re\). For five cases of separate or combined optimization for WGP or/and WKP, given unchanged wing kinematics pattern or WGP or no arguments input, we performed the optimizations of WGP or/and WKP to minimize the energy of flapping wing hovering flight by using hybrid-GA formed by the GA and Nelder-Mead simplex algorithm. The optimization results show that the combined optimization of wing geometry and kinematic parameters can obtain lower flapping frequency, larger wing geometry parameters and lower power density in comparison with those from other cases of optimization. The amplitude of flapping angle for the optimization involving WKP is close to their prescribed upper bound. All of these results arise from the effect of strong coupling relationship between WGP and WKP via \(AR\) and \(Re\) when the minimum power density is obtained under the constraint of lift balancing weight. Moreover, the flapping angle for the solely optimal WKP and combined optimal WGP and WKP manifests harmonic shape profile, and the pitch angle possesses  
Figure 13: The original and dynamically scaled wing planform of fruit fly. round trapezoidal profile with certain faster time scale of pitch reversal controlled by its regulating parameter. The dynamics problem for flapping motion with or without resonant frequency to the drivetrain and pitch motion with constant angle of attack around midstroke needs to be studied deeply in theory, and in the bio-design of FWMAV with active or passive pitch motion through including or excluding the torsional hinge or/and actuator along pitch axis. In addition, no obvious distinction is observed in the optimal visualization results involving WGP for the case of 3D and 2D translational aerodynamic coefficients although there is little difference in amplitude, which directly indicates that the adopted 3D translational aerodynamic force coefficients have little effect on eventual optimization results involving WGP and indirectly shows that the variation of wing shape only brings little influence on lift and power density by probable way of influencing translational aerodynamic coefficients. The advantage of currently proposed combined optimization framework is obvious: it can drastically reduce the computational cost for current complicated optimization problem and quickly provide the fundamental optimized parameters of biomimetic flapping wing micro aerial vehicle at the stage of conceptual design. However, there are two limitations worthy of being noted. Firstly, the extended quasi-steady aerodynamic model presented is unable to take some feeble unsteady effect mentioned above into consideration, thus a much higher fidelity model or efficiency numerical method, such as CFD, need to be set up to perform the combined optimization with higher computation accuracy.