# Wing Geometry and Kinematic Parameters Optimization of Flapping Wing Hovering Flight
## 1 Introduction
They found that flapping angle with sinusoidal profile and pitch angle with profile of rectangular wave could yield maximum lift with given flapping frequency and WGP, while flapping angle with profile of triangular shape and pitch angle also with rectangular profile could realize the minimization of power consumption with given lift constraint and WGP [13]. However, the effect of rotational circulation mechanism and added-mass effect on lift and power consumption has been neglected in their study. According to our best knowledge, these two aerodynamic mechanisms play an important role in wingbeat dynamics, especially in wing pitch motion's realization and maintenance. Thus, they must be considered in the quasi-steady aerodynamic model and eventual power density model of flapping hovering flight. It is noteworthy that an adjoint-based optimization method has been used to explore the combined optimization of wing shape and kinematics pattern of flapping wing hovering flight while maximizing the aerodynamic force coefficients [14]. The result shows that the highest improvement of aerodynamic performance could be obtained by optimal wing shape and kinematic parameters and there is an essential nonlinear dependence between them [14]. Although this method has provided a practical approach to optimize general unsteady aerodynamic flows and has the potential to perform highly efficient and discretely consistent sensitivity analysis for the any complicated problems [15; 16], its successful realization is highly dependent on high-fidelity simulations for complicated model and calculation environments. And a wide range of research topics, such as the techniques of locally optimal, reduced-order model, and checkpointing used to reduce storage requirements, multi-fidelity optimization algorithms and convergence acceleration techniques that impact the computational cost, remain to be explored [16]. Moreover, some other methods, such as gradient-based method [17; 18; 19], sensitivity equations methods [20] and surrogated-based approach [21], also provide thinkable alternatives for optimization of wing shape and kinematics pattern of flapping wings flight. However, all of them suffer from excessive computational cost, which is directly proportional to the dimensionality of design variables. Several attempts based on experimental apparatus have also been made to perform the optimization of flapping-wing flows [22; 23; 24]. The deficiency of optimization based on the experiments is obvious, namely, the optimal results are limited by finite times of experiments and numbers of design variables. Here, taking computational cost and dimensionality of design variables into consideration, we systematically performed the separate or combined optimizations of WGP or/and WKP to minimize the energy of hovering flight on the basis of extended quasi-steady aerodynamic model by using hybrid genetic algorithm (hybrid-GA) [8]. The essential nonlinear relationship between WGP and WKP via flow parameters (_Re_) is explored by minimization of the objective function, which is constructed by power density model with additional penalty items for lift-to-weight ratio, boundary constraints, linear constraint of aspect ratio (_AR_) and nonlinear constraint of _Re_. In this paper, two main highlights are presented. The first highlight is that 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 way for aerodynamic analysis based on insect-scale rigid wings' non-dimensional outline thus a feasibility of optimization involving WGP. The second highlight is that the combined optimizations of WGP and WKP are firstly performed to minimize the energy of flapping wing hovering flight on the basis of analytically extended quasi-steady aerodynamic model by using hybrid-GA.