# Wing Geometry and Kinematic Parameters Optimization of Flapping Wing Hovering Flight
## 1 Introduction  
The development of flapping wing micro aerial vehicles (FWMAV) has become a hot subject in the field of bio-inspired micro-devices engineering because its splendid properties, such as high maneuverability, portability and stealth with small characteristic size, are hopeful to be realized in near future [1; 2; 3; 4; 5]. However, it is far from sufficient to devise the most power-efficient schematic to build the FWMAV based on particular demand, e.g., carrying some certain mass of battery, avionics sensors and other payload to reach certain flight endurance and range [1]. For biomimetic design of FWMAV inspired by flying insects or hummingbirds, the critical starting point is to solve how to efficiently mimic the wing shape and kinematics pattern of the living flier, especially those who can maintain aloft hovering. In the past decade, some attempts have been made to explore the optimal design space of wing geometry parameters (WGP) or wing kinematics parameters (WKP). [PERSON] and [PERSON] explored the unconstrained WKP space controlling the hovering flight of hawkmoth through simulating the hovering dynamic ordinary differential equation based on quasi-steady aerodynamic model to be stabilized at a fixed position and orientation [6]. The optimization simulation that minimized the difference between the simulated and desired locations of the hawkmoth's body in theglobal parameters space was performed by using the micro-genetic algorithm, which appended the Nelder-Mead simplex search algorithm, to find optimal WKP. Later, the sole optimization of WKP, which is reproduced by the neural network to form various function shapes of wing movements, was performed by [PERSON] et al. to maximize the mean lift of the hovering FWMA carrying available payload by using a simulation model called OSCAB (which stands for Outil de Simulation de Concept d'Ailes Battantes, French) and genetic algorithm (GA) [7]. Moreover, [PERSON] and [PERSON] conducted the sole optimization of wing kinematics pattern, which was characterized by specific functional forms determined by 11 parameters for three wing motion angles, to minimize the hovering power density of flapping wing insects on the basis of quasi-steady aerodynamic model by using hybrid-GA formed by the GA and Nelder-Mead simplex algorithm [8]. Subsequently, from the aspects of quasi-steady aerodynamic model and 2D computational fluid dynamics (CFD) numerical simulation separately, they performed a comparative study of aerodynamic efficiency between flapping wing hovering flight with optimal wing motions and fixed wing steady flight with optimal angle of attack under the constraint of lift balancing weight of fruit fly. They found that optimized flapping wing motions were more efficient than optimal steady flight [9; 10]. Following the method of [PERSON] and [PERSON], additionally considering the contribution of elastic strain energies to total mechanical energy, [PERSON] et al. conducted similar study but used a different constrained nonlinear optimization solver to search for optimal wing kinematics formulated by spline interpolation [11]. The result shows that average mechanical power as a function of elastic storage decreased linearly with minimal benefits from the dissipation cost [11]. Recently, the higher order calculus of variation method was adopted by [PERSON] et al. to minimize the aerodynamic power only on the basis of the translational aerodynamic analytical model under the constraint of lift with given WGP. After scrupulous analytical derivation, they found that a flapping angle featuring triangular waveform and pitch angle with constant value during a half-stroke could yield optimal aerodynamic performance [12]. Moreover, in order to acquire maximum lift or minimum power constrained by given lift, [PERSON] and [PERSON] also studied the aero-optimum hovering wing kinematics only on the basis of translational aerodynamic analytical model.