## 2 Materials and Methods  
### Morphological and Kinematic Parameterization of a Wing  
Hummingbird wings are relatively small and narrow, and the plane shape near the wing root is wide. We think that a quarter ellipse can reflect most features of hummingbird wings, so the planar geometry of the ellipse is established, as shown in Figure 1. More importantly, it also simplifies the process of parameterization of wing geometry. The morphological parameters, such as wing length (\(L\)) and wing root chord (\(C_{R}\)), are annotated there. An offset (\(r_{0}\)) exists between the wing root and the flapping axis (\(Z-\)axis). The chord length (\(c(r)\)) of any local cross-section at the radial radius of \(r\) is assumed to be of the following formula:  
\[c(r)= \begin{cases}0&(0\leq r<r_{0})\\ \frac{\pi\overline{c}}{\pi}\sqrt{1-\frac{\left(r-r_{0}\right)^{2}}{L^{2}}}&(r_{ 0}\leq r\leq r_{0}+L)\end{cases} , \tag{1}\]  
where \(\overline{c}\) is the mean chord length and is related to the root chord by \(\overline{c}=\pi C_{R}/4\). The wing area (\(S\)) is expressed as \(S=L\overline{c}\), and the ratio of the wing length (\(L\)) to the mean chord length (\(\overline{c}\)) defines the wing aspect ratio (\(AR\)) as \(AR=L/\overline{c}\). The radius of the second moment (\(R_{2}\)) of the airfoil about the flapping axis can be calculated according to the following formula:  
\[R_{2}=\sqrt{1/S\int_{0}^{r_{0}+L}r^{2}c(r)\mathrm{d}r}. \tag{2}\]Consequently, it is necessary to determine three parameters, \(L\), \(C_{R}\), and \(r_{0}\), to determine the plane shape of the wing. Generally, the reciprocating motion of any wing section can be divided into two progressive movements: flapping and pitching motions. These two degrees of freedom without deviation motions of the wing section are considered in the current design. As shown in Figure 2, two orthogonal coordinate systems (the inertial frame \(O-XYZ\) and the co-rotating frame \(O^{\prime}-\xi\eta\zeta\)) and two Euler angles (the flapping angle \(\varphi\) and the rotation angle \(\theta_{r}\)) are introduced to describe the spatial location of any wing section with a spanwise distance of \(r\) from the flapping axis. With the defined framework in Figure 2, the \(\xi\) axis coincides with the straight leading edge at each instant time, and it rotates back and forth along with the wing. The flapping motion is assumed to be a cosinusoidal oscillation. The flapping angle as a function of time is expressed as  
\[\varphi(t)=-\varphi_{m}\cos(2\pi ft) \tag{3}\]  
where \(\varphi_{m}\) is the flapping amplitude, \(f\) is the flapping frequency, and \(t\) is time. We abstracted the flapping motion as a cosinusoidal waveform because it has been produced widely in many bionic systems and robot experiments [2; 6; 7; 8]. Furthermore, in engineering, the flapping motion of a wing is usually driven by a linkage mechanism that converts the rotational motion output by the motor into the reciprocating motion of the rocker. A time-varying profile approaching cosinusoidal is usually generated by that. Considering that the leading-edge stiffness is usually large enough to limit the variation of the flapping angle along the wingspan, we make this angle independent of the radial radius. At the same instant, this angle has an identical value in all wing cross-sections. In this case, the angular speed, as expressed in \(\varphi(t)=2\pi f\varphi_{m}\sin(2\pi ft)\), can indicate the stroke phase; \(\varphi>0\) and \(\varphi<0\) represent the upstroke and downstroke phases, respectively. Figure 1: The simplified wing-plane geometry. Figure 2: Definitions of the kinematic parameters at a radial cross-section and velocity components used for BET analysis. As for the other motion, the pitching axis is the leading edge of this wing. The rotation angle, defined as the angle between the wing chord and the positive direction of the \(\eta\)-axis, can describe the pitching motion of a two-dimensional wing cross-section. The profile of the rotation angle displays a round-trapezoidal waveform.