# Power Modeling and Experiment Study of Large Flapping-Wing Flying Robot during Forward Flight
## 2 Materials and Methods
### Kinematics of the Flapping-Wing Air Vehicle Flight  
Figure 2 shows the power output and transmission mechanism of the HIT-Hawk. The drive source of the flapping-wing air vehicle is a brushless DC motor, which drives the output shaft to make rotational movement by a gear set with a gear ratio of 81:1. It drives the wing of the FWAV to flap periodically through the space RSSR four-link mechanism. The first and last two spinning pairs and the middle two ball pairs form the spatial RSSR mechanism (see Figure 2). The kinematic equation, detailed in Appendix A, is obtained to establish the relationship between input angle \(\theta\) and flapping angle \(\varphi\). The angular displacement of the pendulum CD is described as:  
\[\varphi=2\text{arctan}\frac{A-\sqrt{A^{2}+B^{2}-C^{2}}}{B-C} \tag{1}\]  
where  
\[ \begin{array}{l}A=2l_{3}(h-l_{1}\sin\theta)\\ B=-2 dl_{3}\\ C=l_{1}^{2}-l_{2}^{2}+l_{3}^{2}+h^{2}+d^{2}+s^{2}-2 sl_{1}\cos\theta-2 hl_{1}\sin \theta\end{array} \]  
Figure 1: The overall structure of the HIT-Hawk. The angular velocity and angular acceleration of the pendulum CD can be calculated by determining the first-order and second-order derivatives from \(\varphi\) to time. Here, \(l_{1}=9\,\mathrm{mm},l_{2}=54.5\,\mathrm{mm},l_{3}=27\,\mathrm{mm}\) and \(s=1.97\,\mathrm{mm},h=48.97\,\mathrm{mm},\) and \(d=18\,\mathrm{mm}\). The flapping motion of birds can be divided into up-and-down flapping, forward and backward swinging, twisting around the spanwise, and folding of the wing surface during upward flapping. During forward flight, forward and backward swinging and folding of the wing are ignored in this work. The variation law of flapping angle \(\varphi\) and twist angle \(\gamma\) with time \(t\) can be established using this basic geometric model of FWAV:  
\[\left\{ \begin{array}{c}\varphi(t)=\varphi\\ \gamma(t)=\frac{\
u}{b}[\gamma_{\mathrm{max}}cos(2\pi ft)+\gamma_{0}]\end{array} \right. \tag{2}\]  
where \(y\) is the coordinate of wing in spanwise, \(f\) is the flapping frequency, and \(b\) is the wingspan of a single wing; \(\gamma_{\mathrm{max}}\) is the maximum twist angle and \(\gamma_{0}\) is the initial twist angle, which are assumed to be \(15^{\circ}\) and \(5^{\circ}\), respectively, in this paper. The object of investigation is a tiny strip along the wing span. When the FWAV flies forward horizontally at speed \(v_{\infty}\), the tiny strip of the wing has three speeds at this time: horizontal forward flight speed \(v_{\infty}\), flapping speed \(v_{\varphi}\), and twisting speed \(v_{\gamma}\). The motion model of the wing is shown in Figure 3. If the wing is relatively stationary, the airflow makes relative motion to the wing, and the aerodynamic center of the wing, i.e., the one-quarter chord position, is used as the reference point for calculation; the wing motion model is shown in Figure 3.  
Figure 3: Velocity model for the motion of wing strips. Figure 2: (**a**) The drive unit mechanism schematic and (**b**) sketch of the RSSR space four-link mechanism. The following formulae can be used to calculate the strip flapping velocity and torsional velocity:  
\[\left\{ \begin{array}{c}v_{\varphi}=-y\varphi(t)\\ v_{\gamma}=-\frac{1}{4}c(y)\dot{\gamma}(t)\end{array} \right. \tag{3}\]  
where \(c(y)\) is the chord length of the strip at position \(y\), determined according to the wing shape. According to the analysis of [PERSON] [20] and [PERSON] [21] on the unsteady aerodynamic mechanism of flying organisms, the downwash velocity can be estimated by the following equation:  
\[w=\frac{2(\alpha_{0}+\theta)}{2+\lambda}v_{\infty} \tag{4}\]  
where \(\alpha_{0}\) is the zero-lift angle of the wing (assumed to be \(0^{\circ}\) here); \(\lambda\) is the aspect ratio of a wing; and \(\theta\) is the static pitch angle, which is equal to the sum of the wing mounting angle \(\theta_{0}\) and the twist angle \(\gamma\).