# Power Modeling and Experiment Study of Large Flapping-Wing Flying Robot during Forward Flight
## 2 Materials and Methods
### Power Model of the Flapping-Wing Air Vehicle in Forward Flight
#### 2.3.1 Aerodynamic Power  
The aerodynamic drag acting on the wings of a flapping-wing air vehicle during steady-state forward flight can be divided into two categories [13]:  
* Induced drag (\(D_{ind}\)): the drag due to lift;
* Profile drag (\(D_{pro}\)): the drag caused by wing-air friction. The parasitic drag (\(D_{par}\)), is defined as the drag caused by friction between the frame and the air, and it only affects the airframe. It is included in the total aerodynamic drag. As a result, overall aerodynamic drag may be written as:  
\[D_{arco}=D_{ind}+D_{pro}+D_{par} \tag{8}\]  
The total aerodynamic power can then be estimated for the steady-state flight process by multiplying the aerodynamic drag by the forward flight speed. The following equation can be used to calculate the different power components, and the three power components follow:  
\[P=D\times V \tag{9}\]Different power components can be obtained from Equation (10). The three power components are induced power (\(P_{ind}\)), profile power (\(P_{pro}\)), and parasite power (\(P_{par}\)). The total aerodynamic power required for horizontal flight can be estimated as follows, similar to the drag component:  
\[P_{areo}=P_{ind}+P_{pro}+P_{par} \tag{10}\]  
1. Induced power  
One of the main focuses of this article is the calculation of induced power. The induced power is primarily due to two factors: the tilt of the left and right lift vectors in the course of the flapping cycle, and the continual change in lift vector amplitude over the flapping cycle [12]. The induced power of each wing strip is calculated as the product of the strip's induced drag and velocity. \[\mathrm{d}P_{ind}=\mathrm{d}D_{ind}\times V \tag{11}\]  
The induced drag caused by the formation of lift is determined by the wing shape, velocity, and air mass, and is computed as follows:  
\[\mathrm{d}D_{ind}=\frac{\rho V^{2}C_{D,ind}c(y)\mathrm{d}y}{2} \tag{12}\]  
where \(\rho\) is the air density, equal to 1.29 Kg/m\({}^{3}\); \(V\) is the relative air velocity; \(C_{D,ind}\) is the induced drag coefficient; \(c(y)\) is the chord length of the strip; and \(\mathrm{d}y\) is the width of the strip. The induced drag coefficient is related to the lift coefficient and the wing parameters [23]:  
\[C_{D,ind}=\frac{k}{\pi\lambda}C_{L}^{2} \tag{13}\]  
where \(C_{L}\) is the lift coefficient; \(k\) is the induced drag factor, usually taken as 1.1-1.2; and \(\lambda\) is the aspect ratio. The total lift of a segment of the wing can be given by the circulatory normal force and an additional normal force contribution, which comes from added mass effect [24]:  
\[\mathrm{d}N=\mathrm{d}N_{c}+\mathrm{d}N_{a}=\frac{\rho V^{2}}{2}C_{n}(y)c \mathrm{d}y+\frac{\rho\pi c^{2}}{4}V_{N}\mathrm{d}y \tag{14}\]  
where \(C_{n}(y)=2\pi(\theta_{0}+\gamma+\alpha^{\prime})\) and \(V_{N}=v_{\varphi}+v_{\infty}\sin\theta-2v_{\gamma}\). The overall lift generated by the flapping-wing vehicle should be more than the force in the vertical direction, which is employed to resist gravity in horizontal level flight due to the presence of the wing-flap angle and angle of attack. This is shown in Figure 4.  
Figure 4: Decomposition of aerodynamic lift in space. As a result, the wing-lift coefficient can be calculated as:  
\[C_{L}=\frac{2\text{d}N\cos\varphi(t)}{\rho c(y)V^{2}\text{d}y} \tag{15}\]  
In turn, the induced drag coefficient can be derived from the equation above. Thus, the induced power of the wings is equal to the total of the induced powers of each wing strip, as shown in the equation below:  
\[P=D\times V \tag{9}\]0  
where \(l\) is the length of the wing. 2. Profile power  
The profile power component is the most difficult to calculate of all the power components. The reason for this is that the profile drag coefficient \(C_{D,pro}\) is dependent on the shape and roughness of the wing surface, and free-stream circumstances and Reynolds number dependence [23].