# Development of an Insect-like Flapping-Wing Micro Air Vehicle with Parallel Control Mechanism
## 2 Prototype Design  
### Flapping Wing Mechanism  
The flapping wing mechanism provides the lift for the FMAV and generates control torque under the manipulation of the control mechanism. Therefore, the flapping wing mechanism requires a small size, low inertia, and high energy efficiency. Many different forms and principles of the flapping wing mechanism have been designed such as various linkage mechanisms [14; 15; 21], string-based mechanism [11; 25], direct drive [20], and their combination or variation. The flapping wing mechanism of this FMAV adopts a string-based mechanism that was first designed by [11] because the string-based mechanism has simpler componentsand a smaller size than other designs. In addition, it has a centrally located drive motor for easy integration with the control mechanism. The schematic of the string-based mechanism is shown in Figure 1. The upper-level strings are the sync strings that join the two pulleys together similar to a synchronous belt, allowing the two pulleys to rotate in reverse synchronously. The lower-level strings are the drive strings, one side of these strings is held to the pulley, and another side is wrapped around the projection of the crank, pulling the left and right pulleys, rotating in turn as the crank rotates. However, one of the problems with the string-based mechanism in the initial design is that the string does not fit perfectly into the pulley's profile every time, resulting in repeated stretching and impact of the string. The string is very susceptible to slackening after a period of operation. In order to make the string and pulley fit better, the pully profile needs to be designed according to the desired flapping trajectory. The schematic diagram of the flapping wing mechanism is shown in Figure 2. The \(xOy\) rectangular coordinate system (RCS) is fixed with the base and point \(O\) is at the center of rotation of the crank. \(OA\) indicates the position of crank. The \(x_{r}O_{r}y_{r},\ x_{l}O_{l}y_{l}\) RCS is fixed with the right and left pullies and points \(O_{r},O_{l}\) are the center of rotation of the pullies, respectively. Assuming a uniform crank rotation and expecting a gentle rotation of the pulley, the rotations of pulleys can be expressed as Equation (1), where the initial angle is set to zero and the flapping amplitude is \(\phi_{m}\). \[\phi_{l}=-\phi_{r}=\phi_{m}\cos(2\pi t/T)+\phi_{m} \tag{1}\]  
Figure 1: The schematic of the string-based flapping wing mechanism. Figure 2: The model of the string-based mechanism. \(xOy,\ x_{r}O_{r}y_{r},\ x_{l}O_{l}y_{l}\) RCS is fixed to the crank, right, and left pulley. The coordinates of \(A\) in the \(xOy\) RCS can be converted to \(x_{r}O_{r}y_{r}\) RCS by Equation (2). \[\mathbf{A_{r}}=\mathbf{Q_{r}}(\mathbf{A}+\mathbf{T_{r}}),\text{ {where} }\mathbf{Q_{r}}=\left[ \begin{array}{cc}\cos\phi_{r}&\sin\phi_{r}\\ -\sin\phi_{r}&\cos\phi_{r}\end{array} \right],\ \mathbf{T_{r}}=\left[ \begin{array}{cc}-d \\ 0\end{array} \right] \tag{2}\]  
Therefore we can obtain the expected trajectory of \(A\) under the \(x_{r}O_{r}y_{r}\) RCS from Equations (1) and (2), and the trajectory is shown in Figure 3. The \(x_{r}O_{r}y_{r}\) RCS is converted into the \(O_{r}x_{r}\) polar coordinate system (PCS) and a fitting curve replaces the desired curve by a polynomial fit. It should be noted that the rotation curve obtained using the fitted results was not a perfect sine curve because the law of motion of \(A\) was not symmetrical with respect to \(O_{r}\) or \(O_{l}\). For the same reason, there were also slight differences in the rotation curves obtained for the left and right pulleys. However, the differences could be effectively reduced compared to the initial design. When the string is free of deformation and slack, Equation (3) should be satisfied, where \(m=\sqrt{\rho^{2}+l^{2}-2\rho lc},c=\cos(\alpha-\theta),s=\sin(\alpha-\theta)\). \[\left\{