## 3 Data Choice Based on the Mechanism Design  
Considering that the overall mechanism of flapping wing vehicles is relatively small, there are size requirements for the flapping mechanism. Assuming that the maximum transverse span allowed is \(D_{span}\), the maximum longitudinal length is \(D_{width}\), and that there is no collision between the roller group and the bearing group, the four parameters  
Figure 3: Coordinate system establishment with point O as the origin, and point A,B,C,D are the intersection points of the common tangent of two circles.  
mentioned above need to meet the design scale constraints of the flapping mechanism as system of Equations (6).  
\[\left\{\begin{array}{c}2(r_{0}+r_{3})\leq D_{width}\\ r_{1}=r_{2}\leq D_{width}\\ 2(d+r_{1})\leq D_{span}\\ r_{0}+r_{3}+r_{1}<d\end{array}\right. \tag{6}\]  
As shown in Figure 4--when considering the inertial effects present during flapping after installing wings, the non-complete structural stiffness, and the stretchability of actual strings--it is necessary to incorporate redundancy into the flapping angle design to avoid the vibrations caused by wingtip collisions. Assuming the maximum allowable flapping angle allowed by the mechanism is \(\theta_{allow\_max}\) and the redundancy ratio is \(k_{r}\), the maximum flapping angle \(\theta_{f\_max}\) that needs to be considered during the design process should satisfy the Inequality (7).  
\[\theta_{f\_max}\leq k_{r}\theta_{allow\_max}, \tag{7}\]  
where \(k_{r}\) is an empirical parameter that is obtained by combining structural parameters such as the flapping frequency \(f_{r}\), wing material, and physical experiments, with a value range of 0.7 to 0.85. Thus, \(\theta_{f\_max}\) can be obtained through the simulation of the design parameters \(r_{0}\), \(r_{1}\), \(r_{3}\), and \(d\), which are detailed in Section 2. \(\theta_{allow\_max}\) is calculated through structural design parameters, and it is given by Equation (8).  
\[\theta_{allow\_max}=2\left(\pi-2 arccos\frac{d}{l_{w}+r_{1}}\right), \tag{8}\]  
where \(l_{w}\) is the wing length. Together with the other parameters related to structural design, it forms the flapping span \(D_{flapping\_span}\), which needs to meet the constraint condition of the whole machine size as Inequality (9).  
\[2(l_{w}+d+r_{1})\leq D_{flapping\_span}. \tag{9}\]  
In summary, the complete design constraint conditions for the string drive flapping mechanism can be expressed as system of Inequality (10).  
\[\left\{\begin{array}{c}2(r_{0}+r_{3})\leq D_{width}\\ 2r_{1}\leq D_{width}\\ 2(d+r_{1})\leq D_{span}\\ r_{0}+r_{3}+r_{1}<0.75d\\ 112.5\frac{l_{0}}{r_{1}}+0.3\leq 2k_{r}\left(\pi-2 arccos\frac{d}{l_{w}+r_{1}}\right) \\ 2(l_{w}+d+r_{1})\leq D_{flapping\_span}\end{array}\right. \tag{10}\]  
where \(\theta_{f\_max}=112.5\frac{l_{0}}{r_{1}}+0.3\) is obtained in the subsequent sections but cited here in advance for the parameter design of the flapping mechanism.  
Figure 4: Schematic diagram of the parameter constraints in the structural design.  
Through analyzing the various constraints in the drive mechanism, as well as by selecting the corresponding design parameters within the obtainable range of values, the corresponding final design parameters could be obtained and a subsequent analysis could be conducted based on the obtained parameter values.