## 4 Data Analysis of Mechanism Design  
Based on the derivations presented above, simulations and physical studies could then be conducted using the parameters listed in Table 2. The flapping angle of the rollers was analyzed under ideal conditions, wherein the string was fully taut through the rotation of the center gear. The total length of the strings was not affected by the flapping angles as their lengths could be adjusted by bypassing the left and right rollers. A string with a length of 34.7 mm was chosen for this study. We assumed that two strings are always in an ideal state (i.e., tight state) without considering the strings used to keep them moving in the same phase. The angles of the right and left rollers are shown in Figure 6.  
Based on the initial parameters in Table 2, we made changes to each parameter to observe its influence on the flapping amplitude and trajectory. Our results, as shown in Figures 7-10, indicate that the radius of the rollers and center gear had a significant impact on the flapping amplitude and trajectory, while changes to the radius of the double-layer bearings and length between the center gear and rollers had little effect. The influence of the parameters on the flapping angle can also be understood from their physical meaning. Firstly, the radius of the double-layer bearing assembly and the installation position of the roller assembly with respect to the center gear had a relatively small impact. Since the strings wrapped around the double-layer bearings, they could be approximated as two tangents. The change in the radius of the double-layer bearing had a minimal impact on the length of these tangents; therefore, it also had minimal impact on the flapping angle. Similarly, the installation position of the roller with respect to the center gear mainly affected the total length of the string, and a change in this value had minimal impact on the length of the tangents. Therefore, both the radius of the double-layer bearing and the installation position of the roller had minimal impact on the flapping angle. On the other hand, when considering both the radius of the roller and the installation radius of the double-layered bearing, we can imagine that the string travels an equal distance around both circles, with the resulting angle change being related to the radius of both circles. \begin{table}
\begin{tabular}{c c} \hline \hline
**Design Parameters (mm)** & **Data** \\ \hline Radius of rollers \(r_{1}\) & 1.2 \\ Radius of bearing \(r_{3}\) & 1.9 \\ Radius of center gear \(r_{0}\) & 1.7 \\ Distance of center of circles \(d\) & 10.15 \\ Length of strings \(l_{w}\) & 65 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Characteristics of the design parameters. Figure 6: Analysis of left roller, right roller, and the flapping angle. Based on the simulation, changing the width of the flapping mechanism during the design process will not cause significant changes to the final flapping angle. Additionally, we do not need to worry about the small size of the double layer bearings, which is a challenging size to achieve in a micro-scale design in structural development. The flapping trajectory and amplitude could be adjusted by changing the radius of the rollers and center gear, which means the ratio of the rollers to center gear had the greatest influence on the design of mechanisms. To visually demonstrate the impact of the ratio of the installation position of the bearing group to the radius of the roller group on the flapping trajectory, we defined the ratio as \(k_{r}\). Figure 11 shows the simulation of the influence of the varying \(k_{r}\) on the  
Figure 8: (a) The influence of the radius of the double-layer bearing on the flapping amplitude. (b) The influence of the double-layer bearing on the flapping trajectory. Figure 7: (a) The influence of the distance of the center of the rollers and the center gear on the flapping amplitude. (b) The influence of the distance of the center of the rollers and the center gear on the flapping trajectory.