## 5 Cam Mechanism  
In previous analyses, it was observed that when both rollers on either side are circular, there exists a discrepancy between the angles generated by the ideal 'unwinding' of the string on one roller and the ideal 'winding' of the string on the other. The presence of a synchronous string necessitates equal rotation angles for both side rollers, which results in a conflict that is manifested as the 'over-tightening' phenomenon of the string and the stagnation phenomenon in physical experiments. To address this issue, the NANO Hummingbird team introduced a patented solution in 2018 [34] involving the use of non-circular rollers, specifically where a cam-like mechanism was referred to as a cam (shown in Figure 12). This design was also adopted by the team of SJTU [27] to effectively mitigate the stagnation phenomenon in string transmission systems with circular rollers. However, their patent and study did not provide insights into the design rationale and parameter analysis of the non-circular cam profile, while [PERSON]'s work on cam [35; 36] and [PERSON]'s research [37] gave us inspiration for deriving cam trajectories. In this section, we start with the conceptual design of the circular roller mechanism, then we elucidate the methodology for designing the non-circular cam mechanism and mathematically model it. In conjunction with the relevant parameters from the circular roller flapping mechanism, we analyze the expression of the non-circular cam mechanism, which is abbreviated as the cam mechanism in the subsequent content. In Figure 13, a detailed motion schematic of the cam mechanism is shown. As illustrated in Figure 13, the diagram depicts the complete cycle of motion for an optimized non-circular cam based on circular rollers. Using counterclockwise rotation as an example, it primarily comprises the following four operational states: in Figure 13b-d, the right roller is driven by the central gear along a non-circular path, while the left roller is pulled by a synchronous string along a circular path;in Figure 13a,b,h, the right roller is driven by the  
Figure 12: Schematic diagram of the switching from the region of the circular roller to the region of the cam roller. central gear along a circular path, while the left roller is pulled by a synchronous string along a non-circular path,in Figure 13f-h, the left roller is driven by the central gear along a non-circular path, while the right roller is pulled by a synchronous string along a circular path; and in Figure 13d-f, the left roller is driven by the central gear along a circular path, while the right roller is pulled by a synchronous string along a non-circular path. This schematic provides a comprehensive overview of the cam mechanism's dynamic behavior during its complete cycle of motion. In the previous section, it was demonstrated, through simulation, that the radius of the double-layer bearings' assembly has a minimal impact on a string-based mechanism. In this section's analysis, we will treat it as a point, disregarding the influence of the bearing assembly's radius, as shown in Figure 12. The total length of the red string and the blue string remained constant. Taking the red string in Figure 12 as an example, its length can be expressed as Equation (15). \[L_{left}=\mathcal{O}_{s}^{*}B+\widetilde{BC} \tag{15}\]  
where \(\mathcal{O}_{s}^{*}B\) represents the winding length of the string on the circular and non-circular tracks, and \(\widetilde{BC}\) represents the length of the string under ideal tension conditions. Under this assumption, in combination with the relevant parameters mentioned earlier, we simulated the change in the deflection angle of the string on the circular roller under ideal tension conditions and obtained a simulation diagram, which is shown in Figure 14. It can be observed that there is a significant difference in the angles between the left and right rollers. This difference also exists in simulations that consider the double-layer bearing radius.