## 2 Theoretical Mechanism Design and Model  
The string-based flapping mechanism was originally used to generate wing flapping for the nano Hummingbird [24], and this was then imitated by the team from SJTU [27]. As the most classical and mature example of flapping wing vehicles that use the string-based mechanism, the nano Hummingbird and its drive mechanism were comprehensively demonstrated in the paper of [24]. This study will replicate and validate the mature platform based on the string-based mechanism of the nano Hummingbird, and the SolidWorks principle diagram is shown in Figure 1, where the main parameters are shown in detail. As shown in Figure 1, drive strings are two independent strings, which wrap around the double-layer bearing on the center gear. Two ends of one drive string are fixed on the left roller while the two ends of the other string are fixed on the right roller. As shown in Figure 2, the rotation direction of the central gear is counterclockwise. In Figure 2a,b, the position of the double-layer bearing is approximately above the connecting line between center of left and right roller. At that time, the right string is pulled by the double-layerbearing and the left string is pulled by the phase keeper strings, and the flapping direction of the left and right wings is downward. In Figure 2c,d, the position of the double-layer bearing is approximately below the connecting line between the center of the left and right roller. At that time, the left string is pulled by the double-layer bearing and the right string is pulled by the phase keeper strings, and the flapping direction of the left and right wings is upward. During the rotation of the center gear, only one side of the string is pulled to maintain a taut state, which causes the rotation of the roller. Meanwhile, another string is not directly pulled by the center gear, which will become loose in the absence of other drives. Therefore, two additional strings are required to connect the left and right rollers to keep the two rollers turning at the same angle, which will result in pulling another string taut (and these are referred to as phase keeper strings). In a theoretical design, the left and right wings need to have the same flapping angle to prevent asymmetry during flapping. Therefore, the rollers on both sides need to have the same radius to ensure they can have the same rotation angle during the rotation of the center gear. As shown in Figure 3, a Cartesian coordinate system was established, and the origin was the center of the center gear. The vector from origin to the center of the right roller, which was named \(O_{r}\), was the positive direction of the x-axis. The center of the left roller was on the negative direction of the x-axis. The center of the double-layer bearing rotated on the center gear with a radius of \(r_{0}\), with coordinates of \((x_{y},y_{y})\), and a radius of \(r_{3}\). We then took the taut left string as an example, which had a total length of \(L\) and was wrapped around the double-layer bearing with an arc of \(BC\), as well as a little arc on the left roller (which was expressed as \(AD\)). There, the point \(ABCD\) was a tangent point of the lines \(AB\) and \(CD\), as well as of the left roller and double-layer bearing. The total length of the left string can be express in Equation (1). \[L=AB+CD+BC+AD+AE \tag{1}\]  
Figure 1: String-based flapping mechanism imitated from the one present in the nano Hummingbird. Figure 2: Schematic diagram of the movement principle of the string-based mechanism. In (**a**,**b**), position of the double-layer bearing is approximately above the connecting line between center of left and right roller. In (**c**,**d**), position of the double-layer bearing is approximately below the connecting line between the center of the left and right roller. Here, point \(E\) is the intersection of the two ends of the left string on the left roller, which controls the rotation of the left roller when the left string is being tautly pulled.