## 2 Wing Deformation Analysis Based on Multi-Plane Simulation Method
The reason for establishing this system of equations is that in the multi-plane assumption, due to the presence of veins, the triangular shape of the wing after deformation remains unchanged when divided by veins. Therefore, the distance between the two points in different coordinate systems is equal. Equations (13) can be expanded as follows. \[\left\{\begin{array}{c}x_{2-W}^{2}+y_{2-W}^{2}+z_{2-W}^{2}=L_{W}^{2}\\ x_{3-W}^{2}+y_{3-W}^{2}+z_{3-W}^{2}=\left(\frac{H_{w}}{\cos(\theta_{\text{roll} })}\right)^{2}\\ (x_{2-w}-L_{w})^{2}+(y_{2-w})^{2}+(z_{2-w})^{2}=\left(L_{w}-\frac{H_{w}}{\tan( \theta_{\text{roll}})}\right)^{2}+(H_{w})^{2}\\ (x_{2-w}-x_{3-w})^{2}+(y_{2-w}-x_{3-w})^{2}+(z_{2-w}-x_{3-w})^{2}=\left(\frac{ H_{w}}{\tan(\theta_{\text{roll}})}-\frac{H_{w}}{\tan(\theta_{\text{roll}})} \right)^{2}\\ (x_{3-w})^{2}+(y_{3-w})^{2}+(z_{3-w}+H_{w})^{2}=\left(\frac{H_{w}}{\tan(\theta _{\textFrom the figure, it can be observed that the feasible range of \(y_{3-W}\) increases with the increase in the slack angle. Meanwhile, \(y_{3-W}\) shows a trend of first increasing and then decreasing with the increase of \(y_{2-W}\). Prior to reaching the maximum value, the angle between plane1 and plane2 is negative, indicating that only when the value of \(y_{2-W}\) starts to decrease does the entire wing begin to exhibit a naturally curved and full shape. Combining with the definition of the wing in an inflated state, the feasible range of \(y_{2-W}\) under different slack angles is derived as shown in Figure 4.  
As shown in Figure 5, under different values of \(y_{2-W}\), the shape of the wing in the three-dimensional space varies. After selecting appropriate values of \(y_{2-W}\), the shape representation of the wing using the multi-plane method closely approximates the wing's installed shape in its natural state, as shown in Figure 6. The main difference lies in the smooth curve of the wing's lower edge in its natural state, while the lower edge represented by the multi-plane method consists of line segments, resulting in some numerical discrepancies between the two. In existing hovering flapping-wing prototypes, a two-degrees-of-freedom flapping mechanism is commonly used to control the flapping and rotation of the wings. During hovering flight, the wings only undergo reciprocating flapping. The flapping process can be divided into a translational stage and a passive wing flipping stage, with the trans  
Figure 4: Influence of the slack angle on the allowable value range of \(y_{2-W}\) when the wing shape is inflated. Figure 3: Effect of changing \(y_{2-W}\) under different slack angles on \(y_{3-W}\). lational aerodynamic force accounting for approximately 70% of the total lift. During the translational stage, the wings are under tension due to the aerodynamic forces. As shown in Figure 5, different wing deformations can be achieved by adjusting the value of \(y_{2-W}\) after determining the slack angle, while the wing's shape at various positions during actual flapping can be obtained by combining the actual flapping angle position. By analyzing the morphological changes of the wing during flapping, aerodynamic forces and drag can be analyzed and various parameters in wing design can be adjusted based on optimization objectives in order to guide the design of the wing.