## 3 Kinematic Analysis of Wings Based on the Multi-Plane Simulation Method  
By adjusting the parameters \(y_{2-W}\), the shape of the wing can be flexibly adjusted. Observing the phenomenon during actual flapping, it is found that the shape of the wing varies at different positions due to factors such as flapping angular velocity and inertia. In addition, there is flexible deformation during wing flipping. As the translational velocity of the wing during flipping is relatively low and the flexible deformation has a significant impact, appropriate adjustments need to be made to the multi-plane method based on the invariant plane area. This section introduces the kinematic analysis of the wing based on the multi-plane simulation method along with the modified data of \(y_{2-W}\) based on the position and flapping angular velocity in order to simulate the deformation of the wing during flapping and prepare for calculating the aerodynamic forces of the wing. Before simulating the deformation of the wing during flapping, it is necessary to describe the shape of the wing in the body coordinate system. In the previous section, we defined the shape of the wing in the right wing coordinate system; as shown in Figure 7, both the right wing coordinate system and the body coordinate system are right-handed co  
Figure 5: Change of wing shape in three-dimensional space when changing the \(y_{2-W}\) value. Figure 6: Comparison between the shapes of real wings and simulated wings: (**a**) shape of real wings when laid flat; (**b**) shape of simulated wings when laid flat; (**c**) shape of real wings when installed; (**d**) shape of simulated wings in three-dimensional space when installed. ordinate systems, with their z-axis directions the same and pointing upwards. The positive direction of the y-axis of the body coordinate system is defined as the forward direction of the body, and the positive direction of the x-axis is defined as the direction of the right wing at the flapping center position, with the coordinate origin located at the center of gravity of the body. The origin of the right wing coordinate system is located at the mounting root point of the right wing, with its x-axis direction the same as the direction of the wing spar. In summary, the position \(P_{W}\) of any point in the right wing coordinate system can be expressed as \(P_{b}\) in the body coordinate system, as follows:  
\[P_{b}=R_{w\to b}P_{w}-O_{b}O_{w}. \tag{15}\]  
The vector from the origin of the body coordinate system to the origin of the wing coordinate system, represented in the body coordinate system, is denoted as \(O_{b}O_{W}\). The rotation matrix from the wing coordinate system to the body coordinate system can be expressed as \(R_{w\to b}\), where \(\psi\) represents the rotation angle around the z-axis. Because the z-axis directions of the two coordinate systems are the same, the rotation matrix can be represented by Equation (16):  
\[R_{w\to b}= \begin{bmatrix}\cos\psi&-\sin\psi&0\\ \sin\psi&\cos\psi&0\\ 0&0&1\end{bmatrix} . \tag{16}\]  
As mentioned above, by combining previous research [52] and actual wing motion images captured under high-speed photograph in Figure 8, it can be inferred that the shape of the wing varies at different positions. Therefore, before analyzing the kinematics of the wing using the multi-plane simulation method, it is necessary to define the shape of the wing at each position. Referring to previous research [47, 48], we consider a linear fitting method in the simulation definition, which defines a linear representation of the wing by defining key points between the leading edge and the trailing edge of the flapping motion. As shown in Figure 9, taking half of a flapping cycle as an example, the flapping angle of the wing ranges from \(-\psi_{max}\) to \(\psi_{max}\). Within the range of angles from \(\psi_{c}\) to \(\psi_{max}\)  
Figure 8: Actual wing motion images captured under high-speed photography.