## 2 Wing Deformation Analysis Based on Multi-Plane Simulation Method  
When conducting aerodynamic analysis of hovering flapping using a quasi-steady model, it is common to simplify it as a single rigid plane and analyze various morphological parameters such as chord length, spanwise length, and the leading and trailing edge shapes. This simplification approach has been widely adopted in existing research on flapping aerodynamics and is a feasible method. However, in actual flapping the wing often undergoes inward concave deformations due to factors such as wing material flexibility, air resistance, and vein position design. To account for such deformation, in this paper we propose a multi-plane fitting method that replaces the flexibly deformed wing with multiple planes by considering the vein positions. Fitting a wing composed of multiple planes and combining it with the quasi-steady model allows the aerodynamic forces to be analyzed. Figure 1 shows the morphological parameters of a wing with two veins in a two-dimensional plane. Due to the presence of a slack angle [50; 51], the wing experiences a passive twist when installed on a flapping aircraft with the leading spar and root spar kept vertical. During high-speed flapping, apart from the flipping that occurs at the two ends,  
the wing surface is mostly under the influence of air resistance and remains in a taut state, forming a full surface in three-dimensional space. At this time, the position of the veins affects the shape of this surface. Although the surface is difficult to express mathematically, it can be approximated as multiple planes separated by veins, then these separate planes can be analyzed in conjunction with the vein positions and slack angle. As shown in Figure 1, the shape of the wing in a two-dimensional plane can be represented using parameters in a Cartesian coordinate system: wing height \(H_{w}\), wing length \(L_{w}\), slack angle \(\theta_{s}\), and design angle of the carbon fiber spar \(\theta_{rod1}\) and \(\theta_{rod2}\). As shown in Figure 1, the points forming the triangular planes in two-dimensional space can be defined as follows:  
\[P_{0-2D},P_{1-2D},P_{2-2D},P_{3-2D},P_{4-2D}, \tag{1}\]  
and, bringing in the parameters, the position of the point can then be expressed as described below. \[P_{0-2D}:(0,0) \tag{2}\]  
\[P_{1-2D}:(L_{w},0) \tag{3}\]  
\[P_{2-2D}:(H_{w}/tan(\theta_{rod1}),-H_{w}) \tag{4}\]  
\[P_{3-2D}:(H_{w}/tan(\theta_{rod2}),-H_{w}) \tag{5}\]  
\[P_{4-2D}:(-H_{w}\cdot\sin(\theta_{s}),-H_{w}\cdot cos(\theta_{s})) \tag{6}\]  
After the wings are mounted, the position of the points in the wing coordinate system in three-dimensional space can be defined as  
\[P_{0-W},P_{1-W},P_{2-W},P_{3-W},P_{4-W} \tag{7}\]  
and the triangular planes Plane1, Plane2, and Plane3 are defined as shown in Figure 2.  
Figure 1: Definition of wing shape in two-dimensional plane. Figure 2: Definition of wing shape in three-dimensional plane. At this point, the wing shape does not consider displacement at the wing root, meaning that it assumes the flapping continues in a hovering state without generating average yaw, roll, or pitch moments. At this stage, the shape of each plane remains unchanged and consistent with its shape in the two-dimensional space. The deformed points in three-dimensional space can be represented as follows:  
\[P_{0-2D}:(0,0) \tag{2}\]0  
\[P_{0-2D}:(0,0) \tag{2}\]1  
\[P_{0-2D}:(0,0) \tag{2}\]2  
\[P_{0-2D}:(0,0) \tag{2}\]3  
\[P_{0-2D}:(0,0) \tag{2}\]4  
where the positions of \(P_{2-W}\) and \(P_{3-W}\) can be solved by the system of equations below. \[ \begin{cases}D_{02-2D}=D_{02-W}\\ D_{03-2D}=D_{03-W}\\ D_{12-2D}=D_{12-W}\\ D_{23-2D}=D_{23-W}\\ D_{34-2D}=D_{34-W}\end{cases}  \tag{13}\]  
The distance between point0 and point2 in the two-dimensional Cartesian coordinate system is represented by \(D_{02-2D}\), while the distance between point0 and point2 in the three-dimensional wing coordinate system is represented by \(D_{02-W}\). The definitions of the other parameters are similar.