## 3 Kinematic and Static Force Analysis  
In this section, to understand and describe the motion characteristics and constraint conditions of the dual-function landing gear, we first analyzed the degrees of freedom (DoF) to ensure that the landing gear does not encounter issues of over constraining during its operation. Subsequently, we performed kinematic modeling of the dual-function landing gear. This involved mapping the rotation angle of the main rod of the landing gear to the motion trajectory of its characteristic points. We also analyzed the changes in the cable length during the morphological transformation of the landing gear. As shown in Figure 5a, the base coordinate system _O-xyz_ is located on the base platform. The second coordinate system \(x_{1i},y_{1i},z_{1i}\) is nested at the revolute joint \(A_{i}\). For ease of calculation, the \(z_{1i}\)-axis direction is parallel to the \(z\)-axis, and the \(y_{1i}\)-axis is parallel to the axis of revolute joint \(A_{i}\). A third coordinate system \(x_{2i},y_{2i},z_{2i}\) is nested at the rotational joint \(B_{i}\), oriented as shown in Figure 5b. Joint \(C_{i}\) is connected by a flexible material and can be considered a spherical joint. The length parameters of each part of the landing gear are shown in the simplified schematic diagram in Figure 5b. The link between joint \(B_{i}\) and joint \(C_{i}\) can be simplified into a straight rod with a certain offset angle. According to the chain rule of coordinate transformation, the relationship between the coordinate systems can be expressed as \({}^{Q}_{B}T\!\!=\!\!A^{Q}_{T}{}^{A}_{B}T\). Based on the connections between the components of the landing gear, we can utilize a mechanism topology diagram to more intuitively represent the relationships between the components. As shown in Figure 6, it can be observed that this mechanism is a general spatial multi-loop mechanism, consisting of 10 rods and 12 kinematic joints, forming three closed loops. Due to the complex motion coupling often occurring between multiple independent loops, we employ screw theory to determine the DoFs of the mechanism. To simplify the calculations, the Plucker coordinates of the screw axis are represented using six scalar elements: \(\mathbf{\xi}=(\mathbf{S};\mathbf{S}_{0})=(\mathbf{S};\mathbf{r}\times\mathbf{S})= \begin{pmatrix}l&m&n;p&q&r \end{pmatrix} \). Here, the unit vector \(S\) denotes the direction of the axis, and \(S_{0}\) refers to the moment of \(S\) about the origin. As illustrated in Figure 5b, the established coordinate systems are used to determine the elements corresponding to the screw axis at each kinematic joint. By opening the loops in the mechanism's topology diagram, three branches are obtained. Each branch is then closed sequentially. First, the screw system \(\mathbf{\hat{\xi}}_{1}\) of the initial fundamental closed loop is established:  
\[ \begin{split}&\mathbf{\xi}_{A_{i}}=\begin{bmatrix}\mathbf{e}_{y_{1}};\bm {p}_{Ai}\times\mathbf{e}_{y_{1}}\end{bmatrix}\\ &\mathbf{\xi}_{B_{i}}=\begin{bmatrix}\mathbf{e}_{y_{2}};\mathbf{p}_{B_{1}}\times\mathbf{e}_{y _{2}}\end{bmatrix}\\ &\mathbf{\xi}_{C_{i},1}=\begin{bmatrix}\mathbf{e}_{I_{1}\xi_{C_{i}}}\times\mathbf{e}_{I_{ 1}\xi_{C_{i}}};\mathbf{p}_{Ci}\times\mathbf{e}_{I_{1}\xi_{C_{i}}}\times\mathbf{e}_{I_{1} \xi_{C_{i+1}}}\end{bmatrix}\\ &\mathbf{\xi}_{C_{i},2}=\begin{bmatrix}(0,-\mathbf{\xi}_{C_{i},1,2}/\mathbf{\xi}_{C_{i},1, x},1);\mathbf{p}_{C_{i}}\times(0,-\mathbf{\xi}_{C_{i},1,x}/\mathbf{\xi}_{C_{i},1,x},1) \end{bmatrix}\\ &\mathbf{\xi}_{C_{i},3}=\begin{bmatrix}\mathbf{\xi}_{C_{i},1}\times\mathbf{\xi}_{C_{i},2}; \mathbf{p}_{C_{i}}\times\mathbf{\xi}_{C_{i},1}\times\mathbf{\xi}_{C_{i},2}\end{bmatrix} \end{split}  \tag{1}\]  
Figure 5: Simplified diagrams of the dual-function landing gear. (**a**) Relationship between the landing gear and drone dimensions. (**b**) Detailed description of the landing gear joints and angles. Figure 6: Topology diagram of the mechanism.