## 3 Kinematic and Static Force Analysis
Circular symbols represent the components of the mechanism; lines represent the types of kinematic pairs connecting the components. The rank of this screw system is determined to be 6, indicating that the first closed loop is not over constrained and that \(\mu_{1}=0\). Next, we constructed the screw system \(\boldsymbol{\hat{\mathfrak{s}}}_{2}\) for the second branch. Similarly, the rank of the screw system \(\boldsymbol{\hat{\mathfrak{s}}}_{2}\) is determined to be 6, with \(\mu_{2}=0\). This indicates that closing the second branch onto the first closed loop does not affect the motion of the first loop. The screw system \(\boldsymbol{\hat{\mathfrak{s}}}_{3}\) for the third branch has a rank of 5, indicating the presence of one redundant constraint, with \(\mu_{3}=1\). Using the modified Grubler-Kutzbach formula, the overall DoF of the landing gear is calculated to be 1. \[M=6(n-g-1)+\sum_{i=1}^{g}f_{i}+\sum_{k=1}^{3}\mu_{k} \tag{2}\]  
Given that the dual-function landing gear has only one DoF, a single input is sufficient to drive the entire mechanism. Furthermore, during the opening and closing process of the landing gear, the motion states of the linkages exhibit symmetry. We will now perform a kinematic analysis based on the structural characteristics of the landing gear. The fully open position of the dual-function landing gear is defined as the initial position. The dimensional parameters of the landing gear are shown in Figure 7a. When the angle \(\theta\) between the cable and the main rod becomes \(0^{\circ}\) in the illustrated position, altering the moment arm acting on the main rod. The angle \(\theta\) is related to the rotational angle \(\theta_{1i}\) of the main rod as follows:  
\[\theta=\arcsin\frac{\h\cos\theta_{1i}}{\Delta s+\s}-\arctan\frac{D_{\rm e}}{ \fff} \tag{3}\]  
Based on the law of cosines, the expression for the change in length \(\Delta s\) of the cable outside the lower compartment is derived as follows:  
When \(5^{\circ}\leq\theta_{1i}\leq\pi/2-\arcsinD_{\rm e}/\h\), the \(\Delta s\) is given by:  
\[\Delta s=\sqrt{\h^{2}+\ffrac{1}{2}+{D_{\rm e}}^{2}+2\hslash\sqrt{\ffrac{2}{2 }+{D_{\rm e}}^{2}\sin(\theta_{1i}-\arctan\frac{D_{\rm e}}{\fffff})}-\s} \tag{4}\]  
When \(\pi/2-\arcsinD_{\rm e}/\h<\theta_{1i}\leq 90^{\circ}\), the angle between the cable and the upper edge of the main rods is zero. The elongation \(\Delta s\) of the cable with respect to \(\theta_{1i}\) is given by:  
\[\Delta s=\sqrt{\hfrac{{D_{\rm e}}^{2}}{\cos^{2}5^{\circ}}-\frac{2\hslash{D_{ \rm e}}}{\cos^{5}5}\cos(\theta_{1i}-5^{\circ})}+\ff+D_{\rm e}\tan 5^{\circ}-\s \tag{5}\]  
Figure 7: Schematic diagram of the dual-function landing gear. (**a**) Including key parameters. (**b**) Theoretical tension variation in the cable during landing gear deployment with different spring stiffnesses. From this, we can derive the relationship between the rotation angle \(\theta_{\rm DC}\) of the DC-geared motor and the rotation angle \(\theta_{1i}\) of the landing gear:  
\[\theta_{\rm DC}=180^{\circ}\times i\times\Delta s/\pi\times{\rm r_{w}} \tag{6}\]  
Here, i represents the transmission ratio of the quick-release mechanism, \(i=0.7\), and \({\rm r_{w}}\) is the radius of the cable drum. Using coordinate transformation, the position of the end point \(D_{i}\) in the base coordinate system is:  
\[\mathbf{p}_{D_{i}}=\mathbf{R}(z,\phi_{i})^{A}\mathbf{p}_{D_{i}} \tag{7}\]  
Similarly, the position of the joint \(C_{i}\) in the base coordinate system can be expressed as follows:  
\[\mathbf{p}_{C_{i}}=\frac{Q}{B}\mathbf{T}\Big{[}{\rm d}\cos\theta_{3i}\cos\theta_{4i} \quad-{\rm d}\cos\theta_{3i}\sin\theta_{4i}\quad-{\rm d}\sin\theta_{3i}\quad 1 \Big{]}^{T} \tag{8}\]  
Since the motion of the joint \(C_{i}\) always lies within the plane yOz, we introduce the constraint \(\mathbf{p}_{C_{i}z}=0\). Thus, we can establish a one-to-one correspondence between \(\theta_{1i}\) and \(\theta_{3i}\). Given any value of \(\theta_{1i}\) or \(\theta_{3i}\) we can determine the position of joint \(C_{i}\) in the base coordinate system.