## 3 Kinematic and Static Force Analysis
\begin{split} F_{\mathrm{ND}_{i}}\cos(\alpha_{\mathrm{D}}+\theta _{1i}+\alpha_{12})\cdot l_{3}+\sqrt{3}aF_{\mathrm{elastic}}\cos(\alpha_{ \mathrm{C}}+\theta_{1i})+F_{\mathrm{ND}_{i}}|A_{i}K_{i}|\\ =\mathrm{G}\cdot(x_{\mathrm{c}}-\mathrm{r})+\sqrt{3}aF_{\mathrm{ T}}\sin\theta_{1i}\end{split}  \tag{17}\]  
As the tensile force \(T\) gradually increases, the landing gear gradually opens, and the sphere moves downward, coming into contact with the landing gear only at point \(D_{i}\). Consequently, we have:  
\[\left\{ \begin{split} F_{\mathrm{ND}_{i}}^{\prime}\cos\alpha_{ \mathrm{D}}=3\mathrm{G}+T\\ F_{\mathrm{ND}_{i}}\cos(\alpha_{\mathrm{D}}+\theta_{1i}+\alpha_{12}) \cdot l_{3}+\sqrt{3}aF_{\mathrm{T}}\sin\theta_{1i}=\mathrm{G}\cdot(\mathrm{r }-x_{\mathrm{c}})\end{split} \right. \tag{18}\]  
To evaluate the impact force during the collision between the landing gear and the object during the grasping process, the force relationship was analyzed for collisions with spherical objects of varying radii. The force distribution is shown in Figure 8. Due to the high-speed collision between the landing gear and the target object, the contact area may experience plastic deformation or fracture, with a significant portion of the energy dissipated as heat or internal losses, characteristic of a typical inelastic collision. The nearly normal collision orientation further results in higher energy dissipation and pronounced plastic deformation characteristics. Under normal temperature conditions, the smooth surface of the target object and the small contact area allow the friction-induced energy dissipation to be neglected. When the landing gear is fully deployed, the spring reaches its maximum stretched position. According to the geometric relationships of the landing gear structure, the spring length at this point is denoted as \(x_{\text{max}}\) and expressed as follows:  
\[x_{\text{max}}=\sqrt{3}(\text{a}\cos 5^{\circ}+\text{r}) \tag{19}\]  
During the closing process of the landing gear, when it collides with the object, the angle \(\theta_{\text{impact}}\) between the lower edge of the main linkage and the horizontal direction is defined as follows:  
\[\theta_{\text{impact}}=\arcsin\frac{\left|{{}^{A}p_{D_{i}}\cdot{e_{z}}} \right|}{\left\|{{}^{A}p_{D_{i}}}\right\|} \tag{20}\]  
According to the law of conservation of energy, \(k(x_{\text{max}}^{2}-x_{\text{impact}}^{2})=I\omega^{2}\), the quantity \(I\) represents the moment of inertia of the main link about the rotational joint \(A_{i}\):  
\[I=\frac{\text{G}_{1}}{3g}\Big{[}(|x_{\text{c}}|-r)^{2}+z_{\text{c}}^{2}\Big{]} \tag{21}\]  
Based on the above formula, the linear velocity \(v_{\text{before}}\) at the contact point \(D_{i}\) can be calculated just before the collision occurs:  
\[v_{\text{before}}=\sqrt{\frac{k(x_{\text{max}}^{2}-x_{\text{impact}}^{2})}{ I}}L_{1} \tag{22}\]  
The impact force \(F_{\text{impact}}\) during the collision is determined by the relationship between impulse and time,  
\[F_{\text{impact}}\cdot\Delta t=\Delta p \tag{23}\]  
where \(\Delta t\) is the collision time; \(\Delta t\) is taken as \(10^{-3}\)s for common materials; \(\Delta p=m(v_{\text{before}}-v_{\text{after}})\) represents the change in momentum at the contact point \(D_{i}\) before and after the collision; and \(m\) is the equivalent mass of the rod, given by \(\text{G}_{1}/3g\). Figure 8: The schematic diagram of the landing gear collision with the target object. The coefficient of restitution \(e\) is taken as 0.24 (almost completely inelastic). \[e=\frac{v_{\text{after}}}{v_{\text{before}}} \tag{24}\]  
where \(v_{\text{after}}\) is the velocity after the impact, and \(v_{\text{before}}\) is the velocity before the impact. Finally, the impact force during the collision can be expressed as follows:  
\[F_{\text{impact}}=\,L_{1}(1-e)\sqrt{\frac{kG_{1}(x_{\text{max}}^{2}-x_{\text {impact}}^{2})}{3g\Big{[}(|x_{\text{c}}|-r)^{2}+z_{\text{c}}^{2}\Big{]}}}/\Delta t \tag{25}\]