## 3 Simulation and Verification  
A reconfigurable legged lander was used as an example to verify the tube model. The lander, in its symmetric configuration, is shown in Figure 4a. The lander has three modes--the adjusting, landing, and walking mode. In the adjusting mode, the lander will hover several hundred meters above the landing site. It will select suitable terrain for landing and avoid hazardous rocks or large craters. At the same time, adjusting links will rotate around the axis \(R_{i}(i=1\sim 3)\) to a suitable position according to the selected landing site. In the landing mode, the lander will descend to several meters above the lunar surface and axis \(R_{i}(i=1\sim 3)\) will be fixed. Then, the engine will be cut off and the lander will fall down freely. When hitting the lunar surface, the impact force will be cushioned by the aluminum honeycomb in the primary and two secondary struts of the lander. After landing, other joint axes will be released. The lander can walk on the lunar surface. For more detail on this reconfigurable mechanism, readers can look at reference [36]. Figure 4: (**a**) The reconfigurable lunar lander mechanism. (**b**) One leg of the lunar lander and its link number and node number. (**c**) The topology graph and connection type of links. Since the landing vibration is crucial to the lander, the elastodynamics of the landing mode are analyzed below. After adjusting the axis \(R_{i}\), the lander goes into the landing mode, and the axis \(R_{i}\) is fixed. Figure 4b shows the symmetric configuration of the lander. From the main body to the footpad are the primary and two secondary struts composed of \(UPS\) limbs, where \(U\) denotes the universal joint, \(P\) denotes the prismatic joint, and \(S\) denotes the spherical joint. Three prismatic joints are formed by the outer and inner tubes. Among the joints, all links are regarded as 3D Euler-Bernoulli beams. The link numbers are shown in Figure 4b. A connection graph of all the links is shown in Figure 4c. The outer tube of the main strut \(L4\) is fixed on the platform \(L13\). The platform \(L13\), the foot pad \(L14\), and the main body \(L15\) are regarded as rigid bodies. All other parts are considered flexible beams. Since the outer tube of the main strut is partly tied to the platform, we defined the tied nodes of the main strut. The tied outer tube nodes \(13,14,15\) are shown in Figure 4b. Their displacement is determined by the platform node \(\mathbf{u}_{p}\). \(\mathbf{t}_{ip}\) is the vector from the tied node \(\mathbf{u}_{i}\) to the platform node \(\mathbf{u}_{p}\). \[\mathbf{u}_{i}=\mathbf{G}\mathbf{u}_{p} \tag{26}\] \[\mathbf{G}= \begin{bmatrix}\mathbf{E}_{3}\ \widetilde{\mathbf{r}_{ip}}\\ \mathbf{0}\ \mathbf{E}_{3}\end{bmatrix} (i=13\sim 15)\]  
The virtual work \(\delta W_{i}\) of the buffer force on the \(i\)th strut is shown as follows. \(\mathbf{n}_{i}\) is the direction vector of the \(i\)th buffer limb. \(\mathbf{u}_{li}\) is the node that buffer force acts on for the \(i\)th inner tube. \(\mathbf{u}_{Oi}\) is the node that buffer force acts on for the \(i\)th outer tube. \(\mathbf{f}_{im}\) is the magnitude of the \(i\)th buffer force. \[\delta W_{i} =\mathbf{F}_{i}^{T}\delta\mathbf{u}_{li}+\big{(}-\mathbf{F}_{i}^{T}\big{)} \delta\mathbf{u}_{Oi} \tag{27}\] \[\mathbf{F}_{i} = \begin{bmatrix}f_{im}\mathbf{n}_{i}\\ \mathbf{0}\end{bmatrix} (i=1\sim 3)\]  
We first verified the statics of the lander. The displacement boundary condition and the magnitude of the buffer force are set below. \(\mathbf{u}_{b}\) is the node of the main body, and \(\mathbf{u}_{f}\) is the node of the footpad. \[\mathbf{u}_{b}=[0,0,0,0,0,0]^{T}\text{m}\] \[\mathbf{u}_{f}=[-5,5,10,0,0,0]^{T}\times 10^{-3}\text{m}\] \[f_{1m}=2000\text{N},\ f_{2m}=10000\text{N},f_{3m}=5000\text{N}\]  
By setting the acceleration terms \(\tilde{\mathbf{u}}_{1}\) and \(\tilde{\mathbf{u}}_{2}^{0}\) in Equations (19) and (20) to \(\mathbf{0}\), the theoretical boundary force \(\mathbf{F}_{f}\) of node \(\mathbf{u}_{f}\) were solved.