# Design and Dynamic Analysis of the Wire-Line Coring Robot for Deep Lunar Rocks
## 3 Dynamic Analysis of the WSC Robot
### Maximum Support Force Analysis of the Anchor Module  
The maximum support force is related to the drilling function of the WSC robot in the subsurface and is critical for the strategy of coring. The anchor module squeezes the drilling hole wall by expanding the anchor plates to provide sufficient friction to support the WSC robot underground. Thus, the higher the squeezing pressure, the stronger the support force of the WSC robot in the subsurface. However, the lunar regolith of the drilling hole wall can often slip when the squeeze pressure is excessive, thus preventing the anchor module from providing a stable positional fixation. Therefore, there is a limit to the support force provided by the anchor module. For determining the maximum support force, the mechanical properties associated with the lunar regolith at different depths should be analyzed. According to NASA's analysis of the returned samples from the Apollo program, the relationship between the lunar regolith packing density \(\rho\) and depth \(z\) is [20]:  
\[\rho=1.92\frac{z+12.2}{z+18} \tag{1}\]  
Additionally, the following relationship that exists between \(\rho\) and the lunar regolith pore ratio \(n\) is [21]:  
\[n=1-\frac{\rho}{G\rho_{w}} \tag{2}\]  
where \(G\) is the average specific gravity of the lunar sample, with a suggested value of 3.1 [22] and \(\rho_{w}\) is the density of pure water at 4 \({}^{\circ}\)C, which is 1 g/cm\({}^{3}\). According to [PERSON]'s failure mechanism, the ultimate bearing capacity is related to cohesion and the angle of internal friction. Additionally, there is an exponential relationship between the lunar regolith pore ratio \(n\) and the lunar regolith cohesion \(c\), as well as a linear relationship with the tangent of the lunar regolith angle of the internal friction \(\varphi\). Thus, the expressions for \(c\) and \(\varphi\) can be obtained by fitting the data from Table 1 [23]:  
\[\left\{ \begin{array}{ll}c=10^{2.9331-6.0035n}&\left(R^{2}=0.9885\right)\\ \varphi=\arctan(3.0982-4.1088n)&\left(R^{2}=0.9519\right)\end{array} \right. \tag{3}\] \begin{table}
\begin{tabular}{c c c} \hline
**Depths \(z\) (m)** & **Cohesion \(c\) (kPa)** & **Angle of Internal Friction \(\varphi\) (\({}^{\circ}\))** \\ \hline
0–0.15 & 0.44–0.62 & 41–43 \\
0–0.3 & 0.74–1.1 & 44–47 \\
0.3–0.6 & 2.4–3.8 & 52–55 \\
0–0.6 & 1.3–1.9 & 48–51 \\ \end{tabular}
\end{table} 
Table 1: Estimated values of cohesion and the angle of internal friction of the lunar regolith at different depths (Data from reference [23]). Bringing Equations (1) and (2) into Equation (3) yields the expressions for \(c\) and \(\varphi\) with respect to \(z\):  
\[\left\{ \begin{array}{c}c=10^{2.93-\frac{3.69\mu+247.2}{z+18}}\\ \varphi=\arctan\left(3.1-\frac{2.53z+169.54}{z+18}\right)\end{array} \right. \tag{4}\]  
A schematic diagram of the slip surface of the anchoring area that was obtained according to [PERSON]'s failure mechanism is shown in Figure 8, in which Zone I is the active Rankine zone, Zone II is the Prandtl zone, and Zone III is the passive Rankine zone. According to [PERSON]'s theory, the ultimate bearing capacity of the hole wall can be obtained as:  
\[p_{u}=cN_{c}+qdN_{q} \tag{5}\]  
where \(N_{c}=\cot\varphi\left[e^{\pi\tan\varphi}\tan^{2}\left(\frac{\pi}{4}+\frac{ \varphi}{2}\right)-1\right]\), \(N_{q}=e^{\pi\tan\varphi}\tan^{2}\left(\frac{\pi}{4}+\frac{\varphi}{2}\right)\), and \(q\) is the uniform load in the passive Rankine zone, which can be considered as 0. Since the anchor module is fixed in position by expanding the anchor plates, there are gaps after unfolding. Additionally, each anchor plate can still form a cylinder-like shape after the anchor module is unfolded.