# Design and Dynamic Analysis of the Wire-Line Coring Robot for Deep Lunar Rocks
## 3 Dynamic Analysis of the WSC Robot
### Dynamic Analysis of the Impact Module
\tag{14}\]  
Substituting the initial conditions into Equation (13) for the solution yields the time from the release of the impact hammer to the return of the initial height as:  
\[t_{1}=\sqrt{\frac{m}{k}}\bigg{[}\arcsin\bigg{(}\frac{mg+kl_{0}}{k\sqrt{a^{2}+ b^{2}}}\bigg{)}-\arccos\bigg{(}\frac{b}{\sqrt{a^{2}+b^{2}}}\bigg{)}\bigg{]} \tag{15}\]  
where \(a=\frac{m\sigma_{lum}}{k}+l_{0}+h\),  
\(b=\frac{\sqrt{\frac{m}{k}}\lambda h}{\alpha}\). Assuming that the impact energy takes time to be fully released and it is set to \(t_{2}\), the constraint that ensures that the impact energy is fully released before loading is:  
\[g_{5}=2\pi R_{c}\lambda(t_{1}+t_{2})+2R_{d}-2\pi R_{c}(1-\alpha)\leq 0 \tag{16}\]  
According to Equations (8), (10)-(12), (16), and the inherent boundaries of each parameter, the numerical model of the constraint problem can be obtained as:  
\[ \begin{array}{ll}\max&f(\mathbf{X})=\frac{1}{2}kh(h+2l_{0})+mg_{lum}h+ \frac{1}{2}m\big{(}\frac{\lambda h}{\alpha}\big{)}^{2}\\ s.t.&g_{1}=\sqrt[3]{\frac{8 KDF}{\pi[\gamma]}}-d\leq 0\\ &g_{2}=l_{0}+h-\frac{D\pi m[\gamma]}{KG_{d}d}\leq 0\\ &g_{3}=\frac{D\pi m[\gamma]}{KG_{d}d}+(n+1.5)d-h-\mathrm{H}_{0}\leq 0\\ &g_{4}=[G_{ham}+k(l_{0}+l)]\frac{h}{2\pi\alpha}-M_{d}\leq 0\\ &g_{5}=2\pi R_{c}\lambda(t_{1}+t_{2})+2R_{d}-2\pi R_{c}(1-\alpha)\leq 0\\ &g_{6}=\alpha-1\leq 0\\ &g_{7}=-\alpha\leq 0\\ &g_{8}=-l_{0}\leq 0\\ &g_{9}=-h\leq 0\\ &g_{10}=h-50\leq 0\\ &g_{11}=0.05-d\leq 0\\ &g_{12}=d-1.5\leq 0\\ &g_{13}=1.5-n\leq 0\\ &g_{14}=n-200\leq 0\end{array}  \tag{17}\]Since the length of the WSC robot was limited by the mission objectives, the working space reserved for the impact module was only 20 mm after the necessary equipment was designed. Additionally, the available installation diameter of the WSC robot was only 38 mm. Considering the working space and working environment of the WSC robot, the parameters were initially determined as follows: the nominal torque and rotation rate of the impact motor were 0.84 Nm and 600 rpm; the energy storage spring was made of oil-quenched and annealed spring steel wire with a shear modulus of 78.5 GPa and tensile strength of 1540 MPa; the mass of the impact hammer was predicted to be 190 g based on its 3D model; \(t_{2}\) was set to 1 ms. If the energy consumption during impact loading, such as friction, is neglected, the global optimal solution of Equation (17) can be obtained by combining the optimization toolbox with the global optimization algorithm in MATLAB. The optimization results after rounding are shown in Table 2. The optimized theoretical single maximum impact energy can reach 3.7 J. The finite element simulation of the impact process on lunar rocks was carried out in LS-DYNA with optimized structural parameters to verify the performance of the impact module. The lunar rocks model used the modified HJC constitutive model in LS-DYNA, and the parameter settings are shown in Table 3. Both the coring tube and the impact hammer use pre-defined isotropic elastic models of aluminum alloy and steel, respectively. The impact stress wave generated by the impact hammer is transmitted to the lunar rocks through the coring tube, and the stress wave transmission simulation effect is shown in Figure 11. The bit module is driven by the impact action to intrude into the lunar rocks and form an impact crushing pit at the contact point. At the same time, the lunar rocks around the bit module generate stress concentration in a short period of time, which leads to cracks inside. During the simulation, the impact stress wave is loaded to the lunar rocks at 0.16 ms, which causes the lunar rock to produce a crushing pit, as shown in Figure 12a. By this time, part of the impact energy in the coring tube has been consumed; however, another part of the impact energy still exists in the coring tube in the form of reflected stress waves. The reflected stress waves are repeatedly returned in the coring tube to enlarge the impact-crushing pit.