# 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  
The impact module converts the kinetic energy of the motor rotation into impact energy by the spring energy storage mechanism. To analyze the dynamics of the impact module, the single impact process can be divided into the energy storage phase and the release phase. For the energy storage phase, the simplified cam curve can be expanded along a straight line, as shown in Figure 10, where the structural parameters are the cam radius \(R_{c}\), spring preload length \(l_{0}\), cam stroke height \(h\), and drive wheel radius \(R_{d}\). For the release phase, a certain length of the gentle section without loading is set on the cam to ensure the effective release of the impact energy and that the proportion of the loaded segment is \(\alpha\). Among them, \(\alpha\), \(l_{0}\), and \(h\) has a large impact on the impact work, so these parameters are used as optimization variables. For coil springs, based on the optimized design of cylindrical compression springs, the spring wire diameter \(d\) and the effective number of turns \(n\) are selected as optimization variables when the maximum energy storage in the restricted space is the optimization objective. The rest of the variables, such as the  
Figure 9: Maximum drilling pressure and rotational torque in relation to depth. spring's mid-range \(D\) and spring shear modulus \(G_{s}\), are constants. Subsequently, the maximum impact energy as the objective function can be expressed as:  
\[f(\textbf{X})=\frac{1}{2}kh(h+2l_{0})+mg_{lunar}h+\frac{1}{2}m(\frac{\lambda h}{ \alpha})^{2} \tag{8}\]  
where \(k=\frac{G_{s}d^{4}}{8D^{3}n}\) is the spring stiffness;  
\(g_{lunar}\) is the lunar gravitational acceleration;  
\(\lambda\) is the impact frequency;  
\(m\) is the mass of the impact hammer. Based on spring design guidelines, the effect of the curvature factor \(K\) should be considered when the spring is subjected to a dynamic load, which can be expressed as:  
\[K=\frac{4C-1}{4C+4}+\frac{0.615}{C} \tag{9}\]  
where \(C=\frac{D}{d}\). Based on the effect of the curvature factor, \(d\) should satisfy the following constraints:  
\[g_{1}=\sqrt[3]{\frac{8 KDF}{\pi[\tau]}}-d\leq 0 \tag{10}\]  
The allowable shear stress \([\tau]\) decreases with the increase in the spring wire diameter; however, it is defined as a constant to simplify the optimization process. For installation space constraints, on the one hand, it is necessary to consider that the spring deformation needs to be smaller than the theoretical full deformation. On the other hand, it is necessary to consider that the height after pre-pressing is less than the installation space of the impact module H\({}_{0}\). Accordingly, the installation dimensions are constrained as follows:  
\[\left\{ \begin{array}{l}g_{2}=l_{0}+h-\frac{D\pi n[\tau]}{KG_{s}d}\leq 0\\ g_{3}=\frac{D\pi n[\tau]}{KG_{s}d}+(n+1.5)d-h-H_{0}\leq 0\end{array} \right. \tag{11}\]  
Figure 10: Simplified cam curve expanded along a straight line. For the energy storage phase, an analysis of Figure 10 yields the constrained relationship between the motor torque and the structure parameters during cam energy storage. \[g_{4}=[G_{ham}+k(l_{0}+l)]\frac{h}{2\pi\alpha}-M_{d}\leq 0 \tag{12}\]  
where \(M_{d}\) is the nominal torque of the motor;  
\(l\) is the current axial displacement of the roller, with the maximum value of \(h\);  
\(G_{ham}\) is the gravity of the impact hammer. For the release phase, if the friction between the impact hammer and the guide wall is neglected, the equation of state for the release phase of the impact hammer is:  
\[\ddot{l}=-\bigg{(}g_{lum}+\frac{k(l_{0}+l)}{m}\bigg{)} \tag{13}\]  
The analysis of the motion of the impact hammer provides the following initial conditions for the motion of the impact hammer at the cam notch:  
\[\left\{ \begin{array}{ll}l(0)=h\\ \dot{l}(0)=\frac{\lambda h}{\alpha}\\ \ddot{l}(0)=-\Big{(}g_{lum}+\frac{k(l_{0}+h)}{m}\Big{)}\end{array} \right.