# Grasping Claws of Bionic Climbing Robot for Rough Wall Surface: Modeling and Analysis
## 3 Design and Analysis of the Miniature Claws
### Calculation of Supporting Shaft Size  
We should calculate the supporting shaft of the hook to determine the cross-sectional diameter of the hook shaft and satisfy the requirements of bending stress. The internal forces on the cross-sectional generally contain moment \(M\), shear force \(F_{S}\), and axial force \(F_{N}\) during the bending deformation of the hook. We used the section method to analyze the stresses on the curved hook. The curved bars were divided into two parts by the cross-sectional (radial section) \(m-m\), with an angle \(\theta\) with a hook top. Stresses on curved bar of hook using section method are illustrated in Figure 4.  
Stressing analysis of \(m-m\) section shows that  
\[\left\{ \begin{array}{l}F_{N}=F_{x}\cos(\theta-\beta)+F_{y}\sin(\theta-\beta )\\ F_{s}=-F_{x}\sin(\theta-\beta)+F_{y}\cos(\theta-\beta)\\ M=F_{x}R(\cos\beta-\cos(\theta-\beta))-F_{y}R(\sin\beta+\sin(\theta-\beta)) \end{array} \right. \tag{5}\]  
where \(F_{x}=F_{y}/n\) and \(F_{y}=F_{s}/n\). \(n\) are the hook number on a single claw. For the whole grasping claw, the claw with minimum hooks (such as two hooks arranged on the upper and lower limbs) is taken as an example to examine the interactions between the grasping claws  
Figure 4: Stresses on curved bar of the hook using section method. and the wall surface. Figure 5 shows the forces exerting on the hooked claw by the climbing models constructed in the previous section. \(L,L_{1},L_{2}\) and \(D\) are the dimensions of the claw. The direction of friction \(F_{f2}\) on the lower claw tip is directly related to the torque magnitude of the torsional spring. When \(\tau_{2}>\tau_{0}\), \(F_{f2}\) points downward along the tangent plane of the micro-protuberance. When \(\tau_{2}<\tau_{0}\), \(F_{f2}\) points upward along the tangent plane of the micro-protuberance. When \(\tau_{2}=\tau_{0}\), \(F_{f2}=0\). In this study, \(\tau_{0}\) is the critical value of the torque of torsional spring. The angles of the upper and lower micro-protuberances are set to be equal to simplify the problem; that is, \(\theta_{1}=\theta_{2}=\theta\), and \(F_{f1}\cos\theta>N_{1}\sin\theta\) for the upper claw. Thus, \(F_{f1}>N_{1}\tan\theta\). Therefore, the angles of micro-protuberances must be smaller than the self-locking angle between the claw tip and the micro-protuberance, i.e., \(\theta<\arctan(\mu)\). Otherwise, the upper claw slides away and does not grasp the micro-protuberances steadily. In Figure 5, \(N_{1}\) and \(F_{f1}\) are the support force and friction exerted on the upper claw tip by the micro-protuberance, respectively; \(F_{s1}\) and \(F_{v1}\) are the tangential and normal forces along the wall surface of the support force and friction exerted on the upper claw tip, respectively; \(N_{2}\) and \(F_{f2}\) are the support force and friction exerted by the micro-protuberance on the lower claw tip, respectively; and, \(F_{s2}\) and \(F_{v2}\) are the tangential and normal forces along the wall surface of the support force and friction exerted on the lower claw tip, respectively. \(C\) is the mass center of the whole hooked claw; \(\tau_{1}\) and \(\tau_{2}\) are the torques of torsional spring for the upper and lower hooked claws, respectively; \(\theta_{1}\) and \(\theta_{2}\) are the angles of the upper and lower micro-protuberances, respectively; and, \(G\) is the gravity of the claw. Therefore, the normal and tangential forces of the hook along the wall surface are directly correlated with the torque of the coiling spring. The following equations are obtained, according to Equations (1)-(5):  
\[\sigma_{\max}=\frac{M_{\max}}{W}=\frac{F_{x}R(\cos\beta-1)-F_{y}R\sin\beta}{ \frac{\pi}{32}d^{3}}=\frac{F_{v1}R(\cos\beta-1)-F_{s1}R\sin\beta}{\frac{\pi }{32}d^{3}}\leq[\sigma], \tag{6}\]  
In Equation (6), \(R=1.5\) mm, \(\beta=80^{\circ}\), and \(n=3\). Figure 6 shows the relationship between the maximum bending stress of the hook and the hook shaft diameter. The hook was made of 45 steel and was subjected to modulation processing.