## 2 Modeling of Wall-Climbing Robot
### Adhesion Part Analysis  
Figure 5a is the cicada, which has the ability to climb trees by using its biometry claws. Figure 5b represents the detailed image of the cicada's claw. From this image, we can see a special bifurcated spine on the end of the claw. The cicada can climb trees by using these bifurcated spines. This paper investigated the use of bifurcated spines in a proposed wall-climbing robot for climbing on rough surfaces. This paper presents a bionic spine design that was inspired by this special hook with an angle, as shown in Figure 5c. The contact angle \(\alpha\) is  
\[\alpha=arcsin(\frac{r+R-h}{r+R})=arcsin(1-\frac{h}{r+R}) \tag{29}\]  
Figure 5e expresses the equivalent force diagram of the bionic hook adhering to the rough surface. The load distribution force angle is \(\beta\) (\(\beta_{1}\), \(\beta_{2}\)), the height of the particle is \(h,f\left(f_{1},f_{2}\right)\) is the friction force between hook and surface, \(F\left(F_{1},F_{2}\right)\) is the comprehensive capsizing force of the hook by the wall and the weight of the hook, and \(N\left(N_{1},\,N_{2}\right)\) is the support force on the hook by the wall. To adhere to the protruding particles, the mechanics must satisfy:  
\[\left\{ \begin{array}{c}\sum\limits_{i=1}^{n}f_{i}sin(\gamma_{i}/2)=\sum \limits_{i=1}^{n}\mu_{i}N_{i}cos(\gamma_{i}/2)\\ =\sum\limits_{i=1}^{n}F_{i}sin(\alpha_{i}+\beta_{i})\\ \sum\limits_{i=1}^{n}N_{i}=\sum\limits_{i=1}^{n}F_{i}cos(\alpha_{i}+\beta_{i} )\end{array} \right. \tag{30}\]  
Figure 4: **(a)** The phase images of the gear 1 and gear 2. **(b)** The Poincare maps of the gear 1 and gear 2. **(c)** The phase images of the gear 1 and gear 3. **(d)** The Poincare maps of the gear 1 and gear 3. In terms of a single hook,  
\[\beta<arctan(2\mu cos(\gamma/2))-\alpha \tag{31}\]  
Here, \(\mu\) is the friction coefficient between the hook and the surface. Integrating with Equations (29)-(31), \(\beta\) is written as  
\[\beta<arctan(2\mu cos(\gamma/2))-arcsin(1-\frac{h}{r+R}) \tag{32}\]  
Figure 5: (**a**) A cicada on the branch of the tree, (**b**) a leg of a cicada, (**c**) the bionic bifurcated spines inspired by the cicada’s claws, (**d**) equivalent force diagram of the bionic hook. (**e**) The equivalent force diagram of the bionic hook adhering to the rough surface. The hooks' adherence on the rough surface must satisfy the condition that \(\beta\) is greater than zero. This means that \(\alpha\) is limited and is less than \(arctan(\mu cos(\gamma/2))\), from Equation (31). Here, \(\alpha\) is defined by the diameter and the height of the particle and \(\alpha\) is the angle of the hook to change the adhere characteristics of the hooks. These parameters are investigated by the following figures. Figure 6a represents the relations of the pressure angle (\(\alpha\)) and the load distribution angle (\(\beta\)) with various hooks' angles \(\gamma\). In this figure, the load distribution angle (\(\beta\)) decreases as the hook angles \(\gamma\) decrease. It means that decreasing the hook angles \(\gamma\) improves the adherence property. The relation between the size of hooks and the particles on the contact surface is provided in Figure 6b. In this figure, the smaller the radius of the hooks' angle, the bigger the load distribution angle (\(\beta\)), as the height and the radius of the particles remain unchanged. Hence, decreasing the size of the hook's end can develop the adherence property. ### Adhesion Part Analysis  
As shown in Figure 7, the tangential adhesive force of the various hook angles of the bionic claws were examined on the adhesive test platform. A rough surface was put on the platform for testing. Figure 7a represents the testing set for adhesive materials. The testing device and the telescopic guideway are on the right; the testing materials are on the left.