# A Worm-Inspired Robot Flexibly Steering on Horizontal and Vertical Surfaces
## 3 Motion Process Analysis and Experiment on Horizontal Platform
### Crawling Process Analysis
The experiment results are basically in accord with the theory. However, the difference between \(\theta_{1}\) and \(\theta_{3}\) shown in the experiments seems to be larger than theoretical expectation, which might be caused by the friction on the bearings and the elasticity of the pipelines. As \(\theta_{2}\) is known, and \(\theta_{1}\) and \(\theta_{3}\) can be solved, we can further solve the length of the left tendon \(L_{1}\). As described above, at this time, the left tendon is in a slack state, and the internal tension is 0. In order to satisfy this requirement, the length of the left tendon must not be less than a geometric minimum \(L_{\text{1 min}}\):  
\[L_{1}\geq L_{\text{1 min}}=GI+AC+CE+EG \tag{29}\]  
In Phase 2, segments 1, 3, and 4 remain fixed, and as shown in Figure 10a, point C, D, and N on Segment 2 advance to the positions of the C\({}^{\prime}\), D\({}^{\prime}\), and N\({}^{\prime}\), respectively. At this time, the length of MN\({}^{\prime}\) (the cylinder length of Segment 1) is d, the length of NN\({}^{\prime}\) is D-d, and the deflection angle of Segment 1 becomes \(\theta_{1}{}^{\prime}\). From the geometric relationship, we can get:  
\[ON^{\prime}=\sqrt{ON^{2}+NN^{\prime}{}^{2}+2 ON\cdot NN^{\prime}\cdot\cos( \theta_{1})} \tag{30}\]  
Figure 10: **(a)** A simplified geometric model of the robot at the end of Phase 2, where ON is the cylinder of Segment 2; **(b)** A simplified geometric model of the robot at the end of Phase 3, where PO is the cylinder of Segment 3. Figure 9: Theoretical relationship and experimental results between the values of \(\theta_{1}\) and \(\theta_{3}\) and the values of \(\theta_{2}\) and \(L_{2}\) at the end of Phase 1. From the figure, we can see that \(\theta_{1}\) is always less than \(\theta_{3}\), because there is a friction force on the tendon. **(a)** When \(\theta_{2}\) is 20\({}^{o}\), the relationship between the values of \(\theta_{1}\) and \(\theta_{3}\) and the value of \(L_{2}\); **(b)** when \(L_{2}\) is 400 mm, the relationship between the values of \(\theta_{1}\) and \(\theta_{3}\) and the value of \(\theta_{2}\). \[\frac{ON}{\sin({\theta_{1}}^{\prime})}=\frac{ON^{\prime}}{\sin(\theta_{1})}=\frac{NN ^{\prime}}{\sin(\varphi)} \tag{31}\]  
According to the simultaneous use of Equations (29) and (30), the following is obtained:  
\[\theta_{1}^{\prime}=\sin^{-1}\!\left(\frac{ON\cdot\sin(\theta_{1})}{\sqrt{ON^{ 2}+{NN^{\prime}}^{2}+2 ON\cdot NN^{\prime}\cdot\cos(\theta_{1})}}\right) \tag{32}\]  
\[\beta_{1}=tan^{-1}\frac{MB-ND\cos(\theta_{1})}{MN-ND\sin(\theta_{1})} \tag{26}\]0  
In Phase 3, segments 1, 2, and 4 remain fixed, and as shown in Figure 10b, points E, F, and O on Segment 3 advance to the positions of the E\({}^{\prime}\), F\({}^{\prime}\), and O\({}^{\prime}\), respectively. At this time, the length of O\({}^{\prime}\)N\({}^{\prime}\) is d, and the deflection angle of Segment 2 becomes \(\theta_{2}^{\prime}\), as shown in Figure 10b. From the geometric relationship, we can get:  
\[\beta_{1}=tan^{-1}\frac{MB-ND\cos(\theta_{1})}{MN-ND\sin(\theta_{1})} \tag{26}\]1  
\[\beta_{1}=tan^{-1}\frac{MB-ND\cos(\theta_{1})}{MN-ND\sin(\theta_{1})} \tag{26}\]2  
\[\beta_{1}=tan^{-1}\frac{MB-ND\cos(\theta_{1})}{MN-ND\sin(\theta_{1})} \tag{26}\]3  
According to simultaneous use of Equations (35) and (34), the value of \(\theta_{2}^{\prime}\) at the end of Phase 3 is obtained:  
\[\beta_{1}=tan^{-1}\frac{MB-ND\cos(\theta_{1})}{MN-ND\sin(\theta_{1})} \tag{26}\]4  
At this point, a motion cycle ends, and the robot returns to the state at the beginning of step 1. Based on the above Equations (11), (24), and (37), the values of \(\theta_{1}\), \(\theta_{2}\), and \(\theta_{3}\) after any of the periodic motions can be solved. The relationship between the three values and the number of cycles T is shown in Figure 11a. It can be seen that regardless of the initial state, after several cycles of the motion, \(\theta_{1}\), \(\theta_{2}\), and \(\theta_{3}\) will tend to a fixed value.