# A Worm-Inspired Robot Flexibly Steering on Horizontal and Vertical Surfaces
## 3 Motion Process Analysis and Experiment on Horizontal Platform
### Crawling Process Analysis
In the figure, assuming that the deflection angle of the left spring (the angle between AC and MN) is \(\alpha_{1}\), and the deflection angle of right spring (the angle between BD and the cylinder axis direction of Segment 1 (MN) is \(\beta_{1}\), then the torque generated by the spring force is:  
\[MN=NO=d \tag{2}\]2  
Assuming that the tendon on the right side has a pulling force of \(F_{t1}\) on Segment 1, the torque generated is:  
\[MN=NO=d \tag{2}\]3  
According to the torque balance and simultaneous use of equations of (12) and (13), we can get:  
\[MN=NO=d \tag{2}\]4  
\[MN=NO=d \tag{2}\]5  
\[MN=NO=d \tag{2}\]6  
Similarly, segments 3 and 4 can be seen as rotating around the P-point hinge. Assuming that the deflection angle of the left spring (the angle between EG and the cylinder of Segment 3 (OP)) is \(\alpha_{3}\), and the deflection angle of the right spring (the angle between FH and OP) is \(\beta_{3}\), then the torque generated by the spring force is:  
\[MN=NO=d \tag{2}\]7  
Assuming that the tendon on the right side has a pulling force of \(F_{t3}\) on Segment 4, the torque generated is:  
\[MN=NO=d \tag{2}\]8According to the torque balance and simultaneous use of Equations (17) and (18), we can get:  
\[MN=NO=d \tag{2}\]9  
\[JA=BK=d^{\prime} \tag{3}\]0  
\[JA=BK=d^{\prime} \tag{3}\]1  
The experiment found that during the movement in Phase 1, due to the large tension in the tendon, it rubs against the pulleys at the corner, and therefore, there is a large resistance. As a result, the actual pulling force of the tendon acting at point B is greater than that at the point H, imposing a greater impact on the actual \(\theta_{1}\) and \(\theta_{3}\). Since the deflection angle of the tendon is circular, the Euler-Eytelwein equation \(T_{\text{load}}=T_{\text{hold}}\text{e}^{it\theta}\) can be used to calculate the resistance, where \(\mu\) is the friction coefficient between the tendon and the reel, and \(\theta\) is the angle difference of the tendon between point B and point H.  
By geometric relationship:  
\[JA=BK=d^{\prime} \tag{3}\]2  
Then, the tension of the tendon to the two points of H and B satisfies:  
\[JA=BK=d^{\prime} \tag{3}\]3  
According to the simultaneous use of Equations (16), (21), and (23), we can get the force constraint:  
\[ \begin{array}{c}\frac{(FH-x_{0})kr\cos(\beta_{3}+\theta_{3})-(EG-x_{0})kr\cos (\theta_{3}-\alpha_{3})}{r\cos(\beta_{3}+\theta_{3})}\\ =e^{-\mu(\theta_{1}+\theta_{2}+\beta_{1}-\beta_{3})\frac{(BD-x_{0})kr\cos( \beta_{1})-(AC-x_{0})kr\cos(\alpha_{1})}{r\cos(\beta_{1})}}\end{array}  \tag{24}\]  
where \(\alpha_{1}\), \(\alpha_{3}\), \(\beta_{1}\) and \(\beta_{3}\) meet:  
\[\alpha_{1}=tan^{-1}\frac{AM-CN\cos(\theta_{1})}{MN+CN\sin(\theta_{1})} \tag{25}\]  
\[\beta_{1}=tan^{-1}\frac{MB-ND\cos(\theta_{1})}{MN-ND\sin(\theta_{1})} \tag{26}\]  
\[\alpha_{3}=tan^{-1}\frac{EO-GP\cos(\theta_{3})}{OP+GP\sin(\theta_{3})} \tag{27}\]  
\[\beta_{3}=\tan^{-1}\frac{OF-PH\cos(\theta_{3})}{OP-PH\sin(\theta_{3})} \tag{28}\]  
In summary, according to the simultaneous use of Equations (11) and (24), the values of \(\theta_{1}\) and \(\theta_{3}\) at the end of Phase 1 can be obtained. Bringing the known related parameters \(D=143.8\) mm, \(d=118.8\) mm, \(d^{\prime}=80\) mm, \(r=82\) mm, \(x_{0}=85\) mm, and \(k=0.3\) N/mm, we calculated the variation of deflection angles \(\theta_{1}\) and \(\theta_{3}\) with different \(L_{2}\) values when \(\theta_{2}=20^{\circ}\) is in the initial phase, which is also experimentally measured, and the results are shown in Figure 9a. We also calculated the variation of deflection angles \(\theta_{1}\) and \(\theta_{3}\) with different \(\theta_{2}\) values when \(L_{2}=400\) mm in the initial phase, which is also experimentally measured, and the results are shown in Figure 9b. As is shown in Figure 9a,b, the theoretical value of \(\theta_{1}\) is smaller than \(\theta_{3}\) at any condition due to the large frictional resistance along the tendons; such a difference may lower the efficiency of the steering motion.