## 2 Analysis of Bending Shape Based on Tendon Insertion Paths and Applied Tensions
### Simulation-Based Bending Shape Analysis  
The main purposes of this section are as follows: (1) the first purpose is to analyze how the bending shape of the overtube is affected by the arbitrarily designed shaping tendon; (2) the second purpose is to validate the proposed mathematical modeling equations by comparing the results with those from physical experiments in Section 2.3. Three different types of overtubes were designed in this analysis (Figure 4). To isolate the effect of the shaping tendons, all the main driving tendons were configured to follow the same path, while only the shaping tendons were designed with varying insertion paths. As described in Section 2.1, the parameter that determines the path of the main driving tendon is \(d\) (\(d_{m1}\) and \(d_{m2}\)), while the parameter that determines the path of the shaping tendon is \(d_{s,i}\). The parameters that determine the tendon paths and overall shape of the joints for each type of overtube are summarized in Table 1. Among the parameters that determine the path of the shaping tendon, negative values indicate that the path is generated below the centerline. Simulations were performed for three different tension cases for each overtube; in all cases, the initial angle of the overtube end tip was set to \(0^{\circ}\). The magnitude of the tension applied to each tendon in each case is as follows: (Case 1) \(T_{m1}=5\) N, \(T_{m2}=5\) N, \(T_{s}=10\) N; (Case 2) \(T_{m1}=5\) N, \(T_{m2}=5\) N, \(T_{s}=15\) N; (Case 3) \(T_{m1}=5\) N, \(T_{m2}=5\) N, \(T_{s}=20\) N. The tension values applied to the tendons in each case were determined by considering the maximum torque the motor can generate, the stiffness of the joints, and the rupture tension of the tendons inserted inside the overtube. All simulations were performed in MATLAB, R2023b.  
Figure 3: The arrangement of tendons within the joint and the moment arms of each tendon when the rotation angle between \(joint_{t}\) and \(joint_{t+1}\) is \(\theta_{i}\).  
Figure 5 shows the simulation results of the bending shape achieved by applying predefined tensions to each tendon. The rotation centers of the bent joints \(O_{i}\) are represented by circles of different colors, and the results in Figure 5 are obtained by connecting these circles with dashed lines. The angle of rotation of the overtube joints and the angle of the end tip for each case are summarized in Table 2.  
\begin{table}
\begin{tabular}{l c c c c c c c c} \hline  & \(d\) & \(d_{s,1}\) & \(d_{s,2}\) & \(d_{s,3}\) & \(d_{s,4}\) & \(R\) (mm) & \(B\) (mm) & \(l_{e}\) (mm) \\  & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & \(R\) (mm) & \(B\) (mm) & \(l_{e}\) (mm) \\ \hline Type-A & & 7 & 7 & –7 & –7 & & & \\ Type-B & 11 & 3 & 3 & 3 & –7 & 15 & 4.80 & 40 \\ Type-C & & 7 & –7 & –7 & 7 & & & \\ \hline \end{tabular}
\end{table}
Table 1: The parameters that determine the tendon paths and overall shape of the joints for each type of overtube.  
Figure 4: (**a–c**) Overall mechanical design of three different types of overtubes for analyzing the effect of the shaping tendon on the bending shape; (**d**) the arrangement of all tendons.