## 2 Analysis of Bending Shape Based on Tendon Insertion Paths and Applied Tensions
### Physical Experiment Design  
The primary objective of this physical experiment is to validate the effectiveness of the proposed mathematical modeling. Considering the main goal of this experiment, a comparison was made between the actual measured bending shape of the designed Type-A overtube and the estimated bending shape obtained from Section 2.2.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Types** & **Cases** & \(\theta_{1}\) (\({}^{\circ}\)) & \(\theta_{2}\) (\({}^{\circ}\)) & \(\theta_{3}\) (\({}^{\circ}\)) & \(\theta_{4}\) (\({}^{\circ}\)) & \(\theta\) (\({}^{\circ}\)) \\ \hline \multirow{4}{*}{A} & \(1\) & \(37.91\) & \(37.91\) & \(-37.91\) & \(-37.91\) & \(0\) \\  & \(2\) & \(44.80\) & \(44.80\) & \(-44.80\) & \(-44.80\) & \(0\) \\  & \(3\) & \(49.21\) & \(49.21\) & \(-49.21\) & \(-49.21\) & \(0\) \\ \hline \multirow{4}{*}{B} & \(1\) & \(16.75\) & \(16.75\) & \(16.75\) & \(-37.91\) & \(12.33\) \\  & \(2\) & \(20.04\) & \(20.04\) & \(20.04\) & \(-44.80\) & \(15.31\) \\  & \(3\) & \(22.21\) & \(22.21\) & \(22.21\) & \(-49.21\) & \(17.41\) \\ \hline \multirow{4}{*}{C} & \(1\) & \(37.91\) & \(-37.91\) & \(-37.91\) & \(37.91\) & \(0\) \\  & \(2\) & \(44.80\) & \(-44.80\) & \(-44.80\) & \(44.80\) & \(0\) \\ \cline{1-1}  & \(3\) & \(49.21\) & \(-49.21\) & \(-49.21\) & \(49.21\) & \(0\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: The angles of rotation of the overtube joints for each different case.  
Figure 5: The bending results estimated based on the mathematical modeling equations proposed in Section 2.1 for **(a)** Case 1, **(b)** Case 2, and **(c)** Case 3.  
Figure 6 provides an overview of the experimental setup environment. Six optical markers were attached to the rotational centers \(O_{i}\) of each joint and to the end tip. A 6-DOF optical tracking device (V120:Trio; OptiTrack, NaturalPoint Inc., Corvallis, OR, USA) was used to measure the overall bending shape of the overtube. Considering that the experiment aims to measure the bending shape of the overtube, the evaluation criteria were defined as the errors between the estimated and actual measured positions of the rotational centers \(O_{i}\) obtained from the mathematical modeling in Section 2.1, as well as the errors between the estimated and actual measured angle \(\theta\) of the overtube end tip. The error between the estimated and actual values of each rotational center was calculated using Euclidean distance and then averaged. Considering the minimum distance that the optical tracking device can measure between markers and the precision of the 3D printer (Form 3L, Formlabs Inc., Somerville, MA, USA) used to fabricate the joints, the joint parameters for the Type-A overtube were determined as follows: \(R=15\) mm, \(B=4.80\) mm, \(d=11\) mm, \(l_{e}=40\) mm, \(d_{s,1}=7\) mm, \(d_{s,2}=7\) mm, \(d_{s,3}=-7\) mm, and \(d_{s,4}=-7\) mm (Table 1).  
The physical experiments were performed in the same conditions for the three different cases outlined in Section 2.2. For each case, the experiment was repeated for a total of 10 trials. Considering the performance of the load cell used in this experiment, an initialtension of approximately 3 N was applied to all tendons to prevent slack in the tendons inserted into the overtube and to set the initial angle of the overtube. In this study, tension control for the tendons inside the overtube joints was achieved using servo motors (Dynamixel XM-540-W270-R: ROBOTIS Co., Ltd., Gangseo-gu, Seoul, Republic of Korea), load cells (333 FDX: KTOYO Co., Ltd., 289-7, Uijeongbusi, Gyeonggido, Republic of Korea), and a Proportional-Integral-Derivative (PID) controller (Figure 6).