## 3 Comprehensive Discussion  
This study proposed a conceptual method for adding a shaping tendon to determine the bending shape and also mathematically analyzed the effect of this shaping tendon, inserted along an arbitrary path in addition to the main driving tendons for constant  
Figure 8: The actual shape of the Type-A overtube when different tensions are applied to the tendons inserted in (**a**) Case 1, (**b**) Case 2, and (**c**) Case 3. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \(O_{2}\) **(mm)** & \(O_{3}\) **(mm)** & \(O_{4}\) **(mm)** & \(O_{5}\) **(mm)** & **End Tip (mm)** & \(\theta\) **(°)** \\ \hline Case 1 & 2.75 & 12.10 & 20.39 & 24.89 & 23.51 & 2.86 \\ Case 2 & 3.03 & 11.40 & 21.78 & 24.74 & 23.15 & 4.29 \\ Case 3 & 3.70 & 13.51 & 20.41 & 24.63 & 23.02 & 5.71 \\ Average & 3.16 & 12.34 & 20.86 & 24.75 & 23.23 & 4.29 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Quantitative comparison of errors between estimated and measured rotational centers \(O_{i}\) and end tip angle \(\theta\) of the Type-A overtube. Figure 7: The actual shape of the Type-A overtube when different tensions are applied to the tendons inserted in (**a**) Case 1, (**b**) Case 2, and (**c**) Case 3.  
curvature bending on the bending shape of the overtube. The proposed method allows for achieving a specific bending shape by simply adding tendons without the need for designing additional segments. Moreover, the bending shape can be estimated based on tendon path and applied tension using the proposed mathematical modeling equations. In Figure 8, the overtubes represented by blue dotted lines are all estimated results based on the mathematical modeling equations from Section 2.1. A key point to note in these equations in the relationship between the parameters that determine the tendon path (\(d_{m1}\), \(d_{m2}\), and \(d_{s,i}\)) and the rotation angles \(\theta_{i}\) of each joint (Table 2). In examining Equations (2)-(4), assuming that the tension applied to the tendons remains constant, the factor that affects the work carried out by the rotation of the joints is the moment arm value of the tendons at each joint. The moment arm values of the tendons at each joint are determined by the design parameters of the joints (Equations (5) and (6)). To effectively change the moment arm values without changing the overall shape of the joint, the parameters representing the tendon path are the main factors (\(d_{m1}\), \(d_{m2}\), and \(d_{s,i}\)). Note that, although it is omitted in Section 2.1, \(r_{s,i}=d_{s,i}\,\cos\left(\frac{\theta_{i}}{2}\right)-(R-B)\sin\left(\frac{ \theta_{i}}{2}\right)\). In other words, since the angle of rotation for each joint varies with changes in the tendon path, inserting the shaping tendon along different paths enables the overtube to achieve a variety of bending shapes. In particular, in the simulation results of Section 2.2, it is important to focus on the relations between the angles of rotation of the joints and the path of the shaping tendon rather than the absolute value of the rotation angles. To focus exclusively on analyzing the effect of the shaping tendon, the initial angle of the overtube was set to \(0^{\circ}\), and the same tension was applied to the two main driving tendons. Therefore, the results summarized in Table 2 are dominated by the parameter \(d_{s,i}\), which represents the path of the shaping tendon. In conclusion, the simulation results summarized in Table 2 indicate that the amount of rotation of each joint is closely related to the ratio of moment arm lengths, which corresponds to the path of the shaping tendon. Therefore, after defining the desired bending shape of the continuum manipulator in terms of the rotation angle ratios of each joint, it is advisable for this ratio to be considered in the moment arm lengths of the shaping tendon when designing the tendon paths in the joints. The system, consisting of rolling joints and tendons, was analyzed from an energy perspective.