## 3 Comprehensive Discussion
The work conducted by the rotation of each rolling joint under the tension applied to the tendon was defined, and the angles of rotation were estimated based on the principle of minimum total potential energy. The results estimated by the mathematical modeling equations were similar in behavior to those obtained from the actual physical experiments; however, errors still remained (Figure 8 and Table 3). The discrepancy between the physical experimental results and the estimated results from the mathematical modeling equations is likely due to friction caused by insufficient tolerance between the tendon inserted into the joint and the joint's hole. In this study, the prototype joint used in the physical experiments was manufactured with a 3D printer to enable preliminary validation of the proposed method. The use of a 3D printer limited the precision of the tendon paths and hole sizes within the joint, and the tendon, made of SUS material, experiences significant friction with the joint made of ABS material. However, in the design intended for actual clinical applications, the joints are made from metal and undergo precision machining, which prevents this type of friction. Therefore, the overtube system developed with metal is expected to yield actual bending results that are more closely aligned with the estimated results based on the mathematical modeling equations proposed in this study. The key to achieving more ideal bending results is to further subdivide the joints of the overtube through precision metal machining and ensure that the tendon path inside the joints has appropriate tolerances. However, if the tolerance is set too loosely, friction may be reduced, but the joints will move freely over the tendon, preventing the desired bending shape from being achieved. Therefore, it is crucial from a practical perspective to determine the appropriate tolerance through physical experiments. In general, in continuum manipulators like the overtube, thin and flexible tendons are used to ensure flexibility. However, unlike rigid pipes, flexible tendons experience elongation,which reduces the tension transmission efficiency. The decrease in tension transmission efficiency can also be a cause of errors, as shown in the experimental results of this study. Therefore, for the proposed method to be reliably utilized from a practical perspective, the tendon should be sufficiently pre-tensioned before being inserted into the robot, ensuring that it no longer elongates once integrated into the system. Considering the limitations of including the effects of friction in the energy-based modeling equations, future research will focus on improving simulation-based estimation performance using methods such as the Modelica language or FEA [27]. In this study, a prototype overtube with an outer diameter of 25 mm and a total length of 160 mm was used to primarily verify the validity of the proposed method. However, for the proposed method to be applied in practical gastrointestinal surgeries, a detailed mechanical design guideline that considers the geometric characteristics of commercial endoscopes is necessary. Considering the overall geometric characteristics of the commercial endoscopes used in the gastrointestinal tract, the diameter of the overtube should be 16 mm or less, the total length should be 100 mm, it should have two degrees of bending freedom, and retroflexion bending (maximum bending angle should be 180\({}^{\circ}\)) should be possible. Additionally, the radius of curvature when bent 180\({}^{\circ}\) should be approximately 24 mm. Since it will be used in the constrained and confined gastrointestinal tract, it should have a sufficiently small radius of curvature when retroflexion bending occurs. It is expected that more meaningful results can be achieved if design guidelines incorporating the proposed method and addressing clinical considerations are proposed.