## 2 Analysis of Bending Shape Based on Tendon Insertion Paths and Applied Tensions  
### Mathematical Modeling  
The purpose of this mathematical modeling is to analyze how the bending shape of a hyper-redundant manipulator with rolling joints changes according to the arbitrarily inserted tendons and the applied tensions. Figure 2 shows the 1-DOF overtube with \(n\) rolling joints bending due to the combined tensions applied to the inserted tendons. In this study, to simplify the problem definition and address practical aspects, a total of three tendons are analyzed: two main driving tendons inserted along parallel paths for constant curvature bending and one shaping tendon inserted along an arbitrary path (Figure 3). Each joint has a rotational center denoted as \(O_{i}\). \(T_{m1}\) and \(T_{m2}\) are the tensions applied to the two main driving tendons, while \(T_{s}\) is the tension applied to the shaping tendon. \(\theta_{n-1}\) is the relative angle between \(joint_{n-1}\) and \(joint_{n}\), and the angle of the overtube end tip after bending \(\theta\) can be summarized as follows:  
\[\theta=\sum\
olimits_{i=1}^{n-1}\theta_{i} \tag{1}\]  
The modeling method for the overtube system used in this study was inspired by my previous work [15]. The overall physical characteristics of a tendon-driven system, such as an overtube, are primarily determined by the tendon inserted inside the overtube. Building on previous studies, which show that the work carried out by external forces is stored as energy in the tendon inside the overtube and that the overtube deflects in the direction where total potential energy is minimized, this study analyzes the bending shape of the overtube. It assumes that the overtube bends in the direction where the total potential energy, resulting from the work conducted on each joint due to the applied tendon tension, is minimized. \(W_{m1}\), \(W_{m2}\), and \(W_{s}\) represent the work carried out by the rotation of the joints due to \(T_{m1}\), \(T_{m2}\), and \(T_{s}\), respectively, and can be defined as follows:  
\[W_{m1}=\sum\
olimits_{i=1}^{n-1}\int r_{m1,\,i}T_{m1}\cdot d\theta_{i} \tag{2}\]  
\[W_{m2}=\sum\
olimits_{i=1}^{n-1}\int r_{m2,\,i}T_{m2}\cdot d\theta_{i} \tag{3}\]  
\[W_{s}=\sum\
olimits_{i=1}^{n-1}\int r_{s,\,i}T_{s}\cdot d\theta_{i} \tag{4}\]Here, \(r_{m1,i}\), \(r_{m2,i}\), and \(r_{s,i}\) are the moment arms of each tendon between \(Joint_{i}\) and \(Joint_{i+1}\). \(W_{m1}\), \(W_{m2}\), and \(W_{s}\) are calculated as the dot products for simplicity. Considering that the tension loss due to friction between the tendons and the internal holes of the joints is negligible, this study assumes that the tension applied at each joint is equal to the input tension (Equations (2)-(4)). Figure 3 shows the moment arm of each tendon when the relative angle between \(Joint_{i}\) and \(Joint_{i+1}\) is \(\theta_{i}\). All tendons inserted into each joint can be classified into two cases: those that assist with bending and those that resist it. The moment arm for each case can be calculated as follows:  
\[r_{m1,i}=\alpha_{m1,i}-\beta_{m1,i}=d_{m1}\cos\left(\frac{\theta_{i}}{2} \right)-(R-B)\sin\left(\frac{\theta_{i}}{2}\right) \tag{5}\]  
\[r_{m2,i}=\alpha_{m2,i}+\beta_{m2,i}= R\tan\left(\frac{\theta_{i}}{2}\right)+\frac{d_{m2}}{\cos\left( \frac{\theta_{i}}{2}\right)}-\frac{\Delta_{m2,i}}{2}\tan\left(\frac{\theta_{i} }{2}\right) \tag{6}\]  
\[\Delta_{m2,i}=2B+2(R-B)\left(1-\cos\left(\frac{\theta_{i}}{2}\right)\right)+2 d\sin\left(\frac{\theta_{i}}{2}\right) \tag{7}\]  
Here, \(R\) represents the radius of the rolling surface, \(B\) is the depth of the tendon hole from the top of the rolling surface, and \(d_{m1}\), \(d_{m2}\), \(d_{s,i}\) are the distances between each tendon and the central axis. Unlike the paths \(d_{m1}\) and \(d_{m2}\) of the main driving tendon for constant curvature bending, the path \(d_{s,i}\) of the shaping tendon, which determines various shapes, can vary for each joint.