## 2 Method  
### Design and Modeling for Bending Shape Prediction of Flat Spring Mechanism  
To manufacture the hyper-redundant manipulator, the segments forming each joint were designed. The specifications of the manipulator targeted in this study are as follows: a 2-DoF bending motion should be possible to realize intuitive movement; elastic fixtures should be placed in the middle of segments; there must be a hollow through which surgical tools can pass. We achieved an optimal design to reduce the diameter of the manipulator for application in MIS. Figure 2a shows one joint of the manipulator, comprising an upper and a lower segment. The joint of the manipulator consists of a protrusion in the upper segment and an indentation in the lower segment. The indentation and protrusion were twisted by 90\({}^{\circ}\). They allowed the manipulator to have two bending directions. A flat spring with a special shape was designed and adopted for the elastic fixture placed at each joint of the manipulator. When placing the elastic fixture on the joint of the manipulator, the position and fixation method can be tried diversely. A typical coil spring or a superelastic rod can be used [28; 29]. However, the coil spring arrangement would not help minimize the manipulator's diameter. It would cause friction by colliding with the cable, which became an obstacle to the operation. Even if the rod-shaped elastic fixtures are suitable enough with thin and simple design, sometimes cause hardship on fixing methods or limit degrees of freedom. In this study, we proposed a special shape of flat spring to meet the aforementioned design requirements. Figure 2b shows a flat spring, which functioned as an elastic fixture between segments. It has a hollow in the middle and does not interfere with cable movement and two bending movements. The wings on both sides maintained balance when the segments were assembled serially. Figure 2c shows the assembly of the segment and the flat spring. Figure 1: Examples of bending motion of a hyper-redundant manipulator with non-uniform curvature: (**a**) The curvature of the distal part is higher as it bends; (**b**)The curvature of the proximal part is higher. The left panel of Figure 2d shows the initial position when two segments and a flat spring were assembled, and the right panel shows the state when it was bent. Looking at the bending state, if we know the length \(x_{Li}\), of the cable after being pulled, we can find the bending angle \(\theta_{i}\), as follows:  
\[\theta_{i}\ =\ 2\cdot sin^{-1}(\frac{x_{Li}}{2y}),\quad y\ =\ \sqrt{B^{2}+(\frac{w}{2})^{2}}. \tag{1}\]  
where \(y\) denotes the length of the hypotenuse cut from the center of rotation of the joint, to the outside of the segment. \(B\) denotes the height from the start of the inclined plane to the axis of rotation. Additionally, \(w\) denotes the width of the segment. The angle of inclined plane could be obtained from \(B\) and \(w\). Fundamentally, when an external force is applied to an elastic object to move the object, the work done at that time is stored as energy in the object. The theory, known as Hooke's law, can be applied to all elastic bodies, including helical coil springs. This theory can also be applied to the case of the manipulator. Most manipulators fix one end-part of the cable at the end of the manipulator and pull and release the cable for a bending motion. A typical manipulator assumes no resistance when pulling the cable. However, as suggested in this study, when an elastic fixture was placed in the joint, a resistance force was generated. Given the pulling force, the length of the cable, and the elastic modulus of the body, the stored energy \(E_{s}\) can be calculated as:  
\[W\ =\ \int_{x_{o}}^{x_{e}}k_{t}xdx\ =\ \frac{1}{2}k_{t}x_{e}{}^{2}\ =\ E_{s}. \tag{2}\]  
where \(x\) is the displacement of the cable from initial position (\(x_{o}\ =\ 0\)) to pulled position \(x_{e}\), and \(k_{t}\) is the total elastic modulus.