## 2 Method
### Design and Modeling for Bending Shape Prediction of Flat Spring Mechanism
The stored energy equation for one joint can be expressed as given in (2), but the manipulator developed in this study had multiple joints. Another important feature was that flat springs with different thicknesses were placed at each joint. Different thicknesses of  
Figure 2: Arrangement of the flat spring between the manipulator’s joints; (**a**) is a segment of the manipulator, (**b**) is a specially designed flat spring, and (**c**) is a joint assembly that has two segments and one flat spring. (**d**) Initially, they were aligned in a line (left panel). When the cable was pulled, the angle of bending varied depending on the stiffness of the flat spring (right panel). these flat springs meant that they had different modulus, which influenced the stiffness. The total moved cable displacement could be expressed as the sum of the cable displacements moved at each joint, and the total energy formula could be derived by using the stiffnesses of the flat spring placed at each joint. The cable displacement, \(\Delta x_{i}\), of each joint is expressed as \(x_{initial}-x_{Li}\), the result of subtracting the cable length, \(x_{Li}\), after bending, from the initial cable length, \(x_{initial}\). When the number of joints was increased to \(n\), the displacement \(\Delta x_{i}\) of the cable moved by compressing the elastic fixture at each joint and the total displacement of the pulled cable \(x_{e}\) could be expressed as follows:  
\[x_{e}\ =\ \Delta x_{1}+\cdots+\Delta x_{n}. \tag{3}\]  
Using Formulas (2) and (3), the total energy stored when driving a manipulator with \(n\) joints can be calculated as follows:  
\[E_{s}(x_{e})\ =\ \frac{1}{2}(k_{1}\Delta x_{1}^{2}+\cdots+k_{n}\Delta x_{n}^{2}) \ =\ \frac{1}{2}\sum_{i\ =\ 1}^{n}k_{i}x_{i}^{2}. \tag{4}\]  
The total energy is the sum of the energies of each joint and the energy of each joint consists of the product of the cable displacement moved at each joint and the stiffness of the flat spring. The cable length for each joint and bending shape of the manipulator is determined by minimizing the energy stored, according to the minimum potential energy theory. Conversely, when the lowest possible stored energy was known, the cable lengths of each joint could be derived, and then the bending shape of the entire manipulator could be predicted. If the stiffness of the flat springs placed on each joint is the same, the overall stiffness can be easily calculated even if the number of joints increases. If the stiffness of the flat springs placed at each joint is different, the formula for deriving the overall stiffness becomes somewhat complicated as the number of joints increases. Moreover, this is only possible under the assumption that stiffness is constant. However, the shape of the flat spring proposed in this study shown in Figure 2b was bizarre and had a different shape from the general helical coil spring. As a result, the stiffness of the flat spring is expressed in the form of a function rather than a constant. This can be obtained through experimentation, and calculated using the following equation using the compressed displacement and the applied external force:  
\[k_{j}(\delta x)\ =\ \frac{F_{ext}(\delta x)}{\delta x},\ j\ =\ 0.08,\ 0.1,\ 0.12,\ 0.14. \tag{5}\]  
where \(k_{j}\) is the stiffness calculated, and \(j\) represents thickness of each flat spring with mm unit, \(\delta x\) is the displacement compressed by an external force, and \(F_{ext}\) is the external force. The stiffness value in the form of a function complicated the calculation to obtain the total minimum energy value. We applied constrained optimization to solve this complex computation.