## 4 Discussion  
In this study, we introduced a hyper-redundant manipulator with elastic fixtures capable of predicting its bending shape with non-uniform curvature for MIS. A specially manufactured flat spring was placed between the hinged-slide joints to alter the non-uniform curvature by reassembling. The experimental results confirmed that the manipulator could have various non-uniform curvatures, by changing the configuration of the flat spring. The manipulator manufactured in this study had four joints, and the number of combinations of flat springs that can be arranged for each joint was quite large. Among them, simulations were performed on four combinations, and experiments were performed on two combinations for comparison. The experimental results confirmed a similar bending shape tendency when analyzed in simulation, proving that the applied minimum energy theory is established. Manipulators with different non-uniform curvatures according to different combinations had various workspaces. It was confirmed that the two combinations of manipulators in the experiment have different workspaces, which could allow the user to choose according to a specific environment or surgical operation. The combination showing the most remarkable difference of the manipulator's bending shape in the simulation was T1 and T2. The experimental result of Figure 11 also showed the same trend as the simulation, and the shape difference between the two combinations was the largest when the final pulled distance was 2 mm. This could be confirmed by the different positions of the joints having the minimum radius of curvature. In the case of the T1 combination, the proximal part had a minimum radius of curvature, and in the T2 combination, the distal part had a minimum radius of curvature. This affected the final position of the endpoints. In the T1 combination, the endpoint was located on the positions 23.19 mm and 13.71 mm, on the horizontal and vertical axes, respectively. Additionally, in the T2 combination, the endpoint was located on the positions 24.57 mm and 8.76 mm, on the horizontal and vertical axes, respectively. The two positions showed a difference of 5.14 mm, which was quite significant, given that the total length of the manipulator is 28 mm. Hence, it could be said that it had a great influence even when it was applied to a surgical target with a small working space. The bending shape also had a reasonably high prediction rate. After matching the starting point and the first joint, the position errors from the second joint to the last joint were measured. As a result, the average error rate of the T1 combination was 3.48%, the average of the T2 combination was 4.16%, and the overall average error rate was 3.82%. It could be confirmed that the bending shape analyzed in terms of the minimum energy was quite effective. However, despite having a low average error rate, it could be confirmed that the maximum error was when the cable was pulled to the maximum of 2 mm. When comparing the experiment and simulation, an error of up to 3.75 mm occurred at the endpoint of the manipulator of the T2 combination, which was about 13.4% when considering the total length of 28 mm. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Config.**} & \multirow{2}{*}{
\begin{tabular}{c} **Pulling** \\ **Length (mm)** \\ \end{tabular} } & \multicolumn{4}{c}{**Difference of ith Joint and Endpoint (mm)**} \\ \cline{3-6}  & & **2 nd** & **3 rd** & **4 th** & **Endpoint** \\ \hline \multirow{4}{*}{T1} & 0.5 & 0.11 & 0.26 & 0.39 & 0.66 \\  & 1.0 & 0.19 & 0.49 & 1.02 & 1.61 \\  & 1.5 & 0.20 & 0.92 & 1.78 & 2.73 \\  & 2.0 & 0.35 & 0.99 & 4.62 & 2.27 \\ \hline \multirow{4}{*}{T2} & 0.5 & 0.09 & 0.23 & 0.53 & 1.18 \\  & 1.0 & 0.14 & 0.43 & 1.30 & 2.54 \\ \cline{1-1}  & 1.5 & 0.47 & 0.99 & 2.12 & 3.75 \\ \cline{1-1}  & 2.0 & 0.26 & 0.56 & 1.38 & 2.66 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Position error of each joint. There were multiple sources of errors in our study.