# Flexible Design of the Connecting Plate in Pneumatic Soft Manipulators
## 4 Structural Analysis  
Taking a manipulator as an example, at any combination of air-pressure values in the two chambers, the manipulator can achieve six bending directions; when the air-pressure values in the three chambers are all different, the manipulator can achieve flexible movement in any direction. Therefore, the manipulator designed in this paper mainly uses the pressure difference in each chamber to change the manipulator's shape. As shown in Figure 6, the air pressure in chamber 1 remained constant. The driving device delivered the gas to increase the air pressure in chambers 2 and 3 to bend the manipulator. The air pressure in the chambers of the manipulator comes from the air pump. Considering the torque balance, the silicone chamber generates a driving torque on the connecting plate through inflation and elongation. Torsion and bending of the chambers cause stress to concentrate on the connecting plate. The plate is also subject to drive torque, making it more difficult to control the manipulator. The rigid connecting plate is made of ABS material, which has high rigidity characteristics and decreases the manipulator's performance in transmitting the driving torque. The novel flexible connecting plate can reduce the influence of the force coupling generated by the movement of the chamber through the bending deformation of its BOM module. We designed a rigid connecting plate with the same-sized radius and thickness via 3D printing, and the relevant parameters and simulation conditions of the rigid and flexible connection plates required for the test are provided in Table 1.  
Based on the finite element method, we used software to simulate and then to verify the detailed stress effect on the movement [30; 31]. Figure 7 shows the stress distribution of the two connecting plates. For the RC plate, EF is the base used to connect two connecting plates and to support the basic shape of the soft manipulator. For radial restriction, the position of CF would be the ideal boundary in the motion. HL is the hole used for fixing another rigid plate in place to complete the modular assembly of the manipulator. According to the model and Equations (3) and (9), we can determine the possible air pressures and loads in \begin{table}
\begin{tabular}{c c c} \hline \hline
**Prototype** & **RC** & **BOM** \\ \hline \(p_{i}\) for simulation & 8 kPa & 8 kPa \\ \(\theta\) for simulation & 45\({}^{\circ}\) & 45\({}^{\circ}\) \\ Thickness & 2 mm & 2 mm \\ Diameter & 130 mm & 130 mm \\ Pure Mass & 100 g & 80 g \\ Material & ABS plastic & silicone \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Basic parameter of the connecting plate in the analysis. Figure 6: The bending state of different chambers when the manipulator moves. the simulation. Therefore, the simulation results of two kinds of connection plates under preset conditions are shown in the following figures:  
According to the simulation test results in Figure 7, the stress in the RC tends to be concentrated in the central area. On the same scale, we easily obtain the conclusion that the stress-concentration effect of BOM is significantly smaller than that of RC. Due to the rigidity of RC, it couples the driving torque from the previous robotic arm module into a total driving force and transmits it to the next robotic arm module during the movement, which causes the control system to fail in effectively responding to the driving torques respectively generated from three chambers. The finite element simulation test can summarize the nonlinear relation via the stiffness characteristics in typical pneumatic soft arms. Although it leads to more significant stresses and strains on the BOM than RC, these stresses and strains are less likely to be transferred to other motion components. The ends of the three chambers are fixed in the same plane by the connecting plate. When the air-pressure values in the three chambers are different, the manipulator extends axially and bends toward the chamber with lower air pressure.