# Flexible Design of the Connecting Plate in Pneumatic Soft Manipulators
## 4 Structural Analysis
When we consider the bending moment on the connecting plate, the air pressure borne by the connecting plate is regarded as a whole. However, considering the flexible material property of the overall flexible manipulator, the soft manipulator is assembled with multi-segment modules bending towards different directions in space: one side of the chamber is driven by air pressure to produce thrust on the connecting plate, and the other side of the chamber produces pressure on the connecting plate. The pressure on both sides of the connection plate is not the same. The relationship between the influence of the deflection angle \(\phi\) and the pressure \(P\) on BOM can be denoted by the Gaussian function as a fitting result. \(\phi\in[0,360^{\circ}]\), and \(P_{t}\) is the total infusion air pressure in the three chambers. \[\left\{ \begin{array}{l}P=2P_{t}\exp(\phi)\\ \exp(\phi)=1.2e^{-(\frac{\phi-180}{90})^{2}}+e^{-(\frac{\phi-252}{7.8})^{2}}+ 1.2e^{-(\frac{\phi-54}{7.2})^{2}}\end{array} \right. \tag{9}\]  
The stresses on the connection plate when the manipulator is bent in different directions are obtained by substituting the air-pressure values in different chambers into Equation (9). In Figure 8, BOM1 represents the stress value between the flexible connecting plate and the chamber numbered 1, RC1 represents the stress value between the rigid plate and the chamber numbered 1, etc. When the soft arm was deflected in the range of 0-360\({}^{\circ}\), the air pressure in the three chambers constantly changed and was not equal. For each 18\({}^{\circ}\) deflection of the manipulator, we recorded the air-pressure parameters of three chambers of a group of manipulators and recorded 20 groups of data in total. The data are shown in  
Figure 7: The simulation result of the structural analysis: (**a**) stress distribution on BOM when two chambers bend; (**b**) results under the same simulation conditions on RC. When the bending angle was \(\theta=45^{\circ}\), we determined the change in stress for three CFs of the connecting plate to different deflection angles \(\phi\) through some experiments. In the graphic, the connecting plate RC was subjected to a maximum stress of about 180 kPa, while the connecting plate BOM was subjected to a maximum stress of about 170 kPa. Under the same deflection angle, we can see that the stress on BOM is always smaller than the stress on RC. This result means that the stresses is less concentrated on BOM. For the flexible pneumatic manipulator, the mapping of air pressure to motion space can be obtained by deriving the relationship between the variation in pressure and the curvature. Combined with the moment-balance equation, the bending curvature of the individual module of the manipulator can be derived from the radius of the chamber-bending curvature; then, the relationship between the curvature \(k\) and the air-pressure parameter \(P\) can be known:  
\[k=\frac{P-P_{t}}{\lambda_{1}}+\frac{\sigma}{\lambda_{1}} \tag{10}\]  
where \(\lambda_{1}\) is the elongation ratio of the chamber along the axial direction. The maximum inflation air pressure of the bending chamber is 170 kPa, and the deflection angle is \(\phi\in(0^{\circ},360^{\circ})\). The maximum elongation ratio of the manipulator is known. The total stress in the module is constant, but the stress-concentration-effect coefficient varies with the deflection angle. Using the total air pressure \(P\) of the bending chamber as the variable, the bending variation law of the module at each deflection angle is fitted to determine the variation range of the maximum bending curvature. In this paper, 24 angles were selected on average among various deflection angles, and the actual maximum bending curvature of the module was deduced by combining the measurement results. BOM represents the curvature of the module assembled with a flexible connection plate, and RC represents the curvature of the module assembled with a rigid connecting plate.