# Flexible Design of the Connecting Plate in Pneumatic Soft Manipulators
## 3 Modeling of the Actuator
### Elastic Bending Moment of BOM  
During movement of the manipulator, the connecting plate is subjected to the air pressure of the pneumatic actuators on both sides simultaneously, and the stress values on both sides of the connecting plate are different. Then, the connecting plate bears the bending moment. CM in Figure 5a represents the area where the connecting plate is subjected to air pressure, and Figure 5b represents the force balance of the connecting plate at the connection of a chamber in a simplified way. The elastic bending moment \(M_{E}\) is the sum of the elastic bending moments produced by the three chambers. Considering the stress-strain relationship, the elastic bending moment of the soft manipulator for BOM can be obtained as follows:  
\[M_{E}=\sum_{i=1}^{3}S_{CF}\sigma_{i}R \tag{3}\]  
where \(S_{CF}\) is the area that CF makes contacts with BOM. \(\sigma_{i}\) is the actual stress generated by the soft actuator on BOM. \(R\) is the radius that we defined as shown in Figure 5.  
This paper assumes that the chambers only separately produce forces in the \(z\) and \(r\) axes on the BOM as an axial-symmetric element. The finite element method has been applied to the nonlinear deformation of flexible materials. Some works model the characteristics using the Yeoh model because it can be a simple stretching motion for some manipulators. However, the BOM would be more complicated, especially if we consider compression, which is different from stretching due to its nonlinear characteristics. Figure 4: Main kinematic parameters of the soft manipulator. Compared with related works, the Navier-Cauchy function was introduced to improve the accuracy of describing stretching and compression on a scale. Based on the Navier-Cauchy equations and illustrated by the Legendre function [27], we combine the modulus with at least four material coefficients and parameters to fit the elastic modulus of the connecting plate:  
\[E=2E_{1}\Bigg{[}1+\frac{v\int_{x_{1}}^{x_{2}}\big{[}C_{1}x^{2}+C_{2}x+1-\delta \big{]}dx}{(1-2\mu)(x_{2}-x_{1})}\Bigg{]} \tag{4}\]  
In Formula (4), there are two task space influence coefficients: \(x_{1}=\cot\theta\) and \(x_{2}=\cot\phi\). With \(E_{1}\) as the initial [PERSON]'s modulus of the silicone material, it can be tested from the tensile experiment before showing nonlinear characteristics in the first tensile stage [28]. Additionally, \(\mu\) is [PERSON]'s ratio. In Equation (6), the material coefficients \(C_{1}\) and \(C_{2}\) are as follows:  
\[\left\{ \begin{array}{l}C_{1}=\frac{C_{1}}{C_{4}}\cdot\frac{n}{4\pi^{2}}\\ C_{2}=-C_{1}\Big{(}\frac{\delta}{2}+\frac{x\cdot d}{x+1}\Big{)}\end{array} \right. \tag{5}\]  
For further fitting, the nonlinear characteristics of the orientation angle exhibit material coefficients \(C_{3}\) and \(C_{4}\), as follows:  
\[\left\{ \begin{array}{l}C_{3}=\left(x_{2}^{2}-1\right)\left[\alpha\big{(}x_{ 2}^{2}-1\big{)}+2x_{2}^{2}(d+1)\right]+2(d+1)\big{(}x_{2}^{2}-1\big{)}\\ C_{4}=\left[\frac{\delta}{2}+\frac{2x_{3}(d+1)(x_{2}-1)}{x_{2}^{2}-1}\right] \cdot\left[\frac{(x_{2}-1)x_{2}(d+1)}{x_{2}^{2}-1}\right]\end{array} \right. \tag{6}\]  
Furthermore, the BOM can be assembled for \(n\) layers in terms of the practical requirement. Additionally, curvature \(k\) as an important factor [29]:  
\[\left\{ \begin{array}{l}\alpha=\frac{n\big{(}\frac{1}{k}+1\big{)}\cdot\ln \big{(}\frac{1}{k}\big{)}}{\frac{1}{k}\big{(}\frac{1}{k}+1\big{)}}\\ \delta=-\frac{n\big{(}\frac{3}{k}+2n\big{)}}{2\big{(}\frac{3}{k}+\frac{3n}{k} +n^{2}\big{)}}\end{array} \right. \tag{7}\]  
Therefore, we can derive the stiffness matrix in Equation (8). With \(\sigma\) and \(\varepsilon\) as the stress and strain, we can separately determine the direction of axis \(z\) and the radius:  
\[ \begin{bmatrix}\sigma_{z}\\ \sigma_{R}\end{bmatrix} =\frac{E}{(1+\mu)(1-2\mu)} \begin{bmatrix}1-\mu&\mu\\ \mu&1-\mu\end{bmatrix}\begin{bmatrix}\varepsilon_{z}\\ \varepsilon_{R}\end{bmatrix}