## 3 Kinematics Model of Soft Gripper  
Soft robots have infinite degrees of freedom. The traditional rigid robot-modeling method is no longer applicable to soft robots. The pneumatic network multi-cavity soft actuator is made of super elastic silica gel material, which presents large deformation and nonlinear characteristics during deformation [12]. Due to the isotropic and incompressible properties of hyperelastic materials, the Yeoh model suitable for describing the large deformation of hyperelastic materials was used to establish the constitutive model of the soft actuator [24]. In addition, the analysis was conducted on the bending characteristics of the actuator under working conditions, and the strain energy density function is expressed as \(W=W(I_{1},I_{2},I_{3})\).  
\[\left\{\begin{array}{c}I_{1}=\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2} \\ I_{2}=\lambda_{1}^{2}\lambda_{2}^{2}+\lambda_{2}^{2}\lambda_{3}^{2}+\lambda_{3} ^{2}\lambda_{1}^{2}\\ I_{3}=\lambda_{1}^{2}\lambda_{2}^{2}\lambda_{3}^{2}\end{array}\right\} \tag{1}\]  
In the formula, \(\lambda_{1}\), \(\lambda_{2}\), and \(\lambda_{3}\) are the elongation ratios of three directions, respectively; \(I_{1}\), \(I_{2}\) and \(I_{3}\) are series expansions.  
Due to the incompressibility of silica gel materials, Formula (1) can be written as two classical expansions of strain energy density function:  
\[W=C_{10}(I_{1}-3)+C_{20}(I_{1}-3)^{2}\]  
In the formula: \(C_{10}\) and \(C_{20}\) are material constants based on the Yeoh model.  
When only the relationship between the axial stress and strain of the soft actuator is considered, the deflection of the principal strain corresponding to the strain energy density function can be obtained; that is, the principal stress strain t, \(\lambda_{1}\), \(\lambda_{2}\), and \(\lambda_{3}\) are as follows.  
\[\text{t}=\frac{2}{\lambda 1}\left(\lambda_{1}^{2}-\frac{1}{\lambda_{1}^{2} \lambda_{2}^{2}}\right)\left(\frac{\partial W}{\partial I_{1}}+\lambda_{2}^{ 2}\frac{\partial W}{\partial I_{2}}\right) \tag{2}\]  
\[\lambda_{2}^{2}=\lambda_{3}^{2}=\frac{1}{\lambda_{1}^{2}} \tag{3}\]  
As the bottom of the limiting layer is covered with silica gel, it is assumed that the limiting layer bottom length, \(l_{0}\), does not change after the bending deformation of the soft actuator. Under the action of air pressure, the bending deformation of the soft actuator is shown in Figure 4. According to the geometric relationship, the axial elongation ratio of  
Figure 3: Structure diagram of pneumatic gripper fixture.  
the actuator, that is, the ratio of the arc length of the middle layer to the arc length of the bottom after the deformation of the actuator, is as follows.  
\[\lambda_{1}=\frac{l}{l_{0}}=\frac{r}{r_{0}}=\frac{360l_{0}+2\pi c\theta+\pi h \theta}{360l_{0}} \tag{4}\]  
\(l\) is the arc length of the actuator center layer; \(l_{0}\) is the arc length at the bottom of the actuator; \(r\) is the bending radius of the middle layer after actuator deformation; \(r_{0}\) is the bottom bending radius after actuator deformation; \(\theta\) is the bending angle of the actuator.  
The axial cross-sectional area and initial cross-sectional area of the soft actuator can be calculated by \(s=S_{0}\lambda_{2}\lambda_{3}\). According to the section equilibrium equation, the relationship between principal stress and driving pressure is obtained as follows:  
\[\mathfrak{t}=\frac{ps_{0}}{s}=\frac{p}{\lambda_{2}\lambda_{3}} \tag{5}\]  
Relationship between bending angle and driving pressure:  
\[\theta=\frac{(2\pi c+\pi h)^{2}p^{2}}{129,600l_{0}}+\frac{360l_{0}p}{\sqrt{(2 \pi c+\pi h)^{3}}}+\frac{(2\pi c+\pi h)^{3}}{3.65l_{0}^{2}} \tag{6}\]