# Kinematic Modelling and Experimental Validation of a Foldable Pneumatic Soft Manipulator
## 2 Kinematic Modelling of the Foldable Pneumatic Soft Manipulator
### Kinematic Model of the Foldable Pneumatic Module  
The foldable pneumatic module, which includes a pneumatic actuator and an inflatable straight arm, is the basic unit of the pneumatic soft manipulator. The difficulty to model the kinematic performance of the pneumatic module lies in the pneumatic actuator. To develop the kinematic model of the pneumatic actuator, a basic inflatable unit is taken and shown in Figure 2. The inflatable unit is composed of three bellows slices that are arranged in an equilateral triangle configuration, and its upper and lower ends are fixed together by rigid constraint plates. It is assumed that the lower end is fixed and the upper end can move freely. Figure 1: Design of the foldable pneumatic module. As is shown in Figure 2, the basic inflatable unit is just a small slice taken from the whole pneumatic actuator. The length of the equilateral triangle configuration is \(a\). When the bellows are not inflated, the bellows slice remains its original length \(L_{0}\). Once the bellows are inflated, the bellows stretches along the axial direction. The elongation of the manipulator is produced by the extension of the bellow structure. Therefore, the strain of the bellow still agrees with the small deformation assumption. Moreover, the axial elongation of the bellows is considered, while the radial inflation is neglected. For the convenience of analysis, the elongated cylindrical bellows is considered as an elastomer. It is assumed that the elastic coefficients of the three bellows are the same, \(k\). The internal pressures of the three bellows are \(P_{1}\), \(P_{2}\) and \(P_{3}\) respectively. The forces produced by the elongation of the three bellows are \(F_{1}\), \(F_{2}\) and \(F_{3}\), and the elongations are \(\Delta L_{1}\), \(\Delta L_{2}\) and \(\Delta L_{3}\) respectively. When the bellows are inflated, the deformations are assumed to be in the elastic range, thus  
\[\left\{ \begin{array}{l}F_{1}=k\Delta L_{1}\\ F_{2}=k\Delta L_{2}\\ F_{3}=k\Delta L_{3}\end{array} \right. \tag{1}\]  
It is assumed that the internal cross-sectional area of the bellows is \(S\). Considering the torque balance of the rigid constraint box at the upper end, it can be obtained as follows:  
\[\left\{ \begin{array}{l}F_{1}=k\Delta L_{1}=P_{1}S\\ F_{2}=k\Delta L_{2}=P_{2}S\\ F_{3}=k\Delta L_{3}=P_{3}S\end{array} \right. \tag{2}\]  
After inflation, if there is no upper rigid constraint plate, the lengths of the three bellows are respectively:  
\[\left\{ \begin{array}{l}L_{1}=L_{0}+\Delta L_{1}=L_{0}+\frac{P_{1}S}{k}\\ L_{2}=L_{0}+\Delta L_{2}=L_{0}+\frac{P_{3}S}{k}\\ L_{3}=L_{0}+\Delta L_{3}=L_{0}+\frac{P_{3}S}{k}\end{array} \right. \tag{3}\]  
The mechanical effects of the rigid constraint plates at the upper end are as follows: (1) The three points A, B and C at the upper end of the three bellows keep still an equilateral triangle, and the length of the side is \(a\); (2) Making the normal direction of the upper plates of the three bellows point to the same direction. The forces produced by the upper rigid constraint plates on the three bellows are \(R_{1}\), \(R_{2}\) and \(R_{3}\) respectively, and their directions are located in the plane where the constraint plate is  
Figure 2: Basic inflatable unit of the pneumatic actuator. located. Taking the fixed-end plane as the XY plane, a cartesian coordinate system is established, as shown in Figure 2. \(A_{0}\), \(B_{0}\) and \(C_{0}\) are the fixed ends of the three bellows respectively. The three points constitute an equilateral triangle, and the length of the side is \(a\). Taking the \(B_{0}\) point as the origin of the coordinate, the axis \(x\) is perpendicular to the \(A_{0}C_{0}\) downward direction, the axis \(y\) is horizontal to the right, and the axis \(z\) is perpendicular to the X-Y plane, forming a right-handed cartesian coordinate system. In this coordinate system, the coordinates of \(A_{0}\), \(B_{0}\) and \(C_{0}\) can be written as,  
\[\left\{