# 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
\begin{array}{c}W=\frac{a\sin\alpha}{M(\Delta L_{1}-\Delta L_{3})}\pm\frac {1}{M}sign(\Delta L_{1}-\Delta L_{3})\times\\ \sqrt{\frac{3a^{2}-3(\Delta L_{1}-\Delta L_{3})^{2}-(2\Delta L_{2}-\Delta L_{3 }-\Delta L_{1})^{2}}{3(\Delta L_{1}-\Delta L_{3})^{2}+(2\Delta L_{2}-\Delta L_ {3}-\Delta L_{1})^{2}}}\end{array}  \tag{13}\]  
When \(\Delta L_{1}=\Delta L_{3}\
eq\Delta L_{2}\), the \(W\) can be expressed as  
\[ \begin{array}{c}W=\frac{-\sqrt{3}a\cos\alpha}{M(\Delta L_{1}-2\Delta L_{2}+ \Delta L_{3})}\pm\frac{1}{M}sign(\Delta L_{1}-2\Delta L_{2}+\Delta L_{3}) \times\\ \sqrt{\frac{3a^{2}-3(\Delta L_{1}-\Delta L_{3})^{2}-(2\Delta L_{2}-\Delta L_{3 }-\Delta L_{1})^{2}}{3(\Delta L_{1}-\Delta L_{3})^{2}+(2\Delta L_{2}-\Delta L_ {3}-\Delta L_{1})^{2}}}\end{array}  \tag{14}\]  
The root \(W\) of the equation contains two positive solutions, the smaller one of which is the actual solution to the equation. This is because the smaller one corresponds to a small deflection which agree with the physical truth. The function of the rigid constraint plates is to ensure that the three central points of the bellows keep an equilateral triangle with a side length of \(a\). All the flexible bellows are identical and have the same inside diameter, outside diameter, length and elasticity coefficient. According to the symmetry, the motion of each bellow unit is the same with respect to its own coordinate system, as shown in Figure 2. Transfer the coordinate \(B_{0}-xyz\) to a new origin point \(B\), and the definition of the new coordinate \(x_{1}y_{1}z_{1}\) in the plane \(ABC\) is the same with the coordinate \(xyz\). Then the transferring method from the coordinate \(x_{0}y_{0}z_{0}\) to the coordinate \(x_{1}y_{1}z_{1}\) without considering the origin shift can be written as,  
\[\left( \begin{array}{c}x_{1}\\ y_{1}\\ z_{1}\end{array} \right)=\mathbf{H}\left( \begin{array}{c}x_{0}\\ y_{0}\\ z_{0}\end{array} \right)=\mathbf{M}_{3}(-\alpha)\mathbf{M}_{2}(\theta)\mathbf{M }_{3}(\alpha)\left( \begin{array}{c}x_{0}\\ y_{0}\\ z_{0}\end{array} \right) \tag{15}\]  
where \(\mathbf{H}\) is the transferring matrix, thus  
\[\left( \begin{array}{c}x_{0}\\ y_{0}\\ z_{0}\end{array} \right)=\mathbf{H}^{\mathcal{I}}\left( \begin{array}{c}x_{1} \\ y_{1}\\ z_{1}\end{array} \right) \tag{16}\]  
When the origin shift \(\vec{B_{0}}\vec{B}_{1}\) is considered, the transferring method from the coordinate \(x_{0}y_{0}z_{0}\) to the coordinate \(x_{1}y_{1}z_{1}\) can be written as  
\[\left( \begin{array}{c}x_{0}\\ y_{0}\\ z_{0}\end{array} \right)=\mathbf{H}^{\mathcal{I}}\left( \begin{array}{c}x_{1}\\ y_{1}\\ z_{1}\end{array} \right)+\vec{B_{0}}\vec{B}_{1} \tag{17}\]where \(B_{0}^{\rightarrow}B_{1}\) can be obtained by combining the equations (4) and (10),  
\[B_{0}^{\rightarrow}B_{1}=\left( \begin{array}{cc}v_{2}\cos\alpha&v_{2}\sin \alpha&L_{2}\end{array} \right)^{T} \tag{18}\]  
By using this transferring method, the motion of the whole pneumatic actuator can be described. Assumed that the total number of the bellow slices is \(N\) in the pneumatic actuator, the coordinate of the top plane center of the pneumatic actuator is \(\mathbf{r}^{\star}\), then the coordinate of \(\mathbf{r}^{\star}\) relative to the fixed coordinate system \(x_{0}y_{0}z_{0}\) can be expressed as  
\[\mathbf{r}_{N}=\left(\mathbf{H}^{T}\right)^{N-1}\mathbf{r}^{\star}+\left[ \left(\mathbf{H}^{T}\right)^{N-2}+\left(\mathbf{H}^{T}\right)^{N-3}+\cdots+ \left(\mathbf{H}^{T}\right)^{1}+\left(\mathbf{H}^{T}\right)^{0}\right]B_{0}^{ \rightarrow}B_{1} \tag{19}\]  
Using the above equation, the directional motion of the pneumatic actuator can be depicted. The inflatable straight arm is fixed at the end of the pneumatic actuator, and the elongation of the straight arm is in the tangent direction of the end of the pneumatic actuator. The stiffness of the straight arm with a certain air pressure can reach to a much higher level than the pneumatic actuator, thus the straight arm can be simplified as a straight line in the kinematic modelling process.