# A Mechatronics-Embedded Pneumatic Soft Modular Robot Powered via Single Air Tube
## 3 Control System and Experimental Model
### Experimental Model between Deformation and Pressure of the Module  
The response characterizations of the soft module are influenced by the chamber's structure and material's properties. Thus, the experimental relationship model was established based on experimental values. As shown in Figure 12, the fixed coordinate system O-xyz is built on point O, which is the center point on one end of the module. Meanwhile, the posture sensor (JY61-MPU6050, Telesky, Shenzhen, China) is fixed on the other end of the module to measure the bending angle. The chambers are numbered Chamber 1, Chamber 2 and Chamber 3 respectively.  
Figure 11: 3D workspace of the soft module: (**a**) The nearest surface of workspace; (**b**) the farthest surface of workspace.  
Figure 8: Comparison of the desired and actual values of the control system.  
Figure 10: 2D workspace of the soft module.  
Figure 9: The hysteretic characteristic of the soft module.  
The experiments were repeated three times with the same module parameters, and the experimental model are shown in Figure 13. The points with different shapes and colors represent the experimental values, which were obtained by inflating air pressure into Chamber 1. The module's bending states vary with the pressure every 10 kPa (from 0 kPa to 60 kPa). The dash lines are the fitting curves of the experimental values. The solid line is the experimental model, which is the average values of three fitting curves. The results showed that the bending angle increases with pressure and the curvature of curves is a substantial increase when pressure values were over 30 kPa. Figure 14 shows the comparison results between the simulation and experimental model. The trend of the experimental results was basically in accord with the trend of simulation values. Moreover, the maximum error of the results between the simulation and experiments was less than 9 degrees.  
The bending angles of the module by inflating air pressure into Chamber 2 are shown in Figure 15. The module bending states vary with the pressure every 10 kPa (from 0 kPa to 60 kPa). The points with the same shape represent experimental values under different pressures. The experiments were repeated three times with the same conditions. As shown in Figure 15, the dash lines are the fitting curves of the experimental values; the solid line represents the experimental model, which is the average values of the fitting curves. The results were three-dimensional curves because the rotate axis was not coincident to the axis of the fixed coordinate system. Chambers 2 and 3 are axial symmetry  
Figure 14: The comparison results between the simulation and experimental model of Chamber 1.  
Figure 12: Coordinate system and deformation analysis of the soft module: (**a**) Coordinate system; (**b**) deformation analysis.  
Figure 13: The experimental model of chamber 1.  
about x-axis. Thus, the bending angle of Chamber 3 can be obtained by the experimental model of Chamber 2. Figure 16 illustrates the comparison of the simulation and experimental model. The maximum error of the results between the simulation and experiment is less than 10 degrees.  
As shown in Figures 14 and 16, there were some deviations of the simulations and experimental results because of the idealization errors. Particularly, the main factor incurring the deviation is the thickness uniformity of the soft module, which cannot be entirely consistent with that in the simulation.