# Studies regarding the Use of Pneumatic Muscles in Precise Positioning Systems
## 3 Results
### Evaluation of the Relation between the Axial Contraction and the Input Pressure  
Of interest for the study of the positioning accuracy of the mechanical systems driven by the pneumatic muscles is the dependence of the pressure necessary for supplying these actuators on the size of the axial contraction that needs to be performed. In this regard, based on the measured values, a short analysis was made in order to deduce the relation between the input pressure and axial contraction. Thus, for the considered cases, the necessary pressures for obtaining some axial contraction were found. These can be seen in Table 6.  
In Figure 11, the evolution of the input air pressure versus pneumatic muscle axial contraction is presented. The dots represent the axial contraction values, and the lines represent the polynomial regression curves. For \(M=0\) kg, the functions that describe the input pressure evolution versus axial contraction at inflation and deflation are:  
\[p_{inflation}=0.001\cdot\Delta l^{3}-0.028\cdot\Delta l^{2}+0.565\cdot\Delta l +0.637 \tag{7}\]  
\[p_{deflation}=0.001\cdot\Delta l^{3}-0.0172\cdot\Delta l^{2}+0.376\cdot\Delta l +0.537 \tag{8}\]  
The graphs for the 2.55 kg and 4.55 kg loads are shown in Figures 12 and 13. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Axial**} & \multicolumn{2}{c}{\(M=0\) **kg**} & \multicolumn{2}{c}{\(M=2.55\) **kg**} & \multicolumn{2}{c}{\(M=4.55\) **kg**} \\  & \multicolumn{2}{c}{\(p\) **[bar]**} & \multicolumn{2}{c}{\(p\) **[bar]**} & \multicolumn{2}{c}{\(p\) **[bar]**} & \multicolumn{2}{c}{\(p\) **[bar]**} \\  & **Inflation** & **Deflation** & **Inflation** & **Deflation** & **Inflation** & **Deflation** \\ \hline
2 & 1.66 & 1.24 & 1.88 & 1.16 & 1.43 & 1.38 \\
4 & 2.54 & 1.78 & 2.72 & 1.95 & 2.91 & 2.18 \\
6 & 3.16 & 2.43 & 3.25 & 2.64 & 3.49 & 2.84 \\
8 & 3.86 & 2.96 & 3.90 & 3.19 & 4.07 & 3.47 \\
10 & 4.53 & 3.4 & 4.62 & 3.72 & 4.59 & 4.10 \\
12 & 4.94 & 4.28 & 5.14 & 4.33 & 5.19 & 4.80 \\
14 & 5.71 & 4.98 & 5.85 & 5.19 & 5.96 & 5.93 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Supply pressure values necessary for axial contraction of the muscle. Figure 10: Polynomial regression. (**a**) Inflation; (**b**) deflation. Figure 11: The evolution of the input pressure versus axial contraction (\(M=0\) kg). Figure 12: The evolution of the input pressure versus axial contraction (\(M=2.55\) kg). Figure 13: The evolution of the input pressure versus axial contraction (\(M=4.55\) kg). The equations that describe the variation of the input pressure versus axial contractions for when a 2.55 kg load is applied to the free end of the muscle are as follows:  
\[p_{inflation}=0.001\cdot\Delta l^{3}-0.016\cdot\Delta l^{2}+0.452\cdot\Delta l+1.06 \tag{9}\]  
\[p_{deflation}=0.002\cdot\Delta l^{3}-0.047\cdot\Delta l^{2}+0.649\cdot\Delta l+0. 024 \tag{10}\]  
The equations that describe the variation of the input pressure versus axial contraction when a 4.55 kg load is applied to the free end of the muscle are presented in Equations (11) and (12). Those equations are validated by the coefficients of determination (R2), which have the values of 0.9972 and 0.9996. \[p_{inflation}=0.003\cdot\Delta l^{3}-0.106\cdot\Delta l^{2}+1.168\cdot\Delta l- 0.461 \tag{11}\]  
\[p_{deflation}=0.002\cdot\Delta l^{3}-0.0503\cdot\Delta l^{2}+0.653\cdot\Delta l +0.244 \tag{12}\]  
In order to obtain precise and predictable contractions at the free end of the pneumatic muscle, Equations (7)-(12) must be known and used for control by means of a proportional pressure regulator of the pneumatic muscle supply. Figure 14 shows a control scheme that includes such a proportional pressure regulator. Compressed air is fed to the muscle through an MPPES-3-1/4-6-010 proportional pressure regulator (produced by Festo AG & Co., Esslingen, Germany). This control diagram enables the slow and uniform charging of the pneumatic muscle without introducing shocks.