# Studies regarding the Use of Pneumatic Muscles in Precise Positioning Systems
## 3 Results
### Evaluation of Hysteresis
#### 3.1.2 Evaluation of Hysteresis for the Case of a 2.55 kg Load Applied  
For the situation of a mass of 2.55 kg load applied to the free end of the muscle, the measurement data are presented in Table 4 and Figure 7.  
It can be observed that, in this case, and similar to previous values, the behavior of the inflated/deflated muscle is not linear. As seen in Figure 7,  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Pressure**} & \multicolumn{5}{c}{**Length [mm]**} & \multicolumn{2}{c}{**Arithmetic**} & \multicolumn{1}{c}{**Standard**} \\ \cline{2-9}
**[bar]** & **1** & **2** & **3** & **4** & **5** & **Mean** & **Deviation** \\ \hline \multirow{9}{*}{Inflation} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\  & 0.5 & 0.2 & 0.3 & 0.4 & 0.2 & 0.4 & 0.3 & 0.08944 \\  & 1 & 0.9 & 1.1 & 1.2 & 1 & 1.3 & 1.1 & 0.14142 \\  & 1.5 & 1.5 & 1.6 & 1.4 & 1.6 & 1.4 & 1.5 & 0.08944 \\  & 2 & 2.3 & 2.5 & 2.4 & 2.2 & 2.6 & 2.4 & 0.14142 \\  & 2.5 & 3.4 & 3.5 & 3.6 & 3.4 & 3.6 & 3.5 & 0.08944 \\  & 3 & 5 & 5.1 & 5.2 & 5.2 & 5 & 5.1 & 0.08944 \\  & 3.5 & 6.9 & 7 & 7.1 & 6.8 & 6.8 & 6.92 & 0.11662 \\  & 4 & 8.3 & 8.5 & 8.4 & 8.3 & 8.4 & 8.38 & 0.07483 \\  & 4.5 & 9.6 & 9.7 & 9.5 & 9.4 & 9.7 & 9.58 & 0.11662 \\  & 5 & 11.5 & 11.8 & 11.7 & 11.6 & 11.5 & 11.62 & 0.11662 \\  & 5.5 & 12.4 & 12.6 & 12.3 & 12.5 & 12.6 & 12.48 & 0.11662 \\ \hline \multirow{9}{*}{Deflation} & 6 & 14.7 & 14.8 & 14.9 & 14.7 & 14.8 & 14.78 & 0.07483 \\  & 5.5 & 14.7 & 14.7 & 14.4 & 14.7 & 14 & 14.5 & 0.27568 \\  & 5 & 13.6 & 13.7 & 13.5 & 13.5 & 13.7 & 13.6 & 0.08944 \\  & 4.5 & 12.4 & 12.5 & 12.3 & 12.4 & 12.4 & 12.4 & 0.06325 \\  & 4 & 11 & 10.8 & 10.8 & 10.9 & 11 & 10.9 & 0.08944 \\  & 3.5 & 9.5 & 9.3 & 9.4 & 9.2 & 9.6 & 9.4 & 0.14142 \\  & 3 & 7.4 & 7.5 & 7.6 & 7.5 & 7.5 & 0.06325 \\  & 2.5 & 5.5 & 5.8 & 5.6 & 5.5 & 5.6 & 0.10954 \\  & 2 & 4 & 4.4 & 4.2 & 4.3 & 4.1 & 4.2 & 0.14142 \\  & 1.5 & 2.5 & 2.7 & 2.8 & 2.9 & 2.6 & 2.7 & 0.14142 \\  & 1 & 1.8 & 1.7 & 1.7 & 1.9 & 1.9 & 1.8 & 0.08944 \\  & 0.5 & 0.8 & 1 & 1 & 0.8 & 0.9 & 0.9 & 0.08944 \\  & 0 & 0.2 & 0.1 & 0.4 & 0.4 & 0.4 & 0.3 & 0.12649 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Muscle axial contraction for a 2.55 kg external load.  
Figure 7: Evolution of axial contraction versus pressure (\(M\) = 2.55 kg).  
evolution of axial contraction are plotted with a blue line and the deflating variation is represented with a red line. There is a distance between the obtained values, the maximum value between inflation and deflation being 2.82 mm at a pressure of 4.5 bar.  
At inflation, with a 2.55 kg mass attached to the free end of the pneumatic muscle, the polynomial regression that best fits the measured data is presented in Equation (3) and for deflation in Equation (4). The evolution of these values can be seen in Figure 8.  
\[\Delta l_{inflation} =0.096+0.079\cdot p+0.718\cdot p^{2}-0.053\cdot p^{3} \tag{3}\] \[\Delta l_{deflation} =0.439+0.180\cdot p+1.092\cdot p^{2}-0.120\cdot p^{3} \tag{4}\]  
For both regressions, the correlation coefficients (\(r_{inflation}=0.998\), \(r_{deflation}=0.999\)) and the coefficients of determination (\(r^{2}_{inflation}=0.997\), \(r^{2}_{deflation}=0.999\)) validate the functions developed in Equations (3) and (4).