## 2 Pneumatic Muscles vs. Pneumatic Cylinders
### Establishing the Dependency of the Actuators Mass on the Length of the Stoke  
The masses of the pneumatic motors depend on the lengths of their strokes and their axial and radial dimensions. In the case of single-acting cylinders based on their technical specifications, the equations of mass versus stroke variations can be determined:  
\[m_{\text{ESNU\_10}}(s)=0.27\times s+0.0373 \tag{3}\]  
\[m_{\text{ESNU\_20}}(s)=0.72\times s+0.1868 \tag{4}\]  
\[m_{\text{ESNU\_40}}(s)=2.4\times s+0.661 \tag{5}\]  
In the three equations, the masses \(m\) are calculated in kg and the stroke length \(s\) is entered in meters.  
Equations (6)-(8) describe the variation in the mass of a pneumatic muscle (in kg) versus the length \(l\) (in meters) of its active part (excluding the end connections). The equations resulted from the information included in the technical specifications of the pneumatic muscles.  
\[m_{\text{DMSP\_10}}(l)=0.094\times l+0.077 \tag{6}\]  
\[m_{\text{DMSP\_20}}(l)=0.178\times l+0.238 \tag{7}\]  
\[m_{\text{DMSP\_40}}(l)=0.34\times l+0.675 \tag{8}\]  
It is known that upon being fed compressed air, the pneumatic muscle contracts by up to 20% of its initial length. The maximum stroke carried out by the free end of the muscle is described by Equation (9):  
\[s=\Delta l=0.2\times l \tag{9}\]  
Thus, the equations describing the variation in the mass of the pneumatic muscles versus the stroke \(s\) becomes:  
\[m_{\text{DMSP\_10}}(s)=0.47\times s+0.077 \tag{10}\]  
\[m_{\text{DMSP\_20}}(s)=0.89\times s+0.238 \tag{11}\]  
\[m_{\text{DMSP\_40}}(s)=1.7\times s+0.675 \tag{12}\]  
Table 3 includes the masses of the six studied pneumatic motors, and Figure 3 shows the variation lines of the masses of the two types of motors versus their strokes.  
\begin{table}
\begin{tabular}{l l c c} \hline \hline \multicolumn{1}{c}{**Motor Type**} & \multicolumn{1}{c}{**Motor Code**} & \multicolumn{1}{c}{**Maximum Exterior Diameter**} & \multicolumn{1}{c}{**Total Stroke**} \\  & & **[mm]** & **Length [mm]** \\ \hline \multirow{3}{*}{\begin{tabular}{l} Round \\ cylinders \\ \end{tabular} } & \begin{tabular}{l} ESNU-10-50-P-A \\ ESNU-20-50-P-A \\ ESNU-40-50-P-A \\ \end{tabular} & \begin{tabular}{l} 16 \\ 30 \\ 50 \\ \end{tabular} & \begin{tabular}{l} 86 \\ 132 \\ 177.6 \\ \end{tabular} \\ \hline \multirow{3}{*}{\begin{tabular}{l} Pneumatic \\ muscles \\ \end{tabular} } & \begin{tabular}{l} DMSP-10-250N-RM-CM \\ DMSP-20-250N-RM-CM \\ DMSP-40-250N-RM-CM \\ \end{tabular} & \
\begin{tabular}{l} 30.1 \\ 45.6 \\ 65.5 \\ \end{tabular} &
\begin{tabular}{l} 318.2 \\ 335 \\ 361 \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table}
Table 2: Dimensions of the studied pneumatic motors.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{1}{c}{**Motor Dimension**} & \multicolumn{1}{c}{**Mass Calculated for a 50 mm Stroke**} \\
**[mm]** & \multicolumn{1}{c}{**[kg]**} \\ \hline \hline \multicolumn{1}{c}{} & **ESNU** & **DMSP** \\ \hline \hline \(10\) & \(0.0508\) & \(0.1005\) \\ \(20\) & \(0.2228\) & \(0.2825\) \\ \(40\) & \(0.795\) & \(0.76\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Masses of the studied pneumatic motors.  
The values in Table 3 and the lines in Figure 3 suggest that for the same diameter of the motors, their masses do not differ significantly. For diameters of 10 mm and 20 mm, single-acting cylinders have a slightly smaller mass, while at 40 mm diameter it is the pneumatic muscle that has the smaller mass. In the case of 40 mm diameter actuators, both cylinders and muscles, as the strokes required to be carried out and implicitly their lengths increase, the difference between cylinder and muscle mass increases, i.e., the pneumatic muscles become increasingly lighter than the cylinders.