## 3 Determination of the Energy-to-Mass Ratio  
The energy-to-mass (\(R_{E/m}\)) ratio is a quantity devised for comparing the energy efficiency of different types of actuators. In [26], [PERSON] proposes the computational relationship of this indicator:  
\[R_{\frac{E}{m}}=\frac{\int\ F(s)\cdot ds}{m(s)} \tag{19}\]  
where \(F(s)\) is the force developed by the pneumatic motor, \(s\) is the stroke and \(m\) is the mass of the motor. For pneumatic muscles, Equation (19) is adapted by taking into account the specific axial contraction \(\varepsilon\) = s/l:  
\[R_{\frac{E}{m}}=\frac{l\cdot\int\ F(\varepsilon)\cdot d\varepsilon}{m(s)}= \frac{\int\ F(s)\cdot ds}{m(s)} \tag{20}\]  
The numerator in Equations (19) and (20) represents the energy developed by the motors over the entire stroke and is obtained by the integration of the force Equations (13)-(18). The integrals in the numerator represent the surfaces between the variation curves of the forces, the abscissa of the graphs and the maximum and minimum limits of the motors' displacements. Figure 6 highlights these surfaces for the ESNU cylinders and the DMSP pneumatic muscles. Equations (21)-(23) represent the calculation formulae of the energy-to-mass ratios for the three studied pneumatic cylinders:  
\[R_{\frac{E}{m},\ \text{ESNU\_10}}=\frac{1.145}{0.27\times\text{s}+0.0373} \tag{21}\]  
\[R_{\frac{E}{m},\ \text{ESNU\_20}}=\frac{4.565}{0.72\times\text{s}+0.1868} \tag{22}\]  
\[R_{\frac{E}{m},\ \text{ESNU\_40}}=\frac{17.95}{2.4\times\text{s}+0.661} \tag{23}\]  
For the pneumatic muscles, first the equations of the energies developed over the working interval are determined. Figure 6b shows the surface that materializes the integrals in Equation (20). \[E_{\text{DMSP\_10}}(s)=\int\limits_{0}^{0.05}F_{\text{DMSP\_10}}(s)\cdot ds =10.0093 \tag{24}\]  
\[E_{\text{DMSP\_20}}(s)=\int\limits_{0}^{0.05}F_{\text{DMSP\_20}}(s)\cdot ds =40.2748 \tag{25}\]  
\[E_{\text{DMSP\_40}}(s)=\int\limits_{0}^{0.05}F_{\text{DMSP\_40}}(s)\cdot ds =153.8121 \tag{26}\]  
The energy-to-mass ratios of the three pneumatic muscles will be:  
\[R_{\frac{E}{m},\ \text{DMSP\_10}}=\frac{E_{\text{DMSP\_10}}(s)}{m_{\text{ DMS\_10}}(s)}=\frac{10.0093}{0.47\times s+0.077} \tag{27}\]  
\[R_{\frac{E}{m},\text{DMSP\_20}}=\frac{E_{\text{DMSP\_20}}(s)}{m_{\text{ DMS\_20}}(s)}=\frac{40.2748}{0.89\times s+0.238} \tag{28}\]  
\[R_{\frac{E}{m}}=\frac{l\cdot\int\ F(\varepsilon)\cdot d\varepsilon}{m(s)}= \frac{\int\ F(s)\cdot ds}{m(s)} \tag{20}\]0Figure 7 shows the variation in the energy-to-mass ratios versus stroke for the six pneumatic motors and Table 6 presents the values of these ratios for a 50 mm stroke. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Motor Dimension** & **Energy-to-Mass Ratio for a 50 mm Stroke** \\
**[mm]** & **[Nm/kg]** \\ \hline  & **ESNU** & **DMSP** \\ \hline
10 & 22.5394 & 99.595 \\
20 & 20.4892 & 142.5657 \\
40 & 22.9834 & 202.3843 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Energy-to-mass ratios of the studied pneumatic motors. Figure 6: Energy developed by: (**a**) ESNU pneumatic cylinders; (**b**) DMSP pneumatic muscles. Figure 7 reveals that in the case of single-acting cylinders, the values of the energy-to-mass ratios are approximately the same regardless of their diameters. This observation does not apply in the case of pneumatic muscles, where the energy-to-mass ratio in a 40 mm diameter muscle is about two-times larger than that in a 10 mm diameter one. The above table shows that for the studied pneumatic actuators with a 50 mm stroke, the energy-to-mass ratios of the pneumatic muscles are about 4- to 9-times larger than those of single-acting cylinders. Consequently, from the viewpoint of energy efficiency, pneumatic muscles are recommended for applications that require the development of a large force while ensuring maximum energy efficiency. Table 7 presents an overview of the results obtained by the comparative study of the two types of linear pneumatic motors. The (+) sign indicates a favorable characteristic while (-) indicates a disadvantage.