## 2 Characterization and Methodology
### Mathematical Model
Dynamic flow forces, gas inertia and pressure loss in tubes can be neglected. 6. Steady state flow forces are considered only. The electromagnetic component can be used to control gas flow through adjusting spool position of the on/off valve. For turbulent flow, the spool orifice equation can be used to represent the gas flow through VCM-DHPV, which includes the subsonic and choked flow regimes. Because the high-pressure air in pneumatic components and pneumatic system is compressible, as the ratio of the upstream pressure \(p_{u}\) to the downstream pressure \(p_{d}\) is larger than the critical value \(\sigma_{cr}\), the flow regime can be regarded subsonic and the mass flow depends nonlinearly on both pressures. However, when the ratio is smaller than the critical value \(\sigma_{cr}\), the flow achieves sonic velocity (choked flow) and depends  
Figure 4: Equivalent electrical circuit of VCM. linearly on the upstream pressure \(p_{u}\)[8; 22]. In these two cases, the air mass flow depends linearly on the spool position of the on/off valve. The standard equation for the mass flow rate through the spool orifice can be modeled as:  
\[\dot{m}_{t}=\alpha\cdot S\cdot p_{u}\cdot\sqrt{\frac{k}{RT}}\cdot\phi(p_{u},p_{ d}) \tag{5}\]  
where \(\alpha\) is the coefficient of contraction, \(S\) is the passage area of the spool orifice, \(S=4C_{d}Dx(t)\), \(C_{d}\) is discharge coefficient, \(D\) is the orifice diameter of the valve, \(x(t)\) is opening of the on/off valve, \(k\) is specific heat ratio of air, \(R\) is gas constant of air, and \(T\) is absolute temperature. \(phi(p_{u},p_{d})\) is given by [24; 25]:  
\[\phi(p_{u},p_{d})=\left\{ \begin{array}{l}\sqrt{\left(\frac{2}{k+1}\right)^{ \frac{k+1}{k-1}}}when\left(\frac{p_{d}}{p_{u}}\leq\sigma_{cr}\right)\\ \sqrt{\frac{2}{k-1}\left(\left(\frac{p_{d}}{p_{u}}\right)^{\frac{2}{k}-\left( \frac{p_{d}}{p_{u}}\right)^{\frac{k+1}{k}}}\right)}when\left(\frac{p_{d}}{p_{u }}>\sigma_{cr}\right)\end{array} \right. \tag{6}\]  
where \(\sigma_{cr}\) is critical pressure ratio, which is equal to 0.5283 and can be defined as:  
\[\sigma_{cr}=\left(\frac{2}{k+1}\right)^{\frac{k}{k-1}} \tag{7}\]  
Based on the momentum conservation principle, the steady gas flow force on valve spool can be calculated as:  
\[F_{s}=\dot{m}_{t}(x(t))v\cos\theta \tag{8}\]  
Besides, the transient gas flow force acting on the valve spool can be written by:  
\[F_{t}=\pm L_{\circ}\frac{\mathrm{d}\dot{m}_{t}}{\mathrm{d}t}=\pm L_{\circ} \alpha\frac{\mathrm{d}S}{\mathrm{d}t}p_{u}\cdot\sqrt{\frac{k}{RT}}\cdot\phi(p_ {u},p_{d})=\pm 4L_{\circ}\alpha C_{d}Dp_{u}\cdot\sqrt{\frac{k}{RT}}\cdot\phi(p_{u}, p_{d})\frac{\mathrm{d}x(t)}{\mathrm{d}t} \tag{9}\]  
where \(L_{\circ}\) is damping length of the on/off valve, \(\theta\) is gas flow jet angle, and \(v\) is gas flow velocity. Therefore, after adopting the Laplace transform for Equations (1) and (3) under the boundary condition \(X(0)=0\) and the initial condition \(X^{\prime}(0)=0\), they can be represented as follows:  
\[ms^{2}X(s)+bsX(s)+k_{s}X(s)+F_{f}+F_{t}+F_{s}=K_{e}I(s) \tag{10}\]  
\[(R+L_{s})I(s)+K_{e}sX(s)=U(s) \tag{11}\]  
Consequently, with Equations (10) and (11), the transfer function between the \(X(s)\) and the \(U(s)\) for the on/off valve driven by the VCM directly can be derived as:  
\[G_{p}(s)=\frac{X(s)}{U(s)}=\frac{K_{e}/(R+L_{s})}{ms^{2}+[K_{e}^{2}/(R+L_{s})+ b]s+k_{s}} \tag{12}\]