# Development and Control of an Electro-Hydraulic Actuator System for an Exoskeleton Robot
## 4 Modeling of the EHA System  
By setting each chamber of the single rod hydraulic cylinder to the control volume and applying the continuity equation for the pressure change, it is derived as follows [27]:  
\[ \begin{array}{l}Q_{1}=\frac{dV_{1}}{dt}+\frac{V_{1}}{\bar{p}_{c}}\frac{dP_{1}}{ dt}-C_{ip}(P_{1}-P_{2})\\ Q_{2}=-\frac{dV_{2}}{dt}+\frac{V_{2}}{\bar{p}_{c}}\frac{dP_{2}}{dt}+C_{ip}(P_{1} -P_{2})-C_{op}P_{2}\end{array} . \tag{10}\]  
\(V_{1}\) and \(V_{2}\) are the cylinder volumes, \(P_{1}\) and \(P_{2}\) are the cylinder pressures, \(C_{ip}\) is the internal leakage coefficient of the piston, and \(C_{op}\) is the external leakage coefficient of the piston. \(\beta\) is the effective bulk modulus of the hydraulic fluid. After rewriting Equation (10) with respect to the change in pressure and arranging it, it is the same as Equation (11):  
\[ \begin{array}{l}\frac{dP_{1}}{dt}=\frac{\bar{p}_{c}}{\bar{p}_{1}}\Big{(}- \frac{dV_{1}}{dt}+Q_{1}-C_{ip}(P_{1}-P_{2})\Big{)}\\ \frac{dP_{2}}{dt}=\frac{\bar{p}_{c}}{\bar{V}_{2}}\Big{(}-\frac{dV_{2}}{dt}-Q_ {2}+C_{ip}(P_{1}-P_{2})-C_{op}P_{2}\Big{)}\end{array} . \tag{11}\]  
The volumes \(V_{1}\) and \(V_{2}\) of both chambers of the cylinder can be expressed as Equation (12) with respect to the initial volume:  
\[V_{1}=V_{01}+A_{1}x_{L} \tag{12}\]  
\[V_{2}=V_{02}-A_{2}x_{L} \tag{13}\]  
when the cylinder moves forward, that is, when the electric motor rotates forward, the total flow rate of the hydraulic circuit can be expressed as follows:  
\[\left\{ \begin{array}{c}Q_{P}=Q_{T}=Q_{1}-Q_{R1}\\ Q_{2}=Q_{PCV2}\end{array} \right.. \tag{14}\]  
In practice, the relief valves were not considered in this model, because they are safety devices and stay below the valve set during operation of the EHA system. So, \(Q_{P}=Q_{T}=Q_{1}\). Additionally,  
Figure 7: The hydraulic fluids circulate in the system: (**a**) is the proposed hydraulic circuit, and (**b**) is a manifold of the hydraulic circuit. \(Q_{T}\) can be expressed as \(D\omega_{m}-k_{\text{\emph{l}ankage}}(P_{1}-P_{2})\). \(D\) is the displacement of the pump, \(\omega_{m}\) is the motor speed, and \(k_{\text{\emph{l}ankage}}\) is the leakage coefficient of the pump. The flow rate \(Q_{PCV}\) through the pilot-operated check valve is expressed as shown in Equation (15), and the flow rate varies depending on the area \(A_{cv}\). As a result, the pressure-flow rate equation for the hydraulic circuit becomes a nonlinear function. The flow rate \(Q_{PCV}\) is determined according to the following equation [28]:  
\[Q_{PCV}=C_{D}\cdot A_{cv}\sqrt{\frac{2}{\rho}}\cdot\frac{\Delta P}{\left( \Delta P^{2}+P_{\sigma}^{2}\right)^{1/4}}, \tag{15}\]  
where  
\[A_{cv}(\Delta P)=\left\{ \begin{array}{ccc}A_{cv,\text{\emph{l}ank}}&\text{ for}&P_{e}\leq P_{\text{max}}\\ A_{cv,\text{\emph{l}ank}}+k\cdot(P_{e}-P_{\text{\emph{c}rack}})&\text{for}&P_{ \text{\emph{c}rack}}<P_{e}<P_{\text{max}}\\ A_{cv,\text{\emph{l}ank}}&\text{for}&P_{e}\geq P_{\text{max}}\end{array} \right.\.\]  
\(C_{D}\) is the flow discharge coefficient, \(A_{cv}\) is the instantaneous orifice passage area, \(A_{\text{\emph{l}ank}}\) is the closed valve leakage area, \(A_{\text{\emph{l}ank}}\) is the fully open valve passage area, \(P_{\sigma}\) is the minimum pressure for turbulent flow, \(P_{e}\) is the equivalent pressure differential across the control member, \(P_{\text{\emph{c}rack}}\) is the valve cracking pressure, and \(P_{\text{max}}\) is the pressure needed to fully open the valve. Using the above Equations (12) and (13) and the Laplace transform, we can obtain Equation (16):  
\[\left(\frac{V}{2\beta}s+C_{ip}\right)P_{1}(s)=-A_{1}sX_{L}(s)+D\omega(s). \tag{16}\]  
when the left chamber in the single rod is equal to the right chamber, \(V\) is the total volume and the volume \(V/2\) is \(V_{01}+A_{1}x_{L}=V_{02}+A_{2}x_{L}\).