## 3 Nonlinear Modeling of Independent Metering Control for Electro-Hydraulic Mechanism  
### Non-Linear Modeling of Dual-Valve Independent Metering Control System  
A dual-valve independent metering control system of electro-hydraulic power mechanism is shown in Figure 7. The pressure and flow rate in inlet/outlet oil chambers of actuator are controlled by a servo valve or proportional valve separately. On the basis of force balance equation and  
Figure 6: Structure of multi-axis coordinated motion system based on adjacent cross-coupling strategy.  
flow rate continuum equation with two working chambers, the nonlinear tracking control law for electro-hydraulic motion control system is designed.  
The nonlinear tracking control law mainly includes speed regulation loop and speed smoothness control loop. In the former control loop, the reference pressure of actuator inlet chamber is obtained from the desired running speed. And the reference backpressure of actuator return chamber is obtained by calculating the balanced negative load. Through the coordinated control with pressure in inlet/outlet oil chamber of actuator, the expected driving force can be tracked by hydraulic driving force, then the tracking control of the motion control system can be realized.  
Dynamic equation of servo valve can be approximated by the first-order, and it is described as:  
\[x_{v}^{\bullet}=-\frac{1}{\tau_{v}}x_{v}+\frac{k_{xu}}{\tau_{v}}u \tag{1}\]  
where \(x_{v}\), \(\tau_{v}\), \(u\), and \(k_{xu}\) are the spool displacement, servo valve time constant, the control input, and the gain of the spool displacement to the input of the servo valve, respectively.  
Flow rate equation of servo valve port can be described as:  
\[q_{1}=\left\{\begin{array}{l}C_{\text{d}}Wx_{v1}\sqrt{\frac{2}{\rho}(p_{\text {s}}-p_{1})},u_{1}\geq 0\\ C_{\text{d}}Wx_{v1}\sqrt{\frac{2}{\rho}(p_{1}-p_{\text{t}})},u_{1}<0\end{array}\right. \tag{2}\]  
\[q_{2}=\left\{\begin{array}{l}C_{\text{d}}Wx_{v2}\sqrt{\frac{2}{\rho}(p_{2}-p _{\text{t}})},u_{2}\geq 0\\ C_{\text{d}}Wx_{v2}\sqrt{\frac{2}{\rho}(p_{\text{s}}-p_{\text{2}})},u_{2}<0 \end{array}\right. \tag{3}\]  
where \(C_{\text{d}}\), \(W\), \(x_{\text{v}}\), and \(\rho\) are respectively throttle valve orifice flow coefficient, servo valve orifice area gradient, servo valve spool displacement, and hydraulic oil density, respectively. \(p_{\text{s}}\), \(p_{\text{1}}\), and \(p_{\text{2}}\) are respectively the outlet pressure of the oil pump, the pressure of rodless chamber and the pressure of rod chamber of the hydraulic cylinder.  
Continuous flow rate equation of hydraulic cylinder inlet oil chamber and return oil chamber:  
\[q_{1}=A_{1}\frac{dx}{dt}+\frac{V_{1}}{\rho_{\text{e}}}\frac{dp_{1}}{dt} \tag{4}\]  
Figure 7: A dual-valve independent metering control system of electro-hydraulic power mechanism.  
\[q_{2}=A_{2}\frac{dx}{dt}-\frac{V_{2}}{\beta_{e}}\frac{dp_{2}}{dt} \tag{5}\]  
where \(V_{1}=V_{01}+A_{1}x\), \(V_{2}=V_{02}-A_{2}x\), \(\beta_{e}\), \(V_{1}\), \(V_{2}\), \(V_{01}\), and \(V_{02}\) are respectively the elastic modulus of hydraulic oil, initial volume of inlet oil chamber and return oil chamber of hydraulic cylinder, respectively. In addition, the impact of system oil leakage is ignored in above equation.  
The following equation represents the dynamic behavior of motion control system:  
\[A_{1}p_{1}-A_{2}p_{2}=m\ddot{x}+B_{m}\dot{x}+F_{L} \tag{6}\]  
where, \(A_{2}\), \(B_{m}\), \(m\), \(x\), and \(F_{L}\) are the effective piston area of the two chambers of cylinder, effective viscous damping coefficient, mass of load, output displacement, and external load interference, respectively. In addition, the impact of system oil leakage is also ignored in above equation.