## 3 Control Strategy Design
### Valve Control
#### 3.1.1 Mathematical Modeling of the Velocity-Load Feedforward on the Metering Modes  
In the PE mode, the \(K_{sa}\) valve is fully opened and the \(K_{bt}\) valve is controlled. The flows across the inlet and outlet orifice of the cylinder are provided by the following orifice equations:  
\[Q_{a}=\left|\dot{x}\right|A_{a}=C_{d}A_{sa}\sqrt{\frac{2(p_{s}-p_{a})}{\rho}} =K_{sa}\sqrt{p_{s}-p_{a}} \tag{3}\]  
\[Q_{b}=\left|\dot{x}\right|A_{b}=C_{d}A_{bt}\sqrt{\frac{2(p_{b}-p_{r})}{\rho}} =K_{bt}\sqrt{p_{b}-p_{r}} \tag{4}\]  
where \(Q_{a}\), \(Q_{b}\) are the flow rate at the bore and rod chamber, \(C_{d}\) is a coefficient of discharge, \(\rho\) is the density of the fluid, \(A_{sa,ab,orbt}\) is the area of orifice, \(K_{sa,ab,orbt}\) is the valve conductance of the proportional valve, and \(p_{s}\), \(p_{r}\) are the pressure at the outlet of the pump and the pressure at the tank. By rearranging Equations (3) and (4), the chamber pressure of the cylinder can be written as:  
\[p_{a}=p_{s}-\left(\frac{\left|\dot{x}\right|A_{a}}{K_{sa}}\right)^{2} \tag{5}\]  
\[p_{b}=p_{r}+\left(\frac{\left|\dot{x}\right|A_{b}}{K_{bt}}\right)^{2} \tag{6}\]  
By substituting Equations (5) and (6) into Equation (2), can be obtained:  
\[\left[p_{s}-\left(\frac{\left|\dot{x}\right|A_{a}}{K_{sa}}\right)^{2}\right]A _{a}-\left[p_{r}+\left(\frac{\left|\dot{x}\right|A_{b}}{K_{bt}}\right)^{2} \right]A_{b}-mg-B_{v}\dot{x}-m\ddot{x}=0 \tag{7}\]  
Following the control strategy and the valve operation in PE mode (shown in Table 1), the signal control of the \(K_{sa}\) valve is always fixed at the maximum value. Meanwhile, the \(K_{bt}\) valve is always adjusted so that the cylinder operates according to the design strategy. Therefore, there exists a ratio factor \(k\) which varies depending on the velocity, pressure, and load of the system during operation. The relationship between valve \(K_{sa}\) and valve \(K_{bt}\) is shown as follows:  
\[k_{PE}=\frac{K_{sa}}{K_{bt}}=\frac{U_{sa}^{VLF}}{U_{bt}^{VLF}} \tag{8}\]where \(k_{PE}\) is the ratio factor of the PE mode. From Equations (7) and (8), the ratio factor \(k_{PE}\) is calculated as the following equation:  
\[k_{PE}=\sqrt{\frac{\left(p_{s}A_{a}-p_{r}A_{b}-mg-B_{v}\dot{x}-m\ddot{x}\right)K _{sa}^{2}-\dot{x}^{2}A_{a}^{3}}{\dot{x}^{2}A_{b}^{3}}} \tag{9}\]  
Using the same approach as in PE mode, the mathematical analysis of HSRE mode can be described as shown in Figure 2b. However, the flow of the bore chamber is the combination between the flow from the rod chamber and the flow from the pump through the \(K_{sa}\) valve. Therefore, the orifice equations of valve \(K_{sa}\) and valve \(K_{sb}\) are difference based on the structure of the hydraulic circuit. Similarly with PE mode, the chamber pressure, and the ratio factor \(k_{HSRE}\) of the HSRE mode are obtained as:  
\[ \begin{split}& p_{a}=p_{s}-\left(\frac{\left|\dot{x}\right|(A_{a}-A_{ b})}{K_{sa}}\right)^{2}\\ & p_{b}=p_{s}-\left(\frac{\left|\dot{x}\right|(A_{a}-A_{b})}{K_{sa }}\right)^{2}+\left(\frac{\left|\dot{x}\right|A_{b}}{K_{sb}}\right)^{2}\\ & k_{HSRE}=\sqrt{\frac{\left(p_{s}(A_{a}-A_{b})-mg-B_{v}\dot{x}-m \ddot{x}\right)K_{sa}^{2}-\dot{x}^{2}(A_{a}-A_{b})^{3}}{\dot{x}^{2}A_{b}^{3}} =\frac{U_{sa}^{ULF}}{U_{sb}^{ULF}}}\end{split}  \tag{10}\]  
For PR mode, the valve control method is the same with the PE mode. However, the flow through the \(K_{bt}\) valve comes from the bore chamber. In this mode, the pressure of the cylinder chamber and the ratio factor \(k_{PR}\) of the PR mode can be calculated as follows:  
\[ \begin{split}& p_{a}=p_{r}+\left(\frac{\left|\dot{x}\right|A_{a}}{K_{bt }}\right)^{2}\\ & p_{b}=p_{s}-\left(\frac{\left|\dot{x}\right|A_{b}}{K_{sa}} \right)^{2}\\ & k_{PR}=\sqrt{\frac{\left(p_{s}A_{b}-p_{r}A_{a}+mg+B_{v}\dot{x}+m \ddot{x}\right)K_{sa}^{2}-\dot{x}^{2}A_{b}^{3}}{\dot{x}^{2}A_{a}^{3}}}=\frac{U _{sa}^{ULF}}{U_{bt}^{ULF}}\end{split}  \tag{11}\]  
The LSRR mode is operated by controlling the \(K_{ab}\) valve and fully opening the \(K_{bt}\) valve.