## 3 Energy-Saving Principle of HPCA
### Pressure Characteristics of HPCA  
#### 3.2.1 Dynamic Pressure Characteristic  
In Figure 4, \(P\), \(V\), \(A\), and \(Q\) represent pressure, volume, area, and flow, respectively, and the subscripts _pis_, \(r\), \(acu\), and \(S\) denote the piston chamber, ring chamber, accumulator, and oil supply port, respectively. The area ratio \(\alpha\) can be described as \(A_{r}/A_{pis}\). \(x_{p}\) is the respective positions of the piston and valve core. \(V_{g0}\) and \(P_{g0}\) represent the initial working volume and inflation pressure, respectively, of the accumulator. The output force and load force are indicated by \(f_{h}\) and \(f_{l}\), respectively. \(Q_{c}\) represents the flow through the check valve.  
Figure 4: Definition of variables for hydraulic schematic diagram of HPCA.  
Figure 3: Foot trajectory and piston movement. (**A**) Foot trajectory; (**B**) initial points; (**C**) piston displacement. Numbers 1 and 2 represent the swing phase and supporting phase, respectively. F/E = flexion and extension.  
For the actuator with the original structure (Figure 1A), regardless of the internal and external leakage, pressure dynamics in working chambers can be expressed as [27]:  
\[\left\{\begin{array}{lcl}\dot{p}_{pis}&=&\frac{(Q_{pis}-A_{pis}\dot{x}_{p}) \beta_{e}}{A_{pis}(x_{p0}+x_{p})}\\ \dot{p}_{r}&=&\frac{(Q_{r}+A_{r}\dot{x}_{p})\beta_{e}}{A_{r}(x_{stroke}-x_{p0}- x_{p})},\end{array}\right. \tag{2}\]  
where \(\beta_{e}\) is the effective bulk modulus of the system; \(x_{p0}\) is the initial position of the piston, and \(x_{stroke}\) is the total stroke of the hydraulic cylinder.  
According to [PERSON]'s law [28], the relationship between the working chamber of the accumulator and the inflation pressure can be expressed as:  
\[P_{g0}\cdot V_{g0}^{n}=P_{accu}\cdot V_{accu}^{n}=CONST, \tag{3}\]  
where \(n\) is determined by the working status of the accumulator. In the isothermal condition, \(n=1\); in the adiabatic condition, \(n=1.4\).  
Differentiating \(P_{accu}\) from Equation (3) gives:  
\[\dot{P}_{accu}=\left\{\begin{array}{lcl}0&(P_{pis}\leq P_{g0})\\ \frac{-nP_{g0}V_{g0}^{n}Q_{accu}}{V_{accu}^{n+1}}&(P_{pis}>P_{g0})\end{array} \right., \tag{4}\]  
where \(Q_{accu}\) is the flow rate at the port of the accumulator.  
For the HPCA, when the gasbag in the accumulator is compressed (\(P_{pis}>P_{g0}\)), the conditions  
\[\left\{\begin{array}{lcl}P_{pis}&=&P_{accu}\\ \dot{P}_{pis}&=&\dot{P}_{accu}\end{array}\right. \tag{5}\]  
are satisfied and the pressure dynamics of the actuator can be described as:  
\[\left\{\begin{array}{lcl}\dot{P}_{pis}&=&\frac{nP_{g0}V_{g0}^{n}(Q_{pis}-A_ {pis}\dot{x}_{p})}{V_{accu}^{n+1}}\\ \dot{P}_{r}&=&\frac{\beta_{e}(Q_{r}+A_{r}\dot{x}_{p}+Q_{c})}{V_{r}}.\end{array}\right. \tag{6}\]  
Figure 5 shows the relationship between the scale factors \(\beta_{e}/V_{pis}\) and \(nP_{g0}V_{g0}^{n}/V_{accu}^{n+1}\) that determine the rate change of the piston chamber pressure as described in Equations (2) and (6). The referenced data are shown in Table 1, among which the working volume of the accumulator \(V_{accu}\) is set to 25% of the initial working volume \(V_{g0}\) that is the minimum value allowed during normal use. Therefore, the scale factor of the HPCA can be regarded as the maximum value at each point. It can be seen from the curves that the addition of the accumulator greatly reduces the rate change of the chamber pressure, which is advantageous for reducing pressure fluctuations.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Description** & **Symbol** & **Value** & \\ \hline piston diameter & \(\backslash\) & 0.025 & m \\ piston rod diameter & \(\backslash\) & 0.012 & m \\ piston stroke & \(x_{stroke}\) & 0.072 & m \\ initial working volume & \(V_{g0}\) & \(1.3\times 10^{-5}\) & m\({}^{3}\) \\ bulk modulus & \(\beta_{e}\) & 1400 & Mpa \\ working volume & \(V_{accu}\) & \(1.3\times 10^{-5}\times\) 0.25 & m\({}^{3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of actuator.