# Delay Analysis and the Control of Electro-Hydrostatic Actuators
## 2 Materials and Methods
### Model of Hydraulic System
Where \(C_{g}\) is the total leakage coefficient of the piston cylinder, \(\beta_{e}\) is the bulk modulus of elasticity of the oil, \(V_{t}\) is the working chamber volume of the piston cylinder, \(F\) is the output force, \(m_{t}\) is the total mass of the load, \(B_{c}\) is the viscous damping coefficient, and \(K\) is the stiffness coefficient of the load. Figure 5: The mechanical structure and the symbol of the hydraulic lock. Figure 6 shows the open-loop model in AMESIM. The hydraulic lock in Figure 6 is built according to the structure of the hydraulic lock in Figure 5 to reflect the characteristics of the hydraulic lock. In order to test the EHA system, it is necessary to build a hydraulic loading system. The loading system in Figure 7 consists of an accumulator 7, an oil tank 8, four electromagnetic valves 1, 2, 3, and 4, and a piston-cylinder 5. The electromagnetic valve controls the flow path. The piston-cylinder of the loading system outputs load force to the EHA system. The accumulator provides energy to the loading system [36]. Figure 6: AMESIM model of the hydraulic system. Figure 7: The schematic diagram of the hydraulic loading system. If electromagnetic valves 1 and 4 are opened and the electromagnetic valves 2 and 3 are closed, a spring-like load to the right will be output. While the electromagnetic valves 2 and 3 are opened and the electromagnetic valves 1 and 4 are closed, a spring-like load to the left will be output. The overflow pressure of the relief valve determines the upper limit of the spring-like force. Figure 8 shows the model of the hydraulic loading system in AMESIM and the component number in Figure 8 is the same as them in Figure 7.  
The AMESIM-SIMULINK interface makes it possible to perform simulations with a combination of the hydraulic system and the motor system. According to the hydraulic device that was used in the experimental facility, the parameters of the hydraulic model are shown in Table 2. \begin{table}
\begin{tabular}{c c} \hline
**Parameter Name** & **Quantity/Unit** \\ \hline Pump delivery & 4 mL/rev \\ Characteristic flow rate of orifice (leakage simulation) & 0.036 L/min \\ The corresponding pressure drop of the orifice & 4 bar \\ Check valve cracking pressure & 0.1 bar \\ Pressure relief valve creaking pressure & 30 bar \\ Pressure relief valve flow rate pressure gradient & 30 L/min/bar \\ The spring rate of the hydraulic lock & 3 N/mm \\ Spool diameter of the hydraulic lock & 7.1 mm \\ Hole diameter of the hydraulic lock & 5 mm \\ Spring preload of the hydraulic lock & 8 N \\ Mass of the spool & 0.2 kg \\ Piston diameter of hydraulic lock & 9 mm \\ Rod diameter of hydraulic lock & 5 mm \\ Piston mass of hydraulic lock & 0.2 kg \\ Piston diameter of Piston-Cylinder & 32 mm \\ Piston-Cylinder’s length of stroke & 80 mm \\ Adiabatic index of air/gas & 1.4 \\ Rated speed of the pump & 1500 rev/min \\ Rod diameter of Piston-Cylinder & 25 mm \\ Mass of load & 0.5 kg \\ Coulomb friction force of piston-cylinder & 25 N \\ Stiction force of piston-cylinder & 25 N \\ Diameter of pipe & 15 mm \\ Length of pipe (Aluminum) & 200 mm \\ Length of pipe (hose) & 430 mm \\ Wall thickness of pipe & 5 mm \\ \hline \end{tabular}
\end{table} 
Table 2: The parameter of the hydraulic system. Figure 8: AMESIM model of hydraulic loading system. The established hydraulic model uses the hydraulic library and the hydraulic component design (HCD) library in the AMESIM library. An orifice simulates the leakage of the pump. When the pressure difference is 4 bar, the leakage is 0.036 L/min. The pump elements with a leakage simulation can reflect the nonlinear characteristics caused by leakage in the experimental facility.