# Parameters Sensitivity Analysis of Position-Based Impedance Control for Bionic Legged Robots' [PERSON]
## 3 HDU Position-Based Impedance Control
### Experiment of Position-Based Impedance Control  
As a typical signal, the sinusoidal response is able to evaluate the impedance control performance under the input of different frequencies and amplitudes. In this paper, a sinusoidal signal is adopted to analyze the system performance and the sensitivity of the main parameters. To study their variation patterns in different conditions, four working conditions are tested in this paper. The details of the working conditions are shown in Table 2.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Initial Value** \\ \hline Gain of servo valve \(K_{xx}/(m/V)\) & 0.0225 \\ Natural frequency of servo valve \(\omega/(\text{rad}/\text{s})\) & 628 \\ Damping ratio of servo valve \(\xi\) & 0.77 \\ Effective piston area \(A_{p}/\text{m}^{2}\) & \(3.368\times 10^{-4}\) \\ Volume of inlet chamber \(V_{g1}/\text{m}^{3}\) & \(6.2\times 10^{-7}\) \\ Volume of outlet chamber \(V_{g2}/\text{m}^{3}\) & \(8.6\times 10^{-7}\) \\ Piston stroke \(L/\text{m}\) & 0.05 \\ Initial position of piston \(L_{0}/\text{m}\) & 0.025 \\ Supply pressure \(P_{s}/\text{Pa}\) & \(1\times 10^{-7}\) \\ Tank pressure \(P_{0}/\text{Pa}\) & \(0.5\times 10^{6}\) \\ Gain of position sensor \(K_{x}/\text{V}/(\text{m})\) & \(54.9\times 10^{-3}\) \\ Outer linkage coefficient of servo valve \(C_{cp}/(\text{m}^{3}/\text{s-Pa})\) & 0 \\ Inner linkage coefficient of servo valve \(C_{ip}/(\text{m}^{3}/\text{s-Pa})\) & 2.38 \(\times 10^{-13}\) \\ Equivalent mass \(m_{t}/\text{kg}\) & 1.1315 \\ Effective bulk modulus \(\beta_{p}/\text{Pa}\) & \(8\times 10^{8}\) \\ Conversion coefficient \(K_{d}\) & \(1.248\times 10^{-4}\) \\ Load stiffness \(K/(\text{N}/\text{m})\) & 0 \\ Viscous damping coefficient \(B_{p}/(N\cdot\text{s}/m)\) & \(54.9\times 10^{-3}\) \\ Proportional gain \(K_{p}\) & 30 \\ differential gain \(K_{i}\) & 10 \\ Desired stiffness \(K_{D}/(N/m)\) & \(1\times 10^{6}\) \\ Desired damp \(C_{D}/(N\cdot\text{s}/m)\) & \(5\times 10^{4}\) \\ Desired mass \(M_{D}/\text{kg}\) & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters and initial values of the simulation model.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**No.** & \multicolumn{2}{c}{**Working Conditions**} \\ \cline{2-3}  & **Frequency \(f\) (Hz) & **Bias, Amplitude \(A\) (N)** \\ \hline
1 & 1 & 1500, 1000 \\
2 & 2 & 1500, 1000 \\
3 & 1 & 2500, 2000 \\
4 & 2 & 2500, 2000 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Four working conditions researched in this paper.  
The mean values of performance indexes in different conditions are shown in Table 3.  
As it can be seen in Table 3, the values of the two performance indexes are close in experiment and simulation, which indicates that the experimental curves fit the simulation curves well. As the position-based impedance control theory in Figure 8 shows, the actual position is greater than the desired one. Thus, the amplitude attenuation is a negative value. It increases with the increase of the disturbance force's amplitude, but has few relationships with the frequency. In contrast, the phase angle delay increases with the frequency, but has few relationships with the amplitude.