# Parameters Sensitivity Analysis of Position-Based Impedance Control for Bionic Legged Robots' [PERSON]
## 3 HDU Position-Based Impedance Control
### Mathematic Model of HDU Position-Based Impedance Control
#### 3.1.2 The Block Diagram and State Space Presentation of Impedance Control System  
The block diagram of HDU position-based impedance control is shown in Figure 7, where the detailed deduction and performance analysis of the inner loop is presented in previous research [5; 15]. In Figure 7, \(m_{t}\) is conversion mass (including the piston, the displacement sensor, the force sensor, the connecting pipe, and the oil in the servo cylinder), \(K_{x}\) is input position, is position sensor gain, \(K_{PID}\) is proportion-integration-differentiation (PID) controller gain including proportional gain \(K_{P}\), integral gain \(K_{I}\) and differential gain \(K_{D}\), \(K\) is load stiffness, \(B_{p}\) is load damping, \(F_{L}\) is load force, \(X_{z}\) is servo valve spool displacement, \(X_{p}\) is servo cylinder piston displacement, \(V_{\xi 1}\) is volume of input oil pipe, \(V_{\xi 2}\) is volume of output oil pipe, \(F_{f}\) is friction, \(U_{r}\) is input voltage, \(U_{g}\) is controller output voltage, \(Q_{1}\) is inlet oil flow, and \(Q_{2}\) is outlet oil flow. Figure 5: Force schematic of HDU. Figure 6: Position-based impedance control schematic. Denote,  
\[K_{servo}=\frac{k_{axv}}{(\frac{\beta^{2}}{\omega^{2}}+\frac{2^{2}_{c}}{\omega} \mathrm{s}+1)} \tag{3}\]  
where \(K_{servo}\) is the transfer function of the servo valve, \(\omega\) is the natural frequency of the servo valve, \(\zeta\) is the damping ratio of the servo valve, and \(K_{axv}\) is the servo valve gain. \[K_{1}=K_{d}\sqrt{\left\{\frac{[1+\mathrm{sgn}(x_{v})]P_{g}}{2}+\frac{[-1+ \mathrm{sgn}(x_{v})]P_{0}}{2}\right\}-\mathrm{sgn}(x_{v})P_{1}} \tag{4}\]  
\[K_{2}=K_{d}\sqrt{\left\{\frac{[1-\mathrm{sgn}(x_{v})]P_{g}}{2}+\frac{[-1- \mathrm{sgn}(x_{v})]P_{0}}{2}\right\}+\mathrm{sgn}(x_{v})P_{2}} \tag{5}\]  
where, \(K_{1}\) and \(K_{2}\) express the transfer function of nonlinear pressure flow, \(K_{d}=C_{d}W\sqrt{2/\rho}\) (\(K_{d}\) is defined as conversion coefficient in this paper), \(C_{d}\) is the orifice flow coefficient of the spool valve, \(W\) is the area gradient of the spool valve, \(\rho\) is the density of hydraulic oil, \(p_{s}\) is the system supply's oil pressure, \(p_{1}\) is the left cavity pressure of the servo cylinder, \(p_{2}\) is the right cavity pressure of the servo cylinder, and \(p_{0}\) is the system return oil pressure. \[K_{3}=\frac{1}{C_{ip}+C_{ep}+\frac{V_{d1}+A_{p}L_{0}+A_{p}X_{p}}{\beta_{e}}s} \tag{6}\]  
\[K_{4}=\frac{1}{-C_{ip}-C_{ep}-\frac{V_{g2}+A_{p}(L-L_{0})-A_{p}X_{p}}{\beta_{e} }s} \tag{7}\]  
where, \(K_{3}\) and \(K_{4}\) express the transfer function of flow continuity, \(L\) represents the total piston stroke of the servo cylinder, \(L_{0}\) is the initial piston position of the servo cylinder, \(C_{ip}\) is the internal leakage coefficient of the servo cylinder, \(C_{ep}\) is the external leakage coefficient of the servo cylinder, \(A_{p}\) is the effective piston area of the servo cylinder, and \(\beta_{e}\) is the effective bulk modulus. The state variables in Figure 7 are expressed as follows:  
\[x_{1}=x_{p},\ x_{2}=\dot{x}_{p},\ x_{3}=x_{v},\ x_{4}=\dot{x}_{v},\ x_{5}= \ddot{x}_{v}-\omega^{2}K_{x}K_{axv}K_{p}\mu_{1},\ x_{6}=P_{1},\ x_{7}=p_{2},\ x_{8}=x_{e}\]  
where the input variables are expressed as follows:  
\(u_{1}=x_{r},\ u_{2}=F_{L}+F_{f}\)  
Disturbance variables are expressed as follows:  
\(w_{1}=F_{L}+F_{f}\)  
Figure 7: Block diagram of HDU position-based impedance control. The state space of the system can be expressed as follows:  
\[\left\{