# Design and Trajectory Tracking Control of a Magnetorheological Prosthetic Knee Joint
## 3 Design of MR Damper Used in the MR Prosthetic Knee Joint
### Establishment of Forward Mechanics Model of the MR Damper  
The Bingham model can effectively reflect the relationship between the output damping force and the structural parameters, but the accuracy of the model is not high when the damper works in low speed. To accurately reflect the mechanical properties of the MR damper, the adjustable sigmoid model shown in Figure 10 is used to establish a positive mechanical model of the MR damper. The adjustable sigmoid model is formed by the parallel combination of the sigmoid function and the viscous damping element, which is expressed as:  
\[F=F_{\text{m}}\frac{1-e^{-a(\dot{x}+kx)}}{1+e^{-a(\dot{x}+kx)}}+C_{0}(\dot{x}+ kx)+f_{0} \tag{13}\]  
where \(F_{\text{m}}\) is the maximum yield force, which affects the length of the hysteresis loop; \(a\) is the parameter related to the damping coefficient of the pre-yield zone; \(k\) is the scale factor of the hysteresis loop width; \(C_{0}\) is the viscous damping coefficient, and \(f_{0}\) is the bias damping force.  
Since the applied current has a great influence on the output damping force, the parameters of the damping force data under different currents are identified. Based on the least square method, the relationship between parameters and the currents is acquired. The parameter identification results are shown in Table 4. It can be seen that the parameters \(a\), \(k\) and \(C_{0}\) do not change significantly with the current, but the parameters \(F_{\text{m}}\) and \(f_{0}\) are greatly affected by the current. Therefore, the average value of the identified \(a\), \(k\), and \(C_{0}\) is used to replace the parameters in the model, that is, the values of \(a\), \(k\), and \(C_{0}\) are 0.32, \(-0.47\) and 0.33, respectively.  
Figure 10: Schematic diagram of adjustable sigmoid model.  
Figure 9: Dynamic performance of MR damper under different currents. (**a**) Damping force-displacement, (**b**) damping force-speed.  
The polynomial fitting method was adopted to fit the relationship between the parameters \(F_{\text{m}}\),\(f_{0}\) and the current, and the results are shown in Figure 11. It can be seen that the parameter \(F_{\text{m}}\) increases with increase of the current, while the parameter \(f_{0}\) decreases with increases of the current. The relationship between parameters \(F_{\text{m}}\),\(f_{0}\) and current can be expressed as follows:  
\[F_{\text{m}}=b_{1}I+c_{1} \tag{14}\]  
\[f_{0}=b_{2}I+c_{2} \tag{15}\]  
where \(b_{1}\), \(c_{1}\), \(b_{2}\) and \(c_{2}\) are fitting parameters.  
According to the results of parameter identification, an adjustable sigmoid model is established and compared with the experimental data, the result is shown in Figure 12. It can be seen that the established adjustable sigmoid model can better reflect the characteristics of damping force with displacement and damping force with velocity. Compared with the Bingham model based on structural parameters, the adjustable sigmoid model can better reflect the damping characteristics of the MR damper at low speeds.