## 4 Displacement Analysis of Bowden Cable-Fabric Transmission
### Knee Motion Impedance Control
In actual discrete control systems, such as a servo motor, the differential can easily amplify the noise; compared with PID control, proportional integration (PI) control is simpler in the modeling of the correction link and can more easily achieve zero pole cancellation. Therefore, the parameter setting of PI control is also more convenient. In order to test the position-based feedforward impedance control feedback system, PI position control was used in the inner loop to track the transmission force and Bowden cable-fabric displacement. The impedance control block diagram based on inner loop PI, as shown in Figure 11, was adopted as a whole, in which the lower leg model is a model of force related to angle and displacement. According to the impedance control block diagram based on the inner loop PI established in Figure 11, the control module of the Bowden cable-fabric force transmission system was built using the Simulink module in MATLAB. It consists of a Bowden cable friction compensation module, fabric friction compensation module, stiffness model module, lower leg swing angle compensation module, servo motor module, impedance module, and lower leg model. The angle of the knee joint was input into the lower leg swing angle compensation module and the fabric friction compensation module. The expected force curve was taken as the input for the other modules, and the output voltage of the motor was limited and protected, as shown in Figure 12.  
The desired force curve is the tension value when the peak value is 40 N, and the joint angle adopts the angle value in Formula (1). Since there is a three-phase motor module in the control system, it needs the Powergui module to discretize it, so the maximum sampling step length was set to 0.00001 s, and the automatic solver was used to solve it, and the influence of different impedance parameters on the control performance of the system was analyzed. When other parameters are unchanged, when the damping parameter is set to a larger value, although the system can quickly produce a response, it will produce a large overshoot and produce a certain jitter when the motor is stable and cannot achieve stability control. The value of the stiffness parameter reflects the degree of rigidity or flexibility of the system in contact with the external environment. When the value of the stiffness parameter is large, the output value will overshoot for a long time near the zero point, and the desired control effect cannot be achieved. Figure 11: Impedance control block diagram based on inner loop PI. Figure 12: Impedance control simulation model based on inner loop PI. It is assumed that the joint angle is a fixed rotation angle, which is the ideal state. However, the actual joint angle in practical application often deviates from the expected angle. After the experimental test and adjustment, the damping parameter \(B\) was set to 0.1 and the stiffness parameter \(K\) was set to 0.3. In order to verify the control effect of the impedance control model on the system output, the simulation comparison of the impedance-free control was conducted based on the simulation model in Figure 12, as shown in Figure 13. We can see that compared with impedance-free control, the output force curve with impedance control is more in line with the expected force curve and is more flexible when the force changes. And there is no large fluctuation or excessive adjustment phenomenon, so the system output is more stable and flexible.