## 4 Displacement Analysis of Bowden Cable-Fabric Transmission
### Knee Motion Impedance Control  
In the power-assisted control of exoskeletons, the application of a Bowden cable-fabric force transmission system should also be combined with impedance control. On the one hand, it can achieve accurate control of force and displacement and reduce the energy consumption of human walking; on the other hand, it can be more compatible with the law of human movement, improve user comfort, and achieve the flexibility of human-computer interaction. In impedance control, the most common mathematical expression in the form of a dynamic equation is as follows:  
\[M\big{(}\ddot{x}_{r}-\ddot{x}\big{)}+B\big{(}\dot{x}_{r}-\dot{x}\big{)}+K(x_{r }-x)=F_{e} \tag{12}\]  
where \(M\), \(B\), and \(K\) are inertial, damping, and stiffness parameters of the impedance model; \(\ddot{x}\), \(\dot{x}\), and \(x\) are the actual acceleration, velocity, and position of the end; \(\ddot{x}_{r}\), \(\dot{x}_{r}\), and \(x_{r}\) are the expected acceleration, velocity, and position of the end; and \(F_{e}\) represents the difference between the expected force and the actual force of the end. In this paper, the impedance control strategy based on the inner loop position control was adopted. In this impedance control, the end force of the Bowden cable-fabric system is controlled by the change in position. If \(x_{r}-x=\Delta x\) is substituted into Formula (12), it can be expressed as follows:  
\[M\Delta\ddot{x}+B\Delta\dot{x}+K\Delta x=F_{e} \tag{13}\]  
Figure 10: The swing model of the lower leg. For position-based impedance control, there are two main components: one part is the position control of the inner loop, which is generally realized by proportional integral differential (PID) control; the other part is the impedance control of the outer loop. When the Bowden cable-fabric force transfer system is applied to the body part, the motor outputs the expected force. Through the end of the force transfer device, the force is output to the lower limb, while also creating a contact force between the body and the device. An impedance controller based on force feedback is used to input the actual force into the impedance model and convert it into the displacement correction of the Bowden cable-fabric. Then, the expected position, actual position, and displacement correction are input into the inner loop position controller, whose main function is to accurately track the input reference position. This can be obtained through the transformation of Formula (13):  
\[F_{e}(s)=\left(Ms^{2}+Bs+K\right)x(s) \tag{14}\]  
The relationship between force and displacement, that is, the transfer function, can be expressed as follows:  
\[H(s)=\frac{\Delta x}{F_{e}}=\frac{1}{(Ms^{2}+Bs+K)} \tag{15}\]  
\(F_{e}\) is the force input into the impedance model, that is, the error between the expected force and the actual output force. And \(\Delta x\) is the amount of displacement that is converted by the impedance control loop to the desired amount, and then, it compensates to the position control of the inner loop. Because the system is slow in actual motion and the inertia parameters are relatively small, the effect in the system is negligible. Simulation and Experimental Research of Bowden Cable-Fabric Force Transfer Control Based on Friction/Displacement Compensation  
In Sections 3 and 4, we described a model which compensates the force transfer loss and displacement loss of an exoskeleton in the process of human walking and introduced the impedance control model to achieve flexible and accurate control. In this section, the effect of the impedance control system on force and displacement control will be evaluated with a simulation and experiment. Simulation of Bowden Cable-Fabric Force Transfer Control Based on Friction/Displacement Compensation  
The assistance of knee joint motion is controlled based on the position inner loop and impedance outer loop control algorithm, and the position inner loop is PID control [24].