## 2 Materials and Methods
### Displacement and Tension Transmission Characteristics of a Closed-Circuit Cable-Pulley System  
The displacement and tension transmission between the driving pulley and the driven pulley are generally closed-circuit cable-pulley systems. To derive the closed-circuit cable-pulley model, we assume that:  
1. The applied moment is always positive.
2. The cable in symmetrical configuration does not go slack.
3. The tension of all the infinitesimal cable segments is the same under initial pretension loads.  
As shown in Figure 5, it is a closed-circuit cable-pulley system consisting of cable \(T\), cable \(B\) and 10 pulleys. The radius of the cable is \(r\), the friction coefficient of the cable moving along the pulleys is \(\mu_{c}\), and the pulley \(P_{0}\) and \(P_{5}\) are, respectively, the driving pulley and the driven pulley, whose radii are, respectively, \(R_{0}\) and \(R_{5}\), the pulley rotates around its support rolling bearing, and the pulley \(P_{i}^{t}\), \(i=1\), \(2\), \(3\), \(4\) are the idler pulleys driven by the cable \(T\), and their corresponding bending radius, sliding angle and wrap angle are, respectively, \(R_{i}^{t}\), \(\theta_{i,i}^{t}\) and \(\theta_{w,i}^{t}\), the tension on both sides of the pulley \(P_{i}^{t}\) are \(T_{i-1,i}^{t}\) and \(T_{i,i+1}^{t}\), respectively, and the tangent length between the pulley \(P_{i}^{t}\) and \(P_{i+1}^{t}\) is \(L_{i,i+1}^{t}\); the pulley \(P_{i}^{b}\), \(i=1\),\(2\),\(3\),\(4\) are the idler pulleys driven by the cable \(B\), their corresponding bending radius, sliding angle and package are \(R_{i}^{b}\), \(\theta_{s,i}^{b}\) and \(\theta_{w,i}^{b}\), respectively, and the tension on both sides of the \(P_{i}^{b}\) are \(T_{i-1,i}^{b}\) and \(T_{i,i+1}^{b}\), respectively, and the tangent length between the pulley and \(P_{i+1}^{b}\) is \(L_{i,i+1}^{b}\).  
The closed-circuit cable-pulley transmission system can be regarded as a double-tendon-sheath transmission system with a constant curvature radius of the sheath. According to the literature [16; 33], the tension and displacement transmission characteristics of the closed-circuit cable-pulley system are shown in Figure 6, mainly by three different phases, which are composed of the following: (1) working phase (III), where the output torque and displacement on the driven pulley is almost linear with the input torque and displacement on the driving pulley, and both cables are in a state of motion. (2) Delay  
Figure 5: Closed-circuit cable-pulley system.  
phase (I), which occurs after the direction of movement of the driving pulley changes, since only the two cables are partially in motion, the tension changes in the driving pulley cannot be immediately propagated to the driven pulley, and the output torque and displacement on the driven pulley does not change with the input position on the driving pulley. (3) Transition phase (II, IV), which refers to the interval from the delay phase to the work phase or from the work phase to the delay phase, in which the output torque and position changes on the driven pulley are substantially linear, but its slope is less than that of the working phase.