*2.2. Cable Drive Transmission*

Cable-driven transmission in the literature of mechanics can refer to more than one type of mechanical transmission [37,38]. In this paper, two types of cable transmission are used, namely pulley-cable transmission [39] and Bowden cable transmission [40]. A Bowden cable is a type of flexible cable used in applications where there is a need to transmit mechanical force or energy, implementing the movement of an inner cable relative to a hollow outer cable housing known as a sheath. In the area of robotics, this form of actuation is usually applied for remote actuation of a robotic joint; force is delivered to the remote joint by means of mechanical displacement between the cable and the outer sheath.

The main factors influencing the cable efficiency are the normal forces on the cable, which are determined by cable tension or preload, the friction coefficients resulting between material combinations, and velocity of the inner wire. Friction between the internal cable and the external sheath usually has an impact on the entire assembly efficiency. Losses and inefficiencies of the Bowden transmission are mainly a result of the complex and non-linear friction phenomena. As described by Kaneko [41], Coulomb friction, viscous friction, stiction, and stick-slip may occur in Bowden cable transmission systems.

The main geometric parameter influencing friction between the sheath and the cable is the total wrap angle of the cable system, illustrated in Figure 3c. A simplified representation of the friction losses of a Bowden cables system can be represented by analogy to sliding a cable over a fixed cylinder at a constant velocity, as indicated in Figure 3a. For this simplified representation, the friction can be expressed by using the expression [42]:

$$\frac{F\_{\rm in}}{F\_{\rm out}} = e^{-\mu\theta} \tag{1}$$

where:


**Figure 3.** (**a**) Simplified equivalent representation of friction in a bowden transmission. (**b**) Representation of friction in a Bowden transmission. (**c**) Total wrap angle θ of the cable system.

The total wrap angle θ of the cable system represented in Figure 3c is defined by the equation [43]:

$$
\theta = \theta\_1 + \theta\_2 = \sum\_{i=1}^{n} \theta\_i \tag{2}
$$

In Figure 4, a model for determining the cable tensions at any point along the mechanism of the Bowden cable system is presented. A model based on Coulomb friction is considered since the system does not use lubricants, and the device can be generally considered on a macro scale. The current version of the prototype is designed to develop speeds and torques higher than needed for rehabilitation purposes. As a result, the frictions generated in the system are not a major challenge for the closed-loop control system [44]. However, at some later point in the project's development and optimization, other friction models may be required. Later optimizations such as reducing the scale of the actuation system may require a more thorough study regarding multiple or even more precise friction models [45,46]. For the current research, the work of Kaneko [41] is considered as the starting point for the Bowden cable transmission mechanism model. The normal force originating from the curved sheath creates friction force between the sheath and the cable. The friction force that appears in the mechanism has a nonlinear tension distribution along the wire. The equations that describe this phenomenon can be expressed using the following equations [41]:

$$T(p) = \begin{cases} T\_{\text{in}} \exp\left(-\frac{c\mu >}{R} p \cdot \text{sign}(\nu)\right) & (p < L\_1) \\\ T\_0 & (L\_1 \le p) \end{cases} \tag{3}$$

$$sign(\nu) = \begin{cases} 1 & (\nu \ge 0) \\ -1 & (\nu < 0) \end{cases} \tag{4}$$

$$L\_1 = \min\{p \in T(p) = T\_0\}\tag{5}$$

$$\begin{cases} T\_{\text{in}} = T(p=0) \\ T\_{\text{out}} = T(p=L) \end{cases} \tag{6}$$

where:


**Figure 4.** Cable tension model parameters of the Bowden cable system.

Numerous studies focus on determining and compensating the friction in a Bowden system. One method utilizes closed-loop feedback of the output tension [47–49]. The compensation is done by monitoring the stress of the cable on the output side, controlling the actuator in such a way that the output tension follows the reference set point. Other studies utilize position-based impedance feedback [41,50,51] combined with a decrease of pre-tension with a slack prevention actuation mechanism to reduce the effect of the friction [52]. For this paper, the friction compensation method relies on the position and current feedback of the system as described in a previous paper [44].
