Bowden Cable
The following section discusses the design of the Bowden cable transmission. This includes
- determining the transmission relations for the Bowden cable
- choosing a suitable Bowden cable
- discussing relevant characteristics of Bowden cables
Bowden cables are used in robotics for remote actuation, so that the actuator can be fixed to the base of a robot to reduce the inertia of the system. Applications of such Bowden cables are described in [LVM_11, VEK_05, APY10, SLL_06, LSA_06].
Figure 1 shows that the cable contraction $\Delta{x}$ equals the partial circumference of the pulley, de- pending on the wrap angle of the cable running on the pulley and its radius $r$. The wrap angle in turn equals the relative knee angle $\psi$. Using this equation yields
;#;
<latex>\begin{align*}
\Delta{\psi} = -2\Delta{\phi}
\end{align*}</latex>
;#;
and therefore the cable contraction $\Delta{u}$ (and likewise the velocity $\dot{u}$ is depending linearly on the
angle $\phi$
;#;
<latex>\begin{align*}
\centering
\Delta{u} &= \Delta{\phi} = -2r\Delta{\phi} \\
\dot{u} &= -2r \dot{\phi}
\end{align*}</latex>
;#;
<imgcaption image1|Relation between the angle and the cable contraction>
</imgcaption>
With this relation, the transmission ratio is
;#;
<latex>\begin{align*}
\centering
i = \frac{\dot{u}}{\dot{\phi}} = -2r
\end{align*}</latex>
;#;
The main drawback of Bowden cables is the high amount of friction. Figure 2 shows the influence of the deflection angle and the friction coefficient on the force efficiency. Because of the exponential relation, it is important to minimize the deflection angle and choose a material pairing which re- duces the friction coefficient.
<imgcaption image2|Theoretical efficiency of Bowden cables, depending on the friction coefficient u and the deflection angle. Source: [SLL_06]>
</imgcaption>
[SLL_06] investigates these effects for the use of Bowden cables in robotics and suggests to use PTFE (Teflon) coated steel cables in combination with a PTFE liner inside the sleeve. This material pairing offers a low friction coefficient and reduces the slip-stick effect because the friction coefficient increases with speed.
In order to maximize the wrap angle of the Bowden cable, it is important to keep a high distance between the actuation box and the leg so as to reduce the change of length which occurs when the leg lifts off. This should allow for a wrap angle of less than 180°, which together with an assumed friction coefficient of 0.1 (with coating) allows for an efficiency of about 0.8.
Another consideration which has to be taken into account is the allowed bending radius of the steel cable. Cable manufacturers suggest a minimum bending radius which must be considered for pulleys, otherwise the cable suffers from increased wear. These radii vary with different types of cable constructions. For example, [dat12] suggests a radius of
;#;
<latex>\begin{align*}
r = \frac{42}{2} * D_{cable}
\end{align*}</latex>
;#;
for 7×7 cable constructions (number of strands times number of wires per strand) and
;#;
<latex>\begin{align*}
r = \frac{25}{2} * D_{cable}
\end{align*}</latex>
;#;
for 7×19 cable constructions.
Choosing a cable with a diameter of 1.5 mm with a 7×7 cable construction allows for a maximum tensile force of about 1800 N [dat12] and a minimum bending/ pulley radius of ;#; <latex>\begin{align*} r = \frac{42}{2} * 1,5\:mm = 31,5\:mm \end{align*}</latex> ;#; For further reference regarding the control, transmission characteristics of Bowden cables are dis- cussed in [APY10] and friction compensation is discussed in [LSA_06]. The compensation uses a Coulomb force model with a deadzone. The actuator must be controlled so as to avoid slack of the cable at all times.