Figure
8c illustrates the experiment set-up for the bending motion test. A measurement equipment to actuate the CRCJ with a single cable using a pulley system is designed. The actuation pulley has a radius of 6 mm, and the actuation cable is connected to a digital force sensor (DIGIFORCE
\(\textregistered \) 9310, burster präzisionsmesstechnik gmbh & co kg, Gernsbach Germany). The single actuation (see Fig.
8b, c, red) tendon for the joint is assembled with a pulley underneath the pulley system. Both pulleys are printed on a monolithic structure. On the top surface of the joint, the tendon is fixed on both sides. The relationship between the cable force
\(F_{\mathrm{c}}\) and actuation force of on the top surface of the manipulator
\(F_{\mathrm{a}}\) can thus be formulated as:
$$\begin{aligned} F_{{\mathrm{c}}} = \kappa \frac{F_{\mathrm{a}}c}{r} . \end{aligned}$$
(6)
The factor
\(\kappa \) is introduced as a factor to capture the effects of friction in the force transmission, e.g. in the pulley system or in the tendon guides due to manufacturing inaccuracy. The bending angle in this experiment is measured using the Navigation camera (Polaris Vicra, NDI, 103 Randall Drive, Waterloo, Ontario, Canada). For this purpose, a form-fitting tracker is attached to the measurement set-up and two markers are mounted on the top of the testing unit, defining the boundary of the top surface (cf. Fig
8a). The position of both markers
p in the coordinate system of tracker
\(CS_{\mathrm{tr}}\) can be derived from:
$$\begin{aligned} ^{\mathrm{tr}} \mathbf{p} =(^{\mathrm{cam}} \mathbf{T} _{\mathrm{tr}})^{\mathrm{T}} \,\, ^{\mathrm{cam}} \mathbf{p} . \end{aligned}$$
(7)
In (
7),
\(^{\mathrm{cam}} \mathbf{T} _{\mathrm{tr}}\) and
\(^{\mathrm{cam}} \mathbf{p} \) represent the homogeneous transformation matrix from camera to tracker coordinate system and the position of marker points in the camera coordinate system, respectively, which are given by the navigation system. The bending angle can be derived using geometry relationships between the initial position and the current position of both markers after the joint is deformed under load. For the bending motion experiment, the relationship between force and bending angle was determined using 3 samples of each joint type. In this work, we define the bending stiffness
K as the quotient of the applied cable force
\(F_{\mathrm{c}}\) and the bending angle
\(\theta \)$$\begin{aligned} K = \frac{F_{\mathrm{c}}}{\theta } . \end{aligned}$$
(8)
Linear correlation of force and bending angle can be shown in the measurement data in Fig.
9a for all joint types. Therefore, linear regression was applied to fit the measurement data of all 9 measurement series. The polynomial coefficients of each sample, which contain the slope
\(P_1\) and y-intercept
\(P_0\) are summarized in Table
2.
Table 2
Polynomial coefficients \(P_0\) and \(P_1\) from linear regression of measurement data of force bending angle
1 | 25.59 | 6.36 | 50.37 | 6.64 | 22.57 | 5.33 |
2 | 23.84 | 6.86 | 48.44 | 10.72 | 21.63 | 2.32 |
3 | 25.72 | 12.18 | 48.75 | 10.86 | 21.93 | 0.44 |
Mean | 25.05 | 8.48 | 49.19 | 9.41 | 22.04 | 2.70 |
The mean values of both polynomial coefficients were determined for each joint type. These were used to plot the force bending angle relation of each joint type in Fig.
9b. Based on the deformation comparison between FEA and experiment,
\(\kappa = 2.13\) in (
6) was determined. While the leaf-type joint 1 has the lowest bending stiffness, leaf-type 2 has the highest. Leaf-type joint reaches the threshold of test break already at about 3.6 N. Due to the high intrinsic elasticity and the contact restriction of the spacing discs, leaf-type joint 2 can only reach a maximum bending angle of around
\(\theta =160^{\circ }\) as shown in Fig.
9. The bending stiffness of CRCJ is slightly lower than the bending stiffness of leaf-type joint 2. This shows the higher flexibility of CRCJ in the bending plane compared to conventional leaf-type joint of similar torsional stiffness. In addition, CRCJ can even surpass
\(180^{\circ }\) bending angle thanks to its rolling-contact mechanism. During this experiment, cable slack was not observed in CRCJ but in both leaf-type joints, which also reflects in the lower bending stiffness of the CRCJ compared to leaf-type 2 due to better force transmission. Result from the FEM analysis also shows a good agreement with the experimental data, which validates the modelling of nonlinear cable force given in (
4) and (
5).