The flexible nature of cables causes CDRs to have relatively low stiffness and thereby significantly affecting the positional accuracy. For stiffness analysis, the pre-tension of cables is assumed to be large enough for the cables to be in tension and in the linear region of the stress–strain curve. In principle, a finite-element analysis will reveal more or less the complete information about the mechanism’s elastic properties. However, such finite-element analyses are not only computationally expensive and time-consuming, but they are also not suitable for design iteration and optimization. Hence, kinematic approaches are preferred without resorting to computationally intensive procedures. One of the first few works in the field of CDRs to address stiffness analysis was by Kawamura et al. (1995), (Morizono et al. 1998). These works mainly investigated the influence of cable pretension on the stiffness of a CDR with the aim of addressing the issue of vibration. Choe et al. (1996) classified the stiffness of a cable in tension into two aspects, one along the cable direction and another orthogonal to the cable direction. The stiffness along the cable direction is contributed solely by the cable stiffness, while the stiffness orthogonal to the cable direction is a function of the cable tension. The stiffness in these two directions is independent of each other. Verhoeven et al. (1998) formulated a simplified stiffness matrix which was a function of the cable stiffness and cable length and analyzed its eigenvalues to determine the nearness to singularity poses. Behzadipour and Khajepour (2006) proposed a four-spring model to represent the stiffness of each cable and calculate the total stiffness matrix of a CDR. This was then subsequently used to determine the stability of the moving platform (adapted from the work done by Svinin et al. (2000; 2001) on rigid-link gough-stewart platforms). Yu et al. (2010) performed stiffness control on cable-driven robots by optimizing the tension distribution of the cables. The stiffness of a CDR is contributed by both the cable stiffness and the internal forces in the system. The internal forces, which are the cable tensions, are primarily used to resist the external wrench. As the CDRs are redundantly actuated closed-loop manipulators, the tension solution will contain a set of homogeneous forces that produce zero resultant wrenches on the moving platform. Hence, the cable tensions can be manipulated to regulate the stiffness of the CDRs.

CDR with Variable Stiffness Characteristics

In order to significantly regulate the stiffness of the CDR, a variable stiffness device should be attached along each cable of the CDR. The variable stiffness device (VSD) refers to an add-on unit with nonlinear stiffness characteristics. As such, the CDR is able to exhibit variable stiffness characteristics. As shown in Fig. 17, the stiffness of the VSD-cable combination is a function of the stiffness of both the cable and the VSD. When the stiffness of the VSD is a function of the cable tension, the stiffness of the combination can thus be controlled by manipulating the cable tensions of the CDR.

Case Study Using a 2-DOF Planar CDR

A 2-DOF planar CDR prototype, as shown in Fig. 18, is used for the experimental studies to verify the ability of the CDR to significantly regulate its stiffness after the VSDs are employed. The moving platform is actuated using four driving cables. Any pose of the 2-DOF planar CDR is represented using the position (in meters) along the X and Y axes, i.e., [PX, PY].
A VSD with the parameters given in Table 4 is designed and built. This VSD is designed to sustain a tension limit of 80 N. The VSD prototype is shown in Fig. 19.

Computer Simulations

The simulations are first performed on the 2-DOF planar CDR with and without the VSDs. For the CDR with VSDs, a VSD is placed on each cable so that the stiffness of the CDR can be significantly regulated by manipulating the cable tensions. The determinant of the stiffness matrix, det(K), is employed to qualify the stiffness of the 2-DOF planar CDR. There are a number of criteria to evaluate the stiffness of a system, such as the trace of the stiffness matrix (Zhang 2009), the eigenvalues of the stiffness matrix (El-Khasawneh and Ferreira 1999), and the determinant of the stiffness matrix (Ceccarelli and Carbone 2002). In this work, the determinant of the stiffness matrix is employed. Let K'cable = 13, 000N/m, tmax = 80 N, and tmin = 3 N, in which tmax and tmin are the upper and lower tension limits. The upper tension limit tmax is restricted by the force/torque capability of the actuators or the loading limit of cables, whichever is lower. The lower tension limit tmin is a positive value that is needed to withstand disturbances.
At a particular pose [0.160, 0.465], three sets of tension solutions, with different levels of cable tensions, are given in Table 5. Different levels of cable tensions help to demonstrate the change in the stiffness (K) of the 2-DOF planar CDR when the tension solutions are manipulated.

Table 5 presents the simulation results on the stiffness analysis of the CDR with and without the VSDs. When the VSDs are not employed, it is observed that the change in the stiffness of the CDR is insignificant at different tension levels. When the VSDs are employed, the determinant of the stiffness matrices has much larger variation, implying that CDR with VSDs is able to produce significant changes to its stiffness. With the VSDs employed, the same tension limit can now produce significant changes to the stiffness of the CDR. As the cable and VSD are connected in series, the net stiffness of the cable-VSD combination is bounded below the stiffness of the cable alone. For applications that demand low stiffness at certain times and high stiffness at other times, such CDRs with variable stiffness can be very useful. If the CDR requires higher stiffness, cables with higher stiffness can always be used.

Experimental Studies

A 2-DOF planar CDR prototype, as shown in Fig. 18, is used for the experimental studies. As the stiffness matrix is a local measure, it should not be employed to calculate deflections for mechanisms or manipulators with large deflections. Hence, the unit load method (Bauchau and Craig 2009) is used to determine the deflection of the moving platform as a result of the external force. The unit load method is a simple and elegant method that can cater to the stiffness nonlinearity and the large deflection of the moving platform. Based on the principle of complementary virtual work, the unit load method applies a virtual unit force along an axis and then determines the deflection caused by the external force. Let the external force along the X axis be FX. Employing the principle of complementary virtual work yields the following:

The 2-DOF planar CDR is set at different cable tensions to induce a change in the stiffness of the VSDs, which in turn changes the stiffness of the CDR. An external force of F = [0 N, 5 N]T is applied at different stiffness levels, and the displacements are measured using dial indicators. The deflections along the X and Y axes are measured individually and are then used to compute the displacements of the moving platform. Load cells are placed along the cables to measure the cable tensions. The nylon-coated steel cables used in the experiments have a stiffness of 13,000 N/m per unit meter. The experiments are performed on two poses. The cable tensions, the computed stiffness determinants, and the analytical and experimental results are given in Table 6. From the analytical and experimental results, it is observed that the displacements reduce as the cable tensions increase. This implies that the CDR with VSDs is able to significantly regulate its stiffness by manipulating the cable tensions. Readers may refer to (Lim et al. 2013) on details of a joint-based hybrid control scheme that can be employed to simultaneously control both the position and the cable tensions of a redundantly actuated CDR, which will allow the stiffness to be regulated via the VSD.