For CDRs, it is important to ensure that positive cable tensions are maintained because of their unilateral driving property. In addition, since CDRs are redundantly actuated, there exist an infinite number of cable tension solutions for a particular pose. Adjustable cable tensions at a given pose can be useful in many CDR applications. For instance, if the user wishes to have lower energy consumption, then lower cable tensions are preferred. If the user wishes to regulate the stiffness, say for instance, achieve higher stiffness, then higher cable tensions are preferred. Hence, planning through optimization of the cable tension solution is an important aspect as well.

Cable Tension Planning of CDRs Without VSDs

For CDRs, the minimum 1-norm and 2-norm tension solutions are commonly used as the performance criteria for tension optimization. Pham et al. (2009) employed linear programming method to determine the minimum 1-norm tension solutions. Fang et al. (2004) derived an analytical method to obtain the minimum 1-norm tension solutions for cable-driven manipulators with one redundant cable. Oh and Agrawal (2005) designed a controller that guarantees positive tension solutions and implemented it on a planar CDR. The controller is able to use linear programming method and quadratic programming method to generate the minimum 1-norm and 2-norm tension solutions, respectively. The minimum 1-norm tension solutions can be efficiently generated by linear programming method, but discontinuity may exist in the tension solutions despite having continuous input variables. Such discontinuity will not occur for the minimum 2-norm solutions (Verhoeven and Hiller 2002). The tension solutions based on minimum 1-norm and 2-norm performance criteria tend to have tensions that are near to the lower tension limit. Hence, it is possible for the cable to lose tension when the CDR is subjected to disturbance. Mikelsons et al. (2008) addressed this issue by generating the safe-zone tension solutions that are away from the lower and upper tension limits. The non-iterative algorithm is shown to produce continuous tension solutions, but the computational efficiency is low when there are many redundant cables. Brogstorm et al. (2009) employed linear programming method to generate the optimally safe tension solutions which are similar to the safe-zone solutions. It allows the users to adjust the tension of all the cables between the tension limits. However, the tension solutions also suffer from possible discontinuity because of the linear programming approach employed. Pott et al. (2009) generated closed-form tension solutions which have tension values closest to the mean feasible tension, i.e., (tmin + tmax)/2. However, the algorithm may miss solutions at some of the poses.
When at least one feasible tension solution exists, a tension optimization scheme can be used to find the optimal feasible solution based on a performance criterion H (T). The tension optimization problem can be written as

where H(T) can be maximized or minimized.
The quadratic programming is employed to generate adjustable tension solution, and the objective function is given as

where td is the desired cable tension.
As an illustration, Fig. 25 shows the distribution of H(T) of the ith cable when tmin = 3 N and tmax = 200 N. It is observed that the objective function can be manipulated by changing the desired cable tension td. Using this objective function, the optimization process will be able to generate the tension solution that is close to the desired cable tension.

Cable Tension Planning of CDRs with VSDs

When the VSDs are employed, the stiffness of the CDR can be regulated by manipulating the cable tensions. Therefore, a stiffness-oriented tension resolution algorithm is proposed. At a particular pose, the algorithm determines the required cable tensions so as to fulfill the stiffness requirement. In literature, Yu et al. (2010) maximized the smallest eigenvalue of the stiffness matrix by optimizing the tension distribution. This method optimizes the magnitude of the stiffness matrix. Liu et al. (2011) optimized the tension distribution to maximize the sum of some diagonal components of the stiffness matrix. For this method, the actual stiffness is affected by the non-diagonal terms. The determinant of the stiffness matrix generally is used to show the change in stiffness when the tension distribution changes. However, the determinant only reflects the magnitude of the stiffness without any information on the direction. For the 2-DOF CDR shown in Fig. 18, a performance measure is proposed to optimize the tension distribution to achieve an isotropic stiffness matrix with high stiffness magnitude. Let the eigenvalues of the stiffness matrix of the 2-DOF CDR be σ1 and σ2. For the isotropic stiffness matrix, σ1 equals to σ2, but this is usually not achievable. Hence, the performance measure will aim to attain eigenvalues that are close to each other. High stiffness magnitude can be achieved by having high eigenvalues. The performance measure to achieve isotropic stiffness matrix with high stiffness magnitude is given as

Figure 26 shows the relationship between the performance measure and the two eigenvalues. It is observed that the performance measure forms a ridge (shown using a dotted line) when the two eigenvalues are equal, and this ridge gets higher when the eigenvalues increase.
Maximizing such performance measure in the objective function allows the optimization process to select a tension distribution that is able to achieve isotropic stiffness matrix with high stiffness magnitude. The optimization problem for the stiffness-oriented tension resolution algorithm becomes as follows:

The “fmincon” function in Matlab can be used to solve the nonlinear optimization problem in Eq. 62. With the proposed stiffness-oriented tension resolution algorithm, users will be able to generate cable tensions that are able to achieve isotropic stiffness matrix with high stiffness magnitude.

Case Study Using a CDR with 2-DOF Orientation

A 2-DOF CDR (without any VSDs), as shown in Fig. 27, is used as a computer-simulated case study to illustrate the proposed cable tension planning algorithm in section “Cable Tension Planning of CDRs Without VSDs.” It has a passive universal joint that connects the moving platform to the base and is actuated using four driving cables. Any pose of this 2-DOF CDR can be represented using two angles θ1 and θ2. The rotation matrix is defined as R = Rx(θ1)Ry(θ2), which is defined with respect to the fixed X and Y axes. At the pose [20o, 20o], the cable lengths and tensions are given in Table 8. Figure 28 shows the tension distribution at [20o, 20o] when the desired cable tensions td are 10 N, 100 N, and 180 N. It is observed that the proposed tension optimization method is able to generate adjustable tension solutions.

With the cable lengths and tensions computed, a hybrid control scheme can be employed to simultaneously control both the position and the cable tensions. For an n-DOF CDR with m cables, the pose of the moving platform can be uniquely determined using n cable lengths. Hence, n cables are placed under displacement control mode to maintain the position of the moving platform. The remaining (m-n) cables are then placed under tension control. In this way, the control scheme can simultaneously control both the position and the cable tensions of the CDR. For the 2-DOF 4-cable CDR, the pose of the moving platform can be uniquely determined by two cable lengths. Hence, if cables 1 and 2 are set in displacement control mode, then cables 3 and 4 will be under tension control mode. In displacement control mode, the cable length is controlled using the encoder feedback from the motor. In tension control mode, the cable tension is controlled using the feedback from a force sensor (or a load cell). In this way, while the moving platform is under position control, the cable tensions can also be changed at the same time, which will allow the stiffness to be regulated (readers may refer to (Lim et al. 2013) for more details on the hybrid control scheme).