# Real-Time Tension Distribution Design for Cable-Driven Parallel Robot
## 7 Discussion: Comparison with Other Tension Distribution Methods
### Continuous Solution  
In practice, continuous solutions are essential. Although non-continuous tendon forces may be a feasible solution, they result in discontinuities in motor torques, which in turn produce vibrations and high mechanical loads. Many existing real-time tension distribution algorithms, such as the closed-form, improved closed-form, Barycentric, and puncture methods, attempt to solve this problem by analyzing the structure matrix \(A^{T}\) and finding an appropriate continuous trajectory in the kernel space of \(A^{T}\) to construct the continuously varying tension solution. The issue of the discontinuity that may be imposed in the calculated control wrench owing to white noise in the sensing process, discontinuous changes of the desired pose, buffering the influence of unexpected external loads, etc., has not yet been considered in the issue of tension distribution. Our approach theoretically guarantees the continuity of the solution of tension distribution based on a previously selected tension function constructed by \(tanh(\cdot)\). Then, tracking the computed wrench based on the selected tension function naturally provides the continuously varying internal tension that previous research attempts to find in the kernel space of \(A^{T}\) and allows for selection of the convergence rate of tracking for the computed wrench by adjusting the parameter \(\alpha_{1},\alpha_{2}\) to filter out any excessive discontinuous changes of the computed wrench that may exist, resulting in an acceptable continuous internal tension. We compared our method with the closed-form method in the following two cases: 1. Measurement noise exists in \(q\), and 2. The desired pose changes discontinuously at some time. Here, we selected closed-form in the comparison because it has the fastest calculation speed, provides an effective way to perform the tension distribution, and is a key part of other real-time algorithms such as the improved closed-form method and the puncture method. The improved closed-form method and the puncture method behave similarly when the control wrench is discontinuous. For the case of existing white noise in sensing \(q\), we compared our method to the closed-form method in Figure 7. Here, white noise is generated using the MatLab Simulink block 'Band-Limited White Noise', with the noise power selected as 0.0000001. As shown in Figure 7, the existence of white noise in the measurement results of \(q\) and the fluctuation of the gray line in the tension results reflect the results of the closed-form  
Figure 7: Comparison of our method and closed-form method (noisy case). Parameters and desired pose are selected as \(\alpha_{1}=0.1\), \(\alpha_{2}=0.1\), \(K_{d}=2\), \(D_{d}=1\), \(K_{i}=0\), \(q_{d}=[0.6,-0.6,0.6,0,0,0]^{T}\). (a) Tracking error of pose. (b) Tension result. method, with a high-frequency vibration that can cause discontinuities in the torque of the motors. In contrast to this result, the blue line, which represents the result of our approach of selecting lower \(\alpha_{1}=0.1\), and \(\alpha_{2}=0.1\), varies smoothly, and the high vibration is eliminated. In addition, compared to the prior closed-form method, although convergence in tracking the desired pose of our method with the selection of \(\alpha_{1}=0.1\), \(\alpha_{2}=0.1\) is a little slower, the difference between the two lines is not significant, and the tracking errors converge to 0. Moreover, we compared our method to the closed-form method in the case where the desired pose changes at some point. Here, the shift times of the desired pose are 5 s and 10 s.  
The gray line of the closed-form method varies at 5 s and 10 s, changing to a maximum of roughly 5 N, as shown in Figure 8, and a significant disturbance in the tension occurs when the desired position is changed. The blue dotted line in the figure shows that our method with the selection of \(\alpha_{1}=0.1\) and \(\alpha_{2}=0.1\) efficiently reduces this disturbance and eliminates the impact of the discontinuous shift of the desired pose.