# Real-Time Tension Distribution Design for Cable-Driven Parallel Robot
## 7 Discussion: Comparison with Other Tension Distribution Methods
### Force Level and Force Level Changing  
The existing tension distribution method may attempt to select the minimum, maximum, average, or any solution of the force level of tension (p99, [18]). In fact, the force level of the cables affects the stiffness of the cable-driven robot [18]. Especially in applications where robots may contact the external environment, the issue of the variable stiffness control of the robot is crucial for completing such sophisticated tasks. As a result, a tension distribution algorithm that is capable of realizing any tension distribution is required, rather than just focusing on achieving a specific minimum or maximum force level.  
In our method, we select \(f_{0}\) in Equation (27) as  
\[f_{0}=A^{+T}\tau_{d}+(\mathbf{U}_{m\times m}-A^{+T}A^{T})s \tag{61}\]  
Figure 8: Comparison of our method and closed-form method (shifting desired pose). Parameters and desired pose are selected as \(\alpha_{1}=0.1\), \(\alpha_{2}=0.1\), \(K_{d}=2\), \(D_{d}=1\), \(K_{i}=0\), \(q_{d}=[0.5,-0.5,0.5,0,0,0]^{T}\) (0 s \(\rightarrow\) 5 s), \(q_{d}=[0.1,0.05,0.5,0,0,0]^{T}\) (5 s \(\rightarrow\) 10 s) and \(q_{d}=[0.5,0.5,0.5,0,0,0]^{T}\) (10 s \(\rightarrow\) 15 s). (**a**) Tracking error of pose. (**b**) Tension result.  
where \(s\in R^{m}\) denotes an optional vector. Here, \(f_{0}\) represents the point in the tension's solution space satisfying \(A^{\mathsf{T}}f=\tau_{d}\) that has the minimum distance with vector \(s\). Therefore, by changing the vector \(s\), selecting \(f_{0}\) in this equation, and using our method, we can obtain an appropriate force level. Notably, when setting the \(s\) as zero (as in Equation (27)), we can obtain the point in the tension's solution space satisfying \(A^{\mathsf{T}}f=\tau_{d}\) that is the closest to the origin. Additionally, this point is necessary for the puncture method.  
The example below shows the ability of our method in terms of selecting the force level and performing force-level changes. Here, \(s\) is selected as \(s=[10,\dots,10]^{T}\in R^{8}\) from 0 s to 5 s and \(s=[15,\dots,15]^{T}\in R^{8}\) is selected from 5 s to 15 s.  
From Figure 9, it can be seen that the gray line in the tension results represents the \(f_{0}\) relating to the selected \(s\) and the blue line represents the tension distribution result of our method. From 0 s to 5 s, all elements of \(f_{0}\) satisfy tension constraints, and the tension calculated by our method can precisely track \(f_{0}\). After 5 s, with increasing \(s\), elements of \(f_{0}\) become bigger than 20 Nm which does not satisfy the tension constraint. In this situation, the tensions calculated using our method converge to an appropriate force distribution, which has the closest distance to \(f_{0}\) in the feasible solutionsm and the force level is enlarged to an extent. Notably, when the force level is changed at 5 s, our method yields satisfactory results for the continuous variation of tension.  
These results demonstrate that our method can be used to achieve any required feasible force level and the change in force level during the robot's movement while maintaining the continuous variation of tension.