# Study on Stiffness-Oriented Cable Tension Distribution for a Symmetrical Cable-Driven Mechanism

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## Abstract

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## 1. Introduction

## 2. Design of the 6-CSJM with VSDs

## 3. Stiffness Model of the VSD

## 4. Stiffness Model of the 6-CSJM

#### 4.1. Kinematic Analysis of the 6-CSJM

#### 4.2. The Stiffness Model of the 6-CSJM

## 5. Stiffness-Oriented Cable Tension Distribution Method

#### 5.1. Formulation of the Optimization Model

#### 5.2. Elimination of Equality Constraint

#### 5.3. Optimization Procedures via Complex Method

- (i)
- Formation of the initial Complex: An initial Complex with six vertices ${\mathit{T}}_{s}^{\left(1\right)}$, ${\mathit{T}}_{s}^{\left(2\right)}$, ⋯, ${\mathit{T}}_{s}^{\left(6\right)}$ is setup in the feasible region randomly.
- (ii)
- Generation of a new complex: The values of the cost function at the vertices are computed. The worst point ${\mathit{T}}_{s}^{\left(W\right)}$, where the cost function obtains the largest value, will be replaced by the mapping point ${\mathit{T}}_{s}^{\left(M\right)}$. In this way, a new Complex is generated. Here, the mapping point ${\mathit{T}}_{s}^{\left(M\right)}$ is computed by$${\mathit{T}}_{s}^{\left(M\right)}={\mathit{T}}_{s}^{\left(C\right)}+\alpha ({\mathit{T}}_{s}^{\left(C\right)}-{\mathit{T}}_{s}^{\left(W\right)}),$$
- (iii)
- Condition of loop stopping: If the error tolerance, $\u03f5\le {\u03f5}_{min}={10}^{-7}$, the iterative procedure will terminate, and we go to (iv). Otherwise, we go back to (i). Here, $\u03f5$ is defined as$$\u03f5=\sqrt{\frac{1}{6}\sum _{j=1}^{6}{[g({\mathit{T}}_{s}^{\left(j\right)})-g({\mathit{T}}_{s}^{\left(B\right)})]}^{2}}.$$
- (iv)
- Finalization of optimal solution: The best point ${\mathit{T}}_{s}^{\left(B\right)}$ is selected as the optimal solution ${\mathit{T}}_{\mathrm{opt}}$. From here, the optimal cable tension distribution is obtained for the desired feasible stiffness.
- (v)
- Validation of stiffness model: The actual stiffness ${\mathit{K}}_{\mathrm{act}}$ is computed by substituting the optimal cable tensions ${\mathit{T}}_{\mathrm{opt}}$ into the stiffness model (21). The error $\eta $ is defined to evaluate the difference between ${\mathit{K}}_{\mathrm{des}}$ and ${\mathit{K}}_{\mathrm{act}}$,$$\eta =\frac{\left|\right|{\mathit{K}}_{\mathrm{des}}-{\mathit{K}}_{\mathrm{act}}{\left|\right|}_{F}}{\left|\right|{\mathit{K}}_{\mathrm{des}}{\left|\right|}_{F}},$$

## 6. Simulation

#### 6.1. Simulation Cases

#### 6.2. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 5.**Optimal cable tensions for the four sub-cases, with the safe tension zones of cables $[{\underline{t}}_{c},{\overline{t}}_{c}]$.

Case | Pose | Load M | Desired Stiffness ${\mathit{K}}_{\mathbf{des}}$ | Actual Stiffness ${\mathit{K}}_{\mathbf{act}}$ | Error $\mathit{\eta}$ |
---|---|---|---|---|---|

Case 1-a | ${\mathit{R}}_{\mathrm{des}1}$ | ${\mathit{M}}_{1}$ | $\left(\begin{array}{ccc}385.7& -3.6& -23.2\\ -3.4& 354.0& -26.1\\ -22.8& -25.5& 133.4\end{array}\right)$ | $\left(\begin{array}{ccc}385.7& -3.6& -23.2\\ -3.4& 354.0& -26.1\\ -22.7& -25.5& 133.4\end{array}\right)$ | $0.001\%$ |

Case 1-b | ${\mathit{R}}_{\mathrm{des}1}$ | ${\mathit{M}}_{1}$ | $\left(\begin{array}{ccc}434.2& -0.6& -24.1\\ -0.5& 416.4& -17.4\\ -23.6& -16.8& 156.5\end{array}\right)$ | $\left(\begin{array}{ccc}434.2& -0.6& -24.1\\ -0.5& 416.4& -17.4\\ -23.6& -16.8& 156.4\end{array}\right)$ | $0.001\%$ |

Case 2-a | ${\mathit{R}}_{\mathrm{des}2}$ | ${\mathbf{M}}_{2}$ | $\left(\begin{array}{ccc}411.5& -12.8& -34.0\\ -12.4& 344.2& -52.6\\ -32.5& -51.9& 147.3\end{array}\right)$ | $\left(\begin{array}{ccc}411.5& -12.8& -34.0\\ -12.4& 344.2& -52.6\\ -32.5& -51.9& 147.3\end{array}\right)$ | $0.001\%$ |

Case 2-b | ${\mathit{R}}_{\mathrm{des}2}$ | ${\mathit{M}}_{2}$ | $\left(\begin{array}{ccc}347.7& 12.4& -54.2\\ 12.8& 353.4& -46.3\\ -52.6& -45.5& 144.6\end{array}\right)$ | $\left(\begin{array}{ccc}347.7& 12.4& -54.2\\ 12.8& 353.4& -46.3\\ -52.6& -45.5& 144.6\end{array}\right)$ | $0.001\%$ |

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**MDPI and ACS Style**

Yang, K.; Yang, G.; Chen, S.-L.; Wang, Y.; Zhang, C.; Fang, Z.; Zheng, T.; Wang, C.
Study on Stiffness-Oriented Cable Tension Distribution for a Symmetrical Cable-Driven Mechanism. *Symmetry* **2019**, *11*, 1158.
https://doi.org/10.3390/sym11091158

**AMA Style**

Yang K, Yang G, Chen S-L, Wang Y, Zhang C, Fang Z, Zheng T, Wang C.
Study on Stiffness-Oriented Cable Tension Distribution for a Symmetrical Cable-Driven Mechanism. *Symmetry*. 2019; 11(9):1158.
https://doi.org/10.3390/sym11091158

**Chicago/Turabian Style**

Yang, Kaisheng, Guilin Yang, Si-Lu Chen, Yi Wang, Chi Zhang, Zaojun Fang, Tianjiang Zheng, and Chongchong Wang.
2019. "Study on Stiffness-Oriented Cable Tension Distribution for a Symmetrical Cable-Driven Mechanism" *Symmetry* 11, no. 9: 1158.
https://doi.org/10.3390/sym11091158