# Analysis of the effect of system parameters for combined nonlinear cable elongation characteristics in CDPR

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## Abstract

The polymer cable is a widely used material in the cable-driven parallel robot (CDPR) system because of its many advantages such as high sensitivity, large workspace and so on. However, the accurate control of CDPR is not easy because of complicated response of the cable. In our previous study, the integrated cable model was derived. Based on the model, parametric studies were progressed in this paper. While operating CDPR, various parameters such as length, applied tension, and tensile rate can be changed and dominantly affect the dynamics of CDPR. For this reason, parametric study was based on these parameters. In this investigation, dynamic creep, hardening factor and short-term recovery were saturated as processing cyclic load. Each saturation rate was dominantly influenced by cable length and applied tension. As the tensile rate was increased, the dynamic creep was decreased. The hysteresis was the characteristic combining all of dynamics. So, the hysteresis also had saturation trends. When the exerted tension was decreased, the length of cable could be reduced or elongated because the creep and recovery occur at the same time.

## Keywords

Dynamic creep Hardening effect Integrated nonlinear dynamic model Short- and long-term recovery## 1 Introduction

In this study, we conduct the parametric study to determine the nonlinear elongation of polymer cable for variations of the length, tension, and tensile rate. The dynamic model of the cable focuses on the hardening effects, dynamic creep, short- and long-term recovery, and hysteresis. In the first section of this paper, we describe the experiment and integrated nonlinear dynamic model. To investigate these characteristics, the cyclic load process is simulated for various samples and parameters because the nonlinear elongation of cable is sufficiently affected in the cyclic load. Next, we report the results of our parametric study under a range of conditions. To investigate the cable recovery characteristics, parametric study for short-term and long-term recovery is carried out by varying cable length, applied tension and recovery time. The effect of parameters for the hysteresis is evaluated with the cyclic process simulation. And, the combined cable nonlinear dynamics are evaluated while operating the arbitrary tension histories.

## 2 Experiment and characteristics of nonlinear cable elongation

### 2.1 Experimental setup

Because of the nonlinearity of our proposal cable model with respect to time, it is required to derive the loading, unloading time. To gain the finally applied loading and unloading time, we perform the cyclic loading test of the Dyneema polyethylene SK78 cable using a SHIMADZU AGS-X PLUS tensile tester. We applied tension from 50 to 150 N to the CDPR system. As the end-effector pose is changed, the cable length is also changed. Thus, it is necessary to perform under various length conditions. Therefore, we used the three kinds of cable lengths (100 mm, 200 mm and 300 mm). And, constant tensile rates are based on the 3 mm/min. Each test is repeated by three times and the mean values of the measurements are calculated.

### 2.2 Dynamic characteristics of polymer cable

*F*

_{f}is currently applied tension,

*F*means the tension over the time,

*E*

_{1,c}and

*E*

_{2,c}is the elastic parameter of elongation,

*E*

_{1,r}and

*E*

_{2,r}means elastic parameter of recovery,

*η*

_{0}is the Newtonian damping parameter for viscous behavior. Each parameter was described in detail and was verified in the [6]. The right-hand side first and second terms of Eq. (1) are related to structural elongation and short-term recovery. To describe the hardening of the cable caused by residual stress and continuous cyclic loads, the hardening factor

*h*(

*ɛ*

_{i − 1}) was introduced. And the right-hand side integral term includes time-dependent dynamic creep and long-term recovery. However, it is difficult to solve the nonlinear dynamic equation for time-dependent applied tension. Therefore, we first did the cyclic test and additionally derived one more model for case of constant tensile rate (see Eq. 2). Because the visco-elastic model has nonlinearity with respect to the loading time

*t*

_{total}, the actual time was derived using the Newton–Raphson method based on the condition that Eq. (1) is equal to Eq. (2) for constant tensile rate. After then, the dynamic strain in the loading and unloading cases could be derived using the predicted loading and unloading time:

Parameters for dynamic cable model

Parameters | Values | Parameters | Values | |
---|---|---|---|---|

| 1.8 Gpa | | 3 Gpa | |

| 8.2 Mpa | | 8.0 × 10 | |

| 5.0 × 10 | | 6.0 × 10 | |

| 62.5 s. | | 1 s. | |

| 0.0035 | | 8.65 × 10 | |

| 3.83 | | 1.0 × 10 | |

\(\dot{\varepsilon }\) | 0.016%/s, 0.032%/s 0.048%/s, 0.064%/s | | 7.06 mm | |

| 100 mm, 200 mm, 300 mm | | 50 N, 100 N, 150 N |

## 3 Integration of nonlinear cable behavior of system parameters

### 3.1 Analysis of dynamic creep

### 3.2 Analysis of cable hardening

### 3.3 Analysis of short- and long-term recovery

The recovery according to the tension (cable length of 300 mm)

Cycle | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

Short-term recovery (mm) | |||||

50 N | 1.55 | 1.18 | 0.93 | 0.89 | 0.87 |

100 N | 1.92 | 1.46 | 1.15 | 1.12 | 1.10 |

150 N | 2.35 | 1.96 | 1.76 | 1.68 | 1.66 |

Long-term recovery (mm) | |||||

50 N | 0.024 | 0.020 | 0.020 | 0.016 | 0.016 |

100 N | 0.034 | 0.034 | 0.028 | 0.027 | 0.027 |

150 N | 0.041 | 0.038 | 0.035 | 0.031 | 0.030 |

Cycle | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|

Short-term recovery (mm) | |||||

50 N | 0.86 | 0.86 | 0.85 | 0.85 | 0.85 |

100 N | 1.10 | 1.09 | 1.09 | 1.09 | 1.09 |

150 N | 1.66 | 1.66 | 1.66 | 1.66 | 1.66 |

Long-term recovery (mm) | |||||

50 N | 0.020 | 0.020 | 0.020 | 0.020 | 0.020 |

100 N | 0.027 | 0.027 | 0.027 | 0.027 | 0.027 |

150 N | 0.030 | 0.030 | 0.030 | 0.030 | 0.030 |

The recovery according to the cable length (applied tension of 100 N)

Cycle | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

Short-term recovery (mm) | |||||

100 mm | 0.62 | 0.50 | 0.45 | 0.41 | 0.37 |

200 mm | 1.21 | 0.96 | 0.86 | 0.78 | 0.68 |

300 mm | 1.79 | 1.36 | 1.04 | 0.97 | 0.96 |

Long-term recovery (mm) | |||||

100 mm | 0.02 | 0.01 | 0.01 | 0.01 | 0.01 |

200 mm | 0.08 | 0.07 | 0.06 | 0.06 | 0.06 |

300 mm | 0.24 | 0.20 | 0.18 | 0.17 | 0.17 |

Cycle | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|

Short-term recovery (mm) | |||||

100 mm | 0.35 | 0.35 | 0.35 | 0.35 | 0.35 |

200 mm | 0.66 | 0.66 | 0.66 | 0.66 | 0.66 |

300 mm | 0.96 | 0.96 | 0.95 | 0.95 | 0.94 |

Long-term recovery (mm) | |||||

100 mm | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 |

200 mm | 0.06 | 0.06 | 0.06 | 0.06 | 0.06 |

300 mm | 0.170 | 0.170 | 0.170 | 0.170 | 0.170 |

### 3.4 Analysis of multi-combined effect of system parameters

Energy dissipation of the hysteresis according to the cable length (maximum tension of 100 N)

Cycle | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

Hysteresis energy dissipation (J) | |||||

100 mm | 0.250 | 0.092 | 0.084 | 0.076 | 0.070 |

200 mm | 0.522 | 0.151 | 0.124 | 0.112 | 0.100 |

300 mm | 0.777 | 0.226 | 0.184 | 0.170 | 0.150 |

Cycle | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|

Hysteresis energy dissipation (J) | |||||

100 mm | 0.067 | 0.063 | 0.062 | 0.060 | 0.060 |

200 mm | 0.093 | 0.087 | 0.086 | 0.086 | 0.086 |

300 mm | 0.149 | 0.141 | 0.138 | 0.138 | 0.138 |

Energy dissipation of the hysteresis according to the tension (cable length of 300 mm)

Cycle | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

Hysteresis energy dissipation (J) | |||||

50 N | 0.135 | 0.119 | 0.110 | 0.099 | 0.090 |

100 N | 0.777 | 0.226 | 0.184 | 0.170 | 0.150 |

150 N | 1.360 | 0.450 | 0.375 | 0.345 | 0.340 |

Cycle | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|

Hysteresis energy dissipation (J) | |||||

50 N | 0.086 | 0.082 | 0.082 | 0.082 | 0.082 |

100 N | 0.149 | 0.141 | 0.138 | 0.133 | 0.129 |

150 N | 0.331 | 0.322 | 0.322 | 0.322 | 0.322 |

## 4 Conclusions

In this paper, we investigated the nonlinear dynamics behavior according to the various properties using the verified model. The dynamic behavior of cable was largely divided into four parts: hardening effect, dynamic creep, short- and long-term recovery, and hysteresis. To investigate the dynamic behavior, the effect of the parameters on the model was investigated. For the parameter study, various tensile rate, cable length and applied tension conditions were used. At first, dynamic creep behavior was investigated. Dynamic creep was saturated by one value as processing the cyclic load. The saturation rate was inversely proportional to the cable length, and proportional to the applied tension. Another factor that determines the dynamic creep was the tensile rate. The simulation is performed at the tension from 0 to 100 N. As the tensile rate increases, the dynamic creep decreases linearly. And, the short- and recovery was investigated based on the cyclic load. As increasing the applied tension, short-term recovery had fast the saturation rate. Moreover, In order to consider the CDPR characteristics in which various tension changes occur, the cable elongation characteristics in cyclic load and arbitrary trajectory were investigated. Because the hysteresis is result of cyclic loading, the characteristics were investigated for integrated characteristics. The hysteresis was the characteristics combining all of dynamic behavior such as dynamic creep, hardening short and long-term recovery. So, the model had same tendency as dynamic creep and hardening. And based on the model, the elongation was predicted in arbitrary tension trajectory. All of cable characteristics were affected by tension history. In particular, when the exerted tension was decreased, the length of the cable could be reduced or elongated because creep and recovery occur at the same time. As mentioned above, the elongation characteristics of the cable are very complex because they occur in a complex form. Therefore, predicting cable characteristics from the analysis of the effects of physical parameters is essential for accurate control of cable-driven parallel robots.

## Notes

### Acknowledgements

This research was supported by Development of Space Core Technology Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (2017M1A3A3A02016340) and Korea Institute of Energy Technology Evaluation and Planning (KETEP) and the Ministry of Trade, Industry & Energy (MOTIE) of the Republic of Korea (20174030201530).

## References

- 1.R. Dekker, A. Khajepour, S. Behzadipour, Design and testing of an ultra-high-speed cable robot. Int. J. Robot. Autom.
**21**, 25–34 (2006)Google Scholar - 2.P. Miermeister, A. Poot, A. Verl, Dynamic modeling and hard-ware-in-the loop simulation for the cable-driven parallel robot IPAnema.
*International symposium on robotics*, pp. 1288–1295 (2010)Google Scholar - 3.R. Chattopadhyay, Textile rope—a review. Indian J. Fibre Text. Res.
**11**, 360–368 (1997)Google Scholar - 4.M. Miyasaka, M aghighipanah, Y. Li, Hysteresis model of longitudinally loaded cable for cable driven robots and Identification of the parameters.
*IEEE International conference on robotics and automation (ICRA)*, pp. 4015–4057 (2016)Google Scholar - 5.V. Schmidt, A. Pott, Increase of Position Accuracy for Cable-Driven Parallel Robots Using a Model for Elongation of Plastic Fiber Ropes, in
*New Trends in Mechanism and Machine Science*, ed. by P. Wenger, P. Flores (Mechanisms and Machine Science, Springer, Cham, 2017), pp. 335–343CrossRefGoogle Scholar - 6.S.H. Choi, K.S. Park, Integrated and nonlinear dynamic model of a polymer cable for low-speed cable-driven parallel robots. Microsyst. Technol.
**24**(11), 4677–4687 (2018)CrossRefGoogle Scholar - 7.J.M. Heo, S.H. Choi, K.S. Park, Workspace analysis of a 6-DOF cable-driven parallel robot considering pulley bearing friction under ultra-high acceleration. Microsyst. Technol.
**23**(7), 2615–2627 (2017)CrossRefGoogle Scholar - 8.S. Kawamura, W. Choe, S. Tanaka, S.R. Pandian, Development of an ultrahigh speed robot FALCON using parallel wire drive systems. J. Robot. Soc. Jpn.
**15**(1), 82–89 (1997)CrossRefGoogle Scholar - 9.T.C. Stamnitz, Electro-opto-mechanical cable for fiber optic transmission systems. U.S. Patent No. 4,952,012 (1990)Google Scholar
- 10.W. Gindl, J. Keckes, Strain hardening in regenerated cellulose fibres. Compos. Sci. Technol.
**66**(13), 2049–2053 (2006)CrossRefGoogle Scholar - 11.S.Y. Lee, H.S. Yang, H.J. Kim, C.S. Jeong, B.S. Lim, J.N. Lee, Creep behavior and manufacturing parameters of wood flour filled polypropylene composites. Compos. Struct.
**65**(3–4), 459–469 (2004)CrossRefGoogle Scholar - 12.P.V. Lade, C.D. Liggio, J. Nam, Strain rate, creep, and stress drop-creep experiments on crushed coral sand. J. Geotech. Geoenviron. Eng.
**135**(7), 941–953 (2009)CrossRefGoogle Scholar - 13.F. Li, R.C. Larock, J.U. Otaigbe, Fish oil thermosetting polymers: creep and recovery behavior. Polymer
**41**(13), 4849–4862 (2000)CrossRefGoogle Scholar - 14.Y.C. Chen, L.W. Chen, W.H. Lu, Power loss characteristics of a sensing element based on a polymer optical fiber under cyclic tensile elongation. Sensors
**11**(9), 8741–8750 (2011)CrossRefGoogle Scholar