# Non-Linear Lumped-Parameter Modeling of Planar Multi-Link Manipulators with Highly Flexible Arms

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## Abstract

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## 1. Introduction

## 2. Dynamic Modeling of the Flexible Robot Manipulator

## 3. Numerical Examples for Some Trajectory-Tracking Cases

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## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 8.**Relative tracking error for the tip with collocated PD control: linear trajectory case (

**a**); parabolic trajectory case (

**b**).

${\mathit{a}}_{0}^{\left(\mathit{i}\right)}$ | ${\mathit{a}}_{1}^{\left(\mathit{i}\right)}$ | ${\mathit{a}}_{2}^{\left(\mathit{i}\right)}$ | ${\mathit{a}}_{3}^{\left(\mathit{i}\right)}$ | ${\mathit{a}}_{4}^{\left(\mathit{i}\right)}$ | ${\mathit{a}}_{5}^{\left(\mathit{i}\right)}$ | ${\mathit{a}}_{6}^{\left(\mathit{i}\right)}$ | |
---|---|---|---|---|---|---|---|

Link 1 | 113.65 | −590.40 | 102.40 | 210.97 | 3612.9 | −3800.3 | −0.00178 |

Link 2 | −0.1364 | 105.65 | −780.47 | 1570.4 | −848.80 | −108.65 | −0.01214 |

${\mathit{a}}_{0}^{\left(\mathit{i}\right)}$ | ${\mathit{a}}_{1}^{\left(\mathit{i}\right)}$ | ${\mathit{a}}_{2}^{\left(\mathit{i}\right)}$ | ${\mathit{a}}_{3}^{\left(\mathit{i}\right)}$ | ${\mathit{a}}_{4}^{\left(\mathit{i}\right)}$ | ${\mathit{a}}_{5}^{\left(\mathit{i}\right)}$ | ${\mathit{a}}_{6}^{\left(\mathit{i}\right)}$ | ${\mathit{a}}_{7}^{\left(\mathit{i}\right)}$ | ${\mathit{a}}_{8}^{\left(\mathit{i}\right)}$ | |
---|---|---|---|---|---|---|---|---|---|

Link 1 | 0.8090 | 1.114 | −11.239 | −1.681 | 1.297 | 19.485 | 14.592 | −6.116 | −15.732 |

Link 2 | 0.2139 | −1.434 | 2.659 | 0.0668 | −9.700 | 10.966 | 3.453 | −5.661 | −0.2856 |

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**MDPI and ACS Style**

Giorgio, I.; Del Vescovo, D. Non-Linear Lumped-Parameter Modeling of Planar Multi-Link Manipulators with Highly Flexible Arms. *Robotics* **2018**, *7*, 60.
https://doi.org/10.3390/robotics7040060

**AMA Style**

Giorgio I, Del Vescovo D. Non-Linear Lumped-Parameter Modeling of Planar Multi-Link Manipulators with Highly Flexible Arms. *Robotics*. 2018; 7(4):60.
https://doi.org/10.3390/robotics7040060

**Chicago/Turabian Style**

Giorgio, Ivan, and Dionisio Del Vescovo. 2018. "Non-Linear Lumped-Parameter Modeling of Planar Multi-Link Manipulators with Highly Flexible Arms" *Robotics* 7, no. 4: 60.
https://doi.org/10.3390/robotics7040060