# Research on Nonlinear Coupling Anti-Swing Control Method of Double Pendulum Gantry Crane Based on Improved Energy

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## Abstract

**:**

## 1. Introduction

## 2. Dynamics of Gantry Crane with Double-Pendulum

**Assumption**

**1.**

**Assumption**

**2.**

## 3. Main Results

#### 3.1. Controller Design

**Property**

**1.**

**Remark**

**1.**

**Remark**

**2.**

#### 3.2. Stability Analysis

**Theorem**

**1.**

**Proof.**

## 4. Simulation Results and Analysis

#### 4.1. Comparative Study

#### 4.2. Robustness Verification

**Case 1.**In this case, we check the proposed controller robustness at different desired locations while keeping the control gain the same as in Table 2. Therefore, the following three transferring distances are set:

- ${p}_{dx}=2$ m;
- ${p}_{dx}=3$ m;
- ${p}_{dx}=4$ m.

**Case 2.**Next, we add the initial swing of hook and swing of payload to disturb the gantry crane control system. The control gains are shown in Table 2. The three groups of initial swing of hook swing and swing of payload are set as:

- Initial ${\theta}_{1}=-3\mathrm{deg},{\theta}_{2}=-5\mathrm{deg}$;
- Initial ${\theta}_{1}=0\mathrm{deg},{\theta}_{2}=\mathrm{deg}$;
- Initial ${\theta}_{1}=3\mathrm{deg},{\theta}_{2}=5\mathrm{deg}$.

**Case 3.**In this case, the control performance of the proposed controller will be tested for different payload masses. At the same time, we use the same control gains in Table 2. There are three different payload masses:

- ${m}_{2}=0.5$ kg;
- ${m}_{2}=1$ kg;
- ${m}_{2}=1.5$ kg.

**Case 4.**In this case, we consider verifying the robustness of the gantry crane with double pendulum control system to different cable lengths. Three groups of different rope lengths are set as follows:

- ${l}_{1}=1{m},{l}_{2}=0.3{m}$;
- ${l}_{1}=1.2{m},{l}_{2}=0.5{m}$;
- ${l}_{1}=1.5{m},{l}_{2}=0.8{m}$.

**Case 5.**There may be external disturbance such as wind resistance and collision during the transportation process of the gantry crane. In this simulation, the disturbances are added to the hook and payload swing to simulate external disturbances throughout the transportation process. We add a sine wave interference of 2 degrees between 10 s and 11 s.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Name | Symbol | Numerical Value | Unit |
---|---|---|---|

Trolley mass | $M$ | 12 | kg |

Hook mass | ${m}_{1}$ | 1.5 | kg |

Payload mass | ${m}_{2}$ | 1 | kg |

Gravity acceleration | $g$ | 9.8 | m/s^{2} |

Cable length 1 | ${l}_{1}$ | 1.2 | m |

Cable length 2 | ${l}_{2}$ | 0.5 | m |

Desired trolley location | ${p}_{dx}$ | 3 | m |

Static friction-related coefficient 1 | ${f}_{r0}$ | 8 | NA |

Static friction-related coefficient 2 | $\xi $ | 0.01 | NA |

Viscous friction-related parameter | ${k}_{r}$ | −1.2 | NA |

Controllers | ${\mathit{k}}_{\mathit{p}}$ | ${\mathit{k}}_{\mathit{d}}$ | ${\mathit{k}}_{\mathit{a}1}$ | ${\mathit{k}}_{\mathit{a}2}$ | ${\mathit{k}}_{\mathit{e}}$ | ${\mathit{k}}_{\mathit{D}}$ | $\mathit{\lambda}$ | $\mathit{\alpha}$ | $\mathit{\beta}$ | $\mathit{K}$ |
---|---|---|---|---|---|---|---|---|---|---|

Proposed method | 5 | 15 | 6 | 1 | NANA | NA | NA | NA | NA | NA |

PD | 6.8 | 2 | NA | NA | NA | NA | NA | NA | NA | NA |

Passivity-based | 6.5 | 20 | NA | NA | 1 | 1 | NA | NA | NA | NA |

CSMC | NA | NA | NA | NA | NA | NA | 0.5 | 17 | −11 | 30 |

Controllers | ${\mathit{\theta}}_{1\mathbf{max}}$ (deg) | ${\mathit{\theta}}_{2\mathbf{max}}$ (deg) | ${\mathit{\theta}}_{1\mathit{r}\mathit{e}\mathit{s}}$ (deg) | ${\mathit{\theta}}_{2\mathit{r}\mathit{e}\mathit{s}}$ (deg) | ${\mathit{p}}_{\mathit{f}}$ (m) | ${\mathit{t}}_{\mathit{s}}$ (s) | ${\mathit{F}}_{\mathbf{max}}$ (N) |
---|---|---|---|---|---|---|---|

Proposed method | 3.47 | 4.29 | 0.03 | 0.02 | 3.005 | 7.46 | 21.27 |

PD | 7.85 | 10.62 | 7.20 | 9.69 | 2.936 | > 20 | 21.40 |

Passivity-based | 6.94 | 9.95 | 1.69 | 2.37 | 2.989 | > 20 | 25.40 |

CSMC | 7.89 | 9.18 | 0.04 | 0.04 | 2.986 | 7.75 | 22.82 |

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**MDPI and ACS Style**

Shi, H.; Li, G.; Ma, X.; Sun, J.
Research on Nonlinear Coupling Anti-Swing Control Method of Double Pendulum Gantry Crane Based on Improved Energy. *Symmetry* **2019**, *11*, 1511.
https://doi.org/10.3390/sym11121511

**AMA Style**

Shi H, Li G, Ma X, Sun J.
Research on Nonlinear Coupling Anti-Swing Control Method of Double Pendulum Gantry Crane Based on Improved Energy. *Symmetry*. 2019; 11(12):1511.
https://doi.org/10.3390/sym11121511

**Chicago/Turabian Style**

Shi, Huaitao, Gang Li, Xin Ma, and Jie Sun.
2019. "Research on Nonlinear Coupling Anti-Swing Control Method of Double Pendulum Gantry Crane Based on Improved Energy" *Symmetry* 11, no. 12: 1511.
https://doi.org/10.3390/sym11121511