# Research on Nonlinear Control Method of Underactuated Gantry Crane Based on Machine Vision Positioning

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- The newly designed machine visual positioning system improves the positioning accuracy and automation efficiency of the gantry crane.
- (2)
- The control law has a concise structure without requiring exact knowledge of system parameters, which is convenient for practical implementation. It is robust against external disturbances, which is validated by experimental results.

## 2. Control Method Design

## 3. Gantry Crane Dynamics Modeling

## 4. Machine Vision Positioning System

#### 4.1. Image Preprocessing

#### 4.2. Target Detection and Positioning

## 5. Trajectory Planning

## 6. Control Law Design and Stability Analysis

#### 6.1. Control Law Design

#### 6.2. Closed-Loop System Stability Analysis

**Theorem**

**1.**

**Proof.**

## 7. Simulation Experiment and Result Analysis

#### 7.1. Machine Vision Positioning Experimental Results and Analysis

#### 7.2. Friction Simulation and Analysis

#### 7.3. Trajectory Planning Simulation and Analysis

#### 7.4. Control Law Experimental Results and Analysis

**Experiment**

**1.**

**Experiment**

**2.**

**Experiment**

**3.**

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Fang, Y.; Ma, B.; Wang, P.; Zhang, X. A Motion Planning-Based Adaptive Control Method for an Underactuated Crane System. IEEE Trans. Control Syst. Technol.
**2012**, 20, 241–248. [Google Scholar] [CrossRef] - Pezeshki, S.; Badamchizadeh, M.A.; Ghiasi, A.R.; Ghaemi, S. Control of Overhead Crane System Using Adaptive Model-Free and Adaptive Fuzzy Sliding Mode Controllers. J. Control Autom. Electr. Syst.
**2015**, 26, 1–15. [Google Scholar] - Sun, N.; Wu, Y.; Chen, H.; Fang, Y. An energy-optimal solution for transportation control of cranes with double pendulum dynamics: Design and experiments. Mech. Syst. Signal Process.
**2018**, 102, 87–101. [Google Scholar] - Sun, N.; Yang, T.; Fang, Y.; Wu, Y.; Chen, H. Transportation Control of Double-Pendulum Cranes with a Nonlinear Quasi-PID Scheme: Design and Experiments. IEEE Trans. Syst. Man Cybern. Syst.
**2019**, 49, 1408–1418. [Google Scholar] [CrossRef] - Lu, B.; Fang, Y.; Sun, N. Modeling and nonlinear coordination control for an underactuated dual overhead crane system. Automatica
**2018**, 91, 244–255. [Google Scholar] - Yasir, M.; Ho, S.W.; Vellambi, B.N. Indoor Positioning System Using Visible Light and Accelerometer. J. Lightwave Technol.
**2014**, 32, 3306–3316. [Google Scholar] [CrossRef] - Hossen, M.S.; Park, Y.; Kim, K.D. Performance improvement of indoor positioning using light-emitting diodes and an image sensor for light-emitting diode communication. Opt. Eng.
**2015**, 54, 035108. [Google Scholar] - Truc, N.T.; Kim, Y.T. Navigation Method of the Transportation Robot Using Fuzzy Line Tracking and QR Code Recognition. Int. J. Hum. Robot.
**2017**, 14, 1650027. [Google Scholar] - Moustafa, K.A.F.; Ebeid, A.M. Nonlinear modeling and control of overhead crane load sway. J. Dyn. Syst. Meas. Control
**1988**, 110, 266–271. [Google Scholar] [CrossRef] - Fang, Y.; Dixon, W.E.; Dawson, D.M.; Zergeroglu, E. Nonlinear coupling control laws for an underactuated overhead crane system. IEEE/ASME Trans. Mechatron.
**2003**, 8, 418–423. [Google Scholar] [CrossRef] [Green Version] - Tuan, L.A.; Kim, J.J.; Lee, S.G.; Lim, T.G.; Nho, L.C. Second-order sliding mode control of a 3D overhead crane with uncertain system parameters. Int. J. Precis. Eng. Manuf.
**2014**, 15, 811–819. [Google Scholar] - Wu, X.; He, X. Partial feedback linearization control for 3-D underactuated overhead crane systems. ISA Trans.
**2016**, 65, 361–370. [Google Scholar] - Zhang, M.; Ma, X.; Rong, X.; Song, R.; Tian, X.; Li, Y. An Enhanced Coupling Nonlinear Tracking Controller for Underactuated 3D Overhead Crane Systems. Asian J. Control
**2018**, 20, 1839–1854. [Google Scholar] - Sun, N.; Fang, Y.; Zhang, Y.; Ma, B. A Novel Kinematic Coupling-Based Trajectory Planning Method for Overhead Cranes. IEEE/ASME Trans. Mechatron.
**2012**, 17, 166–173. [Google Scholar] [CrossRef] - Makkar, C.; Hu, G.; Sawyer, W.G.; Dixon, W. Lyapunov-Based Tracking Control in the Presence of Uncertain Nonlinear Parameterizable Friction. IEEE Trans. Autom. Control
**2007**, 52, 1988–1994. [Google Scholar] [CrossRef] - Di, Y.J.; Shi, J.P.; Mao, G.Y. A QR code identification technology in package auto-sorting system. Mod. Phys. Lett. B
**2017**, 31, 19–21. [Google Scholar] - Canny, J. A Computational Approach to Edge Detection. IEEE Trans. Pattern Anal. Mach. Intell.
**1986**, PAMI–8, 679–698. [Google Scholar] [CrossRef] - Kumar, G.; Umesh, G. Image steganography based on Canny edge detection, dilation operator and hybrid coding. J. Inf. Secur. Appl.
**2018**, 41, 41–51. [Google Scholar] - Zhang, J.; Yu, H.; Deng, H.; Chai, Z.; Ma, M.; Zhong, X. A Robust and Rapid Camera Calibration Method by One Captured Image. IEEE Trans. Instrum. Meas.
**2018**, 1–10. [Google Scholar] [CrossRef] - Li, J.; Liu, Z. Efficient camera self-calibration method for remote sensing photogrammetry. Opt. Express
**2018**, 26, 14213–14231. [Google Scholar] [PubMed] - Kaehler, A.; Bradski, G. Learning OpenCV 3: Computer Vision in C++ with the OpenCV Library; O’Reilly Media: Sebastopol, CA, USA, 2016. [Google Scholar]
- Howse, J.; Joshi, P.; Beyeler, M. OpenCV: Computer Vision Projects with Python; Packt Publishing Limited: Birmingham, UK, 2017. [Google Scholar]
- Lee, H.H. Motion planning for three-dimensional overhead cranes with high-speed load hoisting. Int. J. Control
**2005**, 78, 875–886. [Google Scholar] - Sun, N. Trajectory Planning and Nonlinear Control for Underactuated Cranes: Design, Analysis, and Applications. Ph.D. Thesis, Nankai University, Tianjin, China, 2014; pp. 19–56. [Google Scholar]
- Maghsoudi, M.J.; Mohamed, Z.; Sudin, S. An improved input shaping design for an efficient sway control of a nonlinear 3D overhead crane with friction. Mech. Syst. Signal Process.
**2017**, 92, 364–378. [Google Scholar] [CrossRef] - Xie, X.; Huang, J.; Liang, Z. Vibration reduction for flexible systems by command smoothing. Mech. Syst. Signal Process.
**2013**, 39, 461–470. [Google Scholar] - Uchiyama, N.; Ouyang, H.; Sano, S. Simple rotary crane dynamics modeling and open-loop control for residual load sway suppression by only horizontal boom motion. Mechatronics
**2013**, 23, 1223–1236. [Google Scholar] [CrossRef] - Tuan, L.A.; Lee, S.G.; Ko, D.H.; Nho, L.C. Combined control with sliding mode and partial feedback linearization for 3D overhead cranes. Int. J. Robust Nonlinear Control
**2014**, 24, 3372–3386. [Google Scholar] - Yu, W.; Li, X.; Panuncio, F. Stable Neural Pid Anti-Swing Control For An Overhead Crane. Intell. Autom. Soft Comput.
**2014**, 20, 145–158. [Google Scholar] [CrossRef] - Khalil, H. Nonlinear Systems, 3rd ed.; Prentice-Hall: Englewood Cliffs, NJ, USA, 2002. [Google Scholar]

**Figure 9.**Image preprocessing. (

**a**) Original image; (

**b**) grayscale image; (

**c**) binary chart; (

**d**) Canny edge detection.

**Figure 12.**Results for trajectory planning simulation. (

**a**) X direction; (

**b**) Y direction; (

**c**) compound direction.

**Figure 13.**Results for gantry kinetic energy (GKE) and trolley kinetic energy (TKE) in Experiment 1. (

**a**) X direction; (

**b**) Y direction; (

**c**) compound direction.

**Figure 14.**Results for proposed controller in Experiment 1. (

**a**) X direction; (

**b**) Y direction; (

**c**) compound direction.

**Figure 15.**Results for GKE and TKE in Experiment 2. (

**a**) X direction; (

**b**) Y direction; (

**c**) compound direction.

**Figure 16.**Results for proposed controller in Experiment 2. (

**a**) X direction; (

**b**) Y direction; (

**c**) compound direction.

**Figure 17.**Results for GKE and TKE in Experiment 3. (

**a**) X direction; (

**b**) Y direction; (

**c**) compound direction.

**Figure 18.**Results for proposed controller in Experiment 3. (

**a**) X direction; (

**b**) Y direction; (

**c**) compound direction.

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

Rope length | $l$ | 1 | $\mathrm{m}$ |

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

Gantry preset acceleration | ${a}_{xub}$ | 0.4 | $\mathrm{m}/{\mathrm{s}}^{2}$ |

Gantry preset speed | ${v}_{xub}$ | 0.8 | $\mathrm{m}/\mathrm{s}$ |

Gantry target displacement | ${x}_{ub}$ | 0.548 | $\mathrm{m}$ |

Gantry frame swing angle upper limit | ${\theta}_{xub}$ | 5 | deg |

Trolley preset acceleration | ${a}_{yub}$ | 0.5 | $\mathrm{m}/{\mathrm{s}}^{2}$ |

Trolley preset speed | ${v}_{yub}$ | 1 | $\mathrm{m}/\mathrm{s}$ |

Trolley target displacement | ${y}_{ub}$ | 0.430 | $\mathrm{m}$ |

Trolley swing angle upper limit | ${\theta}_{yub}$ | 5 | deg |

Compound acceleration | ${\ddot{p}}_{ub}$ | 0.5 | $\mathrm{m}/{\mathrm{s}}^{2}$ |

Compound speed | ${\dot{p}}_{ub}$ | 1 | $\mathrm{m}/\mathrm{s}$ |

Compound displacement | ${p}_{ub}$ | 0.697 | $\mathrm{m}$ |

Compound angle 1 | ${\theta}_{1ub}$ | 5 | deg |

Compound angle 2 | ${\theta}_{2ub}$ | 1 | rad |

Control Law | ${\mathit{k}}_{\mathit{p}\mathit{x}}$ | ${\mathit{k}}_{\mathit{d}\mathit{x}}$ | ${\mathit{k}}_{\mathit{a}\mathit{x}}$ | ${\mathit{k}}_{\mathit{e}\mathit{x}}$ | ${\mathit{k}}_{\mathit{v}\mathit{x}}$ | ${\mathit{k}}_{\mathit{p}\mathit{y}}$ | ${\mathit{k}}_{\mathit{d}\mathit{y}}$ | ${\mathit{k}}_{\mathit{a}\mathit{y}}$ | ${\mathit{k}}_{\mathit{e}\mathit{y}}$ | ${\mathit{k}}_{\mathit{v}\mathit{y}}$ |
---|---|---|---|---|---|---|---|---|---|---|

GKE and TKE (66) | 18 | 28 | 1 | NA | NA | 16 | 30 | 1.95 | NA | NA |

proposed controller (38) | 36 | 80 | NA | 1 | 0.5 | 35 | 65 | NA | 1 | 1 |

Control Law | ${\mathit{\theta}}_{\mathit{x}\mathbf{max}}$ | ${\mathit{\theta}}_{\mathit{x}\mathit{r}\mathit{e}\mathit{s}}$ | ${\mathit{t}}_{\mathit{x}\mathit{s}}$ | ${\mathit{F}}_{\mathit{x}\mathbf{max}}$ | ${\mathit{\theta}}_{\mathit{y}\mathbf{max}}$ | ${\mathit{\theta}}_{\mathit{y}\mathit{r}\mathit{e}\mathit{s}}$ | ${\mathit{t}}_{\mathit{y}\mathit{s}}$ | ${\mathit{F}}_{\mathit{y}\mathbf{max}}$ | ${\mathit{\theta}}_{\mathit{p}\mathbf{max}}$ | ${\mathit{\theta}}_{\mathit{p}\mathit{r}\mathit{e}\mathit{s}}$ | ${\mathit{t}}_{\mathit{p}\mathit{s}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|

GKE and TKE (66) | 3.6 | 1.9 | 16.8 | 11.9 | 3.4 | 0.9 | 11.5 | 10.2 | 4.8 | 2.1 | 17.3 |

proposed controller (38) | 2.4 | 0.3 | 6.5 | 10.3 | 1.8 | 0.4 | 7.3 | 4.9 | 2.9 | 0.5 | 7.4 |

Control Law | ${\mathit{\theta}}_{\mathit{x}\mathbf{max}}$ | ${\mathit{\theta}}_{\mathit{x}\mathit{r}\mathit{e}\mathit{s}}$ | ${\mathit{t}}_{\mathit{x}\mathit{s}}$ | ${\mathit{F}}_{\mathit{x}\mathbf{max}}$ | ${\mathit{\theta}}_{\mathit{y}\mathbf{max}}$ | ${\mathit{\theta}}_{\mathit{y}\mathit{r}\mathit{e}\mathit{s}}$ | ${\mathit{t}}_{\mathit{y}\mathit{s}}$ | ${\mathit{F}}_{\mathit{y}\mathbf{max}}$ | ${\mathit{\theta}}_{\mathit{p}\mathbf{max}}$ | ${\mathit{\theta}}_{\mathit{p}\mathit{r}\mathit{e}\mathit{s}}$ | ${\mathit{t}}_{\mathit{p}\mathit{s}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|

GKE and TKE (66) | 4.6 | 3.2 | >30 | 12.0 | 4.8 | 2.3 | 27.2 | 9.1 | >5 | 3.8 | >30 |

proposed controller (38) | 2.5 | 0.3 | 6. | 10.3 | 1.7 | 0.3 | 7.5 | 5.3 | 3.0 | 1.1 | 7.6 |

Control Law | ${\mathit{\theta}}_{\mathit{x}\mathbf{max}}$ | ${\mathit{\theta}}_{\mathit{x}\mathit{r}\mathit{e}\mathit{s}}$ | ${\mathit{t}}_{\mathit{x}\mathit{s}}$ | ${\mathit{F}}_{\mathit{x}\mathbf{max}}$ | ${\mathit{\theta}}_{\mathit{y}\mathbf{max}}$ | ${\mathit{\theta}}_{\mathit{y}\mathit{r}\mathit{e}\mathit{s}}$ | ${\mathit{t}}_{\mathit{y}\mathit{s}}$ | ${\mathit{F}}_{\mathit{y}\mathbf{max}}$ | ${\mathit{\theta}}_{\mathit{p}\mathbf{max}}$ | ${\mathit{\theta}}_{\mathit{p}\mathit{r}\mathit{e}\mathit{s}}$ | ${\mathit{t}}_{\mathit{p}\mathit{s}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|

GKE and TKE (66) | 3.6 | 1.9 | 17.2 | 12.0 | 3.4 | 0.9 | 11.5 | 9.8 | 4.8 | 2.1 | 17.3 |

proposed controller (38) | 2.4 | 0.3 | 6.5 | 11.2 | 1.8 | 0.4 | 7.3 | 11.4 | 3.3 | 0.5 | 8.4 |

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**MDPI and ACS Style**

Shi, H.; Li, G.; Bai, X.; Huang, J.
Research on Nonlinear Control Method of Underactuated Gantry Crane Based on Machine Vision Positioning. *Symmetry* **2019**, *11*, 987.
https://doi.org/10.3390/sym11080987

**AMA Style**

Shi H, Li G, Bai X, Huang J.
Research on Nonlinear Control Method of Underactuated Gantry Crane Based on Machine Vision Positioning. *Symmetry*. 2019; 11(8):987.
https://doi.org/10.3390/sym11080987

**Chicago/Turabian Style**

Shi, Huaitao, Gang Li, Xiaotian Bai, and Jianqi Huang.
2019. "Research on Nonlinear Control Method of Underactuated Gantry Crane Based on Machine Vision Positioning" *Symmetry* 11, no. 8: 987.
https://doi.org/10.3390/sym11080987