# Multi-Objective Motion Control Optimization for the Bridge Crane System

^{*}

## Abstract

**:**

## Featured Application

**The specific application of the research aims to the port transportation, the working efficiency can be improved considering findings of this article.**

## Abstract

## 1. Introduction

## 2. Dynamic Modeling of the Crane System

**Assumption**

**1.**

**Assumption**

**2.**

**Assumption**

**3.**

**Constraint**

**1.**

**Constraint**

**2.**

**Constraint**

**3.**

## 3. Control Design

#### 3.1. Trajectory Planning Control

**Case**

**1.**

**Case**

**2.**

**Case**

**3.**

#### 3.2. LQR Control Based on Trajectory Planning

#### 3.3. Multi-Objective Optimization

^{2}, the maximum crane velocity is 1.5 m/s, the target position is ${x}_{d}=0.4$ m, and the error of two objective functions is limited within $1\text{}\times \text{}{10}^{-4}$.

#### 3.4. Stability Analysis

**Theorem.**

**Proof.**

## 4. Numerical Simulation

^{2}, and its duration as 0.2 s. The responses of the TP, LQR, and proposed algorithm are compared in Figure 9. It can be observed that the maximum payload swing of TP is 7.3° when the external disturbance occurs; obviously, it cannot meet the control demands. In addition, the LQR algorithm can suppress the swing, and the maximum swing is 2.03°. However, the whole duration is still 1.66 s. The performance of the proposed algorithm is superior to that of other control methods, and the settling time is 1.44 s. Thus, it proves that the algorithm can achieve fast, stable ability and realize immune control of external disturbances.

## 5. Experimental Verification

#### 5.1. Constraints Condition

#### 5.2. Different Payload Condition

#### 5.3. Different Cable Length Condition

#### 5.4. Disturbance Condition

^{2}is added, and its duration is 0.2 s. By comparing the responses of three different controllers in Figure 15, the proposed algorithm can realize the optimal control effect compared with the other methods. The proposed algorithm can reach the target position with the shortest time. It can also be observed that the TP algorithm cannot suppress the payload swing, and the LQR algorithm can induce the swing angle by 1.41°. In this manner, conclusions can be made that the proposed method can effectively suppress the external disturbance.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Abdel-Rahman, E.M.; Nayfeh, A.H.; Masoud, Z.N. Dynamics and Control of Cranes: A Review. J. Vib. Control
**2003**, 9, 863–908. [Google Scholar] [CrossRef] - Maghsoudi, M.J.; Mohamed, Z.; Sudin, S.; Buyamin, S.; Jaafar, H.; Ahmad, 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] - Abdullahi, A.M.; Mohamed, Z.; Abidin, M.Z.; Buyamin, S.; Bature, A.A. Output-based command shaping technique for an effective payload sway control of a 3D crane with hoisting. Trans. Inst. Meas. Control
**2017**, 39, 1443–1453. [Google Scholar] [CrossRef] - Zhang, M.; Ma, X.; Gao, F.; Tian, X.; Li, Y. A motion planning method for underactuated 3D overhead crane systems. In Proceedings of the 2015 34th Chinese Control Conference (CCC), Hangzhou, China, 28–30 July 2015; pp. 4286–4291. [Google Scholar]
- Sun, N.; Fang, Y.; Zhang, X.; Yuan, Y. Transportation task-oriented trajectory planning for underactuated overhead cranes using geometric analysis. IET Control Theory Appl.
**2012**, 6, 1410–1423. [Google Scholar] [CrossRef] - Chiu, C.-H.; Lin, C.-H. Adaptive output recurrent neural network for overhead crane system. In Proceedings of the SICE Annual Conference 2010, Taipei, Taiwan, 18–21 August 2010; pp. 1082–1087. [Google Scholar]
- He, W.; Zhang, S.; Ge, S.S. Adaptive Control of a Flexible Crane System with the Boundary Output Constraint. IEEE Trans. Ind. Electron.
**2014**, 61, 4126–4133. [Google Scholar] [CrossRef] - 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] - Rahmani, R.; Karimi, H.; Yusof, R.; Othman, M.F. A precise fuzzy controller developed for overhead crane. In Proceedings of the 2015 10th Asian Control Conference (ASCC), Sabah, Malaysia, 31 May–3 June 2015; pp. 1–5. [Google Scholar]
- Li, C.; Lee, C.-Y. Fuzzy motion control of an auto-warehousing crane system. IEEE Trans. Ind. Electron.
**2001**, 48, 983–994. [Google Scholar] - Nakazono, K.; Ohnisihi, K.; Kinjo, H. Load swing suppression in jib crane systems using a genetic algorithm-trained neuro-controller. In Proceedings of the ICM2007 4th IEEE International Conference on Mechatronics, Tokyo, Japan, 8–10 May 2007; pp. 1–4. [Google Scholar]
- Kimiaghalam, B.; Homaifar, A.; Bikdash, M.; Dozier, G. Genetic algorithms solution for unconstrained optimal crane control. In Proceedings of the 1999 Congress on Evolutionary Computation CEC ’99, Washington, DC, USA, 6–9 July 1999; pp. 2124–2130. [Google Scholar]
- Le, T.A.; Kim, G.H.; Min, Y.K. Partial feedback linearization control of overhead cranes with varying cable lengths. Int. J. Precis. Eng. Manuf.
**2012**, 13, 501–507. [Google Scholar] [CrossRef] - Sun, N.; Fang, Y.; Zhang, X. Energy coupling output feedback control of 4-DOF underactuated cranes with saturated inputs. Automatica
**2013**, 49, 1318–1325. [Google Scholar] [CrossRef] - Yang, B.; Xiong, B. Application of LQR techniques to the anti-sway controller of overhead crane. Adv. Mater. Res.
**2010**, 139, 1933–1936. [Google Scholar] [CrossRef] - Win, T.M.; Hesketh, T.; Eaton, R. SimMechanics Visualization of Experimental Model Overhead Crane, Its Linearization and Reference Tracking-LQR Control. AIRCC Int. J. Chaos Control Model. Simul.
**2013**, 2, 1–16. [Google Scholar] [CrossRef] - Abdullah, J.; Ruslee, R.; Jalani, J. Performance Comparison between LQR and FLC for Automatic 3 DOF Crane Systems. Int. J. Control Autom.
**2011**, 4, 163–178. [Google Scholar] - Choi, S.; Kim, J.; Lee, J.; Lee, Y.; Lee, K. A study on gantry crane control using neural network two degree of PID controller. In Proceedings of the ISIE 2001 IEEE International Symposium on Industrial Electronics Proceedings, Pusan, Korea, 12–16 June 2001; pp. 1896–1900. [Google Scholar]
- Solihin, M.I.; Wahyudi; Legowo, A. Fuzzy-tuned PID anti-swing control of automatic gantry crane. J. Vib. Control
**2010**, 16, 127–145. [Google Scholar] [CrossRef] - Lee, C.H.; Lee, Y.H.; Teng, C.C. A novel robust PID controllers design by fuzzy neural network. In Proceedings of the American Control Conference, Anchorage, AK, USA, 8–10 May 2002; pp. 1561–1566. [Google Scholar]
- Zhang, S.; He, X. Adaptive HJI sliding mode control of three dimensional overhead cranes. In Proceedings of the 2016 Chinese Control and Decision Conference (CCDC), Yinchuan, China, 28–30 May 2016; pp. 5820–5855. [Google Scholar]
- Solihin, M.I.; Kamal, M.; Legowo, A. Objective function selection of GA-based PID control optimization for automatic gantry crane. In Proceedings of the 2008 International Conference on Computer and Communication Engineering (ICCCE), Kuala Lumpur, Malaysia, 13–15 May 2008; pp. 883–887. [Google Scholar]
- Lin, J.; Zheng, Y. Vibration suppression control of smart piezoelectric rotating truss structure by parallel neuro-fuzzy control with genetic algorithm tuning. J. Sound Vib.
**2012**, 331, 3677–3694. [Google Scholar] [CrossRef] - Petrenko, Y.N.; Alavi, S.E. Fuzzy logic and genetic algorithm technique for non-liner system of overhead crane. In Proceedings of the 2010 IEEE Region 8 International Conference on Computational Technologies in Electrical and Electronics Engineering (SIBIRCON), Irkutsk, Russia, 11–15 July 2010; pp. 848–851. [Google Scholar]
- Wang, X.J.; Chen, Z.M. Two-degree-of-freedom sliding mode anti-swing and positioning controller for overhead cranes. In Proceedings of the 2016 Chinese Control and Decision Conference (CCDC), Yinchuan, China, 28–30 May 2016; pp. 673–677. [Google Scholar]
- Wang, X.; Wang, H.; Tian, Y.; Christov, N. Predictive Observer based Lyapunov antiswing control for overhead visual crane. In Proceedings of the 2015 Chinese Automation Congress (CAC), Wuhan, China, 27–29 November 2015; pp. 1670–1675. [Google Scholar]
- Wu, X.; He, X.; Ou, X. A coupling control method applied to 2-D overhead cranes. In Proceedings of the 2016 35th Chinese Control Conference (CCC), Chengdu, China, 27–29 July 2016; pp. 1658–1662. [Google Scholar]
- Chen, H.; Fang, Y.; Sun, N. A swing constraint guaranteed MPC algorithm for underactuated overhead cranes. IEEE/ASME Trans. Mechatron.
**2016**, 21, 2543–2555. [Google Scholar] [CrossRef] - Jafari, J.; Ghazal, M.; Nazemizadeh, M. A LQR Optimal Method to Control the Position of an Overhead Crane. IAES Int. J. Robot. Autom.
**2014**, 3, 252. [Google Scholar] - Blajer, W.; Kołodziejczyk, K. Control of underactuated mechanical systems with servo-constraints. Nonlinear Dyn.
**2007**, 50, 781–791. [Google Scholar] [CrossRef] - Wang, J.; Noda, Y.; Inomata, A. Straight transfer control system using PI control and trajectory planning in overhead traveling crane. In Proceedings of the 2015 IEEE/SICE International Symposium on System Integration (SII), Nagoya, Japan, 11–13 December 2015; pp. 522–527. [Google Scholar]
- Chen, H.; Fang, Y.; Sun, N. Optimal trajectory planning and tracking control method for overhead cranes. IET Control Theory Appl.
**2016**, 10, 692–699. [Google Scholar] [CrossRef] - Summanwar, V.S.; Jayaraman, V.K.; Kulkarni, B.D.; Kusumakar, H.S.; Gupta, K.; Rajesh, J. Solution of constrained optimization problems by multi-objective genetic algorithm. Comput. Chem. Eng.
**2002**, 26, 1481–1492. [Google Scholar] [CrossRef] - Coello, C.A.C. Multi-objective Evolutionary Algorithms in Real-World Applications: Some Recent Results and Current Challenges; Springer International Publishing: Cham, Switzerland, 2015; pp. 3–18. [Google Scholar]
- Konak, A.; Coit, D.W.; Smith, A.E. Multi-objective optimization using genetic algorithms: A tutorial. Reliab. Eng. Syst. Saf.
**2006**, 91, 992–1007. [Google Scholar] [CrossRef] - Cui, Y.; Geng, Z.; Zhu, Q.; Han, Y. Review: Multi-objective optimization methods and application in energy saving. Energy
**2017**, 125, 681–704. [Google Scholar] [CrossRef] - Yeh, W.C.; Chuang, M.C. Using multi-objective genetic algorithm for partner selection in green supply chain problems. Expert Syst. Appl.
**2011**, 38, 4244–4253. [Google Scholar] [CrossRef] - Khalil, H.K. Nonlinear Systems, 3rd ed.; Prentice-Hall, Inc.: Upper Saddle River, NJ, USA, 2002. [Google Scholar]
- Padula, F.; Adamini, R.; Finzi, G.; Visioli, A. Revealing the Hidden Technology by Means of an Overhead Crane. IFAC Papersonline
**2017**, 50, 9126–9131. [Google Scholar] [CrossRef] - 2016 B&R Scholastic Union Competition. Available online: http://www.br-education.com/activities/index.asp?ColumnId=45&Style_ID=3&ID=376 (accessed on 18 March 2018).
- B&R Automation. Available online: https://www.br-automation.com/en-au/about-us/customer-magazine/2013/ 201303 /swift-steady/ (accessed on 18 March 2018).

**Figure 7.**Control response of trajectory planning (TP), linear quadratic regulator (LQR), and the proposed method.

Parameter | Note | Value |
---|---|---|

$L$ | Payload length | 0.122 m |

$g$ | Gravity | 9.81 m/s^{2} |

Parameter | Solution 1 | Solution 2 | Solution 3 |
---|---|---|---|

$Q$ | $\left[\begin{array}{cccc}470.19& 0& 0& 0\\ 0& 219.74& 0& 0\\ 0& 0& 465.33& 0\\ 0& 0& 0& 2.01\end{array}\right]$ | $\left[\begin{array}{cccc}498.57& 0& 0& 0\\ 0& 230.79& 0& 0\\ 0& 0& 503.24& 0\\ 0& 0& 0& 2.01\end{array}\right]$ | $\left[\begin{array}{cccc}474.60& 0& 0& 0\\ 0& 230.78& 0& 0\\ 0& 0& 450.99& 0\\ 0& 0& 0& 2.06\end{array}\right]$ |

$R$ | 2.21 | 2.05 | 2.42 |

$[{k}_{1},{k}_{2},{k}_{3},{k}_{4}]$ | $[14.60,13.13,-14.66,-0.93]$ | $[15.89,13.93,-15.89,-0.95]$ | $[13.99,12.75,-13.75,-0.90]$ |

${K}_{I}$ | 0.02 | 0.02 | 0.04 |

${K}_{D}$ | 166.82 | 169.34 | 175.86 |

**Table 3.**Performance of three solutions. ITAE: integration of the time-weighted absolute value of the errors.

Performance | Solution 1 | Solution 2 | Solution 3 |
---|---|---|---|

Settling time (s) | 4.21 | 2.68 | 1.86 |

Maximum payload swing (deg) | 1.00 | 2.00 | 4.00 |

ITAE of swing | 2469.21 | 2993.82 | 3999.16 |

ITAE of positon | 910.92 | 328.69 | 154.58 |

Parameter | LQR | Proposed Method |
---|---|---|

${K}_{I}$ | NA | 0.02 |

${K}_{D}$ | NA | 169.34 |

$[{k}_{1},{k}_{2},{k}_{3},{k}_{4}]$ | $[1.46,2.56,-3.13,-1.35]$ | $[15.89,13.93,-15.89,-0.95]$ |

Target Position | Settling Time (s) | Maximum Payload Swing (deg) |
---|---|---|

0.2 m | 1.87 | 1.93 |

0.3 m | 2.35 | 1.85 |

0.4 m | 2.67 | 1.93 |

Target Position | Settling Time (s) | Maximum Payload Swing (Deg) |
---|---|---|

0.2 m | 1.63 | 2.0 |

0.3 m | 1.64 | 3.2 |

0.4 m | 1.66 | 4.0 |

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**MDPI and ACS Style**

Xiao, R.; Wang, Z.; Guo, N.; Wu, Y.; Shen, J.; Chen, Z.
Multi-Objective Motion Control Optimization for the Bridge Crane System. *Appl. Sci.* **2018**, *8*, 473.
https://doi.org/10.3390/app8030473

**AMA Style**

Xiao R, Wang Z, Guo N, Wu Y, Shen J, Chen Z.
Multi-Objective Motion Control Optimization for the Bridge Crane System. *Applied Sciences*. 2018; 8(3):473.
https://doi.org/10.3390/app8030473

**Chicago/Turabian Style**

Xiao, Renxin, Zelin Wang, Ningyuan Guo, Yitao Wu, Jiangwei Shen, and Zheng Chen.
2018. "Multi-Objective Motion Control Optimization for the Bridge Crane System" *Applied Sciences* 8, no. 3: 473.
https://doi.org/10.3390/app8030473