# Nonlinear Synchronous Control for H-Type Gantry Stage Used in Electric Vehicles Manufacturing

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## Abstract

**:**

## 1. Introduction

## 2. Nonlinear Model of HGS

#### 2.1. Dynamic Modeling

- The end-effector moves on the horizontal plane, so that the effects of gravity are ignored.
- The mid-beam is modeled as a rigid body.
- The tensile and rotational deformation of the mid-beam are modeled as a tensile spring with stiffness coefficient ${k}_{L}$ and a torsion spring with stiffness coefficient ${k}_{r}$, respectively.
- The end-effector cannot be separated from the mid-beam.

#### 2.2. Friction

## 3. Nonlinear Synchronous Controller Design

#### 3.1. Problem Formulation

#### 3.2. Discontinuous Projection

#### 3.3. Controller Design

#### 3.4. Main Results

**Theorem**

**1.**

**Λ**defined below is positive:

- (1)
- All system signals are bounded, and the Lyapunov function $V(t)$ is bounded by:$$V(t)\le {e}^{-\lambda t}V(0)+\frac{\epsilon}{\lambda}[1-{e}^{-\lambda t}],$$
- (2)
- If there is no lumped uncertainty, i.e., $\mathit{d}=0$, then the asymptotic convergence of system tracking and synchronous error is also achieved, i.e., $\mathit{z}\to 0$, as $t\to \infty $.

**Proof.**

## 4. Experimental Verification

#### 4.1. Experimental Setup

#### 4.2. Experimental Results

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

HGS | H-type gantry stage |

CCC | Cross-couple control |

ARSCR | Adaptive robust synchronous control based on the rigid assumed model |

AC | Adaptive control |

SMC | Sliding mode control |

## Appendix A

## Appendix B

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Model | Rated Power | Maximum Speed | Continuous Force | Continuous Current |
---|---|---|---|---|

Kollmorgen IC44-075 | $10.7$ kW | $7.63$ m/s | 1732 N | $31.8$ A |

Symbol | Description | Value |
---|---|---|

${m}_{1}$ | mass of Linear Motor 1 | 32 kg |

${m}_{2}$ | mass of Linear Motor 2 | 28 kg |

${m}_{b}$ | mass of the mid-beam | $4.46$ kg |

${m}_{h}$ | mass of the end-effector | $2.14$ kg |

${B}_{1}$ | viscous friction coefficient of Motor 1 | $73.03$ N/m/s |

${B}_{2}$ | viscous friction coefficient of Motor 2 | $70.95$ N/m/s |

${A}_{f1}$ | Coulomb friction coefficient of Motor 1 | 50 N |

${A}_{f2}$ | Coulomb friction coefficient of Motor 2 | $65.4$ N |

${k}_{r}$ | equivalent stiffness coefficient of the mid-beam | 11,133.4 Nm/rad |

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**MDPI and ACS Style**

Chen, R.; Jiao, Z.; Yan, L.; Shang, Y.; Wu, S.
Nonlinear Synchronous Control for H-Type Gantry Stage Used in Electric Vehicles Manufacturing. *Energies* **2019**, *12*, 2305.
https://doi.org/10.3390/en12122305

**AMA Style**

Chen R, Jiao Z, Yan L, Shang Y, Wu S.
Nonlinear Synchronous Control for H-Type Gantry Stage Used in Electric Vehicles Manufacturing. *Energies*. 2019; 12(12):2305.
https://doi.org/10.3390/en12122305

**Chicago/Turabian Style**

Chen, Ran, Zongxia Jiao, Liang Yan, Yaoxing Shang, and Shuai Wu.
2019. "Nonlinear Synchronous Control for H-Type Gantry Stage Used in Electric Vehicles Manufacturing" *Energies* 12, no. 12: 2305.
https://doi.org/10.3390/en12122305