# Kinematics Modeling and Analysis of Mid-Low Speed Maglev Vehicle with Screw and Product of Exponential Theory

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## Abstract

**:**

## 1. Introduction

## 2. Forward Kinematics Modeling for a Mid-Low Speed Maglev Train

- (1)
- There is a rotational degree of freedom along the z-direction between base 0 and anti-roll beam 1, whose twist coordinate is ${\xi}_{1}$. This degree of freedom can decouple the motion of the left and right modules of the suspension frame in the running direction.
- (2)
- The ball joint between the anti-roll beam 1 and the hanger rod 2, respectively, includes three rotational degrees of freedom along x, y, and z. Considering the constraint relationship between the two anti-roll beams, the rotational degree of freedom in the z-direction can be ignored. Thus, the rotational degrees of freedom in the x- and y-directions are just considered, and the twists of rotation are ${\xi}_{3}$ and ${\xi}_{2}$, respectively.
- (3)
- The hanger rod 2 has a telescopic translational freedom, and its twist is ${\xi}_{4}$.
- (4)
- The ball joint between the anti-roll beam 1 and the hanger 2 respectively includes three rotational degrees of freedom. The degree of freedom in the z-direction is ignored, and the rotational freedom in the x-direction is retained and the twist is ${\xi}_{5}$.
- (5)
- There is a rotational degree of freedom in the z-direction between the anti-roll beam 3 and the right module 4, whose twist coordinates are ${\xi}_{6}$.

## 3. Solution of Reverse Motion of a Mid-Low Speed Maglev Train

## 4. The Relationship of the Vehicle/Track Position–Posture on the Transition Curve Track

#### 4.1. Parametric Description of the Transition Curve

#### 4.2. The Track Coordinate System on the Transition Curve

#### 4.3. The Posture Matrix of Train Reference System ${V}_{0}$ Relative to Track Reference System ${T}_{0}$

**Note:**The nominal height ${h}_{0}$ of the secondary system with the difference in the z-direction between ${V}_{0}$ and ${T}_{0}$ is directly added to the third term of Equation (48).

## 5. The Motion Analysis of the Maglev Train

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Each Element in Equation (4)

## References

- Youlun, X. Science of Robotics; China Machine Press: Beijing, China, 1992. [Google Scholar]
- Zixing, C. Robotics: Principles and Applications; South and Center University Press: Changsha, China, 1988. [Google Scholar]
- Murray, R.; Zexiang, L.; Sastry, S. A Mathematical Introduction to Robotic Manipulation; CRC Press: Boca Raton, FL, USA, 2005. [Google Scholar]
- Jingjun, D.; Xinjun, L.; Xilun, D. Mathematic Foundation of Mechanisms and Robotics, 2nd ed.; Machine Press: Beijing, China, 2016. [Google Scholar]
- Santolaria, J.; Aguilar, J.-J.; Yagüe, J.-A.; Pastor, J. Kinematic parameter estimation technique for calibration and repeatability improvement of articulated arm coordinate measuring machines. Precis. Eng.
**2008**, 32, 251–268. [Google Scholar] [CrossRef] - Jie, L.; Tijian, S.; Kun, Z. Kinematical Analysis for the Second Suspension System of the Maglev Vehicle. J. China Railw. Soc.
**2007**, 29, 32–38. [Google Scholar] - Fernandez-Gauna, B.; Lopez-Guede, J.M.; Zulueta, E.; Echegoyen, Z.; Graña, M. Basic results and experiments on robotic multi-agent system for hose deployment and transportation. Int. J. Artif. Intell.
**2011**, 6, 183–202. [Google Scholar] - Takács, Á.; Kovács, L.; Rudas, I.; Precup, R.-E.; Haidegger, T. Models for force control in telesurgical robot systems. Acta Polytech. Hung.
**2015**, 12, 95–114. [Google Scholar] - Jie, L.; Guanchun, L.; Kun, Z.; Peng, C.; Danfeng, Z. Kinematical Analysis for the Second Suspension System of the Mid-Low-Speed Maglev Vehicle. In Proceedings of the 2013 China Automation Congress(CAC), Changsha, China, 7–8 November 2013. [Google Scholar]
- Kun, Z.; Jie, L.; Wensen, C. Structure Decoupling Analysis of Maglev Train Bogie. Electr. Drive Locomot.
**2005**, 2005, 22–39. [Google Scholar]

Parameters | Value | Parameters | Value |
---|---|---|---|

Minimum Curve Radius ${R}_{0}$ | 1000 m | ${h}_{0}$ | $0.4625$ m |

Maximum transverse slope angle $\theta $ | ${6}^{\circ}$ | ${h}_{1}$ | $0.501$ m |

Gauge D | 2 m | ${h}_{1}$ | $0.765$ m |

The length of electromagnet l | $2.65$ m | ${l}_{0}$ | $0.431$ m |

The distance between adjacent magnets $\delta x$ | $0.09$ m | ${l}_{1}$ | $0.441$ m |

The distance of the air spring on the same module ${l}_{air}$ | $2.25$ m | ${l}_{2}$ | $0.264$ m |

The distance of adjacent air springs $\delta {x}_{air}$ | $0.49$ m | The length of transition curve ${S}_{0}$ | 36 m |

Number | ${\mathit{\theta}}_{1}$ (degree) | ${\mathit{\theta}}_{2}$ (degree) | ${\mathit{\theta}}_{3}$ (degree) | ${\mathit{\theta}}_{4}$ (mm) | ${\mathit{\theta}}_{5}$ (degree) | ${\mathit{\theta}}_{6}$ (degree) |
---|---|---|---|---|---|---|

1 | 0.42 | −0.33 | −0.00 | −5.58 | 0.01 | −0.42 |

2 | −0.29 | −0.33 | 0.00 | 5.59 | −0.00 | 0.29 |

3 | 0.46 | −0.33 | −0.00 | −5.58 | 0.01 | −0.46 |

4 | −0.33 | −0.33 | −0.00 | 5.59 | −0.00 | 0.33 |

5 | 0.50 | −0.33 | −0.00 | −5.58 | 0.01 | −0.50 |

6 | −0.37 | −0.33 | −0.00 | 5.59 | −0.00 | 0.37 |

7 | 0.54 | −0.33 | −0.00 | −5.57 | 0.01 | −0.54 |

8 | −0.41 | −0.33 | −0.00 | 5.59 | −0.00 | 0.41 |

9 | 0.58 | −0.33 | −0.01 | −5.57 | 0.01 | −0.58 |

10 | −0.44 | −0.33 | −0.00 | 5.59 | −0.00 | 0.44 |

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**MDPI and ACS Style**

Leng, P.; Li, J.; Jin, Y.
Kinematics Modeling and Analysis of Mid-Low Speed Maglev Vehicle with Screw and Product of Exponential Theory. *Symmetry* **2019**, *11*, 1201.
https://doi.org/10.3390/sym11101201

**AMA Style**

Leng P, Li J, Jin Y.
Kinematics Modeling and Analysis of Mid-Low Speed Maglev Vehicle with Screw and Product of Exponential Theory. *Symmetry*. 2019; 11(10):1201.
https://doi.org/10.3390/sym11101201

**Chicago/Turabian Style**

Leng, Peng, Jie Li, and Yuxin Jin.
2019. "Kinematics Modeling and Analysis of Mid-Low Speed Maglev Vehicle with Screw and Product of Exponential Theory" *Symmetry* 11, no. 10: 1201.
https://doi.org/10.3390/sym11101201