# Prediction of Clearance Vibration for Intelligent Vehicles Motion Control

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## Abstract

**:**

## 1. Introduction

## 2. Nonlinear Modeling for Clearance

## 3. Parameter Identification Based on Wavelet Transform

#### 3.1. Time-Varying Parameter of Clearance Structures

#### 3.2. Parameter Identification and Verification

- $a={2}^{-5},{2}^{-6},{2}^{-7},{2}^{-8}$;
- $a={2}^{-6},{2}^{-6.25},{2}^{-6.5},{2}^{-6.75},{2}^{-7}$;
- $a={2}^{-6.5},{2}^{-6.6},{2}^{-6.7},{2}^{-6.8},{2}^{-6.9},{2}^{-7}$.

## 4. Results and Discussion

#### 4.1. Test Model

#### 4.2. Validity of Clearance Model

#### 4.3. Validity of Motion Control

## 5. Conclusions

- With the Hamilton principle and the assumed modal method, a dynamic model of the cantilever beam with double clearance is constructed. A clearance modeling method is proposed to describe clearance dynamics with time-varying parameters. A wavelet transform was used to identify the time-varying stiffness of clearance by estimating the natural frequency and damping ratio. The time-varying damping of clearance was constructed using Rayleigh and complex damping. The clearance model accurately describes the dynamical clearance variation using time-varying damping and stiffness.
- An experimental model was constructed to verify the proposed modeling method. Compared to the constant model, the maximum natural frequency error of the time-varying model was reduced from 5.38% to 1.29%. The maximum MAC also increased from 0.4173 to 0.8492. The MRE and RMSE of each measurement point decreased by 27.67% and 63.41%. The FRAC increased from 0.7971 to 0.9465. CSAC and CSF increased by 6.55% and 12.37%, which is over 90%, particularly near the natural frequency position. Therefore, the above results demonstrate the validity and accuracy of the modeling method.
- The control method based on the clearance model was verified. At 72 km/h, compared with SMC, the tracking error peak of SMC-CM decreased by 21.2% and the reliability increased by 12.7%. Compared to MPC and MSC, the lateral position error peak of SMC-CM decreased by 50% and 14.3%, the heading error peak decreased by 35.7% and 15.6%, and the yaw velocity peak decreased by 23.5% and 3.7%. Peak lateral accelerations decreased by 27.8% and 18.8%. The effectiveness of the control method is verified.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Identification results: (

**a**) is step damping identification results; (

**b**) is step stiffness identification results.

**Figure 3.**Identification results: (

**a**) is continuous damping identification results; (

**b**) is continuous stiffness identification results.

**Figure 4.**Parameter identification based on different sequences: (

**a**–

**c**) represent the comparison of identification results at scales 1, 2, and 3, respectively.

**Figure 7.**FRF of clearance position 1 and 2: (

**a**) is position 1 for sensor 6; (

**b**) is position 2 for sensor 7.

**Figure 8.**FRF of other position: (

**a**–

**f**) represent the comparison of FRF at sensor measuring points 1, 2, 3, 4, 5, 8.

**Figure 9.**Comparison of CSAC: (

**a**–

**h**) represent the comparison of CSAC at sensor measuring points 1, 2, 3, 4, 5, 6, 7, 8.

**Figure 10.**Comparison of CSF: (

**a**–

**h**) represent the comparison of CSF at sensor measuring points 1, 2, 3, 4, 5, 6, 7, 8.

**Figure 11.**Error results at 72 km/h speed: (

**a**) is lateral position error; (

**b**) is yaw velocity error.

Order | Frequency (Hz) | Initial Model Based on Constant Damping | Updated Model Based on Time-Varying Damping | ||||
---|---|---|---|---|---|---|---|

Frequency (Hz) | RE (%) | MAC | Frequency (Hz) | RE (%) | MAC | ||

1 | 9.25 | 9.15 | 1.0811 | 0.5378 | 9.2 | 0.5405 | 0.9422 |

2 | 39.75 | 38.35 | 3.5220 | 0.6283 | 40.25 | 1.2579 | 0.9117 |

3 | 92.85 | 97.85 | 5.3850 | 0.5265 | 91.65 | 1.2924 | 0.8627 |

4 | 188.85 | 191.35 | 1.3238 | 0.4183 | 187.25 | 0.8472 | 0.8492 |

Position | Initial Model Based on Constant Damping | Updated Model Based on Time-Varying Damping | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

MRE | RMSE | FRAC | CSAC | CSF | MRE | RMSE | FRAC | CSAC | CSF | |

1 | 0.8013 | 1.4181 | 0.8643 | 0.8891 | 0.8410 | 0.5763 | 0.7481 | 0.9443 | 0.9633 | 0.9619 |

2 | 0.8078 | 1.3410 | 0.7708 | 0.8921 | 0.8485 | 0.5827 | 0.5902 | 0.9507 | 0.9531 | 0.9608 |

3 | 0.7963 | 1.5589 | 0.8593 | 0.8847 | 0.8363 | 0.5785 | 0.6026 | 0.9465 | 0.9677 | 0.9656 |

4 | 0.8050 | 1.3795 | 0.8680 | 0.8812 | 0.8459 | 0.5810 | 0.5946 | 0.9490 | 0.9487 | 0.9652 |

5 | 0.7970 | 1.6391 | 0.6600 | 0.8973 | 0.8431 | 0.5799 | 0.6574 | 0.9479 | 0.9375 | 0.9599 |

6 | 0.7938 | 1.9037 | 0.8568 | 0.8956 | 0.8392 | 0.5777 | 0.0976 | 0.9457 | 0.9567 | 0.9568 |

7 | 0.8007 | 1.0062 | 0.6373 | 0.8870 | 0.8371 | 0.5778 | 0.2419 | 0.9458 | 0.9532 | 0.9521 |

8 | 0.7969 | 1.3674 | 0.8599 | 0.8912 | 0.8457 | 0.5743 | 0.7174 | 0.9423 | 0.9372 | 0.9657 |

Means | 0.7999 | 1.4517 | 0.7971 | 0.8898 | 0.8421 | 0.5785 | 0.5312 | 0.9465 | 0.9522 | 0.9610 |

Parameter | SMC-CM | SMC |
---|---|---|

Lateral position error (m) | 0.025 | 0.033 |

Yaw velocity (deg/s) | 2.83 | 3.24 |

Parameter | SMC-CM | SMC | MPC |
---|---|---|---|

Lateral position error (m) | 0.06 | 0.07 | 0.12 |

Yaw velocity (deg/s) | 0.27 | 0.32 | 0.42 |

Parameter | SMC-CM | SMC | MPC |
---|---|---|---|

Lateral acceleration (deg/s^{2}) | −5.2 | −5.4 | −6.8 |

Yaw velocity (deg/s) | −2.6 | −3.2 | −3.6 |

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**MDPI and ACS Style**

Zhang, Y.; Zhang, F.; Wang, W.; Meng, F.; Zhang, D.; Wang, H.
Prediction of Clearance Vibration for Intelligent Vehicles Motion Control. *Sustainability* **2022**, *14*, 6698.
https://doi.org/10.3390/su14116698

**AMA Style**

Zhang Y, Zhang F, Wang W, Meng F, Zhang D, Wang H.
Prediction of Clearance Vibration for Intelligent Vehicles Motion Control. *Sustainability*. 2022; 14(11):6698.
https://doi.org/10.3390/su14116698

**Chicago/Turabian Style**

Zhang, Yunhe, Faping Zhang, Wuhong Wang, Fanjun Meng, Dashun Zhang, and Haixun Wang.
2022. "Prediction of Clearance Vibration for Intelligent Vehicles Motion Control" *Sustainability* 14, no. 11: 6698.
https://doi.org/10.3390/su14116698