# A Model Predictive Control Method for Vehicle Drifting Motions with Measurable Errors

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Vehicle Dynamics Model

#### 2.1. Vehicle Dynamics Equations

_{tf}-y

_{tf}and the x

_{tr}-y

_{tr}coordinate systems, respectively, and the tire revolution directions are expressed by the x

_{tf}-axis and the x

_{tr}-axis.

#### 2.2. Tire Force

#### 2.3. Roll Safety Analysis

## 3. Vehicle Drifting Based on Model Predictive Control

## 4. Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Roll dynamics model: (

**a**) before the wheel lift-off; (

**b**) after the wheel lift-off. The vehicle postures with gray broken lines express the vehicle motions without rolling.

**Figure 4.**The UniTire tire model with an affine approximation: (

**a**) the longitudinal tire force at ${\widehat{\alpha}}_{i}$; (

**b**) the lateral tire force at ${\widehat{\kappa}}_{i}$.

**Figure 5.**The velocity limitations in steady-state drifting motions in different road conditions. The vehicle can drift at greater velocities with better road conditions.

**Figure 6.**The simulation result with the 6 m/s original velocity and the target group (c): (

**a**) the velocity; (

**b**) the sideslip angle; (

**c**) the yaw rate; (

**d**) the steering angle; (

**e**) the TYDEX longitudinal slip ratios of the front and rear tires; (

**f**) the radius of the motion curve; (

**g**) the motion curve of the vehicle. All continuous lines suggest the simulation results, while the dashed lines suggest the targets.

**Figure 7.**The simulation result with the 10 m/s original velocity and the target group (c): (

**a**) the velocity; (

**b**) the sideslip angle; (

**c**) the yaw rate; (

**d**) the steering angle; (

**e**) the TYDEX longitudinal slip ratios of the front and rear tires; (

**f**) the radius of the motion curve; (

**g**) the motion curve of the vehicle. All continuous lines suggest the simulation results, while the dashed lines suggest the targets.

**Figure 8.**The simulation result with the 8.1 m/s original velocity and the target group (c): (

**a**) the velocity; (

**b**) the sideslip angle; (

**c**) the yaw rate; (

**d**) the steering angle; (

**e**) the TYDEX longitudinal slip ratios of the front and rear tires; (

**f**) the radius of the motion curve; (

**g**) the motion curve of the vehicle. All continuous lines suggest the simulation results, while the dashed lines suggest the targets.

**Figure 9.**The simulation result with the 9 m/s original velocity and the target group (d): (

**a**) the velocity; (

**b**) the sideslip angle; (

**c**) the yaw rate; (

**d**) the steering angle; (

**e**) the TYDEX longitudinal slip ratios of the front and rear tires; (

**f**) the radius of the motion curve; (

**g**) the motion curve of the vehicle. All continuous lines suggest the simulation results, while the dashed lines suggest the targets.

**Figure 10.**The simulation result with the 13.7 m/s original velocity and the target group (e): (

**a**) the velocity; (

**b**) the sideslip angle; (

**c**) the yaw rate; (

**d**) the steering angle; (

**e**) the TYDEX longitudinal slip ratios of the front and rear tires; (

**f**) the radius of the motion curve; (

**g**) the motion curve of the vehicle. All continuous lines suggest the simulation results, while the dashed lines suggest the targets.

Parameter Symbol | Unit | Value |
---|---|---|

$m$ | kg | 1126.7 |

${m}_{b}$ | kg | 1111 |

${I}_{z}$ | kg·m^{2} | 2038 |

${l}_{f}$ | m | 1.265 |

${l}_{r}$ | m | 1.335 |

${h}_{b}$ | m | 0.136 |

${h}_{g}$ | m | 0.518 |

${A}_{f}$ | m^{2} | 1.6 |

$g$ | N/kg | 9.8 |

$\mu ={\mu}_{f}={\mu}_{r}$ | 0.65 |

$\mathit{R}$ (m) | $\mathit{v}$ (m/s) | $\mathit{\beta}$ (rad) | $\mathit{\gamma}$ (rad/s) | ${\mathit{\delta}}_{\mathit{f}}\text{}\left(\mathbf{rad}\right)$ | ${\mathit{\kappa}}_{\mathit{r}}$ | |
---|---|---|---|---|---|---|

(a) | 12 | 8.73 | −0.06 | 0.73 | 0.06 | 0.03 |

(b) | 12 | 6.82 | −0.90 | 0.91 | −0.7 | 0.95 |

(c) | 12 | 8.05 | −0.51 | 0.77 | −0.26 | 0.61 |

(d) | 20 | 10.5 | −0.49 | 0.6 | −0.26 | 0.51 |

(e) | 42 | 15 | −0.51 | 0.41 | −0.26 | 0.53 |

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## Share and Cite

**MDPI and ACS Style**

Xu, D.; Han, Y.; Ge, C.; Qu, L.; Zhang, R.; Wang, G.
A Model Predictive Control Method for Vehicle Drifting Motions with Measurable Errors. *World Electr. Veh. J.* **2022**, *13*, 54.
https://doi.org/10.3390/wevj13030054

**AMA Style**

Xu D, Han Y, Ge C, Qu L, Zhang R, Wang G.
A Model Predictive Control Method for Vehicle Drifting Motions with Measurable Errors. *World Electric Vehicle Journal*. 2022; 13(3):54.
https://doi.org/10.3390/wevj13030054

**Chicago/Turabian Style**

Xu, Dongxin, Yueqiang Han, Chang Ge, Longtao Qu, Rui Zhang, and Guoye Wang.
2022. "A Model Predictive Control Method for Vehicle Drifting Motions with Measurable Errors" *World Electric Vehicle Journal* 13, no. 3: 54.
https://doi.org/10.3390/wevj13030054