# Adaptive Sliding Mode Control for Yaw Stability of Four-Wheel Independent-Drive EV Based on the Phase Plane

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## Abstract

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## 1. Introduction

## 2. Vehicle Dynamic Model

#### 2.1. Reference Model

#### 2.2. Tire Model

_{i}(i = 0, 1, …,13) is the fitting coefficient; $\xi $ is the wheel camber; $\alpha $ is the wheel sideslip angle; ${S}_{h}$ is the horizontal offset; ${F}_{z}$ is the wheel vertical load.

## 3. Establishment of Phase Plane

#### 3.1. Establishment of $\beta -\dot{\beta}$ Phase Plane and Division of Stability Region

#### 3.2. Stable Boundary Equation of $\beta -\dot{\beta}$ Phase Plane

## 4. Design of Yaw Stability Controller

#### 4.1. Design of ASMC Controller for Yaw Rate

#### 4.2. Design of ASMC Controller for Sideslip Angle

#### 4.3. Stability Analysis of Control System

#### 4.4. Particle Swarm Optimization of Sliding Mode Control Parameters

#### 4.5. Joint Controller

#### 4.6. Drive Torque Distribution

## 5. Simulation Analysis

#### 5.1. Simulation and Verification of Yaw Moment Controller

#### 5.2. Lane Change Simulation

#### 5.3. Serpentine Test

#### 5.4. Lemniscate Test

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Table 1.**The $\beta -\dot{\beta}$ phase plane stability boundary parameters of different values of $\mu $.

µ | c | k |
---|---|---|

0.3 | 0.08 | −1.68 |

0.4 | 0.10 | −2.02 |

0.5 | 0.13 | −2.39 |

0.6 | 0.15 | −2.83 |

0.7 | 0.18 | −2.86 |

0.8 | 0.20 | −3.03 |

0.9 | 0.23 | −3.36 |

1.0 | 0.27 | −3.79 |

Controller Parameters | c_{ω} | ε_{ω} |
---|---|---|

Optimization results | 93.2007 | 9.9821 |

Controller Parameters | c_{β} | ε_{β} |
---|---|---|

Optimization results | 46.1308 | 3.0325 |

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

Vehicle weight | kg | 1530 |

Distance between the front axle and the center of mass | m | 1.2 |

Distance between the front axle and the center of mass | m | 1.4 |

Inertia of vehicle around z-axis | kg m^{2} | 2500.6 |

Front and rear axle wheel track | m | 1.65 |

Height of vehicle center of mass | m | 0.6 |

Effective radius of wheel | m | 0.33 |

Longitudinal cornering stiffness | N/rad | 40,000 |

Lateral cornering stiffness | N/rad | 50,000 |

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

${c}_{\omega}$ | 93.2007 |

${c}_{\beta}$ | 46.1308 |

${\epsilon}_{\omega}$ | 9.9821 |

${\epsilon}_{\beta}$ | 3.0325 |

$\gamma $ | 0.43 |

$\tau $ | 12 |

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**MDPI and ACS Style**

Zhou, Z.; Zhang, J.; Yin, X.
Adaptive Sliding Mode Control for Yaw Stability of Four-Wheel Independent-Drive EV Based on the Phase Plane. *World Electr. Veh. J.* **2023**, *14*, 116.
https://doi.org/10.3390/wevj14050116

**AMA Style**

Zhou Z, Zhang J, Yin X.
Adaptive Sliding Mode Control for Yaw Stability of Four-Wheel Independent-Drive EV Based on the Phase Plane. *World Electric Vehicle Journal*. 2023; 14(5):116.
https://doi.org/10.3390/wevj14050116

**Chicago/Turabian Style**

Zhou, Zhigang, Jie Zhang, and Xiaofei Yin.
2023. "Adaptive Sliding Mode Control for Yaw Stability of Four-Wheel Independent-Drive EV Based on the Phase Plane" *World Electric Vehicle Journal* 14, no. 5: 116.
https://doi.org/10.3390/wevj14050116