# Optimal Coordinated Control of Active Front Steering and Direct Yaw Moment for Distributed Drive Electric Bus

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## Abstract

**:**

## 1. Introduction

^{∞}control strategy to improve the lateral stability and handling performance of vehicles. It applied the Takagi-Sugeno fuzzy modeling method to solve nonlinear problems of lateral dynamics of vehicles [27].

## 2. Vehicle System Modeling

#### 2.1. Linear 2-DOF Vehicle Model

_{x}is the longitudinal speed; ν

_{y}is the lateral speed; r is the yaw rate; δ is the front wheel steering angle; a is the distance from the center of gravity of the vehicle to the front axle; b is the distance from the center of gravity of the vehicle to the rear axle; I

_{z}is the yaw moment of inertia; F

_{yi}is the lateral force of the i-th wheel (i = 1, 2); F

_{y}is the resultant lateral force; M

_{z}is the resultant yaw moment.

_{i}is the lateral tire stiffness of the i-th wheel (i = 1, 2); α

_{i}is the sideslip angle of the i-th wheel (i = 1, 2).

#### 2.2. Nonlinear 7-DOF Vehicle Model

_{xi}is the longitudinal force of the each wheel (i = fl, fr, rl, rr); F

_{yi}is the lateral force of the each wheel (i = fl, fr, rl, rr).

_{w}is the wheel moment of inertia, R

_{e}is the effective radius of wheels, w

_{i}and T

_{i}denote the angular velocity and wheel torque of the i-th wheel (i = 1, 2, 3, 4), respectively.

_{zi}is the vertical force of the i-th wheel (i = 1, 2, 3, 4), h

_{g}is the height of the center of gravity.

## 3. State Observer

#### 3.1. The Observer Design

#### 3.2. The Observer Simulation Verification

## 4. Hierarchical Coordinated Controller Design

#### 4.1. AFS Controller Design for Handling

#### 4.1.1. Sideslip Angle Controller

_{y}. Therefore, the ITSMC method can be employed to minimize sideslip angle.

_{β}is an unknown but bounded function. It is used to represent uncertainty and disturbance in practical nonlinear vehicle models and defined as:

_{y}as the control output, we can obtain:

**Proof:**

_{1}with respect to time:

#### 4.1.2. Yaw Rate Controller

_{z}. The NFTSMC method can be used to design the yaw rate controller.

_{AFS}is an additional yaw moment that improves vehicle handling stability. g

_{ψ}is an unknown but bounded function which satisfies the following equation.

_{AFS}as a control input can be obtained thus:

**Proof:**

_{2}with respect to time:

_{1}and w

_{2}are weighting coefficients.

#### 4.2. DYC Controller Design for Stability

#### 4.2.1. Adaptive Fuzzy Sliding Mode Controller

_{1}> 0, k

_{2}> 0.

**Proof:**

_{β}and e

_{ψ}are respectively used as the fuzzy control inputs, and the weight coefficient λ is used as the fuzzy control output, shown in Table 2.

#### 4.2.2. Torque Distribution Controller

_{l}and c

_{r}are the ratio of the vertical load on the left wheel and the ratio of the vertical load on the right wheel.

#### 4.3. Coordinated Controller Design

#### 4.3.1. Overall Structure of the Coordinated Controller

#### 4.3.2. Determination of the Stability Boundary

#### 4.3.3. Distribution of Additional Yaw Moment

## 5. Coordinated Controller Simulation Verification

#### 5.1. Maneuver 1: Single Lane Change under Peak Steering Angle 180°

#### 5.2. Maneuver 2: Single Lane Change under Peak Steering Angle 240°

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 6.**Estimated parameters at high adhesion coefficient and high speed. (

**a**) Lateral speed, (

**b**) Sideslip angle.

**Figure 7.**Estimated parameters at low adhesion coefficient and low speed. (

**a**) Lateral speed, (

**b**) sideslip angle.

**Figure 11.**The parametric simulation results under maneuver 1. (

**a**) Sideslip angle, (

**b**) Yaw rate, (

**c**) Driving trajectory, (

**d**) Longitudinal speed, (

**e**) Additional yaw moment, (

**f**) Additional front wheel steering angle.

**Figure 13.**The parametric simulation results under maneuver 2. (

**a**) Sideslip angle, (

**b**) Yaw rate, (

**c**) Driving trajectory, (

**d**) Longitudinal speed, (

**e**) Additional yaw moment, (

**f**) Additional front wheel steering angle.

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

Vehicle mass | m | 7620 kg |

Distance from CG to front axle | a | 3105 mm |

Distance from CG to rear axle | b | 1385 mm |

Wheelbase | d | 2030 mm |

Yaw moment of inertia | I_{z} | 30,782.4 kg·m^{2} |

Effective radius of wheel | R_{e} | 510 mm |

Wheel moment of inertia | J_{w} | 14 kg·m^{2} |

Height of CG | h_{g} | 1200 mm |

Acceleration of gravity | g | 9.8 m/s^{2} |

e_{ψ} | e_{β} | ||||
---|---|---|---|---|---|

NB | NS | ZO | PS | PB | |

NB | ZO | PS | PB | PS | ZO |

NS | NS | ZO | PB | ZO | NS |

ZO | NB | NB | NB | NB | NB |

PS | NS | ZO | PB | ZO | NS |

PB | ZO | PS | PB | PS | ZO |

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**MDPI and ACS Style**

Lin, J.; Zou, T.; Su, L.; Zhang, F.; Zhang, Y.
Optimal Coordinated Control of Active Front Steering and Direct Yaw Moment for Distributed Drive Electric Bus. *Machines* **2023**, *11*, 640.
https://doi.org/10.3390/machines11060640

**AMA Style**

Lin J, Zou T, Su L, Zhang F, Zhang Y.
Optimal Coordinated Control of Active Front Steering and Direct Yaw Moment for Distributed Drive Electric Bus. *Machines*. 2023; 11(6):640.
https://doi.org/10.3390/machines11060640

**Chicago/Turabian Style**

Lin, Jiming, Teng Zou, Liang Su, Feng Zhang, and Yong Zhang.
2023. "Optimal Coordinated Control of Active Front Steering and Direct Yaw Moment for Distributed Drive Electric Bus" *Machines* 11, no. 6: 640.
https://doi.org/10.3390/machines11060640