# Path Tracking Control of Commercial Vehicle Considering Roll Stability Based on Fuzzy Linear Quadratic Theory

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## Abstract

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## 1. Introduction

- (1)
- To solve the problem of driving safety under extreme working conditions, a path-tracking controller considering roll stability was designed based on the LQ theory.
- (2)
- The weight of the classical LQ controller’s cost function is fixed, and the function’s adaptability to the driving scenario is poor. Therefore, a fuzzy LQ controller with a self-adjusting weight coefficient was designed. The dynamic performance of the system can be improved effectively by optimizing the weight coefficient of the cost function online.

## 2. Methodology

#### 2.1. Linear Quadratic Theory

#### 2.2. Fuzzy Control Theory

## 3. Steering and Braking Cooperative Control Model

## 4. Cooperative Controller

## 5. Fuzzy Control of Self-Adjusting Weight Coefficient

## 6. Co-Simulation and Hardware-in-Loop Experiments

## 7. Conclusions

- (1)
- To solve the problem of driving safely under extreme working conditions, a path-tracking controller that considers roll stability was designed based on the LQ theory.
- (2)
- The weight of the classical LQ controller’s cost function was fixed, and the function’s adaptability to the driving scenario was poor. Therefore, a fuzzy LQ controller with a self-adjusting weight coefficient was designed. The dynamic performance of the system can be improved effectively by optimizing the weight coefficient of the cost function online.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Fuzzy controller of ${\Gamma}_{y}$. (

**a**) Membership function of $\overline{e}$; (

**b**) Membership function of $\overline{\varphi}$; (

**c**) Membership function of ${\varsigma}_{y}$; (

**d**) Three-dimensional map.

**Figure 4.**Fuzzy controller of ${\Gamma}_{\varphi}$. (

**a**) Membership function of $\overline{e}$; (

**b**) Membership function of $\overline{\varphi}$; (

**c**) Membership function of ${\varsigma}_{\varphi}$; (

**d**) Three-dimensional map.

**Figure 9.**The control effect of single line—change condition. (

**a**) Lateral displacement (m); (

**b**) Lateral error (m); (

**c**) Front-wheel angle (deg); (

**d**) Yaw angle (deg).

**Figure 10.**The control effect of single line—change condition. (

**a**) Roll angle (deg); (

**b**) Yaw rate (deg/s); (

**c**) Active yaw moment (Nm); (

**d**) $\dot{\beta}-\beta $ phase diagram.

**Figure 11.**The control effect of double-line—change condition. (

**a**) Lateral displacement (m); (

**b**) Lateral error (m); (

**c**) Front wheel angle (deg); (

**d**) Yaw angle (deg).

**Figure 12.**The control effect of single-line—change condition. (

**a**) Roll angle (deg); (

**b**) Yaw rate (deg/s); (

**c**) Active yaw moment (Nm); (

**d**) $\dot{\beta}-\beta $ phase diagram.

Symbol | Description | Value [Unit] |
---|---|---|

${l}_{f},{l}_{r}$ | Distance from the center of mass | 1.95 m; 1.54 m |

${l}_{w}$ | Half of the wheelbase | 1.3 m |

$m$ | ehicle mass | 10,690 kg |

${m}_{s},{m}_{u}$ | Sprung and unsprung mass | 9360 kg; 1330 kg |

${b}_{s}$ | Suspension roll damping factor | 8.26 × 10^{4} Nms/rad |

${C}_{f},{\mathrm{C}}_{r}$ | Equivalent cornering stiffness of front and rear axle | 3.80 × 10^{5} N/rad;6.84 × 10 ^{5} N/rad |

${k}_{u}$ | Roll stiffness of unsprung mass | 5.39 × 10^{6} Nm/rad |

${v}_{x}$ | Longitudinal velocity | 70 km/h |

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

${k}_{s}$ | Roll stiffness of suspension | 1.06 × 10^{6} Nm/rad |

${I}_{x}$ | Roll inertia of sprung mass | 7.70 × 10^{3} kg m^{2} |

${I}_{z}$ | Yaw inertia of sprung mass | 3.01 × 10^{4} kg m^{2} |

${I}_{xz}$ | Yaw-roll inertia product of the sprung mass | 0 kg m^{2} |

$r$ | Height of roll axis from the ground | 0.63 m |

${h}_{u}$ | Height of CG of unsprung mass from ground | 0.51 m |

$\psi $ | Yaw angle | - |

$\beta $ | Sideslip angle | - |

$\varphi $ | Roll angle | - |

${\delta}_{f}$ | Front wheel angle input | - |

$\Delta M$ | Active anti-roll moment | - |

$\overline{\mathit{e}}$ | $\overline{\mathit{\varphi}}$ | ||||||
---|---|---|---|---|---|---|---|

NB | NM | NS | NO | PS | PM | PB | |

NB | NO | PS | PS | PM | PM | PB | PB |

NM | NO | NO | PS | PS | PM | PM | PM |

NS | NS | NO | NO | PS | PS | PM | PM |

NO | NM | NS | NS | NO | NO | PS | PS |

PS | NM | NM | NS | NS | NO | NO | PS |

PM | NB | NM | NM | NS | NS | NO | NO |

PB | NB | NB | NM | NM | NS | NS | NO |

$\overline{\mathit{e}}$ | $\overline{\mathit{\varphi}}$ | ||||||
---|---|---|---|---|---|---|---|

NB | NM | NS | NO | PS | PM | PB | |

NB | NO | NO | NS | NM | NM | NB | NB |

NM | PS | NO | NO | NS | NM | NM | NB |

NS | PS | PS | NO | NS | NS | NM | NM |

NO | PM | PS | PS | NO | NS | NS | NM |

PS | PM | PM | PS | NO | NO | NS | NS |

PM | PB | PM | PM | PS | NO | NO | NS |

PB | PB | PM | PM | PS | PS | NO | NO |

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## Share and Cite

**MDPI and ACS Style**

Fan, Z.; Yan, Y.; Wang, X.; Xu, H.
Path Tracking Control of Commercial Vehicle Considering Roll Stability Based on Fuzzy Linear Quadratic Theory. *Machines* **2023**, *11*, 382.
https://doi.org/10.3390/machines11030382

**AMA Style**

Fan Z, Yan Y, Wang X, Xu H.
Path Tracking Control of Commercial Vehicle Considering Roll Stability Based on Fuzzy Linear Quadratic Theory. *Machines*. 2023; 11(3):382.
https://doi.org/10.3390/machines11030382

**Chicago/Turabian Style**

Fan, Zhixian, Yang Yan, Xiangyu Wang, and Haizhu Xu.
2023. "Path Tracking Control of Commercial Vehicle Considering Roll Stability Based on Fuzzy Linear Quadratic Theory" *Machines* 11, no. 3: 382.
https://doi.org/10.3390/machines11030382