# Vehicle Rollover Warning and Control Based on Attitude Detection and Fuzzy PID

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## Abstract

**:**

## 1. Introduction

## 2. Vehicle Dynamics Modeling

#### 2.1. Vehicle Rollover Dynamics Model

_{s}denotes the spring mass, kg; β indicates the lateral deflection angle; ω

_{r}refers to the angular velocity of the transverse sway; h indicates the difference between the heights of the center of mass on the spring and the center of lateral camber of the vehicle; p indicates the lateral camber velocity; and ${F}_{y1}=2{k}_{f}{\delta}_{1}\mathrm{and}{F}_{y2}=2{k}_{r}{\delta}_{2}$ represent the lateral forces of the front and rear wheels. k

_{f}and k

_{r}denote the lateral deflection stiffness of the front and rear wheels, N/rad; δ

_{1}denotes the effective side deflection angle of the front wheels (including the side deflection angle of the front tires and the elastic deformation of the front suspension and steering system); δ

_{2}denotes the effective lateral deflection angle of the rear wheels (including the lateral deflection angle of the rear tires and the elastic deformation of the rear suspension).

_{z}represents the rotational inertia of the z-axis of the vehicle, kg·m

^{2}; I

_{xz}indicates the inertia product of the mass on the spring around the x and z axes, kg·m

^{2}; a and b denote the distances from the center of gravity of the vehicle to the front and rear axles, m.

_{x}represents the rotational inertia of the mass on the spring about the x-axis, kg·m

^{2}; v denotes the vehicle speed; D

_{f}and D

_{r}denote the front and rear suspension lateral camber damping; φ symbolizes the body roll angle; C

_{φ1}and C

_{φ2}denote the front and rear lateral camber rigidity.

_{1}and N

_{2}denote the positive torque coefficient for the front and rear-wheel return, N·m/rad; ${E}_{f}=\frac{\partial {\delta}_{1}}{\partial \varphi}$ and ${E}_{r}=\frac{\partial {\delta}_{2}}{\partial \varphi}$ correspond to the front and rear-wheel lateral steering coefficients.

#### 2.2. Validation of the Vehicle Dynamics Model

#### 2.3. Rollover Stability Evaluation Index

_{l}, and F

_{r}denote the sums of the vertical loads on the left-hand side and the right-hand side of the wheels of the vehicle, respectively. F

_{l}and F

_{r}satisfy ${F}_{l}+{F}_{r}=mg$. LTR takes values in the range of [–1,1]. When the value of LTR is zero, the car is stable, and all four wheels are in contact with the ground; the larger the value of |LTR|, the greater the tendency of the car to roll over. When LTR = −1 or LTR = 1, the left or right wheel of the car is completely off the ground, at which point the vehicle is considered to have rolled over. To establish a sufficient safety margin, the rollover threshold is generally set to 0.8, that is, when |LTR| ≥ 0.8, the vehicle is considered to have the risk of rollover, and the vehicle should perform the corresponding anti-rollover intervention to be restored to a stable state.

_{y}denotes lateral acceleration at the center of mass; T refers to wheelbase of the vehicle; h denotes the distance between the center of mass and the ground.

_{y}and ϕ can be obtained from onboard sensors and inertial guidance systems, respectively. In the subsequent simulation, the relevant parameters in Equation (11) can be directly output from the software and are more suitable for simulation control.

## 3. Selection of the Control Strategy

#### 3.1. Upper-Level Control Strategy

#### 3.1.1. Selection of the Control Algorithm

_{P}, T

_{i}, and T

_{d}denote the scale factor, integration time, and differentiation time, respectively. The effectiveness of the output curve can be improved by adjusting the magnitudes of the K

_{P}, T

_{i}, and T

_{d}, values. The optimal solution for the control parameters of this system can be obtained by rectifying the PID parameters to achieve a better PID control effect.

_{p}, ΔK

_{i}

_{,}and ΔK

_{d}, based on which the parameters are adjusted. The output parameter of the control system is the braking torque of the target wheel.

_{p}, ∆K

_{i}

_{,}and ΔK

_{d}were selected as Gaussian affiliation functions, as shown in Figure 8. Fuzzy inference rules form the core of a fuzzy controller, which consists of many different combinations of fuzzy conditions [2]. A dual-input and triple-output controller was used, which has seven fuzzy linguistic variables for each parameter, and its number of control rules per parameter is listed in Table 2, Table 3 and Table 4. In the tables, the row is the deviation E, the column is the deviation rate of change EC, and the intersection of the rows is a rule; then, each parameter has 49 rules.

#### 3.1.2. Controller Results Analysis

#### 3.2. Lower Actuators

#### 3.2.1. Judgment Logic for the Braking Target Wheel Cylinder

#### 3.2.2. Overall Design of the Wire-Controlled Auxiliary Braking System

## 4. Co-Simulation Analysis and Result Verification

#### 4.1. Fish Hook Steering Test

#### 4.2. J-Turn Test

## 5. Conclusions

- In this paper, an anti-rollover wire-controlled auxiliary braking system was designed for the rollover problem of forest fire trucks. The proposed three-degrees-of-freedom dynamics model of the forest fire truck was first built in MATLAB/Simulink, and the dynamics model was simulated and verified in CarSim under the steering wheel angle step input condition and double-shift line condition. The auxiliary braking control system consisted of a vehicle rollover warning module and a wire-controlled auxiliary braking system. The vehicle rollover warning module was intended to use the real-time roll angle and its angular velocity obtained by the inertial guidance system as input variables, and the lateral load transfer rate (LTR) as the rollover evaluation index and determine whether the current body attitude is in the rollover critical state according to the algorithm. The wire-controlled auxiliary braking system adopted the electro-hydraulic wire-controlled actuation system (EHB) to achieve vehicle speed reduction by differential braking technology in the rollover critical state. The braking system consisted of a DC reducer motor, plunger pump, accumulator, solenoid valve, hydraulic brake line, brake sub-pump caliper assembly, brake pressure sensor, electronic control unit and so on. When the wire-controlled auxiliary braking system worked, it first obtained the current body attitude parameters, calculated the required target brake wheel cylinder pressure through PID/Fuzzy-PID control, and then applied differential braking to the front left/right wheels of the vehicle to provide intervention transverse moment and reduce the rollover tendency of the vehicle. To verify the effectiveness of the designed braking system, two extreme operating conditions were selected to analyze the control effects of PID control and fuzzy-PID. For both, the maximum lateral acceleration improved by at least 42.8% and the maximum slide slip angle improved by 32%, while for the maximum roll, the fuzzy PID was able to advance the control to reduce the maximum roll by 20%, while the PID control was only able to control within 5%. The results showed that both PID and fuzzy-PID control algorithms can reliably calculate the target brake wheel cylinder pressure. The latter control effect is better than the former, PID control is at least 2 s delayed than fuzzy PID with faster response, shorter adjustment time and smaller overshoot;
- In order to design a more perfect braking system in future work, and to verify the execution capability of the hardware of the EHB system, it is also necessary for it to perform hydraulic simulation to verify whether it can realize the process of rapid increase and slow decompression, as well as to design the appropriate brakes and other related components according to the selected vehicle.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

Parameter Notation | Parameter Definition | Unit |

m | The overall vehicle mass | kg |

m_{s} | The spring mass | kg |

β | The lateral deflection angle | rad |

ω_{r} | The angular velocity of the transverse sway | rad/s |

h | The difference between the heights of the center of mass on the spring and the center of lateral camber of the vehicle | m |

F_{y1} | The lateral forces of the front wheel | N |

F_{y2} | The lateral forces of the rear wheel | N |

k_{f} | The lateral deflection stiffness of the front wheel | N/rad |

k_{r} | The lateral deflection stiffness of the rear wheel | N/rad |

δ | The nominal front-wheel deflection angle | rad |

δ_{1} | The effective side deflection angle of the front wheels | rad |

δ_{2} | The effective side deflection angle of the rear wheels | rad |

I_{x} | The rotational inertia of the mass on the spring about the x-axis | kg·m2 |

I_{z} | The rotational inertia of the z-axis of the vehicle | kg·m2 |

I_{xz} | The inertia product of the mass on the spring around the x and z axes | kg·m2 |

a | The distances from the center of gravity of the vehicle to the front axle | m |

b | The distances from the center of gravity of the vehicle to the rear axle | m |

v | the vehicle speed | m/s |

D_{f} | The front suspension lateral camber damping | N·m·s/rad |

D_{r} | The rear suspension lateral camber damping | N·m·s/rad |

φ | The body roll angle | rad |

C_{φ}_{1} | The front lateral camber rigidity | N·m/rad |

C_{φ2} | The rear lateral camber rigidity | N·m/rad |

L | The length of wheelbase | m |

N_{1} | The positive torque coefficient for the front wheel return | N·m/rad |

N_{2} | the positive torque coefficient for the rear wheel return | N·m/rad |

E_{f} | The front wheel lateral steering coefficient. | |

E_{r} | The rear wheel lateral steering coefficients. | |

d | Wheelbase | m |

l_{f} | The distance from center of mass to front axle | m |

l_{r} | The distance from center of mass to rear axle | m |

k_{s} | Suspension Sway Stiffness | N/rad |

g | Gravitational acceleration | m/s^{2} |

a_{y} | The lateral acceleration at the center of mass | m/s^{2} |

T | Wheelbase of the vehicle | m |

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**Figure 2.**Step-input response curve of steering wheel angle: (

**a**) steering angle curve; (

**b**) yaw rate; and (

**c**) lateral acceleration.

**Figure 3.**Response curve of double shifting line: (

**a**) road offset routes; (

**b**) yaw rate; and (

**c**) lateral acceleration.

**Figure 14.**Co-simulation model. (

**a**) PID whole vehicle control model; (

**b**) Fuzzy PID whole vehicle control model.

**Figure 15.**Response curve of fishhook steering condition: (

**a**) LTR; (

**b**) yaw rate; (

**c**) lateral acceleration; (

**d**) side slip angle; (

**e**) roll; and (

**f**) braking torque.

**Figure 16.**Response curve of J-turn condition: (

**a**) LTR; (

**b**) yaw rate; (

**c**) lateral acceleration; (

**d**) side slip angle; (

**e**) roll; and (

**f**) braking torque.

Name, Symbol/Legal Unit of Measure | Value |
---|---|

Overall vehicle mass m/kg | 2515 |

Spring mass m_{s}/kg | 1780 |

Wheelbase d/m | 1.478 |

Distance from center of mass to front axle l_{f}/m | 1.485 |

Distance from center of mass to rear axle l_{r}/m | 1.265 |

Distance from center of mass to rear axle h/m | 0.484 |

Inertia around z-axis I_{z}/kg·m^{2} | 3300 |

Inertia around the x-axis I_{x}/kg·m^{2} | 1000 |

Front-wheel lateral deflection stiffness k_{f}/(N/rad) | 44,400 |

Rear-wheel lateral deflection stiffness k_{r}/(N/rad) | 43,600 |

Suspension equivalent damping coefficient c | 2000 |

Suspension Sway Stiffness k_{s}/(N·m/rad) | 44,500 |

Gravitational acceleration g/(m/s^{2}) | 9.8 |

ΔK_{P} | EC | |||||||
---|---|---|---|---|---|---|---|---|

NB | NM | NS | ZO | PS | PM | PB | ||

E | NB | PB | PB | PM | PM | PS | ZO | ZO |

NM | PB | PB | PM | PS | PS | ZO | NS | |

NS | PM | PM | PM | PS | ZO | NS | NS | |

ZO | PM | PM | PS | ZO | NS | NM | NM | |

PS | PS | PS | ZO | NS | NS | NM | NM | |

PM | PS | ZO | NS | NM | NM | NM | NB | |

PB | ZO | ZO | NM | NM | NM | NB | NB |

ΔK_{i} | EC | |||||||
---|---|---|---|---|---|---|---|---|

NB | NM | NS | ZO | PS | PM | PB | ||

E | NB | NB | NB | NM | NM | NS | ZO | ZO |

NM | NB | NB | NM | NS | NS | ZO | ZO | |

NS | NM | NM | NS | NS | ZO | PS | PS | |

ZO | NM | NM | NS | ZO | PS | PM | PM | |

PS | NM | NS | ZO | PS | PS | PM | PB | |

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

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

ΔK_{d} | EC | |||||||
---|---|---|---|---|---|---|---|---|

NB | NM | NS | ZO | PS | PM | PB | ||

E | NB | PS | NS | NB | NB | NB | NM | PS |

NM | PS | NS | NB | NM | NM | NS | ZO | |

NS | ZO | NS | NM | NM | NS | NS | ZO | |

ZO | ZO | NS | NS | NS | NS | NS | ZO | |

PS | ZO | ZO | ZO | ZO | ZO | ZO | ZO | |

PM | PB | NS | PS | PS | PS | PS | PB | |

PB | PB | PM | PM | PM | PS | PS | PB |

Steering | LTR | Brake Wheel |
---|---|---|

Turn left | $<0$ | Right front wheel |

Turn right | $>0$ | Left front wheel |

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## Share and Cite

**MDPI and ACS Style**

Wang, R.; Xu, X.; Chen, S.; Guo, N.; Yu, Z.
Vehicle Rollover Warning and Control Based on Attitude Detection and Fuzzy PID. *Appl. Sci.* **2023**, *13*, 4339.
https://doi.org/10.3390/app13074339

**AMA Style**

Wang R, Xu X, Chen S, Guo N, Yu Z.
Vehicle Rollover Warning and Control Based on Attitude Detection and Fuzzy PID. *Applied Sciences*. 2023; 13(7):4339.
https://doi.org/10.3390/app13074339

**Chicago/Turabian Style**

Wang, Ruiyang, Xiangbo Xu, Shao Chen, Ningyan Guo, and Zhibin Yu.
2023. "Vehicle Rollover Warning and Control Based on Attitude Detection and Fuzzy PID" *Applied Sciences* 13, no. 7: 4339.
https://doi.org/10.3390/app13074339