# Rollover Stability of Heavy-Duty AGVs in Turns Considering Variation in Friction Coefficient

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## Abstract

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## 1. Introduction

## 2. The Mechanism of Steering Rollover of AGV

#### 2.1. Model of AGV Rollover

_{zi}= 0, the AGV starts to tip over and can no longer maintain balance within the rolling plane. The threshold of tipping over when the AGV starts to tip over is given by the following Equation (3):

_{s}is the total mass of the AGV; B is the wheelbase of the AGV; a

_{y}is the lateral acceleration of the AGV; h

_{g}is the height of the centre of mass from the ground; h

_{r}is the height of the roll centre from the ground; F

_{zi}is the vertical reaction force received by the inner wheel of the AGV; and $\Phi $ is the roll angle of the centre of mass of the AGV.

#### 2.2. Dynamic Rollover Model for Heavy-Duty AGV

_{1}is generated between the helm wheel and the ground due to the increase of the wheel side deflection, and the lateral force F

_{1}generates a transverse moment to the centre of mass of the AGV, which causes the AGV to swing sideways. The lateral force F

_{2}is generated between the helm wheel and the ground to balance the tendency of the AGV to swing sideways, and the lateral acceleration a

_{y}causes the AGV to roll sideways. Through the analysis of the above heavy-duty AGV motion state, the following differential equations of motion can be established.

_{y}is the lateral acceleration; V

_{x}and V

_{y}represent the moving speed of AGV along the X-axis and Y-axis respectively. r represents the yaw rate of AGV; ϕ represents the sideslip angle of AGV centroid; ψ represents the yaw angle of AGV; ϕ represents the AGV roll angle; δ represents the wheel angle; m represents AGV quality; h represents the distance from the centre of mass to the roll centre; k

_{s}and C

_{s}represent the suspension roll stiffness and roll damping, respectively.

_{y}. and there is a lateral acceleration a

_{y}of the AGV, will produce a certain lateral roll angle $\varphi $, in the lateral, transverse, lateral roll three directions of motion at the same time, with the increase of lateral force and lateral acceleration, the moment increases, at this time the AGV makes the lateral rollover motion.

_{1}, F

_{2}can be calculated by the following equation:

_{1}and k

_{2}are the equivalent lateral deflection stiffness of the front and rear wheels, respectively, and are taken as positive values. When the wheel lateral deflection angle is small, the lateral deflection angles a

_{1}and a

_{2}satisfy the following relations.

_{s}and roll damping C

_{s}of the AGV suspension can have a certain resistance effect on the roll motion of the AGV, but when the lateral acceleration increases to a certain limit value, the AGV roll angle will increase to exceed the limit value and cause a rollover. This dynamic model provides a theoretical basis for the mathematical model of omnidirectional mobile AGV.

#### 2.3. The Omnidirectional Kinematics Model of Heavy-Duty AGV

#### 2.3.1. Mathematical Model of a Heavy-Duty AGV with Four Helm Wheels

#### 2.3.2. Mathematical Model for Positive Kinematics Analysis of Heavy-Duty AGV

- (1)
- The heavy-duty AGV is a rigid body.
- (2)
- The running ground of the heavy-duty AGV is horizontal and of suitable smoothness.
- (3)
- The motion speed of heavy-duty AGV is low and there is no air resistance.
- (4)
- The driving wheel of heavy-duty AGV has good contact with the ground, and the driving wheel does pure rolling.

_{r}axis, then translate along the Y

_{r}axis to move to the centre of wheel i, rotate β

_{i}angular value at the centre of wheel i, and the AGV coordinate system coincides with the helm wheel coordinate system, then translate Fr

_{Ii}along the helm wheel coordinate system respectively, translate the AGV coordinate system to the centre of rotation I, reverse β

_{i}angular value around the centre of rotation, and in X

_{r}and Y

_{r}axes translations to move the helm wheel coordinate system to the origin Or, then the coordinate transformation equation of the motion of the AGV coordinate system along wheel i is expressed as:

_{I}along the X-axis coordinates of the virtual wheel to the centre of rotation, reverses β angle, and then translates along the X

_{r}and Y

_{r}axes respectively to return to the initial position, then the coordinate transformation equation of the motion of AGV coordinate system along the virtual wheel is expressed as:

#### 2.3.3. Inverse Kinematics Analysis of the Model

_{1}, V

_{2}, V

_{3}, and V

_{4}, and their angles are β

_{1}, β

_{2}, β

_{3}, and β

_{4}, respectively. then it is described in Figure 4.

## 3. Heavy-Loaded AGV Rollover Simulation Analysis

#### 3.1. Simplification of AGV Model

- (1)
- Simplify the AGV into a mass point, so that the AGV mass is concentrated in one point;
- (2)
- Ignoring the influence of suspension characteristics when considering the AGV model as a rigid object to perform force analysis;
- (3)
- Ignoring the asymmetry of the left and right tires and the front and rear axles;
- (4)
- Ignoring the effect of longitudinal motion of the AGV on rollover during steady-state steering and driving;
- (5)
- The influence of the dynamic characteristics of the AGV in the pitch direction on rollover is not considered.

#### 3.2. Simulation Modelling

#### 3.3. Simulation Process of Heavy-Duty AGV

#### 3.4. Analysis of Factors Affecting the Rollover Stability of AGV

#### 3.4.1. Impact of Turning Speed on the Stability of Rollover in AGV

#### 3.4.2. The Impact of Centre of Gravity Position on the Stability of AGV Rollover

#### 3.4.3. The Influence of Road Friction Coefficient on the Stability of AGV Rollover

## 4. Comprehensive Evaluation of Factors Affecting Rollover

#### 4.1. Rollover Risk Metrics

_{1}is the vertical ground reaction force acting on the outer wheel of the AGV, and F

_{2}is the vertical ground reaction force acting on the inner wheel of the AGV.

#### 4.2. Orthogonal Test Analysis of Rollover Influencing Factors

#### 4.3. Influence of Rollover Factors on the Load Transfer Rate

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**Simplified model of heavy-duty AGV. (1) AGV chassis, (2) helm mechanism, (3) AGV carrier plate, (4) Braking system, and (5) Mass centre ball.

**Figure 6.**Simplified simulation model of heavy-duty AGV. (

**a**) Simplified model position at 0 s. (

**b**) Simplified model position at 2 s. (

**c**) Simplified model position at 4 s. (

**d**) Simplified model position at 6 s. (

**e**) Simplified model position at 8 s. (

**f**) Simplified model position at 10 s.

**Figure 7.**Graph of the Effect of Turning Speed Variation on the Sideslip Angle of the Centre of Mass.

**Figure 10.**Curves of the effect of tangential direction change on the sideslip angle of the centre of mass.

**Figure 11.**The lateral force curve of the outer wheel of AGV with the different road friction coefficients.

**Figure 12.**Variation of the side deflection angle of AGV with the different road friction coefficients.

Symbols | Implication | Symbols | Implication |
---|---|---|---|

V_{1} | Right front helm wheel speed | V_{2} | Left front helm wheel speed |

V_{3} | Left rear helm wheel speed | V_{4} | Right rear helm wheel speed |

β_{1} | Angle between right front helm wheel and AGV coordinate system | β_{2} | Angle between left front helm wheel and AGV coordinate system |

β_{3} | Angle between left rear helm wheel and AGV coordinate system | β_{4} | Angle between right rear helm wheel and AGV coordinate system |

W_{1} | Right front helm wheel angular velocity | W_{2} | Left front helm wheel angular velocity |

W_{3} | Left rear helm wheel angular velocity | W_{4} | Right rear helm wheel angular velocity |

r_{1} | Right front helm wheel rotation radius | r_{2} | Left front helm wheel rotation radius |

r_{3} | Left rear helm wheel rotation radius | r_{4} | Right rear helm wheel rotation radius |

V | Heavy-duty AGV speed | R | Heavy-duty AGV rotation radius |

$\omega $ | Heavy-duty AGV angular velocity | I | Heavy-duty AGV rotary centre |

X_{i} | The lateral distance of the helm wheel from the centre position | Y_{i} | The longitudinal distance between the helm wheel and the centre position |

α | Heavy load AGV rotary centre abscissa | b | Heavy load AGV rotary centre ordinate |

Variable Names | Symbols | Numerical Values | Units |
---|---|---|---|

AGV quality | M | 5 | t |

Duty | M_{S} | 10 | t |

Total height | h | 5 | m |

Rotational inertia around the x-axis | I_{X} | 21,300 | kg×m^{2} |

Distance from the centre of mass to the front axle | a | 3.5 | m |

Distance from the centre of mass to the rear axis | b | 3.5 | m |

Sway stiffness | ${K}_{\varphi}$ | 90,672 | N/rad |

Sway damping | ${C}_{\varphi}$ | 5677 | N/rad |

Rotational inertia around the z-axis | ${I}_{Z}$ | 58,893 | kg×m^{2} |

Turning radius | r | 6 | m |

Wheelbase | l | 3.590 | m |

Sum of lateral deflection stiffness of two front wheels | k_{1} | −60 | kN/rad |

Sum of lateral deflection stiffness of both rear wheels | k_{2} | −60 | KN/rad |

The inertia of rotation around the centre of the lateral roll | I_{xeq} | 64,952 | kg×m^{2} |

Level | Turning Speed V°/s | Centroid Height h/mm | Road Friction Coefficient f |
---|---|---|---|

1 | 50 | 1200 | 0.2 |

2 | 60 | 1400 | 0.4 |

3 | 70 | 1600 | 0.6 |

4 | 80 | 1800 | 0.8 |

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**MDPI and ACS Style**

Fu, W.; Wang, X.; Zhang, X.
Rollover Stability of Heavy-Duty AGVs in Turns Considering Variation in Friction Coefficient. *Lubricants* **2023**, *11*, 119.
https://doi.org/10.3390/lubricants11030119

**AMA Style**

Fu W, Wang X, Zhang X.
Rollover Stability of Heavy-Duty AGVs in Turns Considering Variation in Friction Coefficient. *Lubricants*. 2023; 11(3):119.
https://doi.org/10.3390/lubricants11030119

**Chicago/Turabian Style**

Fu, Weijie, Xinyu Wang, and Xinming Zhang.
2023. "Rollover Stability of Heavy-Duty AGVs in Turns Considering Variation in Friction Coefficient" *Lubricants* 11, no. 3: 119.
https://doi.org/10.3390/lubricants11030119