# Analysis and Roll Prevention Control for Distributed Drive Electric Vehicles

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Generating Mechanism of the Rolling Moment of DDEV

#### 2.1. Rolling Moment of DDEV

_{x}is the rotational inertia of the vehicle around the x axis; φ is vehicle roll angle; m

_{s}is vehicle sprung mass; h

_{s}is the distance from the center of the sprung mass to the roll center of the car; a

_{y}is lateral acceleration at the center of mass; g is gravitational acceleration; K

_{φ}is the equivalent roll stiffness of the vehicle; C

_{φ}is equivalent roll damping of automobile and ΔM

_{X}is additional roll moment.

#### 2.2. Analysis of the Rolling Effect for the Additional Vertical Force of DDEV

_{f}, the force balance equation and the moment balance equation can be obtained as

_{x}

_{1}and T

_{1}represent the ground driving force and motor torque transmitted from the front wheel to the vehicle body via the suspension; P

_{Li}(i = 1, 2) is the force exerted by the left body on the front suspensions; z

_{i}(i = 1, 2) and θ

_{i}(i = 1, 2) are the corresponding distance and angle in the Figure 2.

_{L1}, P

_{Li}

^{′}and P

_{L2}, P

_{Li}

^{′}can be expressed as

_{Li}(i = 1, 2) and P

_{Li}

^{′}(i = 1, 2) are acting force and reaction force; P

_{Li}

^{′}(i = 1, 2) is the force exerted by the front suspensions on the left body.

_{X1}is the roll moment the left front wheel; M

_{X2}is the roll moment of the left rear wheel; M

_{X2}is the roll moment of the right front wheel; M

_{X4}is the roll moment of the right rear wheel; K

_{i}(i = 1, 2, 3, 4) is the corresponding coefficient.

_{i1}and P

_{i2}represent the force exerted by the vehicle body on the inner side of the front suspension; z

_{i1}and z

_{i2}are the corresponding distance; θ

_{i1}and θ

_{i2}are the corresponding angle.

_{i1}, P

_{i1}

^{′}and, P

_{i2}, P

_{i2}

^{′}can be expressed as

_{ij}(j = 1, 2) and P

_{ij}(i = 1, 2) are a pair of acting force and reaction force, and P

_{ij}(i = 1, 2) are the force exerted by the front-inner suspension on the vehicle body.

_{in}is the distance from the roll center to the instantaneous center of roll motion on the inside of the front suspension.

_{o1}

^{′}and P

_{o2}

^{′}represent the force exerted by the front-outer suspension on the vehicle body, l

_{out}is the distance from the roll center to the instantaneous center of roll motion on the outside of the front suspension, θ

_{o1}and θ

_{o2}are the corresponding angle.

_{5}and K

_{6}are the coefficient term of the roll moment generated by the front-inner wheel and front-outer wheel.

## 3. Roll Stability Control Algorithm

#### 3.1. Yaw and Roll Decoupling Control Algorithm

#### 3.1.1. Yaw Stability Control

_{x}is the vehicle’s longitudinal speed, k

_{r}is the cornering stiffness of the rear wheel, k

_{f}is the cornering stiffness of the front wheel, L is the vehicle wheelbase and K is the stability factor, $K=\frac{m}{{L}^{2}}(\frac{a}{{k}_{\mathrm{r}}}-\frac{b}{{k}_{\mathrm{f}}})$.

_{rd}and ideal side slip angle β

_{d}as state variables described as

_{z}is the moment of inertia of the vehicle around the Z axis.

_{r}and side slip angle β re-track the change of the ideal value. The relationship between the vehicle’s steering characteristics and the compensated additional yaw moment is shown in Table 1.

_{r}and the actual side slip angle β as state variables, the equation of state of automobile motion described as

_{z}is additional direct yaw moment, and

**B**

_{1}= [0 1/I

_{z}]

**.**

^{T}_{r}is the difference between the actual yaw velocity and the ideal yaw velocity.

**K**is the feedback matrix, and

**K**= [k

_{1}k

_{2}]

**.**

^{T}#### 3.1.2. Roll Stability Control

_{y}is the vehicle lateral speed.

_{X}to recover the roll stability.

#### 3.1.3. Torque Distribution Strategy for Decoupling Control

_{z}, and the roll moment to be compensated is ΔM

_{x}, and the yaw moment and roll moment that can be compensated by adjusting the driving torque of each wheel are shown in Table 2 respectively.

**K**= [ΔF

_{x}

_{1}ΔF

_{x}

_{2}ΔF

_{x3}ΔF

_{x4}]

**$A=\left[\begin{array}{cccc}{K}_{1}+{K}_{5}& 0& {K}_{3}& 0\\ -\frac{1}{2}B\mathrm{cos}\delta +\mathrm{sin}\delta \cdot a& 0& -\frac{1}{2}B& 0\\ 0& {K}_{2}+{K}_{6}& 0& {K}_{4}\\ 0& \frac{1}{2}B\mathrm{cos}\delta +\mathrm{sin}\delta \cdot a& 0& \frac{1}{2}B\end{array}\right]$, $B={\left[\begin{array}{cccc}\Delta {M}_{\mathrm{X}}/2& \Delta {M}_{\mathrm{Z}}/2& \Delta {M}_{\mathrm{X}}/2& \Delta {M}_{\mathrm{Z}}/2\end{array}\right]}^{T}$.**

^{T}_{xi}should meet the limits as following:

_{xi}is the increment of each driving force. r is the radius of the wheel; T

_{max}is the maximum driving moment of the motor; μ is the road adhesion coefficient.

#### 3.2. Anti-Rollover Control Algorithm Based on Differential Brake

_{b}represents the braking force exerted on the front outer wheel, a

_{y}is the vehicle lateral acceleration, F

_{yf}and F

_{yr}are the side force of vehicle front and rear wheels, ΔM

_{Z}represents the compensated additional yaw moment.

_{Z}will be calculated by fuzzy control algorithm.

#### 3.2.1. Evaluation Index of Vehicle Rollover

_{z1}, F

_{z2}, F

_{z3}and F

_{z4}are the vertical load of each driving wheel.

#### 3.2.2. Fuzzy Control Algorithm

_{Z}who is entered into torque distribution controller. The torque distribution controller outputs the braking pressure applied on the front outer wheel.

#### 3.2.3. Distribution Strategy of Yaw Moment

_{Z}. When the vehicle is in danger of rollover, a braking torque will be applied to the right front wheel of the vehicle separately. The relationship between compensated yaw moment and braking force is

_{w}is the rotating inertia of front outer wheel; ω is the angular velocity of the wheel; T

_{b}is the vehicle braking torque.

## 4. Simulation and Verification

#### 4.1. Vehicle Model

_{m}to the target electromagnetic torque T

_{m}

^{*}. The transfer function is

#### 4.2. Simulation Verification of Yaw and Roll Decoupling Control Algorithm

#### 4.2.1. Angular Step Input Condition

#### 4.2.2. Sine Input Condition

#### 4.2.3. Fish Hook Test Condition

#### 4.3. Simulation Verification of Anti-Rollover Control Algorithm

#### 4.3.1. J-Turn Condition

#### 4.3.2. Fish Hook Test Condition

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Symbol | |

Iₓ (kg·m^{2}) $\varphi $ (deg) | rotational inertia of the vehicle around the x axis vehicle roll angle |

m_{s} (kg) | vehicle sprung mass |

h_{s} (m) | distance from the center of the sprung mass to the roll center of the car |

b (m) | distance from center of mass to rear axle |

a (m) | distance from center of mass to front axle |

m (kg) | vehicle mass |

v_{y} (km/h) | vehicle lateral speed |

v_{x} (km/h) | vehicle longitudinal speed |

a_{y} (m/s^{2}) | vehicle lateral acceleration |

ω (rad/s) | angular velocity of the wheel |

k_{r}, k_{f} (N/rad) | cornering stiffness of the rear and front wheel |

L (m) | vehicle wheelbase |

r (m) | wheel rolling radius |

B (m) | track width |

$\delta $ (deg) | front wheel angle |

K | stability factor |

g_{y} (m/s^{2}) | gravitational acceleration |

F_{y}_{f}, F_{y}_{r} (N) | side force of vehicle front and rear wheels |

K^{ϕ} (N/rad) | equivalent roll stiffness of the car |

C^{ϕ} (N/(km/h)) | equivalent roll damping of automobile |

ΔM_{X} (N·m) | additional roll moment. |

F_{x}_{1} (N) | ground driving force |

T_{1} (N·m) | motor torque transmitted from the front wheel to the vehicle body via the suspension |

P_{Li} (i = 1, 2, 3, 4) (N) | force exerted by the left body on the front suspensions |

P_{Li} (i = 1, 2, 3, 4) (N) | force exerted by the front suspensions on the left body |

P_{ij} (j = 1, 2) (N) | force exerted by the car body on the side of the front-inner suspension |

P_{ij} (j = 1, 2) (N) | force exerted by the front-inner suspension on the car body |

P_{oi}^{′} (j = 1, 2) (N) | force exerted by the front-outer suspension on the car body |

K_{i} (i = 1, 2, 3, 4, 5, 6) | the corresponding coefficient of the roll moment |

z_{i} (i = 1, 2, 3, 4) (m) z _{ij} (j = 1, 2) (m) l _{out}, l_{in} (m) | the corresponding distance |

θ_{i} (i = 1, 2, 3, 4) (rad)θ _{ij} (j = 1, 2) (rad) θ _{oi} (i = 1, 2) (rad) | the corresponding angle |

M_{Xj} (j = 1, 2, 3, 4) (N·m) | roll moment generated by the suspension to the vehicle body |

M_{Xi}, M_{Xo} (N·m) | roll moment generated by the front wheel via the suspension |

ω_{r} (dge/s) | vehicle yaw rate |

ω_{rd} (dge/s) | ideal yaw rate |

ω_{rmax} (dge/s) | the maximum values of yaw rate |

β_{d} (dge) | ideal side slip angle |

β_{rmax} (dge) | the maximum values of side slip angle |

$\xi $ | weight coefficient between the roll angle and roll angular velocity |

e (dge) | the error of roll angle |

$\eta $ | switching gain |

$\mu $ | road adhesion coefficient |

sat (s) | the saturation function |

ΔF_{x}_{i} (i = 1, 2, 3, 4) (N) | increment of each driving force |

ΔT_{i} (i = 1, 2, 3, 4) | wheels drive torque |

T_{max} (N·m) | maximum driving moment of the motor |

F_{b} (N) | braking force exerted on the front outer wheel |

I_{w} (kg·m^{2}) | rotating inertia of front outer wheel |

T_{b} (N·m) | vehicle braking torque. |

F_{z}_{i} (i = 1, 2, 3, 4) (N) | wheels vertical load |

P | braking pressure |

S | braking efficiency coefficient. |

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**Figure 5.**Simulation results of angular step input condition. (

**a**) Steering wheel angle. (

**b**) Vehicle track. (

**c**) Yaw velocity. (

**d**) Side slip angle. (

**e**) Lateral acceleration. (

**f**) Roll angle.

**Figure 6.**Simulation results of sine input condition. (

**a**) Steering wheel angle. (

**b**) Vehicle track. (

**c**) Yaw velocity. (

**d**) Side slip angle. (

**e**) Lateral acceleration. (

**f**) Roll angle.

**Figure 7.**Simulation results of fish hook test condition. (

**a**) Steering wheel angle. (

**b**) Vehicle track. (

**c**) Yaw rate. (

**d**) Side slip angle. (

**e**) Lateral acceleration. (

**f**) Roll angle.

**Figure 8.**Simulation results of J-turn condition. (

**a**) Steering wheel angle. (

**b**) LTR. (

**c**) Lateral acceleration. (

**d**) Roll angle. (

**e**) Wheel vertical load before control. (

**f**) Wheel vertical load after control.

**Figure 9.**Simulation results of fish hook test condition. (

**a**) Steering wheel angle. (

**b**) LTR. (

**c**) Lateral acceleration. (

**d**) Roll angle. (

**e**) Wheel vertical load before control. (

**f**) Wheel vertical load after control.

Steering Condition | Yaw Velocity | Additional Yaw Moment |
---|---|---|

left understeer | ${\omega}_{r}$ > 0 | $\Delta {M}_{z}$ > 0 |

left oversteer | ${\omega}_{r}$ > 0 | $\Delta {M}_{z}$ < 0 |

light understeer | ${\omega}_{r}$ < 0 | $\Delta {M}_{z}$ < 0 |

light oversteer | ${\omega}_{r}$ < 0 | $\Delta {M}_{z}$ > 0 |

Wheel | Compensated Roll Moment | Compensated Yaw Moment |
---|---|---|

front left wheel torque | $\Delta {M}_{\mathrm{X}1}$ | $\Delta {M}_{\mathrm{Z}1}$ |

rear left wheel torque | $\Delta {M}_{\mathrm{X}3}$ | $\Delta {M}_{\mathrm{Z}3}$ |

front right wheel torque | $\Delta {M}_{\mathrm{X}2}$ | $\Delta {M}_{\mathrm{Z}2}$ |

rear right wheel torque | $\Delta {M}_{\mathrm{X}4}$ | $\Delta {M}_{\mathrm{Z}4}$ |

ΔM_{Z} | e | |||||||
---|---|---|---|---|---|---|---|---|

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

ec | PB | ZO | ZO | NW | NS | NM | NB | NB |

PS | PM | PS | ZO | NW | NS | NB | NB | |

ZO | PB | PM | PW | ZO | NW | NM | NB | |

NS | PB | PB | PS | PW | ZO | NS | NM | |

NB | PB | PB | PM | PS | PW | ZO | ZO |

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

Vehicle masssprung massun-sprung mass | 1380 900 480 |

Distance from center of mass to front axle | 1.05 |

Distance from center of mass to rear axle | 1.57 |

front wheel tread | 1.4 |

rear wheel tread | 1.4 |

height of centroid | 0.6 |

Tire diameter load radius | 0.33 |

tire type | 255/75 R16 |

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## Share and Cite

**MDPI and ACS Style**

Chang, X.; Zhang, H.; Yan, S.; Hu, S.; Meng, Y.
Analysis and Roll Prevention Control for Distributed Drive Electric Vehicles. *World Electr. Veh. J.* **2022**, *13*, 210.
https://doi.org/10.3390/wevj13110210

**AMA Style**

Chang X, Zhang H, Yan S, Hu S, Meng Y.
Analysis and Roll Prevention Control for Distributed Drive Electric Vehicles. *World Electric Vehicle Journal*. 2022; 13(11):210.
https://doi.org/10.3390/wevj13110210

**Chicago/Turabian Style**

Chang, Xiaoyu, Huanhuan Zhang, Shuai Yan, Shengli Hu, and Youming Meng.
2022. "Analysis and Roll Prevention Control for Distributed Drive Electric Vehicles" *World Electric Vehicle Journal* 13, no. 11: 210.
https://doi.org/10.3390/wevj13110210