# Stability Control for Electric Vehicles with Four In-Wheel-Motors Based on Sideslip Angle

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Vehicle Model

_{f}, longitudinal forces, lateral forces, and the yaw moment at the vehicle center of mass are as follows.

_{xi}and F

_{yi}are calculated according to the magic formula tire model.

_{z}is the inertia moment of vehicle. V

_{x}is the longitudinal velocity of vehicle. V

_{y}is the lateral velocity of vehicle. γ is the yaw rate. δ

_{f}is the front-wheel angle. l

_{f}is the distance from the center of gravity to front axle. l

_{r}is the distance from the center of gravity to rear axle. d is the wheelbase. F

_{xi}is the longitudinal forces of each wheel and F

_{yi}is the lateral forces of each wheel. Subscripts 1, 2, 3, and 4 indicate the left-front wheel, right-front wheel, left-rear wheel, and right-rear wheel respectively. λ

_{i}is the tire slip rate. ${\alpha}_{i}$ is the sideslip angle of the tire. x

_{1x},x

_{2x},x

_{3x}, and x

_{4x}are the parameters determined by road conditions to get F

_{xi}. x

_{1y},x

_{2y},x

_{3y}, and x

_{4y}are the parameters determined by road conditions to get F

_{yi}.

_{z}is the yaw moment. The other letters are shown above.

## 3. Stability Criterion Based on $\beta -\dot{\beta}$ Phase Planes

#### 3.1. Boundary Equation of $\beta -\dot{\beta}$ Phase Planes

#### 3.2. Influence of Driving Conditions on $\beta -\dot{\beta}$ Phase Planes

#### 3.3. Parameters Determination for Boundary Lines of $\beta -\dot{\beta}$ Phase Planes

## 4. Estimation of Sideslip Angle

_{y}is the lateral acceleration, k

_{f}is the cornering stiffness of front axle, k

_{r}is the cornering stiffness of rear axle, F(t) is the Jacobian matrix of the partial derivatives of function f(x(t),u(t),w(t)) to state x(t), and Δt is the sampling time. The other letters are shown above.

^{-}(t) is given an initial value of I

_{3×3}, and ${\widehat{x}}^{-}$(t) is given an initial value of [0,0,0]

^{T}, the estimation of sideslip angle can be obtained by forming a circulative process continuously of the prediction step and the update step of the extended Kalman filter.

## 5. Design of Stability Control System for Electric Vehicle with Four IWMs

#### 5.1. Calculation of Yaw Moment

_{β}).

_{β}is the parameter of exponential approaching rate based on $\beta -\dot{\beta}$ decision. The letters are shown above.

_{β}) which is replaced by the saturation function sat(s

_{β}/ϕ). The saturation function is as follows:

#### 5.2. Distribution of Wheel Longitudinal Forces

_{1}, T

_{2}, T

_{3}, and T

_{4}represent the output torque of left-front wheel, right-front wheel, left-rear wheel and right-rear wheel, respectively; R is the wheel radius; T

_{xi}is torques of each in-wheel-motor; F

_{zi}is the vertical forces of each wheel; and i = 1–4; T

_{max}is the peak torque of in-wheel-motors. The other letters are shown above.

_{xi}is the longitudinal forces of each wheel. The other letters are shown above.

_{x1}F

_{x2}F

_{x3}F

_{x4}]

^{T},C is a null matrix, and the expression of H is as follows:

## 6. Cosimulation Model Based on Matlab/Simulink and Carsim

#### 6.1. Vehicle Model

#### 6.2. IWM Model

_{max}is the maximum torque at the current speed. When the motor speed is lower than the base speed, T

_{max}is a constant. When the motor speed is higher than the base speed, T

_{max}is a function of the motor speed.

## 7. Simulation Evaluation

#### 7.1. Evaluation of Sideslip-Angle Estimation

#### 7.2. Evaluation of Torque Distribution

#### 7.3. Evaluation of Vehicle Stability Control

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**Phase plane in different driving conditions (

**a**) Vx = 120 km/h, μ = 0.8; (

**b**) Vx = 80 km/h, μ = 0.8; and (

**c**) Vx = 80 km/h, μ = 0.5.

**Figure 6.**Estimation for sideslip angle (when V

_{x}is 40 km/h and μ is 0.8) (

**a**) Lateral acceleration. (

**b**) Sideslip angle.

**Figure 7.**Estimation for sideslip angle (when V

_{x}is 100 km/h and μ is 0.8) (

**a**) Lateral acceleration. (

**b**) Sideslip angle.

**Figure 8.**Verification of optimal torque distribution (

**a**) trajectory, (

**b**) yaw rate, (

**c**) sideslip angle, (

**d**) $\beta -\dot{\beta}$ phase plane, (

**e**) wheel torque when average distribution, (

**f**) wheel torque when optimal distribution, (

**g**) vertical load of each wheel, and (

**h**) sum of wheel load rates.

**Figure 9.**Simulation results of sinusoidal delay condition (

**a**) front-wheel angle, (

**b**) $\beta -\dot{\beta}$ phase plane of vehicle, and (

**c**) yaw rate of vehicle.

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## Share and Cite

**MDPI and ACS Style**

Yang, K.; Dong, D.; Ma, C.; Tian, Z.; Chang, Y.; Wang, G.
Stability Control for Electric Vehicles with Four In-Wheel-Motors Based on Sideslip Angle. *World Electr. Veh. J.* **2021**, *12*, 42.
https://doi.org/10.3390/wevj12010042

**AMA Style**

Yang K, Dong D, Ma C, Tian Z, Chang Y, Wang G.
Stability Control for Electric Vehicles with Four In-Wheel-Motors Based on Sideslip Angle. *World Electric Vehicle Journal*. 2021; 12(1):42.
https://doi.org/10.3390/wevj12010042

**Chicago/Turabian Style**

Yang, Kun, Danxiu Dong, Chao Ma, Zhaoxian Tian, Yile Chang, and Ge Wang.
2021. "Stability Control for Electric Vehicles with Four In-Wheel-Motors Based on Sideslip Angle" *World Electric Vehicle Journal* 12, no. 1: 42.
https://doi.org/10.3390/wevj12010042