Research of smoothness on torque coordination of two-wheel independently driven hub electric vehicle based on fuzzy control

Abstract

Aiming at the problems of drive torque coordination, steering vibration and asymmetry of two-wheel independent drive hub electric vehicle, a new torque coordination control algorithm of driving wheel is proposed based on fuzzy control theory. The algorithm takes yaw rate deviation and centroid side deviation as inputs of fuzzy controller, outputs additional yaw moment needed by vehicle, and distributes it to each driving wheel reasonably according to the requirements of each driving wheel. Under the coordination of the driving anti-skid control algorithm, the vibration of driving wheels was reduced and the vehicle can run smoothly. The algorithm is verified by Simulink-CarSim joint simulation, and a fast prototype experiment platform based on dSPACE is built. The results of experiment and simulation show that the control strategies are effective and feasible.

Keywords

Fuzzy control electric vehicle independent drive torque control dSPACE

Introduction

In recent years, the development of motor control technology, vehicle technology, energy management technology, and vehicle-mounted power supply technology paved the way for the development of multi-wheel independent driving electric vehicle,^5-7 and became the four key technologies of the development of electric vehicle. Many scholars, enterprises, and universities in domestic and foreign have conducted extensive research in this field.^8-9

Since the driving torque of each wheel of the independently driven electric automobile can be independently controlled and freely distributed, the torque coordination control based on the direct yaw moment control can ensure that the vehicle can provide the most sufficient driving force for the whole vehicle under the condition of no slipping, and it is more flexible and efficient in implementation compared with the electronic stability program (ESP) of the traditional vehicle.^10-11 Meanwhile, under the precise control of torque distribution, for the two front steering wheels, the problem of steering asymmetry caused by the mechanical vibration and signal noise could be effectively improved.^12-14

Jin et al.¹ proposed the use of back-propagation (BP) neural network and proportional–integral–derivative (PID) control method to coordinate the driving torque; Hu² proposed a control allocation method using a hierarchical mechanism to optimize the allocation of drive/brake torque to improve vehicle handling stability; Ren³ proposed a torque distribution strategy aiming at improving the energy efficiency of the drive system; and Xu et al.⁴ proposed an additional yaw moment decision-making fuzzy PID controller for driving torque distribution.This article presents a control strategy based on fuzzy control theory, which takes torque coordination as the main line and anti-skid as the auxiliary line to reduce the instability of the vehicle by decreasing the vibration of steering wheels caused by the asymmetric mechanical structure and uncoordinated driving wheel control, so that we could achieve the whole vehicle power distribution and steering coordination control, which ultimately improve the stability and smoothness of the whole vehicle.

Design of control system

Design of torque coordination control system

The steering wheel signal and accelerator pedal signal are input into the system through a driver model, the information of vehicle running state acquired by a vehicle body sensor are calculated through a control algorithm, the yaw moment required by the steering of the whole vehicle is regularly calculated with the expected driving torque $T d$ , and the additional yaw moment ∆ $M z$ which is distributed to each driving wheel as required. On this basis, the structure diagram of driving control as shown in Figure 1.

Figure 1.

Drive coordination control structure diagram.

Determination of yaw moment based on fuzzy control theory

According to the experience of the team, we would make sure that the maximum deviation of centroid side is 5°, and the maximum deviation of yaw rate is 6°/s. In conclusion, the yaw rate deviation $Δ γ$ and the centroid side deviation $Δ β$ are used as inputs of the fuzzy controller. After the input variables are processed by fuzzy operation, the fuzzy value of additional yaw moment can be obtained by querying the fuzzy reasoning decision table. Then, after the fuzzy value obtained by reasoning is de-fuzzified, the controller can obtain the actual value of the additional yaw moment ∆ $M z$ .

Yaw rate deviation $Δ γ$ and centroid side deviation $Δ β$ as the controller inputs, and its membership function of the basic domain is [–1,1], which are divided into five grades ((NB), (NS), (ZO), (PS), (PB)). According to the input changes, real-time output additional yaw moment ∆ $M z$ , its basic domain is [–1,1], in order to more accurately control the motion state of the vehicle, is divided into seven grades ((NB), (NM), (NS), (ZO), (PS), (PM), (PB)), and for fuzzy processing, the triangular membership function is selected.

Established fuzzy rules: the following rules should be met when adjusting control parameters:

When the vehicle has insufficient steering, apply a forward yaw moment to the outer wheels, so that the vehicle can turn smoothly;

When the vehicle turns too much, reduce the inboard wheel by a yaw moment, so that it can turn smoothly.

The design of fuzzy inference rules table is shown in Table 1.

Table 1.

Fuzzy rules for fuzzy controllers.

$Δ β$	$Δ γ$
$Δ β$	NB	NS	ZO	PS	PB
NB	PB	PB	NS	NB	NB
NS	PB	PM	NS	NM	NB
ZO	PM	PS	ZO	NS	NM
PS	PB	PM	PS	NM	NB
PB	PB	PS	PS	NS	NB

$Δ γ$ : yaw rate deviation; $Δ β$ : center of mass deflection angle deviation.

Distribution of driving torque

After determining the yaw moment and the desired drive torque, the torque needs to be distributed. The drive torque distribution system requires good adaptability, and can make corresponding adjustments to specific roads, and it is able to reasonably distribute the torque of each wheel to meet the control objectives.^15-16 The accelerator pedal opening degree is linearly proportional to the driver’s desired driving torque, and the desired driving torque is worked out according to the pedal opening degree $T d$ and then combined with the real-time yaw moment value ∆ $M$ , the design of driving torque distribution is completed. Due to tire friction circle limitations, drive torque distribution is a constrained torque distribution. The distribution of driving torque should consider the constraints of driving motor, vehicle model itself, tire friction circle, and road adhesion. On this basis, a distribution algorithm targeting stability is defined.

Referring to the vehicle dynamics model, equations (1)–(3) can be obtained

T d = K S a c

(1)

2 T d = {T 1}^{″} + {T 2}^{″}

(2)

Δ M = (F x 2 - F x 1) \cos δ \cdot \frac{d f}{2} + (F x 4 - F x 3) \cdot \frac{d r}{2}

(3)

Among them, the limit of yaw moment should also be less than the maximum yaw moment value that can be provided by the ground. When the driving wheel does not slip, the anti-slip control strategy is not executed, and at this time, $T 1 = {T 1}^{'}, {T 2}^{″} = {T 2}^{'}$ . $S a c$ is the pedal opening degree (m), K is the coefficient, and T₁ and T₂ are the driving torques.

At the same time, torque distribution also subjects to the constraints of the tire friction circle, so that the vector sum of the longitudinal and lateral forces shall be less than or equal to the maximum adhesion that the road surface can provide

\sqrt{F x i^{2} + F y i^{2}} \leq μ i F z i

(4)

In equations (3) and (4), $F x i$ is the longitudinal force of each wheel (N), $F y i$ is lateral force of each wheel (N), and R is the wheel radius (m).

Design of drive anti-skid control system

The drive anti-skid control module is shown in Figure 2. First, according to the running speed of the vehicle $V$ collected by the data acquisition and calculation module and the speed of each driving wheel $V i$ , the slip ratio $λ i$ of the driving wheel is calculated. In this article, the optimal slip ratio is selected according to the experience, and the slip ratio limit is set to 0.2. When the slip ratio of the wheel is larger than this value, the driving wheel is considered to slip and the driving anti-slip control strategy is executed. When the slip ratio is less than this value, the wheel is considered not to slip and the drive anti-slip control system is not activated.

Figure 2.

Diagram of driving anti-skid control.

The slip ratio can be estimated by the following equation (5)

λ i = \frac{V i - V}{V}

(5)

V i = \sqrt{u^{2} + v^{2}}

(6)

where $V i$ is the driving wheel speed, $V$ is the vehicle running speed, $u$ is the longitudinal speed, and $v$ is the lateral speed.

The input $Δ λ$ of the drive anti-slip control is the difference between the actual slip ratio $λ i$ and the optimum slip ratio $λ d$ ; the output ∆ $T 1$ is the adjustment torque of the motor; the PID controller block diagram is shown in Figure 3.

Figure 3.

Diagram of drive anti-skid controller.

The drive correction torque is calculated as follows

Δ T 1 = K p Δ λ + K i Δ λ + K d \frac{d Δ λ}{d t}

(7)

where ∆ $T 1$ is the adjusting torque of driving anti-skid control strategy (N m), $K p$ is the proportion coefficient, $K i$ is the integral coefficient, and $K d$ is the differential coefficient.

By modifying the torque of each wheel, the motion attitude of the vehicle body is improved, and the actual slip ratio approaches the optimal slip ratio, thus ensuring the running stability of the vehicle.

The whole vehicle test platform

According to the expected functional requirements of the electric vehicle driven by the hub motor, in order to meet the requirements of its dynamic control, the structure of the whole vehicle is first arranged, and then the electronic control system structure of the whole vehicle is designed. The electronic control system architecture of the whole electric vehicle can reflect the design objectives of the experimental platform, and the electronic control system architecture of the test sample vehicle is shown in Figure 4. The hardware layout of the test sample vehicle and the interface definition of each component are also developed around the architecture.

Figure 4.

The Electronic Structure of Electric Vehicle of Target Test Vehicle

The whole vehicle controller adopts dSPACE company’s MicroAutoBox rapid prototype controller, and uses it to verify the control strategy. This only needs to have a feasible control algorithm, but does not need to work on the controller hardware development. The fast prototype controller has high-speed computing ability, wide input/output (I/O) interface, and auto signal conditioning function, which facilitates reading information collected by each sensor. Pc software is a graphic-based development interface, which is simple and easy to operate, and do not need to be manually program debugging. It can realize the seamless docking with MATLAB/Simulink, and can create a control model in the MATLAB/Simulink environment, through the automatic generation of code into the controller can identify and execute the code, which simplifies the control algorithm development process and improve the development efficiency. Table 2 is the actual vehicle parameters.

Table 2.

Basic parameters of test sample vehicle.

Projects	Parameter
Outline dimensions (length × width × height) mm³	3600 × 1440 × 1780
Wheelbase (mm)	1860
Wheel pitch (mm)	1210
Minimum turning diameter (mm)	4500
Quality of preparation (kg)	1000
Tire type and specification	Diagonal tire 238-12/2310-12
Tire outer diameter (mm)	580
Tire section width (front/rear) (mm)	203.2/254
Hub diameter (mm)	304
Rated voltage (V)	72
Rated power of motor (kW)	3

Modeling simulation and analysis

In order to better analyze the stability of the vehicle, the 7-degree-of-freedom vehicle model is simplified, and the 2-degree-of-freedom vehicle models of lateral motion and yaw motion are established. Based on this model, the yaw rate and the sideslip angle of the center of mass are calculated under ideal driving conditions. The 2-degree-of-freedom vehicle model is established considering only lateral motion and yaw motion.

When the car is running steadily, the center of mass side deflection angle $β$ is too small to consider, and the front and rear axis side deflection angle are

α 1 = β + \frac{a γ}{u} - δ

(8)

α 2 = β - \frac{b γ}{u}

(9)

Differential equation of automobile 2-degree-of-freedom model

I z \overset{•}{γ} = (a C f - b C r) β + \frac{1}{u} (a^{2} C f + b^{2} C r) γ - a C f δ

(10)

m (\overset{•}{v} + u γ) = (C f + C r) β + \frac{1}{u} (a C f - b C r) γ - C f δ

(11)

where $m$ is the automobile quality (kg); $C f$ is the lateral stiffness of front and rear axles (N/rad); $δ$ is the front wheel angle (degree); $I z$ is the automobile moment of inertia (kg m²); a, b are the distance from the front and rear axle to the center of mass (m); $u$ is the longitudinal speed (km/h); $v$ is the lateral speed (km/h); $γ$ is the yaw rate (°/s); and $β$ is the center of mass side declination (degree).

At the steady state, yaw rate is

γ d = \frac{u}{L (1 + K u^{2})} δ

(12)

K = \frac{m}{L^{2}} (\frac{a}{C r} - \frac{b}{C f})

(13)

To ensure steady-state driving of the vehicle, the steady-state yaw rate shall meet the following conditions

| γ d | \leq | \frac{μ \cdot g}{u} |

(14)

The calculation formula of expected yaw rate is

γ d = \min {| \frac{u}{L (1 + K u^{2})} \cdot δ |, | \frac{μ \cdot g}{u} |} s i g n (δ)

(15)

The desired center-of-mass side-slip angle is used to reflect the motion trajectory of the vehicle model. The closer the desired center-of-mass side-slip angle is to 0, the closer the actual motion trajectory is to the target motion trajectory. This article sets the desired center-of-mass yaw angle to 0, that is $β d = 0$ .

$K$ is the stability coefficient, $L$ is the axial distance of two axes (m), $μ$ is the tire road surface adhesion coefficient, and $g$ is the gravity acceleration (m/s²).

The obtained ideal yaw rate value is compared with the actual value of the vehicle, and then the control model is built. The vehicle model is built in the CarSim, and the effectiveness of the control algorithm is verified by the joint simulation with the control strategy designed in Simulink. Figure 5, which is built in Simulink, shows the drive anti-skid control strategy, and Figure 6 shows the Simulink-CarSim joint simulation model.⁶

Figure 5.

Drive anti-skid control strategy.

Figure 6.

Simulink-CarSim joint simulation model.

Torque coordination simulation validation—angular step input

The given steering wheel angle input is shown in Figure 7 for angular step simulation analysis. When the steering wheel angle is positive, the vehicle turns left, and when the angle is negative, the vehicle turns right. At this time, the road surface adhesion coefficient is selected to be 0.85, no slip occurs, and no slip control is performed.

Figure 7.

Steering angle.

As shown in Figures 8 –12, the simulation results under low-speed angular step condition show that the curve without control is the operation curve of the CarSim reference vehicle, and the curve after control is obtained by executing the control strategy. Figure 8 is a graph comparing the actual vehicle speed with the target vehicle speed, and the vehicle speed is maintained at approximately 30 km/h. Figure 9 is the output torque of the left and right drive wheel motors after the control strategy is executed. The output torque changes correspondingly with the change of the steering wheel angle; the maximum torque can reach up to 37.8 N m; the change of the torque corresponds to the change of the speed of the drive wheel.

Figure 8.

Vehicle speed.

Figure 9.

Torque of motor.

Figure 10.

Yaw rate.

Figure 11.

Center of mass side declination.

Figure 12.

Lateral acceleration.

As shown in Figure 10, the simulation results show that the yaw rate deviation of the CarSim is 1.5°/s, which has good tracking effect. Figure 11 is a graph showing the change of the center of mass side angle with time, and the deviation of the center of mass side angle after control is about 0.5°, which is within a small error range. Figure 12 is the lateral acceleration curve with time. The control effect is consistent with the trend of CarSim reference model, with small deviation and good tracking effect. From the stable trend and expected changes of the curves in Figures 8 –12, the stability control effect of the whole vehicle can be obtained from the trends and changes of the curve, the amplitude and error of torque and speed are acceptable.

In conclusion, it is shown that the coordinated control strategy can make the two front wheels independent driving vehicles run smoothly under the low-speed angular step condition, and simulations under vehicle sharp turn is acceptable as well.

Simulation verification of serpentine condition

The given steering wheel angle is shown in Figure 13 for the simulation analysis of the serpentine condition. During 0–2 s, the steering wheel angle is 0°; at 2–6 s, the steering wheel angle is stepped from 0° to 100°; at 6–14 s, from 100° to −100°, at 14–18 s, from −100° to 0°; and after 18 s, the angle of 0° is maintained unchanged. Under this kind of steering wheel angle input, the simulation analysis was carried out. When the steering wheel angle is positive, the vehicle turns left, or the vehicle turns right. At this time, the road surface adhesion coefficient is selected to be 0.85, no slip occurs, and no slip control is performed.

Figure 13.

Steering angle.

The simulation results are shown in Figures 14 –18 for high-speed moving line above. Figure 14 is a comparison of the actual vehicle speed and the target vehicle speed. The target vehicle speed is 80 km/h, and the actual vehicle speed remains basically floating around it. Figure 15 is the output torque of the left and right drive wheel motors after control. The output torque changes correspondingly with the change of the steering wheel angle; the torque can reach up to 89.5 N m; the change of the torque corresponds to the change of the speed of the drive wheel. Figure 16 is a graph of yaw rate versus time. The simulation results are very close to the yaw rate of the CarSim with good tracking effect. Figure 17 is a plot of centroid side declination over time, with the controlled centroid side declination closer to 0. Figure 18 is a curve of lateral acceleration with time, which is consistent with the trend of CarSim model after control; it has a small deviation and a good tracking effect. From Figures 15 –18, the curves change steadily and smoothly, especially the yaw rate of the whole vehicle and the driving torque of the two wheels; the simulation of strategies proposed in this article could be proved that the improvement of vibration of the two front steering wheels.

Figure 14.

Vehicle speed.

Figure 15.

Torque of motor.

Figure 16.

Yaw rate.

Figure 17.

Sideslip angle.

Figure 18.

Lateral acceleration.

The simulation results of low-speed angular step and high-speed sinusoidal conditions show that the proposed control strategy can be applied to low-speed and high-speed driving conditions to ensure the smooth running of the vehicle under different conditions.

Simulation verification of driving anti-skid on bisectional road

In order to verify the anti-skid control function of the two-wheel independent drive electric vehicle, the bisectional road was simulated and tested. The road surface with different adhesion coefficient on both sides of the left wheel and the right wheel of the vehicle is bisectional road; the set longitudinal length of the road surface is 100 m, the transverse width is 10 m, the adhesion coefficient on the road surface where the left wheel is located is 0.85, and the adhesion coefficient on the road surface where the right wheel is located is 0.2. Set the vehicle to run in the middle of the road and accelerate the straight line at an initial speed of 10 km/h. Combined with the road surface selected in this article, “the maximum adhesion coefficient and the optimal slip ratio” of the road surface are found. When the slip ratio is between 0.05 and 0.15, the comprehensive stability is high. The optimal slip ratio of the right front wheel is 0.15. The road information is shown in Figures 19 and 20.

Figure 19.

Pavement adhesion coefficient.

Figure 20.

Three-dimensional pavement information.

The simulation results are shown in Figures 21 and 22 for driving anti-skid control simulation of bisectional road surface. Figure 21 is the slip rate after the anti-slip control strategy is executed, and the slip rate of the left front wheel remains approximately 0.04, closely to 0, because the left front wheel enters the high-adhesion road surface without slip. When slip occurs, right front wheel slip rate increases rapidly and starts the anti-slip control strategy. Under the action of the control strategy, the slip ratio of the right front wheel is kept at about 0.12, and the slip ratio is kept near the optimal slip ratio. Figure 22 is a graph of vehicle speed and wheel speed. After the control strategy is implemented, the vehicle speed and wheel speed are basically close to each other without slipping.From the simulation results, strategies proposed here can avoid the wheel slip and realize the smooth running of the vehicle.

Figure 21.

Drive pulley slip rate.

Figure 22.

Vehicle speed and wheel speed.

dSPACE experimental verification

The dSPACE hardware system used in this article is a Microsoft box II (DS1401/DS1511) real-time simulation system, which is small in size, easy to use in real-time vehicles, and has I/O interfaces, which can be used for signal conditioning and data recording. In the hardware-in-the-loop simulation, the established real-time simulation model is downloaded to the Microsoft box II controller, which calculates the vehicle state and transmits the real-time data to the computer. Meanwhile, driver signals, feedback signals, and the like are accessed through the I/O interface.⁷

As shown in Figures 23 and 24, the whole simulation experiment platform mainly includes the host computer, controller, and controlled object. The host computer software control desk collects data and detection signals in real time, and can monitor the parameters such as driver input and vehicle state quantity in real time. The host computer and the controller are realized through the network transmission control protocol/Internet protocol (TCP/IP).

Figure 23.

Structure diagram of experimental platform.

Figure 24.

dSPACE simulation experiment platform.

The experiment mainly includes linear starting acceleration driving test and constant speed turning test.

Linear start acceleration test steering wheel corner input is shown in Figure 25, and there is a jitter between 0 and 10, but the error is within the acceptable range. The torque output through the control system is shown in Figure 26. During 0–9 s, no acceleration, torque remains near 0. The acceleration torque starts to increase after 9 s, and there is a small difference between the left and right front wheel drive torque due to steering wheel jitter. With the release of the accelerator pedal, the drive torque is reduced to 0. The results show that the control strategy designed in this article can achieve linear smooth operation and real-time control.

Figure 25.

Steering angle.

Figure 26.

Drive torque.

The input angle of steering wheel angle sensor is shown in Figure 27 for the simulation of vehicle in sharp turn condition. First, it is a straight line, then there is a sharp turn and maintains a certain angle. There is some jitter in the data, and the error is within the acceptable range because there is jitter in the actual operation. Ensure that the throttle pedal opening remains the same, making the system turn at a constant speed. The torque curve output by the control system after control is shown in Figure 28. As the steering wheel rotation angle changes, the driving torque changes accordingly, and the torque of the left front wheel decreases with the increase of the rotation angle, whereas the right front wheel keeps the same effect as the off-line simulation.

Figure 27.

Steering angle.

Figure 28.

Drive torque.

Operate the steering wheel angle sensor to control the change of the angle input. At 0–5 s, the steering wheel angle remains unchanged, and then operate it to make a reciprocating change; after the angle is 0, it remains constant, as shown in Figure 29. The drive wheel torque output curve is shown in Figure 30. As the steering wheel angle changes, the drive torque changes accordingly, keeping the same effect as the off-line simulation.

Figure 29.

Steering angle.

Figure 30.

Drive torque.

Through the analysis of the above experimental results, strategies proposed in this article can reasonably distribute the driving torque to driving wheels. Meanwhile, results of hardware-in-the-loop experimental simulation are consistent with the simulation results under co-simulation of Simulink and CarSim.

Conclusion

Based on the existing two-wheel independent driving electric vehicle experimental platform, a set of control algorithms based on fuzzy control theory were designed to coordinate and control the two independent hub motors to realize the whole vehicle driving anti-skid and driving wheel torque coordination. Meanwhile, from curves in the result of simulation, the vibration of steering wheels caused by the uncoordinated control had been greatly improved. Under the verification of Simulink-CarSim joint simulation and dSPACE hardware in-loop simulation, the effectiveness of the two-wheel independent drive electric vehicle control algorithms designed in this article were proved, the control of the two front wheels which would provide a theoretical reference for exploring the technology of multi-wheel independent drive electric vehicle.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Natural Science Foundation of Shaanxi Province of China (2016GY-007), the Service Local Special Program of Shaanxi Province Education Department (15JF023), and Key Research and Development Project of Shaanxi (Grant No. #2016KTZDGY4-05). Meanwhile, it was financially supported by Xi’an University of Science and Technology.

References

Jin

Gao

Jiang

et al . Research on control and coordination method of four-wheel independent driving/steering electric vehicle. Adv Mech Eng 2017; 9: 698877.

. Stability control of four—wheel independently driven electric vehicle with driver in ring [D]. Hefei University of Technology, 2017.

Ren

Torque coordination optimization control of four—wheel—drive electric vehicles [D]. Jilin University, Changchun, China, 2017.

Zhang

et al . Fuzzy control algorithm for independently driven electric vehicle yaw moment. J Xi’An Jiaotong Univ 2014; 48(7): 83–89.

Zhang

Xiang

Research on torque distribution strategy of four—wheel independently driven electric vehicle based on stability. Highw Traffic Sci Technol 2016; 33(5): 141–146.

Sun

Duan

Simulation model of front wheel independent drive electric vehicle based CarSim. Automot Eng 2015; 8: 27–31.

Zheng

Zhang

SS.

Experimental platform of brushless motor drive control system for four-wheel electric vehicle based on dSPACE. Equip Manuf Tech 2013 (2): 42–44.

Hind

Yaw moment Lyapunov based control for in-wheel-motor-drive electric vehicle. IFAC PapersOnLine 2017; 50: 13828–13833.

Wang

Xue

et al . Design and control strategy optimization for four-wheel independently actuated electric vehicles. Enrgy Proced 2017; 105: 2323–2328.

10.

Alipour

Sabahi

Sharifian

MBB

. Lateral stabilization of a four wheel independent drive electric vehicle on slippery roads. Mechatronics 2015; 30: 275–285.

11.

Chen

et al . Estimation of longitudinal force, lateral vehicle speed and yaw rate for four-wheel independent driven electric vehicles. Mech Syst Signal Pr 2018; 101: 377–388.

12.

Shuai

Zhang

Wang

et al . Lateral motion control for four-wheel-independent-drive electric vehicles using optimal torque allocation and dynamic message priority scheduling. Control Eng Pract 2014; 24(1): 55–66.

13.

Wang

Zhou

et al . Research on straight line stability control strategy of four wheel drive vehicle based on the sliding mode variable structure control and optimization algorithm. Enrgy Proced 2016; 104: 342–347.

14.

Wang

ZF.

Direct yaw-moment control based on fuzzy logic of four wheel drive vehicle under the cross wind. Enrgy Proced 2017; 105: 2310–2316.

15.

Zhao

Zhang

Zhao

Stability control system for four-in-wheel-motor drive electric vehicle. In: Proceedings of the sixth international conference on fuzzy systems and knowledge discovery, Tianjin, China, 14–16 August 2009, pp. 171–175. New York: IEEE.

16.

Cao

Zhai

Sun

et al . Straight running stability control based on optimal torque distribution for a four in-wheel motor drive electric vehicle. Enrgy Proced 2017; 105: 2825–2830.