Acceleration slip regulation control for four-wheel independently drive electric vehicle based on fuzzy control

Abstract

To improve the acceleration performance and stability of the four-wheel independent drive (4WID) electric vehicle on low-adhesion road, a fuzzy control that doesn’t depend on accurate vehicle models is proposed. Taking the driving torque of one side wheel as a reference the slip rate is controlled by controlling the torque errors between the left and right wheels to a certain range. Carsim-Simulink co-simulation is used to analyze the acceleration stability of 4WID electric vehicle on low-adhesion road and $\mu$ -split road. The simulation results show that the wheel slip rate can be controlled within a reasonable range through proposed method, and the stability and safety of the vehicle can be effectively improved on the basis of ensuring the power performance of the vehicle.

Keywords

4WID electric vehicle acceleration slip regulation fuzzy control Carsim-Simulink co-simulation

1. Introduction

Four-wheel independently drive electric vehicle is usually driven by in-wheel motors. It not only has the environmental and economic characteristics of pure electric vehicles, but also has the advantages of large chassis space and sufficient power. Nowadays, many scholars paid the attention to 4WID electric vehicles [1, 2, 3, 4]. In the driving system control of 4WID-EV, it is necessary to consider the control of motor and the integrated control of the entire vehicle. In the control of motor, many scholars have improved the response speed and the accuracy of motor control through the advanced control algorithms [5, 6, 7, 8, 9]. In the integrated control of the entire vehicle, the control of slip rate is closely related to the handling and safety, especially on low-adhesion road with ice and snow [10, 11]. Slip rate control of wheels plays an important role in improving the safety and stability of vehicle.

At present, a lot of researches have been done on the slip rate control of 4WID electric vehicles in the world. A novel model predictive controller-based multi-model control system (MPC-MMCS) was proposed to solve the longitudinal stability problem of 4WID-EV [12]. A new longitudinal control strategy which combines acceleration slip regulation and antilock braking system was proposed using an observation algorithm of effective radius in order to prevent wheels slipping in acceleration [13]. Aiming at 4WID electric vehicle are driven by the front and rear axles simultaneously an acceleration slip regulation system was proposed, in which there are including three control modes: average distribution of inter-axle torque, optimal distribution of inter-axle torque and independent control of optimal slip rate, respectively [14]. Under unknown and complicated road conditions it is very important to ensure the stability and safety of vehicle. Joa et al. proposed a control method without any tire-road friction information [15]. Liu et al. proposed a forgetting factors recursive least square algorithm to get an optimal slip rate according to adhesion coefficient under current road conditions [16]. Aiming at the different wheel speed results in high complexity of control method a slip ratio observer was built based on acceleration of four wheels to control the output torque of drive motors and keep slip rate near the optimal slip rate [17]. These control methods have achieved good results in slip rate control and improved the stability and safety for 4WID-EV. However, these control methods need a vehicle model which has a proper accuracy. The effectiveness of control is decided by the accuracy of the vehicle model. Therefore, in order to improve accuracy of controlling the complex vehicle models were used lead to these control methods are difficult to apply in practice.

To simplify vehicle model and improve acceleration performance on low-adhesion road an improved fuzzy control that doesn’t depend on accuracy vehicle model is proposed in this paper. Taking the driving torque of one side wheel as reference the slip rate is controlled by controlling the torque errors between the left and right wheels to a certain range. The structure of this study is as follows: 4WID electric vehicle model with drive model of motor is established in Section 2. In Section 3, Observer of slip rate, control strategy, and fuzzy controller are designed. The co-simulations with acceleration simulation of low-adhesion road and $\mu$ -split road are carried out in Section 4. Finally, there are the conclusions in Section 5.

2. Vehicle modeling

2.1 Vehicle model

For state estimation and control, the vehicle bicycle model is usually employed because of its simplicity. In a 4WID-EV the differential torque generated by the left and right wheels should also be included in the model. The vehicle model is shown in Fig. 1. The governing equations are shown as [18]:

$\displaystyle m\cdot V_{x}\cdot\left({\dot{\beta}+\gamma}\right)=2\cdot C_{f}% \cdot\left({\delta_{f}-\frac{l_{f}}{V_{x}}\gamma-\beta}\right)+2\cdot C_{r}% \cdot\left({\frac{l_{r}}{V_{x}}\gamma-\beta}\right).$ (1) $\displaystyle I\cdot\dot{\gamma}=2\cdot l_{f}\cdot C_{f}\cdot\left({\delta_{f}% -\frac{l_{f}}{V_{x}}\gamma-\beta}\right)-2\cdot l_{r}\cdot C_{r}\cdot\left({% \frac{l_{r}}{V_{x}}\gamma-\beta}\right)+N_{z}.$ (2)

Figure 1.

Vehicle model.

As in-wheel-motors can generate positive and negative torques easily, a simple torque distribution law can be defined as:

$\displaystyle T_{fl}=F_{fl}\cdot r=\frac{m\cdot r\cdot\dot{V}_{x}}{4}+\frac{r% \cdot N_{z}}{2d_{r}}\cdot\cos\delta_{f}.$ (3) $\displaystyle T_{fr}=F_{fr}\cdot r=\frac{m\cdot r\cdot\dot{V}_{x}}{4}-\frac{r% \cdot N_{z}}{2d_{r}}\cdot\cos\delta_{f}.$ (4) $\displaystyle T_{rl}=F_{rl}\cdot r=\frac{m\cdot r\cdot\dot{V}_{x}}{4}+\frac{r% \cdot N_{z}}{2d_{r}}.$ (5) $\displaystyle T_{rr}=F_{rr}\cdot r=\frac{m\cdot r\cdot\dot{V}_{x}}{4}-\frac{r% \cdot N_{z}}{2d_{r}}.$ (6)

2.2 Tyre model

The Magic Formula tyre model was proposed by Professor Pacejka [19]. The longitudinal force and lateral force simultaneously were expressed in a trigonometric function. According to different input parameters and fitting parameters, the tyre longitudinal force or lateral force can be calculated. The Magic Formula tyre model has a high fitting accuracy regardless of lateral force or longitudinal force. The equations of Magic Formula type model are shown as follow:

$\displaystyle F=(D\cdot\sin(C\cdot\arctan(B\cdot X-E\cdot(B\cdot X-\arctan(B% \cdot X)))))+S_{v}.$ (7) $\displaystyle X=S{+}S_{h}.$ (8)

$\displaystyle X=\beta+S_{h}.$ (9)

Figure 2.

Longitudinal and lateral forces of the tyre: a. Tyre longitudinal force, b. Tyre lateral force.

where $F$ is the longitudinal force or lateral force of the wheel; $S$ is the slip rate of the wheel; $C$ , $D$ , $B$ , $E$ , $S_{v}$ , $S_{h}$ are the corresponding fitting parameters, $X$ is the calculation variable.

Equation (8) is used to calculate longitudinal force and Eq. (9) is used to calculate lateral force. The relationship between the tyre longitudinal force and slip rate is shown in Fig. 2a. The relationship between lateral force and side slip angle is shown in Fig. 2b. When the slip rate or side slip angle is smaller, the relationship between longitudinal force and slip rate or between lateral force and side slip angle is approximate proportional.

The single tyre force is shown as Fig. 3. The tyre force is analyzed as follows:

$\displaystyle\sum{F_{i}}=m\cdot\dot{V}_{x}.$ (10) $\displaystyle J\cdot\dot{w}=T_{d}-F_{x}\cdot r-T_{b}-M_{f}.$ (11) $\displaystyle M_{f}=F_{z}\cdot f\cdot r.$ (12)

Figure 3.

Single tyre force.

2.3 Driving system

4WID-EV is usually driven directly by in-wheel motors without any transmission mechanism. Therefore, the permanent magnet synchronous motor (PMSM) with simple structure, small size and high efficiency is chosen as driving motor of 4WID-EV. According to the fast response characteristics of synchronous motor the transfer function $G_{i}(s)$ between target value $T_{Li}$ and actual value $T_{i}$ of motor output torque can be shown as [20]:

$\displaystyle G_{i}(s)=\frac{T_{i}}{T_{Li}}=\frac{1}{1+2\xi\cdot s+2\xi^{2}% \cdot s^{2}}.$ (13)

where $\xi$ is the damping ratio related to motor parameters.

3. Controller design

3.1 Observer of slip rate

In the course of vehicle running, the wheel slipping will not only lead to worsening tyre wear, but also affect the stability of the vehicle. If the slip rate exceeds a certain range the wheel slipping will cause serious traffic accidents. The slip rate of each wheel $S_{i}$ is defined as:

$\displaystyle S_{i}=\frac{w_{i}\cdot r-V_{x}}{w_{i}\cdot r}.$ (14)

where $w_{i}$ is the rotating speed of each wheel.

The slip rate is obtained by setting observer considering the measurement problem of the vehicle’s longitudinal velocity at CoG. The derivative equation of Eq. (14) is shown as follow:

$\displaystyle\dot{S}_{i}=\frac{V_{x}\cdot\dot{w}_{i}-w_{i}\cdot\dot{V}_{x}}{w_% {i}^{2}\cdot r}.$ (15)

Considering the resistance force is smaller and ignoring the braking force in the driving process of the vehicle, Eq. (11) is simplified as follow:

$\displaystyle J\cdot\dot{w}_{i}=T_{i}-F_{i}\cdot r.$ (16)

The force of each wheel is assumed to be equal in initial state. Equations (10) and (16) are used instead of the relevant variables of Eq. (15). The observer equation of slip rate is as follow:

$\displaystyle\dot{S}_{i}=\frac{m\cdot r^{2}\cdot\left({1-S_{i}}\right)\cdot% \dot{w}_{i}-4\left({T_{i}-J\cdot\dot{w}_{i}}\right)}{m\cdot r^{2}\cdot w_{i}}.$ (17)

According to Eq. (17), the observer of slip rate was built in Simulink software as shown in Fig. 4.

Figure 4.

Slip rate observer.

Figure 5.

Relationship between adhesion coefficients and slip rate.

The relationship between adhesion coefficient and slip rate is shown in Fig. 5. When the slip rate is between 0.1 and 0.25, the longitudinal and lateral adhesion coefficients of vehicles are higher, which can provide greater longitudinal and lateral adhesion [21]. Therefore, in order to ensure the stable running of vehicles the slip rate must be controlled within an appropriate range.

Figure 6.

Control strategy.

3.2 Control strategy

According to the definition of slip rate its theoretical value is between 0 and 1. In this paper a fuzzy control method is designed to control the slip rate. The control strategy is shown in Fig. 6. In Fig. 6 $V_{d}$ is the set speed, $T_{i}^{\prime}$ is the torque regulation of each wheel, $T_{ei}$ is the motor torque required by acceleration pedal, $S_{\min}$ and $S_{\max}$ are the minimum threshold and maximum threshold of slip rate. When the vehicle speed $V_{x}$ is less than or equal to the set speed $V_{d}$ , which is usually set to about 5 km/h, the vehicle is relatively stable, and the control of slip rate is not carried out. Therefore, the torque regulation of each wheel $T_{i}^{\prime}$ is zero. When the vehicle speed $V_{x}$ is more than the set speed $V_{d}$ , there are two cases. Case 1: when the slip rate $S_{i}$ is between $S_{\min}$ and $S_{\max}$ , the slip rate meets the requirement and the control of slip rate is not carried out. Case 2: when the slip rate $S_{i}$ is less than $S_{\min}$ or more than $S_{\max}$ , the fuzzy control method is used to adjust the motor torques. The slip rate is controlled by increasing or decreasing the motor torques, so that the slip rate can be kept within the set range. Considering the deviation of the vehicle running in straight line, it is necessary to limit the wheel torque error on both sides to a certain range. If the error is larger than a certain value, the smaller value of motor torques is taken as a reference, and the motor torque on the other side is limited to the set range to ensure the wheel torque errors on both sides are acceptable and the lateral stability of the vehicle.

3.3 Fuzzy controller

Fuzzy control is a computer digital control technology based on fuzzy set theory, fuzzy linguistic variables and fuzzy logic reasoning. It belongs to an intelligent control algorithm and has strong robustness and is suitable for linear or non-linear control [22]. The fuzzy control schematic diagram is shown in Fig. 7.

Figure 7.

Fuzzy control schematic diagram.

Figure 8.

Membership functions: a. Membership function of $e$ , b. Membership function of $e_{c}$ , c. Membership function of $u$ .

According to the requirement of slip rate control, the fuzzy controller is designed as Mamdani controller with double input and single output. The input is the error $e$ between slip rate and set slip rate and the error variation ec. The output is the torque regulation $u$ . The fuzzy fields of error $e$ is represented by five fuzzy subsets: NS (negative small), ZE (zero), PS (positive small), PM (positive middle), PB (positive big). The actual range of slip rate is [0, 1]. The slip rate is set as 0.15 when it is controlled. The fuzzy field of $e$ and fuzzy quantitative factors $k_{e}$ are set as [ $-$ 0.15, 0.85] and 1, respectively. The fuzzy subsets of variation ec are NB (negative big), NS (negative small), ZE (zero), PS (positive small), PB (positive big). The fuzzy field of $e_{c}$ and fuzzy quantitative factors $k_{ec}$ are set as [ $-$ 1.5, 1.5] and 0.03, respectively. The fuzzy subsets of torque regulation $u$ are NB (negative big), NM (negative middle), NS (negative small), NO (negative zero), ZE (zero), PO (positive zero). The fuzzy field of $u$ and proportionality factor are set as [ $-$ 1, 0.1] and 350, respectively. When the input/output of fuzzy controller is over the boundary value, it is transformed to boundary value for calculation by limited module. The membership function is shown in Fig. 8.

Fuzzy control rules are the relationship between input and output described by fuzzy language, and they are the core part of the fuzzy controller. The design of control rules does not depend on accurate mathematical model, but it needs some experience. The controller is getting better and better by debugging the parameters of the controller over times.

Figure 9.

Overall frame diagram of simulation.

4. Simulation and analysis

4.1 Co-simulation model

Carsim is a software for vehicle dynamics simulation. It can simulate the handling of vehicle and set flexibly the road, experimental environment, and simulation conditions. However, Carsim currently does not support the pure electric vehicle modeling. Therefore, in order to model the 4WID electric vehicle it is necessary to set the power system as the four-wheel drive. The external motors were used as the power source to drive the vehicle by external power input model. After setting the parameters in Carsim, the vehicle model needs to be transferred to Simulink library, and the Carsim model will become a module in the Simulink library. By connecting the module with the motor model established in Simulink, the co-simulation model of the 4WID electric vehicle will be completed. In order to simplify the simulation, the response speed of the motor is not considered in the simulation process. The overall frame diagram of simulation is shown in Fig. 9, and the key parameters of the vehicle are as listed in Table 1. According to the vehicle parameters and vehicle performance indicators, the maximum driving torque and peak power of the single motor were matched as 350 N $\cdot$ m and 14 kW, respectively.

Table 1
Key simulation parameters

Variables	Notation	Value	Unit
Vehicle mass	$m$	1450	kg
Distance from CoG to front axle	$l_{f}$	1.2	m
Distance from CoG to rear axle	$l_{r}$	1.4	m
Tire effective rolling radius	$r$	0.31	m
Vehicle yaw inertia	$I$	1536	kg $\cdot$ m ${}^{2}$
Vehicle width	$d_{r}$	1.54	m

Figure 10.

The variation curve of wheel torque on low-adhesion road: a. Left front wheel, b. Right front wheel, c. Left rear wheel, d. Right rear wheel.

Figure 11.

The variation curve of wheel slip rate on low-adhesion road: a. Left front wheel, b. Right front wheel, c. Left rear wheel, d. Right rear wheel.

4.2 Acceleration simulation of low-adhesion road

Vehicles need sufficient adhesion coefficient when the vehicle starts to accelerate from static condition. On the one hand, low-adhesion road will affect the power performance of the vehicle. On the other hand, the slipping of wheel will affect the lateral stability of the vehicle. Therefore, it is very suitable to verify the control strategy through low-adhesion road.

The road friction coefficient is set as 0.2 and the vehicle accelerates linearly from zero initial speed with maximum drive torque in simulation. The simulation results are shown in Figs 10 and 11. In the figures, no means without fuzzy control, control means with fuzzy control. Figure 10 shows the variation curve of wheel torque and Fig. 11 shows the variation curve of wheel slip rate in the simulation process. From Fig. 10, it can be seen that the driving torques of the four wheels reach nearly the maximum output torque of the in-wheel motor without fuzzy control. From Fig. 11, the corresponding slip rate curve shows that the slip rate of each wheel reaches about 0.9. All of wheels have serious slip, the driving force is in vain, and the vehicle tends to be seriously unstable. In the case of fuzzy control, the driving torque of each wheel is rapidly controlled from the initial 350 N $\cdot$ m to about 200 N $\cdot$ m, and the motor works in the effective area. The slip rate of each wheel is also rapidly controlled from 0.9 to 0.2, which is conducive to the stable running of vehicles on the low-adhesion road according to Fig. 5.

These results show the proposed strategy can effectively improve the acceleration performance and stability of vehicle on low-adhesion road.

4.3 Acceleration simulation of $\mu$ -split road

The lateral instability is easy to occur on $\mu$ -split road due to the difference of adhesion coefficient between the left and right wheels, which lead to the $\mu$ -split road simulation is a very good condition to test the reliability of control strategy. In the simulation process, the right side is set as high-adhesion road, its adhesion coefficient is 0.8, and the left side is set as low-adhesion road, its adhesion coefficient is 0.2. The vehicle accelerates linearly from zero initial speed with maximum drive torque and the simulation results are shown in Figs 12 and 13.

From Fig. 12, it can be seen that the driving torque of each wheel reaches its maximum value without fuzzy control, and when there is fuzzy control the driving torque of each wheel is beginning to stabilize after a short period of instability, and the torque difference between the left and right wheels can be kept within a set range. At the same time, the driving torque of the left wheel with low adhesion coefficient is less than that of the right wheel with high adhesion coefficient, which is consistent with the road condition.

It can be seen from Fig. 13 that in the absence of fuzzy control, the slip rates of the left wheels on the low-adhesion road are greater than 0.9, and that of the right wheels on the high adhesion road are changing from less than 0.1 to about 0.9. The main reason is that the vehicle shifts to the left side with the low adhesion coefficient during acceleration, which results in a reduction in the adhesion coefficient of right wheels and increases the vehicle instability. Under the condition of fuzzy control, the driving torques of the wheel are adjusted rapidly according to the slip rate of wheel. So that the slip rates of left wheel are adjusted to about 0.2. At the same time, the slip rates of right wheel are close to 0 due to the limitation of the wheel torque difference. The adhesion coefficient on both sides ensures the stable running of the vehicle.

Figure 12.

The variation curve of wheel torque on $\mu$ -split road: a. Left front wheel, b. Right front wheel, c. Left rear wheel, d. Right rear wheel.

Figure 13.

The variation curve of wheel slip rate on $\mu$ -split road: a. Left front wheel, b. Right front wheel, c. Left rear wheel, d. Right rear wheel.

These results show the proposed strategy can effectively avoid the instability of vehicle due to different adhesion coefficient between both side wheels and guarantee the acceleration performance in various conditions roads.

5. Conclusion

In this paper, a fuzzy control strategy is proposed to improve the acceleration performance and stability of 4WID electric vehicle in various conditions roads. The designed controller guarantees that the torque error between the left and right wheels is limited to a certain range. Simulation results are provided to demonstrate the effectiveness in improving the acceleration performance and stability of 4WID electric vehicle in various conditions roads. In the future, the experiment platform will be built and the effectiveness of proposed control strategy will be further verified.

Footnotes

Acknowledgments

This work was support by the Natural Science Foundation of Fujian Province (Project No. 2016J01204), the Project of Fujian Collaboration Innovation Center for R&D of Coach and Special Vehicle (Project No. 2016BJC010), and the Scientific Research Foundation of Fujian University of Technology (Project No. GY-Z19010).

Glossary

References

Khan

M.A.

Aftab

M.F.

Ahmed

and Youn

, Robust differential steering control system for an independent four wheel drive electric vehicle, Int. J. Automot. Techn 20 (2019), 87–97.

Singh

A.K.

and Potluri

, Comments on model-independent adaptive fault-tolerant output tracking control of 4WS4WD road vehicles, IEEE T. Intell. Transp 16 (2015), 1588–1593.

Alipouri

and Alipour

, Attenuating noise effect on yaw rate control of independent drive electric vehicle using minimum variance controller, Nonlinear Dynam 87 (2017), 1637–1651.

and Li

, Fault-tolerant control of electric vehicles with in-wheel motors using actuator-grouping sliding mode controllers, Mech. Syst. Signal Pr 72–73 (2016), 462–485.

and Lin

, Adaptive fuzzy dynamic surface control for induction motors with iron losses in electric vehicle drive systems via backstepping, Inform. Sciences 376 (2017), 172–189.

Luo

Lin

Liu

Zhao

and Ma

, Finite-time dynamic surface control for induction motors with input saturation in electric vehicle drive systems, Neurocomputing 369 (2019), 166–175.

Niu

Lin

and Zhao

, Adaptive fuzzy output feedback and command filtering error compensation control for permanent magnet synchronous motors in electric vehicle drive systems, J. Franklin I 354 (2017), 6610–6629.

Shi

Chen

and Lin

, Approximation-based discrete-time adaptive position tracking control for interior permanent magnet synchronous motors, IEEE T. Cybernetics 45 (2015), 1363–1371.

Zhao

Lin

and Ma

, Barrier lyapunov function-based adaptive fuzzy control for induction motors with iron lossed and full state constraints, Neurocomputing 287 (2018), 208–220.

10.

Alipour

Sabahi

and Sharifian

M.B.B.

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

11.

Zhang

Liu

Zhou

and Zhao

, Adaptive sliding mode fault-tolerant coordination control for four-wheel independently driven electric vehicles, IEEE T. Ind. Electron 65 (2018), 9090–9100.

12.

Shi

Yuan

and Liu

, Model predictive controller-based multi-model control system for longitudinal stability of distributed drive electric vehicle, ISA T 72 (2018), 44–55.

13.

Hartani

Khalfaoui

Merah

and Aouadj

, A robust wheel slip control design with radius dynamics observer for EV, SAE Int. J. Veh. Dyn., Stab., and NVH 2 (2018), 135–146.

14.

Peng

Xiong

and Fan

, An acceleration slip regulation strategy for four-wheel drive electric vehicles based on sliding mode control, Energies 7 (2014), 3748–3763.

15.

Joa

Sohn

and Bae

, Four-wheel independent brake control to limit tire slip under unknown road conditions, Control Eng. Pract 76 (2018), 79–95.

16.

Liu

Jin

and Chen

, Vehicle traction control algorithm based on optimal slip ratio under complicated road conditions, Journal of Jilin University (Engineering and Technology Edition) 46 (2016), 1391–1398.

17.

Guo

and Su

, Acceleration slip regulation of distributed driving off-road vehicle, Mechanical Science and Technology for Aerospace Engineering 38 (2018), 608–612.

18.

Wang

Nguyen

Fujimoto

and Hori

, Multirate estimation and control of body slip angle for electric vehicles based on onboard vision system, IEEE T. Ind. Electron 61 (2014), 1133–1143.

19.

Pacejka

H.B.

and Bakker

, The magic formula tyre model, Vehicle Syst. Dyn 21 (1992), 1–18.

20.

Tahami

Kazemi

and Farhanghi

, A novel driver assist stability system for all wheel-drive electric vehicles, IEEE T. Veh. Technol 52 (2003), 683–692.

21.

Liu

Jin