A Study of Coordinated Vehicle Traction Control System Based on Optimal Slip Ratio Algorithm

Abstract

Under complicated situations, such as the low slippery road surface and split- μ road surface, traction control system is the key issue to improve the performance of vehicle acceleration and stability. In this paper, a novel control strategy with engine controller and active pressure controller is presented. First and foremost, an ideal vehicle model is proposed for simulation; then a method for the calculation of optimal slip ratio is also brought. Finally, the scheme of control method with engine controller and active brake controller is presented. From the results of simulation and road tests, it can be concluded that the acceleration performance and stability of a vehicle equipped with traction control system (TCS) can be improved.

[ProQuest: [...] denotes non US-ASCII text; see PDF]

Gang Liu 1,2 and LiQiang Jin 1

Academic Editor:Luis J. Yebra

1, State Key Laboratory of Automotive Simulation and Control, Jilin University, Changchun, Jilin 130025, China

2, Department of Automatic Control, Henan Institute of Technology, Xinxiang, Henan 453000, China

Received 5 March 2016; Revised 31 May 2016; Accepted 21 June 2016

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Vehicle traction control system plays an important role in electronic stability control (ESC) system. In order to improve the vehicle traction performance on low μ surface road and maximize the longitudinal friction coefficient, TCS is used to control the slip ratio within the range of optimal slip ratio by regulating engine output torque or braking pressure.

In recent years, many theoretical studies based on wheel slip rate are conducted all around the world. In literatures [1, 2], the logic threshold controller is realized by regulating the braking pressure. It is easy to implement this method, but the development cycles for different vehicles are long. Reference [3] introduces the throttle actuator system with the time delay scheme, and the vehicle test result shows the good performance for the TCS. In [4], the proportion integration differentiation (PID) controller is implemented to adjust slip ratio. Due to high nonlinearity of the vehicle dynamic system, these conventional linear controllers cannot meet the requirements of TCS under complicated road conditions. In [5], a fuzzy controller is proposed to control the driving wheel slip for robust traction control. The simulation indicates that the controllers can further maximize the traction force. Reference [6] introduces a coordinated cascade control strategy which includes two sliding mode controllers. Both the engine torque and the active brake pressure are tuned by sliding mode controllers. Reference [7] proposes a sliding mode observer for road adhesion coefficient estimation, and then the fuzzy controller is used to regulate the engine output torque by adjusting the fuel supply. The control strategy is implemented on Volvo 76 Turb. Reference [8] presents a model predictive control strategy for TCS. Reference [9] introduces a second-order sliding mode of TCS controller, and the TCS is coupled with a nonlinear observer to estimate the road condition.

From the researches mentioned above, we learn that there are two requirements for TCS controller algorithm. One is robustness. Wheel slip should be maintained at around target slip ratio under any complicated road surface conditions. The other one is that the TCS controller strategy should meet different demands in different road surface conditions. When the vehicle is accelerating on the low μ road surface, the stability should be considered as a key point. When it is driving on μ split (one driven wheel on the ice), the traction issue should be considered first. All of these references mentioned above are on single controller to regulate the slip ratio. However, the single controller cannot maintain the vehicle stability under complicated road conditions. It is necessary to combine the active brake pressure controller with the engine torque controller. This paper presents a traction control system based on the engine output torque adjustment and brake pressure control. The optimal slip ratio is estimated via Varied Forgetting Factor Recursive Least Squares (VFFRLS). And the optimal slip ratio is considered as the desired value. The engine output torque is adjusted by fuzzy PID controller. The active brake pressure controller is based on sliding mode control theory. The control strategy is to coordinate these two controllers according to road surface conditions.

Section 2 introduces the vehicle dynamic model. Section 3 introduces the estimation algorithm of desired slip ratio. Section 4 explains the coordination strategy between the output torque and the brake pressure. Section 5 presents the results of simulation and the winter road experiments. Finally, conclusions are introduced in Section 6.

2. Vehicle Model

2.1. Vehicle Model

As shown in Figure 1, a nonlinear 7-DOF (degree of freedom) vehicle model is used to predict vehicle response under different road conditions during various driving maneuvers [10, 11]. Some factors, such as the air drag resistance and the resistance of slope, are not considered. The vehicle model includes longitudinal motion, lateral motion, and yaw motion. The vehicle dynamic equations can be described as follows: [figure omitted; refer to PDF] where _{v x} is the longitudinal velocity; _{v y} is the lateral velocity; r is the yaw angle; δ is the steer angle; m represents the vehicle mass; _{I v} is the yaw inertia moment; _{F x} denotes the longitudinal forces of wheels; _{F y} denotes the lateral forces of the wheels; m is the vehicle mass; a is the distance between front axle and center gravity; b is the distance between rear axle and center gravity; W is width of vehicle.

Figure 1: 7-DOF vehicle model.

[figure omitted; refer to PDF]

Wheel rotational equations can be expressed as [figure omitted; refer to PDF] where _{I w} denotes wheels inertia moment; _{T q} is the longitudinal driving torque; _{T b} is the breaking torque; _{r w} is the wheel radius; _{F d} is driving force between road and tire; w is wheel speed.

The equations for each wheel can be described as [figure omitted; refer to PDF] where _{F z} denotes the vertical force of each wheel; _{h g} denotes the distance between the gravity center and the ground; L is the distance between the front axle and the rear axle.

Other parameters of the vehicle model used for simulation are listed in Table 1.

Table 1: Values of the vehicle parameters.

Parameter	Value
Vehicle mass	1545 m/kg
Gravity	9.81 m/s2
a	1.14 m
b	1.63 m
Wheel radius	0.32 m
Wheel width	1.56 m
Height of center gravity	0.52 m
Inertia moment	2300 kg [low *] m2
Wheel inertia moment	1 kg [low *] m2

2.2. Tire Model

Pacejka's tire model is used to simulate the contact forces between the road and the tire [12]. The equation of the tire longitudinal forces _{F x} and _{F y} can be defined as follows: [figure omitted; refer to PDF] where B is the tire stiffness factor, C represents the shape factor, D represents the peak factor, E represents the curvature factor, and λ is the slip ratio, which can be expressed as follows: [figure omitted; refer to PDF]

2.3. Engine Model

In this paper, the dynamics model of the engine can be depicted as [figure omitted; refer to PDF] where _{T e} is engine output torque; _{T s} denotes the static output torque of engine which is according to engine map chart; d α / d x denotes the function of the differential of throttle angle α ; _{ω e} denotes the engine rotation speed; _{J e} denotes crank's inertia moment. The engine output torque depended on the throttle angle and the engine speed of rotation.

3. Parameter Estimation

The precondition of slip ratio control is to estimate the optimal slip ratio exactly. The method of estimating the optimal slip ratio is based on seeking extreme value of μ - λ curve ( μ - λ curve denotes the relation between the road friction coefficient and the slip ratio). The relationship between μ and λ can be expressed by Kiencke model [13]. The equation for Kiencke model is as follows: [figure omitted; refer to PDF] where _{p 1} and _{p 2} denote the parameters for different road conditions. Table 2 shows the values of _{p 1} and _{p 2} in different road surface conditions. Optimal slip ratio is available by seeking the extreme value of Kiencke model (8). Optimal slip ratio _{λ p} is shown as follows: [figure omitted; refer to PDF] It is obvious that optimal slip ratio is available by determination of _{p 1} and _{p 2} on different road surfaces. Considering that the coefficients of friction μ ( λ ) and slip λ are available instantly, _{p 1} and _{p 2} mentioned in (8) can thus be formulated by the Variable Forgetting Factors Recursive Least Square (VFFRLS) algorithm [14, 15]: [figure omitted; refer to PDF] where Θ ( k ) , ^{Φ T} ( k ) , and y ( k ) are given as [figure omitted; refer to PDF] The recursive process of VFFRLS algorithm is shown as follows: [figure omitted; refer to PDF] where I is the identity; p ( k ) denotes the covariance matrix; G ( k ) denotes the gain matrix; Λ denotes the variable forgetting factor, which is in the range ( 0.9 , 1 ) . Seen from (11), the variable forgetting factor Λ is directly related to the dynamic tracking ability of Variable Forgetting Factors Recursive Least Square (VFFRLS) algorithm [16]. A low value of Λ means the low weight of the past data. Owing to the different road conditions (i.e., varying- μ road), the value of Λ is to decrease during road transient conditions and then restore back to its original value under steady state. The value of Λ varies as follows: [figure omitted; refer to PDF] where _{Λ 0} denotes the value of Λ under steady state; τ denotes a sensitive coefficient which regulates the adjustment speed of forgetting factor; [straight epsilon] denotes the differential equation of longitudinal acceleration.

Table 2: The values of Kiencke mode on different road conditions.

Road condition	p 1	p 2
Dry asphalt	10.5104	34.5987
Wet asphalt	18.3410	58.4155
Dry concrete	11.2732	39.0633
Dry cobblestone	14.5401	6.2497
Wet cobblestone	58.2343	51.0124
Snow	118.3411	277.8144
Ice	536.0750	1010.8

Figure 2 is the simulation result of road friction coefficient estimation. The proposed method can estimate the coefficient quickly, and the error between the true value and the estimation value is less than 0.1.

Figure 2: Simulation results of road friction coefficient.

[figure omitted; refer to PDF]

4. Controller Design

4.1. Control Scheme

The overall structure for TCS can be depicted as in Figure 3. When the driving vehicle accelerates on a complicated road surface, if the driving force of driving wheels exceeds the friction force, the wheel begins to skid. The TCS begins to estimate the optimal slip ratio. The value of optimal slip ratio is considered as the desired value of TCS. The engine torque controller and the brake torque controller are activated to adjust the output torque and the brake pressure.

Figure 3: The structure of TCS.

[figure omitted; refer to PDF]

Generally speaking, vehicles may get into acceleration trouble when driving on two typical road conditions. One is the low μ slippery road, and the other one is the split μ road surface. This paper proposes a control scheme to solve the traction problem by maintaining the slip ratio of wheels.

Under the low μ slippery road condition, engine torque controller is used to regulate the slip ratio of driving wheels. Under the split-μ road condition, the engine torque controller and the active brake controller are coordinately used to avoid the phenomenon of spin in two steps. In the first step, the active brake controller is used to regulate the slip ratio of driving wheels on the low μ road surface. The desired pressure can be calculated by the fuzzy controller. In the second step, the engine output torque is chosen to increase the output torque and the wheel angular acceleration. The engine output torque is calculated by the sliding mode controller.

4.2. Engine Output Torque Controller

As known to all, the throttle is used to adjust the output torque by regulating the amount of air. The adaptive sliding mode controller is chosen as the control algorithm of engine output torque. The error between the desired slip ratio and the current slip ratio can be defined as follows: [figure omitted; refer to PDF] where _{λ i} denote the current slip ratio; _{λ p} is the desired slip ratio described in Section 3.

The sliding surface is as follows: [figure omitted; refer to PDF] where c is a constant and meets the Hurwitz theorem.

And the differential version of (14) is as follows: [figure omitted; refer to PDF] Then choose the control law: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] [varphi] is the thickness about the boundary layer of the saturation function and the value of [varphi] is greater than zero; _{s i} = 0 can be achieved in finite time; both _{k 0} > 0 and _{[straight epsilon] 0} > 0 are the sliding coefficients, which can be tuned according to the effectiveness of control.

The differential version of (2) can be defined as [figure omitted; refer to PDF] Then based on (5), (18), and (16), the following formula can be deduced: [figure omitted; refer to PDF] Based on (18) and (19), the control torque can be deduced as [figure omitted; refer to PDF]

4.3. Active Brake Controller

As for the design of active brake controller, the fuzzy control algorithm is adopted to keep the slip ratio within the range of optimal slip ratio. There are two input signals and one output signal for fuzzy logic controller. One input signal is e which denotes the error between the current slip ratio and the optimal slip ratio. The other one is ec which represents the first derivative of e . The output signal of fuzzy logic controller regulates the opening of the solenoid valve by the PWM of ECU. Then the target brake pressure is available.

e is divided into six fuzzy subsets: [NB, NM, Z, PS, PM, and PB]; ec is divided into seven fuzzy subsets: [NB, NM, NS, Z, PS, PM, and PB]; the output signal is divided into four fuzzy subsets: [NB, Z, PS, and PB].

As shown in Table 3, the rule base of active brake controller is based on the experience knowledge and expert knowledge. Then the gravity center method is used for defuzzification. If the output signal is zero, the brake pressure is in pressure-hold condition; if the output signal is NB, the brake pressure is in pressure-increasing condition; if the output signal is PB, the brake pressure is in pressure-decreasing condition.

Table 3: Table of fuzzy control rules.

e	[...] e c
NB	NM	NS	Z	PS	PM	PB
NB	PB	PB	PB	PS	PS	PS	PS
NM	PB	PB	PB	PS	PS	Z	Z
Z	PS	PS	PS	Z	NS	NS	NS
PS	PS	Z	Z	Z	NS	NS	NS
PM	NS	NB	NB	NB	NB	NB	NB
PB	NB	NB	NB	NB	NB	NB	NB

5. Experiment and Results

5.1. Simulation Result

For validation, some typical simulations were carried out by using Matlab/Simulink software. The basic simulation conditions are listed as follows:

(i) Target path: straight, no steering angle input,

(ii) Accelerator pedal opening: 100%.

Case 1 (low μ slippery road surface).

Figure 4 shows the relation between the opening degree of accelerator pedal and the vehicle speed. The opening degree of accelerator pedal is proportional to the vehicle speed. However, the vehicle speed ceases to increase when the opening degree reaches 90%.

Figure 5 shows the simulation result of the vehicle driving under a low μ slippery road surface condition. The friction coefficient of road surface is 0.2. From Figures 5(a) and 5(b), from the comparison between the noncontrolled one and the controlled one, we can find that the speed of uncontrolled vehicle is 50 km/h at 11 s, but that of the controlled one is 50 km/h at 7 s. We can also find that there is about 45% improvement in the performance of longitudinal acceleration. From Figure 5(c), the one under control keeps the slip ratio within the range of optimal slip ratio (about 0.16). However, the driving wheels of the uncontrolled one slipped excessively.

Figure 4: Control input.

[figure omitted; refer to PDF]

Figure 5: Simulation result of Case 1.

(a) Velocity versus time in Case 1 with no control

[figure omitted; refer to PDF]

(b) Velocity versus time in Case 1 under control

[figure omitted; refer to PDF]

(c) Comparison of slip ratio between the one under control and the noncontrolled one

[figure omitted; refer to PDF]

Case 2 ( μ -jump road surface).

Figure 6 shows the simulation result of the vehicle driving under a μ -jump road surface condition. The phenomenon of road transition from high μ ( μ = 0.8 ) to low μ ( μ = 0.2 ) happens at 4 s. From Figure 6(c), the slip ratio of uncontrolled one is soaring after 4 s. It is obvious that the controller could identify the change of road conditions and the slip ratio is controlled to the new desired value.

Figure 6: Simulation result of Case 2.

(a) Velocity versus time in Case 2 with no control

[figure omitted; refer to PDF]

(b) Velocity versus time in Case 2 under control

[figure omitted; refer to PDF]

(c) Comparison of slip ratio between the one under control and the noncontrolled one

[figure omitted; refer to PDF]

5.2. Road Test Result

As shown in Figure 7, the TCS controller and a suite of standard sensors are added to the vehicle. The controller is a 32-bit microcontroller unit which is produced by Freescale Semiconductor. The C code is obtained from the TCS control strategy of Simulink model; then C code is downloaded to microcontroller unit. And the microcontroller is mounted to the hydraulic unit. The testing system consists of Vector CANalyzer and a Speedbox. The CANalyzer captures sensory information and engine information from CAN bus. The Speedbox is used to measure the speed of the test vehicle.

Figure 7: The MCU controller.

[figure omitted; refer to PDF]

Case 1 (ice road test).

Figure 8 shows that the road test was carried out on an ice road. The friction coefficient of the ice road was approximately 0.2. Figure 9 shows the test result of noncontrolled vehicle. The velocity of the vehicle was only 17 km/h at the end of the road test. Furthermore, the slip ratio increased excessively.

Figure 10 shows the test result of the controlled one on the ice road surface. From Figure 10(a), the angular velocity of the driving wheel was reduced effectively by the engine output torque controller. The improvement of vehicle acceleration was 43%. The gear changed at 12 s. The optimal slip ratio was approximately 0.1. Figure 10(b) shows that the slip ratio was controlled within the range of 0.1. From Figure 10(c), when the vehicle began to accelerate at 2 s, the current engine torque decreased quickly to the desired one. Then, the engine output torque followed the tendency of vehicle speed change. To conclude, the engine torque controller played an important role in restraining the slip ratio of the driving wheels on the icy road surface.

Figure 8: Winter test.

[figure omitted; refer to PDF]

Figure 9: The winter test result with no TCS controller.

[figure omitted; refer to PDF]

Figure 10: The winter test results obtained on the ice road.

(a) Wheel speed versus vehicle speed

[figure omitted; refer to PDF]

(b) Slip ratio

[figure omitted; refer to PDF]

(c) Torque

[figure omitted; refer to PDF]

Case 2 (split- μ road test).

Figure 11 shows the vehicle test on the split- μ road surface. The right side was ice road ( μ = 0 . 2 ), and the left side was dry concrete pavement ( μ = 0.8 ). Figure 12 shows the test results of parameters. From Figure 12(a), the angular velocity of the wheel on the slippery side was close to vehicle speed.

When the vehicle began to accelerate at 3 s, the slip ratio of front right wheel increased excessively. Then the value was kept close to the desired slip ratio (approximately 0.1) as shown in Figure 12(b). From Figure 12(c), when the front right wheel began to skid at 3.2 s, the engine torque controller reduced the torque quickly. Then the slip ratio was controlled by the brake pressure properly. To sum up, in the spilt- μ road test, the proper switch between the engine controller and the active pressure controller can improve the performance of acceleration and stability.

Figure 11: The vehicle test under the split- μ road surface.

[figure omitted; refer to PDF]

Figure 12: The winter test results obtained on split- μ road surface.

(a) Wheel speed versus vehicle speed

[figure omitted; refer to PDF]

(b) Slip ratio

[figure omitted; refer to PDF]

(c) Torque

[figure omitted; refer to PDF]

6. Conclusions

To sum up, this paper presents a novel traction control system, which consists of an engine torque controller and a pressure controller, for the evaluation of vehicle acceleration and stability. The characteristics of the TCS can be described as follows:

(1) A method for real-time calculation of optimal slip ratio is realized by Variable Forgetting Factors Recursive Least Square (VFFRLS) algorithm. And then the optimal slip ratio is considered as the desired value of slip control.

(2) The cascade control method with fuzzy control algorithm and sliding mode control algorithm can be effectively adapted to the complicated road surface conditions.

(3) The algorithm takes 2 ms to run a time and runs once every 20 ms, so that the TCS controller can discover and correct the vehicle wheel slipping phenomenon in time.

(4) Simulation, which is based on Matlab software, and typical road tests were carried out. The results indicate that the control scheme is fit for complicated working circumstances.

Acknowledgments

This work was supported by China Postdoctoral Science Foundation (2013M540248).

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