2010 8th IEEE International Conference on Control and Automation Xiamen, China, June 9-11, 2010
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A New Fuzzy Logic Road Detector for Antilock Braking System Application M. Habibi, A. Yazdizadeh
Abstract— Antilock braking system (ABS) is capable of stopping a vehicle wheel without locking while decreasing the stopping distance. In this paper, fuzzy logic is applied in two different ways. In the first strategy, a hybrid controller is proposed which is a combination of a sliding mode controller (SMC) with a novel fuzzy controller to improve sliding mode controller efficiency. A Conventional SMC design is improved by using a saturation function in control input to reduce chattering and then it is combined with the new fuzzy controller to increase controller performance by eliminating the time responses oscillation. As a second application, a new fuzzy road detector is designed to detect three different road conditions. Simulations under these selected road conditions are performed to demonstrate the effectiveness of the proposed hybrid controller. Simulation results show good performance of the proposed controller in different road detected states.
I. INTRODUCTION
T
HE locking phenomenon during emergency braking on slippery surfaces causes long distance braking and also has bad effects on stability of vehicles. Antilock braking system (ABS) prevents the wheel slip problem, helps the driver to keep safely control of the vehicle, minimizes the stopping distance and eventually, enhances the ability of steering the vehicle. Since ABS introduction in the 1950s, various control methodologies have been developed. Achieving satisfactory performance is the main goal of all control methods for ABS which has been developed or is under research. Because of nonlinearity in the vehiclebraking dynamics, variation of model parameters over a wide range due to variation of road surface and vehicle conditions, operation of controller at unstable equilibrium point in an optimal performance and uncertainty of sensor signals, many difficulties arise in design of a controller. Therefore, robustness of the controller is an important issue which is to be addressed in solving these problems. Sliding mode controller is a good candidate that because of its effectiveness in a nonlinear system, has widely been investigated in recent decades [1]-[4]. Another suitable control strategy to tackle these problems is fuzzy control (FC). Despite the absence of
analytical modeling information, systems governed by fuzzy controllers are often highly robust [6] and because of their effectiveness at handling the uncertainties and nonlinearities associated with complex systems such as antilock braking systems, they are another suitable option to be chosen [5]-[7]. However, in these controllers the large number of fuzzy rules makes the analysis complex. The other approach that is proposed in the design of ABS controller is fuzzysliding-mode control (FSMC) design method which is known as a hybrid control strategy and is the combination of fuzzy control and sliding mode control method [8]-[11]. The main advantage of the FSMC is its requirement for fewer fuzzy rules than FC does and also FSMC system has more robustness against parameter variation [10]. This article has two aims. The principle aim of this research is to propose a hybrid controller, comprised of a sliding mode controller and a new fuzzy controller. The behavior of the antilock braking system with conventional sliding mode controller is not suitable, because when the vehicle tends to stop and speed is approaching zero, oscillation in output signals is observed. For overcoming this problem a fuzzy controller is designed so that the overall performance in emergency braking maneuver improved. Most of the researches consider the optimal slip ratio a constant value or defined it in a specified range but there is a strong dependency between road condition and optimal slip ratio, so the kind of road must be identified. To achieve it, in this study the second aims is devoted to design a new road condition detector by use of fuzzy logic method. In this method, the effects of road condition changes and the vehicle speed on the target slip ratio are considered. This paper is organized as follows: Section II describes the tire/road friction model used in the study, the following section presents mathematical vehicle model. In the fourth section, the road condition detection technique is discussed. In section V, the new Fuzzy-Sliding Mode Controller is presented. The simulation results are presented in section VI and conclusion constitute the last part of the paper. II. TIRE/ROAD FRICTION COEFFICIENT
M. Habibi is with the Department of Electrical Engineering, Power and Water University of Technology, TEHRAN, IRAN (e-mail:
[email protected]). A. Yazdizadeh is with the Department of Electrical Engineering, Power and Water University of Technology, TEHRAN, IRAN (corresponding author, fax: +982177312553; e-mail:
[email protected]).
978-1-4244-5196-8/10/$26.00 ©2010 IEEE
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Antilock braking system maximizes the tire/road friction force ( Fx ) which is proportional to the normal load of the vehicle ( Fz ). Road adhesion coefficient is the coefficient
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of this proportion and is denoted by ȝ. Road coefficient of adhesion is a nonlinear function of wheel slip ratio, which is a well known parameter to represent slippage. The tire slip ratio is defined as: V - RZ V
O
(1)
Where Ȝ is the slip ratio, V is the velocity of the vehicle, R is the wheel radius and Ȧ is the angular speed of the wheel. The objective of ABS control system is to increase tire-road friction force by keeping the operating point of the car near the peak value of the ȝ-Ȝ curve during the ABS maneuvers. In the nonlinear ȝ-Ȝ curve, there is only one zone where the maximum friction will be achieved so the desirable slip ratio is restricted in this zone, while from the definition (1), the slip ratio can change in the range of one to zero. As the slip ratio becomes to be one, the tire is completely locked and when the slip ratio is zero, the wheel is in rolling motion without slipping. In this paper the following formula, which is introduced by Burckhardt [12] has been used as the tire friction model:
P x (Ox , Vx )
(C1 (1 e
C2Ox
) C3O ) e
C4OxVx
(N.M), Fx is the road friction force, F f is the rolling resistance force of the tire and Fa is the aerodynamic drag force (N). To simplify the model, the relationship between caliper pressure P and the braking torque W b is assumed to be linear:
Wb
Kb P
(5)
The expressions of different forces are given as follows: Ff
f 0 3.24 f s ( k mphV )
2.5
(6)
2
Fa
0.5 U Cd A f V
Fx
P (O ) Fz
(8)
Fz
mtot . g ( mcar heg x) / l
(9)
(7)
(2)
Table. I shows friction model parameters for different road conditions. TABLE I FRICTION MODEL PARAMETERS FOR THREE ROAD CONDITIONS Road Condition C3 C1 C2 Dry asphalt Wet asphalt Snowy
1.2801 0.8750 1.1973
23.9900 33.8220 25.1680
Fig. 1. Diagram of a quarter vehicle model.
0.5200 0.3470 0.5373
All the variables and parameters used in (5)-(9) are described as follow:
C4 is in the range of 0.02-0.04.
mcar : the total mass of vehicle, l : wheel base, Fz : normal load of the tire, Pb : output hydraulic pressure
III. SYSTEM DYNAMICS
(kpa), Z : wheel angular speed (rad/s), V : vehicle linear
Fig. 1 shows the diagram for a quarter vehicle model. Based on this diagram a simplified longitudinal vehicle model considering the rotational dynamics of the wheel and the linear vehicle dynamic can be derived as follows:
speed (m/s), Cd : aerodynamic drag coefficient, U : air
J Z
(3)
between (m/s) and mile per hour (mph), ( k mph =2.237),
(4)
heg : center of gravity height, K b : braking constant.
mtot x
W b RFx RF f mtotV
Fx Fa F f
Where, J is the moment of inertia of wheel (kg.m2), mtot is the total mass of quarter vehicle ( mtire 0.25mcar ) ,
R is the wheel radius, W b is the brake torque on the wheel
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density, A f : frontal area
of the vehicle,
f 0 : basic
coefficient, f s : speed effect coefficient, k mph : scale factor
IV. ROAD CONDITION DETECTION Based on the equation (2), the adhesion coefficient depends on velocity and road conditions. Fig. 2 shows adhesion coefficient versus wheel slip curves for three
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different road surfaces and different vehicle speeds. This figure clearly shows that in different road conditions and in vehicle speeds, optimal speed ratio changes too (the maximum point moves downward when speed increases from 0 to 100 km/h). Thought most of researches considered optimal slip ratio as a constant or keep it only in a specific range, this study investigates the effects of vehicle speed changes and also different road conditions in the control strategy during stop maneuvering. The type of road is detected by use of fuzzy based method. According to this method, the value of speed V (t ) is obtained by integrating acceleration, which is assumed to be specified. After that the value of longitudinal slip ratio Ȝ and adhesion coefficient ȝ are obtained by applying (1), (2). Knowing these 2 values, a point of adhesion characteristic curve is determined. Finally, by use of a fuzzy system, we determine to which curve this point is belonged.
logic output according to Mamdani’s method. The output value relates to road type and can be considered as the road condition. The output value is between 0 and 1, meaning 0 a low adherence road and 1 as a high adherence road. Multiplying this value by the slip ratio, related to maximum adhesion coefficient in that road condition, the reference slip ratio is obtained. The fuzzy rules are established according to adhesion characteristic curve and performance point of vehicle on this curve. Table 2 shows the rules table for the fuzzy system and the membership functions values are given in Fig. 3. In this study the proposed fuzzy detector, detects 3 different road types: wet asphalt, dry asphalt and snowy asphalt and its performance is compared with reference slip ratio which is obtained directly from ȝ-Ȝ curves and stored in a look-up table.
Adhesion Coefficient
1.2 1 0.8 0.6 0.4 0.2 0 0
(a)
0.2
0.4
0.6
0.8
1
Slip
Fig. 2. ȝ-Ȝ curves for three different road surfaces and different vehicle speeds. The maximum point moves downward when speed increases from 0 to 100 km/h. TABLE II RULE TABLE OF FUZZY DETECTOR ȝ S M Ȝ Small Wet Wet Medium Snow Wet Large Snow Wet
L Wet Dry Dry
(b)
Fuzzy logic systems are rule-based systems in which a set of fuzzy rules represents a control decision mechanism to adjust the effects of certain causes coming from the system [8]. The performance of the proposed fuzzy detector would be the following: Adhesion coefficient and slip ratio enter to fuzzy system as crisp data. These data are changed to linguistic data via the fuzzy membership functions. Then the resulting outputs go through an interference engine, consists of a set of fuzzy “if-then” rules. The fuzzy output from interference engine is then changed back to a crisp value by the deffuzification process and is nominated as “road condition”. The center of area method is used to generate the fuzzy
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(c) Fig. 3. Membership functions of fuzzy road detector (a) First input: Adhesion coefficient, (b) Second input: Slip ratio, (c) Output: Maximum Slip ratio.
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V. CONTROLLER DESIGN
A. Sliding Mode Controller (SMC ) Sliding mode control is a kind of robust control that is used to control nonlinear plants. As it is known to us, ABS is a nonlinear plant, so SMC provides an effective method to control it. In this section the common sliding-surface is used. This sliding surface is defined as: s O Od . The control objective is to find a control brake torque such that the slip ratio tracks the desired slip ratio Od .
The switching control gain K can be selected by the following reaching condition where Ș is a strictly positive constant: ss d K s
(15)
Using (10) and substituting (11) into (15), results: Ka2 s t K s
(16)
The control brake torque ( W b ) in the SMC consists of two parts: a) The equivalent control torque ( W beq ) that forces the systems states to move along the desired sliding surface, b) The switching control torque ( W bsw ) that ensures the trajectory of the system approaches to the desired sliding surface. In this sliding-surface design, the equivalent control is determined from the condition of
s
0.
Differentiating the sliding-surface is expressed as follow: s O Od 0 (10) Rewriting (1) and the differentiating respect to time yields:
O
1
RZ
O
V
V
V N a1
O
R VJ N a2
Wb
R
2 ( Fx Ff )
a1O a2W b a3 Od
a3
(11)
0
(12)
Finally the equivalent control brake torque W beq , is obtained:
W b ,eq
( O 1) / R R ( F F ) (VJ O ) / R VJ x f d
(13)
The second part of the controlled brake torque is switching control torque which is defined as: W bsw K sgn( s ) where K is a switching control gain and
sgn(s) is a sign function. So the controlled brake torque becomes:
Wb
W beq W bsw
W beq K sgn( s )
(VJ K ) / R
(17)
Chattering phenomena is an undesirable effect of discontinuous control commands during sliding mode control. To reduce the chattering phenomena, different methods are introduced. In this paper, the sign function is replaced by saturation function in control input of sliding surface as follow:
Wb
W beq Ksat (s / I )
(18)
Where I is the boundary layer.
Substituting (11) in to (10) gives:
O Od
K t K / a2
V
VJ V
a1O a2W b a3
O
Solving for the switching control gain, the following inequality is obtained:
(14)
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B. Fuzzy Logic Control Applying the previous sliding mode controller to ABS model, results good responses but not satisfactory. By investigating output results of SMC design, the responses consist of input control (brake torque command) and slip ratio tracking error, e O Od are undesirable and full of oscillation. This effect happens while speed is approaching zero. On the other hand when the vehicle is stopping, high chattering is observed in control signals. Fuzzy control is proposed in order to overcome this problem in this paper. Fuzzy logic systems are rule-based systems in which a set of fuzzy rules represents a control decision mechanism to adjust the effects of certain causes coming from the system [8]. The performance of the proposed fuzzy controller is as follows: The input signal (vehicle speed) enters to fuzzy system as crisp data. This data is changed to linguistic data via the fuzzy membership functions. After that the resulting output goes through an interference engine, consists of a set of fuzzy “if-then” rules. The fuzzy output from interference engine is then changed back to a crisp value by the deffuzification process and is nominated as “switching gain”. The center of area method is used to generate the
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switching gain, K as the output, according to Mamdanis’ method. The membership functions are illustrated in Fig. 4. The fuzzy rules describe as follow: - If (Velocity is very small (ss)) or (Velocity is small (s)), then (gain is small (s)). - If (Velocity is medium (m)), then (gain is medium (m)). - If (Velocity is large (b)), then (gain is very large (bb)). When vehicle speed is decreasing, the fuzzy controller decreases the switching gain too. Therefore, stability condition in sliding mode controller is satisfied (Equation (17)). Results of simulation in next section will show the effectiveness of this proposed method to avoid chattering in low speed while stopping.
(a)
N.M. A time delay of 0.12ms for the hybrid system is selected. In first step the fuzzy road detector is simulated and its efficiency is compared to the reference wheel slip ratio in the look-up table which was described in section 3. During this step the conventional SMC with boundary layer is selected as the controller. The boundary-layer thickness I in the sliding mode controller with saturation function is set to 0.01 to prevent chattering problem. The design parameters like switching gain K and constant are as follow: K 1000, K =0.3 The road surface changes from wet asphalt to snowy after 1 second and then changes from snowy to dry asphalt after 0.5 second. Fig. 5a shows the vehicle and wheel speed during the braking maneuvering. The exact road condition (desired slip ratio) from look-up table can be compared with detected road condition by proposed fuzzy road detector in Fig 5b. The road detector, estimates road condition near desired values in snowy and wet asphalt and it’s near average of desired values in dry asphalt. In the next step of simulation, the proposed hybrid controller is investigated. The time response of the brake torque and reference (selected) slip ratio whit real slip ratio by use of conventional SMC are represented respectively in Figs. 6a, 6b. Also, these characteristics are shown in Figs. 7a, 7b for the proposed hybrid controller. It is shown that the oscillation in the time responses of the conventional sliding mode controller tend to increase when vehicle tends to stop but in the case of hybrid controller, the vibration vanishes so that vehicle has good stability and steerability. VII. CONCLUSION
(b) Fig. 4. Membership functions of fuzzy controller.
VI. SIMULATION To investigate the effect of the proposed controllers in previous section, the results of proposed hybrid controller and conventional sliding mode controllers are compared through MATLAB simulation software. The vehicle parameters used in simulation are selected as follow: heg J fs
0.5 m , l
2.5 m , U
2 13.75 kgm , k mph 0.005, A f
3 1.3kg / m , mtire 2.237, C d
2 2.04 m , g
40 kg , 0.25mcar
0.539, R
9.81, Vx 0
0.326 m , f0
375 kg , 0.01,
60 km / h
It is assumed that vehicle is moving in a straight-line maneuver at 60 km/h. All tests are run for a 1ms sampling period and the maximum braking torque is limited to 1500
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In this study, fuzzy logic concept is applied to the ABS through two different ways. In the first step, a new fuzzy road detector for Antilock braking system is proposed. In this fuzzy road detector, different road states can be detected through fuzzy rules. In the second step, a novel hybrid controller is proposed that consists of a sliding mode controller and a fuzzy controller. The performances of the proposed controller and the common sliding mode controller based on the detected road condition are compared through simulation. As an important conclusion, it is shown that the time response oscillations in ABS with hybrid controller is much less than ABS with the common sliding mode controller, so the vehicle has adequate lateral stability and good steerability in various road conditions and road transitions.
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15000
70 60
Tb (N.m)
R*W V
V,R*W (km/h)
50 40
10000 5000 0 0
30
0.5
1
10 0.5
1
1.5 Time (s)
2
Slip, Estimated Slip
0.2
0 0
2.5
(a) Estimated Slip Desired Slip
Desired Slip, Estimated Slip
0.16
2.5
Estimated Slip Slip
0.1 0.05
Dry
Wet Snow
0.5
1
1.5
2
2.5
(b) Fig. 7. Time response of (a) brake torque and (b) reference slip ratio with estimated slip ratio by use of the proposed hybrid controller.
0.1 0.08 0.06
REFERENCES
0.04
[1]
0
0.5
1
1.5 Time (s)
2
2.5
(b) Fig. 5. (a) Vehicle and wheel speed during the braking maneuvering in wet, snowy and dry condition, (b) desired and estimated slip ratio. 15000
Tb (N.M)
2
Time (s)
0.12
0
0.15
0 0
0.14
0.02
10000 5000 0 0
0.5
1
1.5
2
2.5
Time (s) (a) 0.2 Slip, Estimated Slip
1.5
Time (s) (a)
20
0.15
Estimated Slip Slip
Dry
Wet
0.1 0.05 Snow 0 0
0.5
1
1.5
Time (s)
2
2.5
(b) Fig. 6. Time response of a) brake torque and b) reference slip ratio with estimated slip ratio by use of conventional SMC.
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