行政院國家科學委員會專題研究計畫 期中進度報告
子計畫二：追蹤動態最佳滑差之防鎖死剎車系統之智慧型控
制(1/2)
計畫類別： 整合型計畫 計畫編號： NSC94-2213-E-030-011- 執行期間： 94 年 08 月 01 日至 95 年 07 月 31 日 執行單位： 輔仁大學電子工程學系(所) 計畫主持人： 王偉彥 計畫參與人員： 李宜勳, 高弘霖, 陳銘滄, 錢立軒, 謝濬鴻 報告類型： 精簡報告 報告附件： 出席國際會議研究心得報告及發表論文 處理方式： 本計畫可公開查詢中 華 民 國 95 年 6 月 1 日
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先進車輛控制及安全系統之設計與模擬-子計畫二：追蹤動態最佳滑差之防
鎖死剎車系統之智慧型控制(1/2)
計畫類別：□ 個別型計畫 □ 整合型計畫 計畫編號： NSC 94-2213-E-030-011 執行期間：94 年 8 月 1 日至 95 年 7 月 31 日 計畫主持人：王偉彥 共同主持人： 計畫參與人員：李宜勳, 高弘霖, 陳銘滄, 錢立軒, 謝濬鴻 成果報告類型(依經費核定清單規定繳交)：□精簡報告 □完整報告 本成果報告包括以下應繳交之附件： □赴國外出差或研習心得報告一份 □赴大陸地區出差或研習心得報告一份 □出席國際學術會議心得報告 □國際合作研究計畫國外研究報告書一份 處理方式：除產學合作研究計畫、提升產業技術及人才培育研究計畫、列管計畫及下列情形者外，得 立即公開查詢 □涉及專利或其他智慧財產權，□一年□二年後可公開查詢 執行單位：輔仁大學電子工程系 中 華 民 國 95 年 5 月 31 日行政院國家科學委員會專題研究計畫期中報告
先進車輛控制及安全系統之設計與模擬-子計畫二：追蹤動態最佳滑差之防
鎖死剎車系統之智慧型控制(1/2)
計畫編號：NSC 94-2213-E-030-011 執行期限：94 年 8 月 1 日至 95 年 7 月 31 日 主持人：王偉彥 輔仁大學電子工程系 參與人員：李宜勳, 高弘霖, 陳銘滄, 錢立軒, 謝濬鴻 AbstractThis paper proposes an antilock braking system (ABS), in which unknown road characteristics are resolved by a road estimator defined by LuGre friction model with a road condition parameter. For precisely defining the road condition parameter and further reducing the braking time, two strategies are used: the first is the
µ λ− curves are built from the real simulation data and second is the wheel velocity is introduced into the road estimator. The three-dimension mapping function are proposed to transfer the road condition parameter and wheel velocity to a slip ratio of the wheel. The slip ratio controller is used to maintain the slip ratio of the wheel at the reference values for various road surfaces. In the controller design, an observer-based direct adaptive fuzzy-neural controller (DAFC) for an ABS is developed to on-line tune the weighting factors of the controller under the assumption that only the wheel slip ratio is available. Finally, this paper gives simulation results of an ABS with the road estimator and the DAFC, and is shown to provide good effectiveness under varying road conditions.
I. Introduction
Braking under critical conditions, such as those encountered with wet or slippery road surfaces, panicked driver reactions, or mistakes committed by other drivers and pedestrians can lead to wheel lock. This
phenomenon is strongly undesirable, since the friction force on the locked wheel is usually considerably less as it slides on the road. Furthermore, while the wheels are locked, steering becomes impossible, leading to loss of control of the vehicle. Hence, preventing locking-up during the braking process is important for driver and passenger safety. For these reasons, anti-lock braking systems (ABS) [15-17] have become one of the most common automotive technologies. The wheel slip ratio, which is the difference between the vehicle speed and the wheel speed, is regarded as one of the most important process parameters affecting the quality of vehicle control. The main goal of most ABS system is to ensure that the wheel slip ratio remains within the range of about 0.1 to 0.3, which is suitable for most road conditions. Most traditional ABS control strategies [1, 2] aim to maintain the wheel slip ratio at 0.2, although this value is a compromise. Hence, one main problem of these methods is determining how to find the optimal slip ratio if the road characteristics are unknown.
In general, optimal slip ratios are highly dependent on the particular road characteristics, such as whether the road is dry or wet. A tire/road friction estimator using only angular wheel velocity has been proposed to monitor the road characteristics [4]. In [4], a LuGre model [9, 10], which effectively models tire/road friction based on the relative contact velocity, is used to form a tire/road friction estimator. Otherwise, for
further reducing the braking time, a wheel velocity is introduced into the road estimator. We build a family of the µ λ− curves for different ω ’s value, and then define the optimal slip ratio from this family curves. In this paper, the LuGre tire/road estimator is applied to estimate the road characteristics and then supply the reference slip ratio to an ABS controller through a mapping function. A feedforward neural networks [12] using back-propagation learning algorithm proposed to identify the mapping function from road characteristics and the wheel velocity to reference slip ratios. Hence, this ABS, based on the LuGre tire/road friction estimator, called LuGre-based ABS, can recognize the road characteristics and obtain a reference slip ratio.
Regarding controllers, researchers have greatly improved the performance of ABS by utilizing many algorithms, such as sliding mode control (SMC) [5], fuzzy logic control (FLC) [6], adaptive control, genetic neural control [7], etc. All these methods assume that the system states are available for measurement. In traditional anti-lock braking systems, certain states, such as the wheel acceleration and the brake-line pressure, are difficult to obtain, or are prone to noise and other measurement errors. To resolve this problem, a state observer is required. Thus, the observer-based direct adaptive fuzzy-neural controller (DAFC) [14] is applied to ABS control, and its stability can be guaranteed by the universal approximation theorem. The DAFC can overcome system uncertainties and disturbances. In this paper, we propose an antilock-braking system based on a road estimator, using the DAFC to force the wheel slip ratio to follow a reference slip ratio obtained by the road estimator.
II. Problem formulation
The wheel speed can be written as:
I R F P K I T Tb − t = b i − x = ω (1)
and the differential equation of vehicle longitudinal dynamics is q x m F vD= (2)
where ω is the angular velocity of the wheel,
vis the vehicle velocity, Fx is the longitudinal reactive force, Tb is the braking torque, mq
is the mass of the quarter of the vehicle supported by the wheel, R is the tire rolling radius, I is the moment of inertia, and Kb is the gain between the pressure of the ABS, P , i
and the braking torque Tb . The control objective of ABS is to regulate wheel slip to maximize the coefficient of friction between the wheel and the road for any road condition. The coefficient of friction during braking can be described as a function of the slip ratioλ, which is defined as v R v ω λ= − . (3) A. Friction Model
The tire force includes the normal force, the longitudinal force, and the lateral force. The normal force F is a function of the z
weight of the vehicle, or the component of the weight acting in a vertical plane relative to the road surface. The longitudinal force
x
F is effective at road-surface level; it
allows the driver to apply throttle and then brakes to accelerate and slow the vehicle. As the vehicle is being steered, the lateral force
y
F controls its maneuverability and stability in changing the vehicle’s forward direction. The normalized friction force is defined as
z x F
F
=
µ . In previous research [4], a road condition θ introduced into the LuGre friction model and the normalized friction force is shown as follows:
z R z v g v v z r r r κ ω δ θ − − = ) ( | | 0 (4) z r x z z v F F =(δ0 +δ1+δ2 ) (5) where | | ( ) ( vs r v c s c r F F F e v g = + − ) − , (6) 0
δ is the normalized rubber longitudinal lumped stiffness, δ1 is the normalized
rubber longitudinal lumped damping, δ2 is the normalized viscous relative damping,
c
F is the normalized Coulomb friction,
s
F is the normalized static friction, v is s
the Stribeck relative velocity, z is the internal friction state, and v is a relative r
velocity defined as rω−v, which is equal to −λv. The parameter θ is introduced to capture the changes in the road conditions. The constant κ ≈L/2>0 captures the effect in the lumped model of the force distribution. The L is patch length.
A feedback control law is proposed to force the normalized friction force equation shown in (5) into steady-state characteristic, so that we can get the fixing the velocity v or wheel velocityω. Then a family of curves λ vFx for different value of ω’s is shown in Fig. 1. Using the same way, Fig. 2 shows that there are various µ vsλ curves under different values for θ, where µ is the normalized friction force and λ is the slip ratio. From Fig. 2 and 3, we can find the maximum normalized friction force in
different θ ’s and ω ’s. The
three-dimensional mapping surface can be defined and is shown in Fig. 4. The reference slip ratio is approximating to an optimal slip ratio when the parameterθ effectively captures the real road
characteristics. B. Second order dynamic system
In traditional anti-lock braking systems, certain states, such as the wheel acceleration, and the brake-line pressure, are difficult to obtain, or are prone to noise and other measurement errors. To resolve this problem, we will use an observer-based direct adaptive fuzzy-neural controller to control the ABS following the reference slip ratio. First differentiating equation (3) along with equation (1), we find the differential of λ to be
(
)
− + − = FR K P v I R v λ x b i (1 ) 1 λ (7)Defining λ=[λλ] and z=[z z z], then the time derivative function of (7) can be expressed as
Fig. 1.Steady-state profiles computed for the uniform force distribution.
sec / 1 33 . 133 = ω sec / 1 33 . 103 = ω sec / 1 33 . 73 = ω sec / 1 33 . 43 = ω µ slip slip λ µ 425 . 0 :θ= A 505 . 0 :θ= B 65 . 1 :θ= C 55 . 2 :θ= D ) (θ λd=g slip λ µ 425 . 0 :θ= A 505 . 0 :θ= B 65 . 1 :θ= C 55 . 2 :θ= D slip λ µ 425 . 0 :θ= A 505 . 0 :θ= B 65 . 1 :θ= C 55 . 2 :θ= D ) (θ λd=g
Fig. 2. Static view of the distributed LuGre model under different values for θ .
w
θ
slip
Fig. 4. Three-dimensional mapping surface which is a smooth mapping function from known θ and ω to a reference slip ratio
) , , , , ( ) ( ) , , , ( ~ ) , , , , ( )] ) 1 ( ( [ 2 z z z i i i b v v X P v v d u v b v f P v v d Iv P RK v a a Iv F R λ λ λ λ λ λ λ + + ≡ + − − − + = (8) where ) ) 1 ( ( ) , , , ( ~ 2 v a a Iv F R v f= λλ z = X + −λ v−λ v , Iv RK v b( )=− b , and i P u= . d(λ,v,v,Pi,z)
represents the nonlinearities obtained by differentiating λD with respect to v on the
premise that other parameters are treated as constants.
III. Estimator for Tire/Road Contact Friction
By substituting equation (5) into (1) and (2) and adding viscous rotational friction δw, we can derive a one-wheel vehicle model with the LuGre friction model used in previous studies [8-12], called the LuGre-based ABS, r z z qv F z z F v m = (δ0 +δ1)+ δ2 (9) r r z z z v u RF Iω =− (δ0 +δ1+δ2 )−δωω+ (10) z R z v g v v z r r r κ ω δ θ − − = ) ( | | 0 (11) 0 = θ (12)
where u is an input variable for the road r
estimator, and we have neglected the term
2
δ in equation (10). In equations (9)-(12), we assume that onlyωis measurable. To set our systems in the same framework as the classical dynamic system, the following change of coordinates is introduced:
z RF I I v Rm z q 1 δ ω χ ω η + = + = (13)
Substituting equation (13) into (9)-(10), we obtain the following equations:
2 0 1 2 2 2 1 0 1 0 0 0 0 0 0 [ ( , ) ( , )] 1 1 0 0 1 1 0 1 [0 ] z q r r q z z q r r q z r F m u y R m F R F I m I y u I R Rm -R F y I I ω ω δ δ θφ ψ δ δ δ δ δ δ δ δ − = − + + − − + − + − + + = x x x x x (14)
where the state vector x=[η χ z]T, output
variable yr =ω , z v Ry g v Ry u r r r ) ( ) , ( 0 − − =δ φ x and ψ(yr,x)=κRωz.
Under the hypotheses from previous methods [4], an estimator structure is proposed for the LuGre-based ABS shown in equation (14) and it can be expressed as: x x x x x ˆ ] 1 0 [ ˆ ) ˆ ( ˆ )] ˆ ( ) ˆ ( | ˆ | [ ˆ ) ˆ ( 0 1 1 }] ˆ }{ ˆ 1 ){ ( ) , ( ) , ( ˆ [ 1 0 0 ˆ 0 0 1 0 0 0 0 ˆ 1 0 2 2 1 0 1 2 2 2 1 0 2 I -RF I y y z y v g v y y k k k u y Rm I R I m F I F R y y y y u Rm m F z r r r r r r r r r q q z z r r r r r q q z δ ω ω α δ γ θ δ δ δ δ δ δ θ α ψ φ θ δ δ δ ω ω = − − + = − + + + − − + + − + + + − + − − − = (15) with ) ) ( ( 2 )) ˆ ( ˆ ( ) ( min 2 2 max ε λ ρ ρ θ α φ ψ − + = Q r r r y y y and )) ( ; 0 ( λmin Qr ε∈ .
Lipschitz Properties of φ and ψ : It is immediately domination function and constant max , ) (yr κRyr L κRyr ρψ = ψ = (16)
A tedious but straight forward derivation show that the φ function is also Lipschitz, with the constant given by
}] 1 { [ max max max 0 s c r c s r c v v Rm z v L µ µ µ µ δ φ − + = (17)
r y z ˆˆ, , ˆ , ˆ χ
η , and vˆr are the estimated values of η,χ,z, yr , and vr . The estimator gain vector of the road estimator is ]K =[k1 k2 k3 .
IV. Observer-Based Direct Adaptive Fuzzy-Neural Control
In this paper, the control objective is to design a DAFC such that the slip ratio,
λ
=
y , follows a reference slip ratio, d
m
y =λ . First, we convert the tracking problem to a regulation problem, so equation (8) is rewritten as λ C λ B Aλ λ T y d bu f = + + + = ( ( ) ) (18) where = 0 0 1 0 A , = 1 0 B , = 0 1 C , and 2 2 1 ] [ ] [ T = T ∈R = λ λ λ λ λ is a vector of
state. Define the output tracking error
λ λ − = −
= ym y d
e , the reference vector
T d d T m m m=[y yC ] =[λ λC ] y and the
tracking error vector [e e]T [e e ]T
2 1
=
= C
e .
Based on the certainty equivalence approach, an optimal control law is
] ˆ ) ( [ 1 * Te c m K y f b u = − λ + + (19)
where eˆ denotes the estimate of e, and
T c c c =[k2 k1]
K is the feedback gain vector,
chosen such that the characteristic
polynomial of T
c
BK
A− is Hurwitz
because (A, B) is controllable. Since only the system output (the slip ratio), y=λ, is assumed to be measurable, and f(λ) is assumed to be unknown, the optimal control law (19) cannot be implemented. Thus, we suppose a control input u is
f sv
u u= +u (20)
where an observer-based direct adaptive fuzzy-neural controller (DAFC) uf is designed to approximate the optimal control law (19), and the control term u sv
is employed to compensate for the
external disturbances and modeling error. The adaptive control laws uf and u sv
can be expressed as:
2 1 1 2 1 1 ˆ [ ( )] ˆ ( | ) ˆ ( ) ˆ ( ) i j i j h i j A i j T f c h j A i j T c p e u e µ µ ϕ = = = = = =
∑
∏
∑
∏
e θ θ e (21) and 1 1 1 1 1 1 0 and | |0 and | | , where α is a positive constant | | sv if e e u if e e e if e ρ α ρ α ρ α α ≥ > = − < > > > > > > > > (22)
with the adaptive law
< < ≥ < < = 0 ) ˆ ( ~ and || || )), ˆ ( ~ ( ) 0 ) ˆ ( ~ and || (|| || || ), ˆ ( ~ 1 1 1 1 e θ θ e P e θ θ θ e θ θ r θ θ ϕ ϕ κ ϕ ϕ κ T c c T c c c c e m if e e m or m if e c c c I (23) where c c T c e e e θ θ e θ e e Pr 2 1 1 1 || || ) ˆ ( ~ ) ˆ ( ~ )) ˆ ( ~ (κ ϕ =κ ϕ −κ ϕ , [ 1 2 h T] c = p p p
θ is adjusted vector, ˆe
denote estimate of the error vector
1 2
[e e ]T =
e and ~e=e−eˆ. Thus, using control law u , we can prove that e t converges 1( ) to zero as t→ ∞.
V. Overall Control System and Simulation Results
Slip ratio control is a well-known method in ABS [3]. In general, a given optimal slip ratio with respect to various road conditions is used as a reference signal. One main disadvantage of this method is the difficulty in getting the optimal slip ratio when the road conditions are unknown.
Optimal slip ratios vary according to the individual road surfaces, such as a dry or wet road. The ABS with the LuGre friction model (LuGre-based ABS) proposed in this paper can recognize the road condition and then supply the corresponding reference slip ratio to the vehicle system through a mapping function. To monitor the road situation in the LuGre-based ABS, it is only necessary to have the angular wheel velocity
ω , which is easily measured by actual sensors. The overall control system is shown in Fig. 4.
The mapping function λd =g(θ) is shown in Fig. 3. It maps the road condition parameter θ and wheel velocity to the reference slip ratio λd. Neural networks are well-known nonlinear approximators [12, 13], especially for function approximation. Thus, to quickly and accurately learn the mapping function g(θ), we adopt NNs as the learning mechanism.
This section presents the simulation results of the proposed LuGre-based ABS to show that the tracking error of the closed-loop system can approach an arbitrarily small value. The data used for the simulation are shown in Table 1. To identify the mapping
function )λd =M(θ,ω , the
back-propagation learning algorithm is applied to multilayer feedforward neural networks consisting of processing elements with continuous differentiable activation functions. An examples are used to test the proposed LuGre-based ABS. The first assumes the road condition θ is 0.425 (dry road) and the parameters relating to the road estimator are θmax =0.5 and γ =100000 .
Figure 5 shows the curve for the dry road condition, θ =0.425, and the estimate, θˆ . Figure 5 shows that the road estimator is not effective when the relative velocity is too small. Figure 6 shows the curves of the vehicle speed as it is reduced from 33.3
m/sec in18.54 seconds and the corresponding
velocity of the wheel. In Figure 7, it can be observed that after a very short period of transient response, the wheel slip λ has approached the reference slip ratio. Figure 8 shows the control signal for the dry road.
Figure 5 Estimated parameter θˆ and evolution of θ.
θ
θˆ
t θ
Figure 8. control input u.
u
Figure 7. Estimated slip ratio λd and tracking slip ratio λ. λ d λ λ v ω vorω
Figure 6. Real vehicle velocity(v) and rotation rate (ω).
) (θ λd=f ABS Vehicle Model B T uˆ=− w θ uˆ d λ λ e TB Observer based Direct Adaptive Fuzzy-Neural Control Road Estimator θ ˆ λ f(θˆ) d= r u B r T u= ω
Figure 6. The overall system of LuGre-based
) ˆ (θ λd=g
VI. Conclusion
In this paper, we introduce a road estimator based on the LuGre friction model into the ABS system, called the LuGre-based ABS. This solves the traditional problem of slip ratio control when the road condition is unknown. Otherwise, for further reducing the braking time, a wheel velocity is introduced into the road estimator. In anti-lock braking systems, certain states are difficult to obtain by real physical sensors, or are prone to noise and other measurement errors. To resolve this problem, the observer-based direct adaptive fuzzy-neural controller (DAFC) is applied to ABS control, and its stability is guaranteed by the universal approximation theorem. The simulation results show that the road estimator effectively and quickly captures the road conditions, and the DAFC can force the output to track the reference slip ratio obtained by the road estimator.
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Parameter Value
m Vehicle mass 275kg
0
v Initial vehicle velocity 33.33(m/sec)
0
ω Initial wheel velocity 133.33(1/sec)
R Wheel radius 0.25 m
b
K Gain between piandTb 25
g Gravitational constant 9.8 (m/ s2)
I Moment of inertia of wheel 12.891(kg−m2)
z
F Normal force 2695 N
0
δ Normalized rubber longitudinal lumped stiffness
181.54 (1/m)
1
δ Normalized rubber
longitudinal lumped damping 1 (s/m)
2
δ Normalized viscous relative damping 0.0018 C F Normalized Coulomb friction 0.6 S
F Normalized static friction 1.55
s
v Stribeck relative velocity 12.5 m/s
