Variable Structure Control for EmergencyBraking Systems using LuGre Tire Model

Li Kaijun, Deng Kun, and Xia Qunsheng

designed controller is found to give better results than theAbstract-In this paper, a Variable Structure Control (VSC) conventional controllers, based on the assumption that vehicle

controller (also known as Sliding Mode Control (SMC)) is and angular velocities are both available online by eitherproposed for emergency braking control with a dynamic tire/road measurement or estimations. Two other methods are usingfriction model called LuGre model. After investigating the factorswhich may have influence on the vehicle longitudinal dynamic magnetic markers imbedded in the pavement and an

response, a complex quarter vehicle model of the vehicle accelerometer to calculate velocity by integration (see [6]),longitudinal dynamic system is obtained with LuGre model. To which have drawbacks of requiring an accurate sensing systemfacilitate the controller design, a simplified quarter vehicle model and infrastructure and needing updating because ofis established, and using this simplified model, the VSC controller accumulation of integration errors. An observer wasis designed. Furthermore, its stability is proved using Lyapunov constructed to detect the tire/road conditions under normalstability theory in this paper, and it has better performance of . . .convergence compared with conventional PID controller which is tractio ncoi onsrnlaer [7], whl cine[8 anadtieshown by simulation results. emergency braking controller design was considered using the

observer-based dynamic friction model.Index Terms- LuGre model, Quarter vehicle model, Variable In this paper we derive a complex quarter vehicle model

Structure Control (VSC) using a dynamic friction tire model, taking into account manyactual factors which may have influence on the vehiclelongitudinal behavior. Later the complex model is simplified

I. INTRODUCTION for the facility of the sliding mode controller design. UsingV EHICLE motion and vehicle dynamics are mainly Lyapunov method, a sliding mode controller is developed with[governed by the forces exerted by the road surface on the favorable simulation results, and meanwhile the stability of the

tires, and therefore tire/road friction plays a vital role in designed controller is testified.vehicle behavior. Due to this fact, accurate modeling of tiremechanics is of special importance for the development of II. QUARTER VEHICLE MODELcontrol systems which act on the tire. Traditionally, theroad/tire friction was modeled using the static slip/force map, A. Complex Quarter Vehicle Modeland one of the most famous is the "magic formula" tire model In this part, we first establish complex quarter vehicle modelwhich depended on many parameters acquired from static-state using a dynamic friction tire model. In the so-called complexexperiment (see [1]). In recent papers [2]-[4], a novel dynamic model we consider the load transfer caused by the decelerationtire model, called LuGre model, was proposed. The superiority of vehicle mass under braking condition. Considering theof LuGre model over conventional "magic formula" lies in its bicycle model we know that the vertical load of the front wheelability to describe closely some of the physical phenomena increases, and that of the rear wheel decrease simultaneously.found in road/tire friction (i.e. hysteresis loops, pre-sliding One of the primary task of the controller is to prevent the frontdisplacement, etc.), which in turn depend on parameters wheel from locking, otherwise the vehicle would lose itsdirectly related with the phenomena to be observed, for

s

instance the change on the road surface characteristics (i.e. dry, steering ability and it would be very dangerous.wet, etc.). A quarter-vehicle model with the average lumped LuGre

In [5], a sliding mode controller to prevent wheel from dynamic tire friction model was shown in Fig. 1. The equationsskidding at various road conditions is designed, and the for the model are derived here starting with the LuGre friction

model and ba_Manuscript received September 15, 2006. l!Li Kaijun is with the State Key Laboratory ofAutomotive Safety and Energy,M

Tsinghua University, Beijing 100084, P.R.China (Tel: 8610-6278-6257; e-mail:L r'itlikjO4@mails.tsinghua.edu.cn). /r

Deng Kun is with the State Key Laboratory of Automotive Safety andEnergy, Tsinghua University, Beijing 100084, P.R.China (e-mail:X ddk05@mails.tsinghua.edu.cn). F± F.,

Xia Qunsheng is with the State Key Laboratory of Automotive Safety andEnergy, Tsinghua University, Beijing 100084, P.R.China (e-mail: xiaqs@mail.tsinghua.edu.cn). Fig. 1. The quarter-vehicle model.

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wxw-Ff-a bx LJ2f b LJ X2l X+J2x3fl+532x3 - -324Kf _ -ph2

S100 g21 df

mv-F -F -F (K

the friction internal state, Vr is the relative velocity,

wherez is r) 2 Ldox+f m ))

0 is an unknown parameter of the tire/road condition, which -(calxl +c52 x-Jf -A45 )g-]1x2can suitably describe the road ctAaracteristics.(7

f(Vr)=Rc +(t )e v *2 is the normalized 2-( 2

static friction coefficient, tx is the normalized CoulombA/

friction, Vs is the Stribeck relative velocity, where 51' = -0-g(X3 ),52' =cr1 +cr2,

~~~ 25 ~ ~ 10

The braking force Fx can be expressed as a function of A=3.24f[(mph) ,B= PC A,andvertical load: 5to d

(2)=F(z+z5)= m (-51oxl -2Xj3 + +x Ax )(g + -34

wherez 0 is the normalized rubber longitudinal lumpedlo/t, )

stiffness, c1 iS the normalized rubber longtudinal lumped T

damping,n 2 the normalized viscous relative damping. The 3 2

parameter's values in the LuGre model are referred to [9]. -q2__ ____________________

braking maneuvers, vertical load for the real axle can be 1+o+.x+expressed as:

F=m(g+ v (3) +< mg(-I'xl -2'X3 +fO +Ax2)- RK2 (8)

where h is the height of mass of vehicle C.G., and / is thewheelbase. B. Simplified Quarter Vehicle Model

Rolling resistance is given by (as in [10]): The complex model needs to be simplified in order tostatc f nct facilitate the primary investigation of control strategy. To

Ff=!fto+.24fKj Fz (4) derive the simplified quarter vehicle model, it is assumed thatt t 100 ) ) ~~~~~~~~thereis no load transfer and the mass of vehicle body is

distributed evenly on four wheels, and that the rollingAerodynamic drag force can be modeled as:F1~ can be expCessedAasVa function o(5) A=3e24s ph as,

a 2 df x1 F X3-g23 X1where p is air density, Cd is the drag coefficient, A is

the front area. /Define the state variables as: x3is:=+-fo)g-Bx (9)

Thepin, u tho=rmX2l=vVX3s=Vo r + Jo) + B.d- RWKgKTeLuGre friction model in (1) can be expressed as:|9X3= OC ( Xl-52 X - 2_ h g

xlzx3O)t(x3 3 O(3)1() where a= g+Jw mgj

Jfo AC31where g(X3)l= , and By defining that:

1 (X3) f (X) = X3,7g1 (X) = - (Xh

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The sigmoid-like function

f2(X) = g(50x1 +(o +o2))x3) -Bx -f0g, 2

g2 (X)= 0-glg(x3)X1, 1_13(X) = -a(Xox, +(U1 +c72)X3 -f0)+Bx2, 0

g3 (X) = aoX1g(x3)xl, -1

0 0;=]1 ,2 -10 -5 0 5 10

wi = l R ff i=3Fig. 2. The sigmoid-like function with 6=0.02.

J The following Lyapunov function is chosen.the above (5) can be rearranged as: I- 2 (17X,= (X) +Og (X) + wu, i=P1, 2, 3 (10) 2 Sb (1)

and u = Pb is the control input. The time derivative of V is:V=Kbss =s-Pb + OAF X bA2 (X))

III. VARIABLE STRUCTURE CONTROLLER DESIGN

The longitudinal slip can be expressed with respect to state s(-i,i<s)-f(X)%(X)-/(X) 2(X)+L(X)+kji2(X))variables as: (1 1 ) =-qv8 (s)s-;|s/ (X) -k, |s/3 (x)l +9s/s(X) +k;,S2 (X)

§3< -17v,5 (s)s < O (18)

Hence2x3 =-)Lx2, and then

X3d (0)=-Amax2 (12) Therefore, the designed controller meet the stabilitywhere Amax = arg max T~ requirement as is shown by the previous derivation.

weeSmax =agmxFThe sliding mode surface can be chosen as: IV. SIMULITION RESULTS

S = X3- X3d (0) = X3 + AmaxX2 (13) In last section, stability of designed VSC controller is provedusing Lyapunov stability theory. The analysis of section 2

where SZmax is the peak value ofthe longitudinal slip which can shows that the simplified quarter vehicle model is more suitablebe evaluated from (8), and differentiate s: for designing VSC controller. In order to test the performance

._3 +, . +, 14) of designed controller, a comparison is done between them-X3 lax 2 mIaxX2 (14) conventional PID controller and designed VSC controller whenSubstituting (6) into (10), yield they are both applied in the simplified quarter vehicle model.

= f3 (x) +0g3 (x) +w3I + [12(x) +0g2 (X)]+<x2 By defining the tracking error for PID controller as

=,w3Pb+OA(X)+A (X) (15)= A - Ad where Ad is the desired slip assumed to be

known beforehand. Therefore, PID controller output is

where/31(X) =9g2 (X) + g3 (X), u = KP8+ Kdj + K, J , with Kp =5000, K, =300,A (X) = 'maxf2 (X) + A3 (X) + kmax 2 Kd =10.KdtOLet the control input be: An emergency braking maneuver was simulated with aU =Pb= 7/V(s) +A/3(X) i1 (X) + A' (X) i/2(X) (16) vehicle initial velocity of 30m/s, and the primary controlwhere objective is to prevent the wheel from locking and obtain theqi1(X) = sgn(s,l (X)), maximum deceleration.

V/'2 (X) = Kb sgn(s,2 (X)) Velocitybh =--= , and select; >OS= K0S. VSisCa E 20

control gain with 77 > 0, and v8 s = is the >9

sigmoid-like function, 3 is a small positive scalar. 0 1 2 3 4 5 6

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Relative velocity Velocity0 ~~~~~~~~~~~~~~~~30

-2~~ ~ ~~~~~~2E 0

Time (s) Time (s)Cotrolled braking pressure Relative velocity

Vsc-4~~~~~~~~~~~~~~~~~~~~~-PID

0

0 1 2 3 4 5 6 O 1 2 3 4 5 6Time (s) Time (s)

Braking deceleration Cotrolled braking pressure

3000 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 200

Vsc

-PID

mO 1 2 3 4 5 6 0 1 2 3 4 5 6Time (s) Time (s)

Fig. 3 Comparison of simulation results between VSC and PID controller. Cotrolled sliding surface error0

In order to test controller's ability to adapt to the road surfacecharacteristics, one scenario of parameter 0 is used insimulation, and the responses of vehicle speed and controlled :variables are shown in Fig. 4. Fig. 4-(a) shows the 0 ( __

which is related to four different road surface condition, 0 1 2 3 4 5 6namely dry, wet, a smooth variation from wet to snow, and Time (s)snow conditions.(referred to [7]). Fig. 4. Simulation results corresponding to the variation of parameter 0

0

4 SnowV. CONCLUSION

In this paper, controller for emergency braking is designed.

|Dry Before designing the controller two vehicle models with LuGremodel, one is a complex quarter vehicle model and the other is

0 1 2 3 4 5 6 a simplified one, are established.Time (s) Based on the simplified quarter vehicle model, a Variable

Internal state x Structure Control (VSC) controller for emergency brakingcontrol is designed. Theoretical analysis and simulation results

-0.2 T show that the VSC controller has the following advantages:1) The controller proposed in this paper determines the required-0.4

m ~~~~~~~~~~~~pressurein the master cylinder to achieve maximumQ -0.6 deceleration during braking maneuvers.

-0 .8 ~~~~~~~~~~~~~2)Using Lyapunov stability theory, the contro ller's stability is0 1 2 3 4 5 6 proved.

Time (s) 3)The controller can adapt to the variation of parameters suchas 0 which has been shown in the simulation results above.

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[3] Canudas de Wit, and P. Lischinsky, "Adaptive friction compensationwith partially known dynamic friction model," International Journal ofAdaptive Control and Signal Processing, vol. 1 1, pp.65-85, 1997.

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[5] Kachroo and Tomizuka, "Sliding mode control with chattering reductionand error convergence for a class of discrete nonlinear systems withapplication to vehicle control," ASME Int. Mech. Eng. Congr. Expo, SanFrancisco, CA, 1995.

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[7] Canudas de Wit, C. and Horowitz, R., "Observers for tire/road contactfriction using Only Wheel Angular Velocity Information," Proceedings of38th IEEE Conference ofDecision and Control, Phoenix, AZ, 1999.

[8] Yi, J., Alvarez, L., Claeys, X., and Horowitz, R., "Emergency BrakingControl with an Observer-based Dynamic Tire/Road Friction Model andWheel Angular Velocity Measurement," Vehicle System Dynamics,vol.39, No.2, pp.81-97, 2003.

[9] Deur, J., Asgari, J. and Hrovat, D., "A 3D Brush-type Dynamic TireFriction Model," Vehicle System Dynamics, vol.42,No.3,pp. 133-173,2004

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