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Adaptive vehicle skid control

E. Faruk Kececi a,*, Gang Tao b

a Department of Mechanical Engineering, Izmir Institute of Technology, Urla, Izmir 35430, Turkeyb Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA 22903, USA

Received 21 September 2004; accepted 15 October 2005

Abstract

In this paper, adaptive vehicle skid control, for stability and tracking of a vehicle during slippage of its wheels without braking, isaddressed. Two adaptive control algorithms are developed: one for the case when no road condition information is available, andone for the case when certain information is known only about the instant type of road surface on which the vehicle is moving. Thevehicle control system with an adaptive control law keeps the speed of the vehicle as desired by applying more power to the drive wheelswhere the additional driving force at the non-skidding wheel will compensate for the loss of the driving force at the skidding wheel, andalso arranges the direction of the vehicle motion by changing the steering angle of the two front steering wheels. Stability analysis provesthat the vehicle position and velocity errors are both bounded. With additional road surface information available, the adaptive controlsystem guarantees that the vehicle position error and velocity error converge to zero asymptotically even if the road surface parametersare unknown. 2006 Elsevier Ltd. All rights reserved.

Keywords: Active steering control; Adaptive vehicle skid control; Driver assistance system; Non-linear adaptive control design; Vehicle stability andtracking

1. Introduction

Initially, active control of vehicle dynamic systems isused to reduce fuel consumption and emissions [1]. In cur-rent automotive technology, active control is also used toincrease the safety and the reliability of the vehicle rideand handling. Special control systems are designed for dif-ferent parts of the vehicle dynamics [24]. Anti-lock brakesystems (ABS) are designed to prevent skidding of the

wheels during braking [5,6], while traction control systems(TCS) are accomplishing the same objective, preventing theslipping of the wheels, during acceleration [6]. When thevehicle is on a slippery surface, because of the drop atthe coefficient of road adhesion, the drive wheels may slip.The traction control system reduces the engine torque or

applies brakes to the slipping wheels and brings the slip-ping wheels into the desirable skid range. In order toincrease the stability of the vehicle during cornering, activeyaw control systems are designed, where either by brakingor by transferring the torque between the wheels, the speedof the vehicle is decreased to critical speed to turn the cor-ner and yaw control is achieved [7,9,10]. Vehicle state esti-mation [11,12] and road condition estimation [13] methodsare used to improve the performance of ABS, TCS and

active yaw control systems. In [1416] controller algo-rithms are designed to compensate for the error in thelateral and yaw motion where sliding mode control andminimization of cost function are used.

In order to increase the vehicle handling performance,advanced hardware designs are also used. In four-wheelsteering vehicles, the rear wheels can also be steered[6,17] which increases the steering ability of the vehicle.The steer by wire mechanism which has replaced the con-ventional steering mechanism with electrical motors,rotates the steering wheels (which are the two front wheels

0957-4158/$ - see front matter 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.mechatronics.2005.10.005

* Corresponding author. Tel.: +90 232 750 6618; fax: +90 232 750 6515.E-mail addresses: [email protected] (E.F. Kececi), gt9s@virgi-

nia.edu (G. Tao).URL: http://robotics.iyte.edu.tr (E.F. Kececi).

Mechatronics 16 (2006) 291301
mailto:[email protected]:[email protected]:[email protected]://robotics.iyte.edu.tr/http://robotics.iyte.edu.tr/mailto:[email protected]:[email protected]:[email protected]

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of the vehicle) according to the drivers intentions [1820].This system gives the ability to implement the automaticsteering concepts, which are used in driver assistancesystems.

Direct yaw moment control systems, driver assistancesystems, for vehicle handling at the high speed steering,

are employed to stabilize the motion of the vehicle by brak-ing the slipping wheel and bringing it to a skid range sothat it can start applying driving force again [21]. Thesesystems are used in applications such as lane change [14]and lane keeping [22].

In existing vehicle skid control systems, the skiddingwheel is detected and the power applied to this wheel isreduced until the traction is regained. However, when thetotal vehicle dynamic is considered, after the skiddingoccurs, in order to regain control of the vehicle, the totalspeed of the vehicle is reduced, which can be a problemin traffic when it causes some vehicles to come to a suddenstop.

When a vehicle with an existing skid control technologyis considered, where the total speed of the vehicle isdecreased to regain traction at the skidding wheel, if thevehicles behind this slowing down vehicle do not need toslow down, because they do not need to have any tractionproblem, they can hit the slowing down vehicle. The big-gest advantage of the proposed design is that it does notrequire for the vehicle to decrease its speed, so the vehiclewill not cause any danger to the other vehicles in the traffic.

In this paper, the concept of controlling the stability ofthe vehicle during slippage of the wheels without braking,which can be applied to electrical vehicles [7] as well as

the all wheel drive vehicles, where the drive wheels andsteering wheels can be controlled separately, has beendeveloped. An adaptive control algorithm is developedfor the case when no road condition information is avail-able. Stability analysis proves that the vehicle positionand velocity errors are both bounded. If the additionalroad surface information available, the adaptive controlsystem guarantees that the vehicle position error and veloc-ity error converge to zero asymptotically even if the roadsurface parameters are unknown.

This research is the design of adaptive fault tolerant con-trol algorithm. The application is the stability and trackingcontrol of a vehicle and the failure is described as the slip-page of the wheels of the vehicle. The control algorithm isdesigned to overcome the faults arising from the slippageand to keep the vehicle at the desired speed and directionintended by the driver.

Another application of this control algorithm is autono-mous vehicles in a platoon system, where the followingvehicles should stay at the same speed, acceleration, anddirection with the lead vehicle. Preventing the slippage ofthe wheels of the both the leading and the following vehiclecan stop any error at the speed acceleration and direction,and as a consequence any possible collusion.

This paper is organized as follows: The driver assistance

systems are introduced and the new concept, adaptive vehi-

cle skid control, is explained in Section 1. System structureof an electrical vehicle is studied in Section 2 and dynamicequations of the system are formulated. In Section 3 anadaptive control approach is used to compensate the un-certainties arising from the road conditions. Section 3.1explains the operation of the system and in Section 3.2

the problem, changing road friction coefficient, is modeled.In Section 3.3 adaptive control schemes are developedfor the unknown and known road condition cases andwith Lyapunov stability analyses, the boundedness of theclosed-loop signals are proved. Simulation results showthe effectiveness of the control algorithm in Section 4.Conclusions and future work for this research are givenin Section 5.

2. System structure and dynamics

In order to demonstrate the adaptive controller designfor the wheel slip/skid problem, an electrical vehicle is

designed, which is shown in Fig. 1. The back wheels ofthe vehicle have in-wheel electrical motors and the speedof the back wheels can be controlled separately. The frontwheels are used for the purpose of steering only and areaccompanied by steer by wire system.

For low speed applications, kinematic steering (a.k.a.Ackermann steering) is used to model the motion of thevehicle where the sideslip angles are small so that, thewheels are assumed to be pure rolling. A single-track model(a.k.a. bicycle model) is used to simplify the analysis. Thewheels on an axle are represented by a single wheel withthe double of cornering stiffness.

In this new study, dynamic steering is used, consideringthe sideslip angles are big enough to create corneringforces. A single-tack model is not applicable since the backwheels are controlled separately and the cornering forceseffecting each front and back wheel can be different becauseof the possible different road conditions acting on eachwheel.

When the model shown in Fig. 2 is considered, the fol-lowing symbols are utilized as: V is the velocity vector; uis the longitudinal velocity; v is the lateral velocity; b isthe sideslip angle; r is the yaw rate; CM is the center ofmass; lf and lr are the distances between the CM locationand the front and the back axles respectively; ar is the backwheel slip angle; lt is the wheel tread; Fdl is the driving forceapplied by the left back wheel; Fdr is the driving forceapplied by the right back wheel; Fbl is the lateral force atthe left back wheel; Fbr is the lateral force at the right backwheel; Ffl is the lateral force at the left front wheel; Ffr is thelateral force at the right front wheel; afl is the left frontwheel slip angle; dl is the left front wheel steering angle;afr is the right front wheel slip angle and dr is the right frontwheel steering angle.

Moreover, the vehicle is assumed to be a rigid bodymoving on a flat surface and the vehicle has back wheeldrive and the front wheels are free rolling. It is also

assumed that air resistance, load transfer between the axles,

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suspension system dynamics, caster effect and tire dynamicsare not included in the dynamics of the vehicle. Interestedreaders can find more detailed analysis of the vehicledynamics in [8], since this research is more focused on

the design of an adaptive controller, these simplifieddynamic equations will be used. With these assumptions,dynamic equations of the system are formulated as

m _u rv Fdl Fdr Ffl sindl Ffr sindr 2:1

m_v ru Fbl Fbr Ffl cosdl Ffr cosdr 2:2

J_r lfFfl sindl Ffr sindr lrFbl Fbr ltFdr Fdl

2:3

where m is the vehicle mass and J is the moment of inertiaabout the vehicle mass center. The first and second equa-tions show the longitudinal and lateral dynamics respec-

tively, where the third equation formulates the yaw

motion dynamics. Since the roll and pitch motions areslower than yaw motion [23], they are not included intothe dynamic equations.

From the drive wheel dynamics, the driving forces, Fdland Fdr, are formulated as

Fdj

sj

Rif

sj

R6

Wj

lbjr

Wj

lbjrif

sj

R>

Wj

lbjr

8>>>>>:

2:4

where b indicates the back axle, jindicates the wheel, left orright, sj is the driving torque applied to the wheel and R isthe wheel radius, Wj is the vehicle weight on the wheellbj(r) is the road surface friction coefficient which is a func-tion ofr wheel slip ratio.

The maximum driving force can be applied to the wheel

is formulated asW

jlbjr and if the applied driving force value

Fig. 2. Dynamical steering of the vehicle.

Fig. 1. Prototype electrical vehicle.

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sjR

is bigger than the maximum applicable force, the drivingforce will be equal to the maximum value.

The lateral force effecting each wheel is formulated asthe function of the cornering stiffness Cij of the wheeland sideslip angle aij as

Fij Cijaij 2:5

where i indicates the axle: i= f for front or i= b forback, j indicates the wheel: j= l for left or j= r forright.

Cornering stiffness Cij is a function of the road surfacefriction coefficient and formulated as

Cij Clij 2:6

where C is the cornering coefficient of the tire and lij is thefriction coefficient effecting the jth wheel at the ith axle.

Sideslip angle aij is formulated as

aij arctanv rli

u rlt dij 2:7

This study assumes equal slip angles on the left and rightwheels and only the front wheels are steered. For the sim-plification of the system steering angles at the front rightand left wheels, dfr and dfl, are assumed to be the samesteering angle d = dfr = dfl. Linear tire model is used andthe cornering stiffness coefficient C is assumed to be thesame for each tire. It should also be noted that the sideslipoccurs only when the longitudinal motion exists, ensuringthat cornering forces are finite. Under these assumptions,the cornering forces are formulated as

Fbl Clbl arctan

v rlr

u rlt

2:8

Fbr Clbr arctanv rlru rlt

2:9

Ffl Clfl arctanv rlfu rlt

d

2:10

Ffr Clfr arctanv rlfu rlt

d

2:11

Substituting Eqs. (2.8)(2.11) into (2.1)(2.3), thedynamic equations of the system are found as

m _u mrv sr

Rsl

R

Csind arctanv rlfu rlt

d

lfl lfr 2:12

m _v mru Carctanv rlru rlt

lbl lbr

Ccosd arctanv rlfu rlt

d

lfl lfr 2:13

J_r lfCsind arctanv rlfu rlt

d

lfl lfr

lrCarctanv rlru rlt

lbl lbr

lt

Rsr sl

2:14

which can be rewritten as

Dq f _q;lij g _q; s;lij 2:15

where lij is the friction coefficient effecting the jth wheel atthe ith axle, i indicates the axle, front or back, j indicatesthe wheel, left or right, the sate vector _q and the controlinput vector s are formulated as respectively, _q u; v; r

T

and s = [d,sr,sl]. The positive-definite mass matrix D, theterm containing the unaffected dynamics from the controlinput f _q; lij and the term contains the controllabledynamics g _q; s;lij are formulated as respectively

D

m 0 0

0 m 0

0 0 J

264

375 2:16

f _q;lij

mrv

mru Carctanv rlru rlt

lbl lbr

lrCarctan

v rlru rlt

lbl lbr

2666664

3777775

2:17

g _q;s;lij

Csind arctanv rlfu rlt

d

lfl lfr

sr

Rsl

R

Ccosd arctanv rlfu rlt

d

lfl lfr

lfCsind arctanv rlfu rlt

d

lfl lfr

lt

Rsr sl

266666664

377777775

2:18

3. Adaptive wheel skid control

3.1. System operation

Driver assistance systems can be categorized into threedifferent groups by their application area [4]. Longitudinalcontrol involves keeping the vehicle at a designated dis-tance from the prior vehicle maintaining a relatively con-stant speed. This is accomplished by using cruise controland adaptive cruise control systems, where radar is addedto a standard cruise control to measure the distance andthe speed of the prior vehicle to adjust the following dis-tance and the speed of the vehicle. Anti-lock brake systems(ABS) and traction control systems (TCS) can be cate-gorized in the longitudinal vehicle control systems, wherethese systems are designed to prevent skidding in the longi-tudinal dynamics.

Lateral control of the vehicle is achieved by look-downand look-ahead reference systems and accomplishes thelow speed steering of the vehicle. Lane keeping, lane chang-ing, turning, avoiding obstacles, steering over correctionbecause of a flat tire and lane departure problems aresolved by lateral control algorithms.

The direct yaw moment control system, a combinationof longitudinal and lateral control systems, is used to con-

trol the vehicle at high speed steering. This system stabilizes


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the motion of the vehicle by braking the slipping wheel andbringing it to a skid range that it can start applying drivingforce again [21]. In the current direct yaw moment systemtechnology, the skidding wheel is detected and the powerapplied to this wheel is reduced, until the traction isregained. However, when the total vehicle dynamic is con-sidered, after the skidding occurs, in order to regain controlof the vehicle, the total speed of the vehicle is reduced,which can be a problem in traffic, because accidents occurwhen any vehicles speed suddenly decreases.

In the proposed system, the speed and the direction ofthe vehicle are designed to be maintained as desired bythe driver. A motion stabilizer system whose block diagramis shown in Fig. 3 is proposed. The operation of the motionstabilizer is as follows: The speed and the direction of thevehicle are determined by the driver. In case of a wheel slip-page, an adaptive control algorithm keeps the speed of thevehicle as desired by applying more power to the drivewheels where the additional driving force at the non-skid-ding wheel will compensate for the loss of the driving forceat the skidding wheel, and the controller also arranges thedirection of the vehicle by changing the steering angle ofthe steering wheels.

3.2. Problem formulation

While a ground vehicle is travelling on the road, theforces acting on the vehicle are the functions of the roadfriction coefficient and these driving forces are formulatedas

Fdj

sj

Rif

sj

R6

Wj

lbjr

Wj

lbjrif

sj

R>

Wj

lbjr

8>>>>>:

3:1

where b indicates the back axle, jindicates the wheel, left or

right, sj is the driving torque applied to the wheel and R is

the wheel radius, Wj is the vehicle weight on the wheellbj(r) is the road surface friction coefficient which is a func-tion ofr wheel slip ratio. Lateral forces are formulated as

Fij Clijaij 3:2

where C is the cornering coefficient of the tire and lij is thefriction coefficient, aij is the sideslip angle effecting the jthwheel at the ith axle.

On different road surfaces, the road friction coefficients,which are functions of slip ratio, vary and for an exampletire are shown in Fig. 4. The road friction coefficient isfound experimentally, but also can be approximated by

an analytical expression. A general expression in the rangeof 1 < r < 1 is

l a1 ebr cr2 dr 3:3

where the coefficients a, b, c and d are obtained from theexperimental curves and r is the slip ratio and is defined as

Fig. 3. Block diagram of the proposed driver assistance system.

0 0.5 10

0.4

0.8

1.2

DryWetSnowIce

Fig. 4. Tire friction coefficient vs. wheel slip.


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r Rx v

Rx3:4

where R is the wheel radius, x is the angular speed of thewheel and v is the linear speed of the wheel center. TheABS system will not allow the wheel to get locked, thismeans x will not be zero and r, slip ratio, is always defined.

At different surfaces (i.e., dry asphalt, wet asphalt or ice)the a, b, c, d coefficients will differ. The uncertainty in theroad friction confection is expressed as

lij lij1kij1t lij2kij2t lijkkijkt 3:5

where lij is the actual road friction coefficient at the ith axlejth wheel, lijp, p = 1, . . . , k, are road friction coefficients fordifferent surfaces, which last a certain period of time, andkijp(t) are time functions indicating which values of lijpare taken by lij at a given time. lij is a function of timeshowing the ith axle jth wheel road friction coefficient.

kijp 1 ifl

ij l

ijp

0 otherwise

3:6

Function kijp(t) indicates only one value can be active atany time, that is kijm(t)kijn(t) = 0, for m5 n, and kij1 +kij2 + + kijk = 1.

The following two cases are possible for kijp(t)

(i) The road condition information is not available, thatis, kijp(t) are unknown;

(ii) The road condition is measured, that is, kijp(t) areknown.

While the real values oflijp are not known, these boundscan be found experimentally. When the road condition ismeasured, kijp(t) are known.

During the field test the friction coefficients will requiretuning, because there is no exact friction coefficient valuefor an icy road. However, the adaptive control algorithmensures that with the parameter update laws, arbitrarilyguessed unknown parameter values are converged to thevalues that makes the system error converge to zero, wherethe system is stable.

The controller objective is to design a feedback controllaw s for the system (2.15) to ensure that all closed-loopsystem signals and parameter estimates are bounded, andthat the system output q(t) asymptotically tracks a givenreference output qd(t). In other words, the controller objec-tive is to keep the speed and the direction of the vehicle asdesired by the driver, despite the uncertainties arising fromthe road conditions.

3.3. Adaptive controller design

Because of computing difficulties there were someassumptions made in the previous section during the for-mulation of the mathematical model of the system. Thereare also measuring difficulties in the system, which causes

more assumptions, such as the mass and the inertia of

the vehicle, since these values would change with the num-ber of passengers. These assumptions cause uncertaintiesand in order to compensate for these uncertainties in thesystem, adaptive control methods are used. The adaptivecontrol system uses the knowledge from plant dynamicsand updates the control parameters on-line to eliminate

the tracking error.

3.3.1. Design with unknown road condition

When there is no information available about the roadcondition, the adaptive control scheme can be derived inthe following procedure. The term, including the controlinputs in the dynamical equations of the system,

g _q; s;lij in (2.18), is expressed as

g _q; s;lij Kss 3:7

where s = [d,sr,sl]T and

Ks

Csind A d lfl lfr

d

1

R

1

RCcosd A d lfl lfr

d0 0

lfCsind A d lfl lfr

d

lt

R

lt

R

26666664

377777752 R33

3:8

with A arctanvrlfurlt

.

Let ^Ks be the estimate ofKs and choose the control lawas

s ^K1s sd 3:9

where sd is a feedback control law designed based on theideal system

Dq f _q;lij sd 3:10

The existence of ^K1s is to be ensured by a projectionalgorithm in the adaptive law for updating the estimate^Kst. From (2.15), (3.7) and (3.9), the closed-loop dynamicequation is

Dq f _q;lij Ks ^Kss sd 3:11

The ideal matching parameter for ^Kst is Ks.To design an adaptive control scheme to handle param-

eter uncertainties, for _q u; v; rT and _qd ud; vd; rd

T, the

tracking error e and the filtered tracking errors s; _s aredefined as

e q qd; _e _q _qd 3:12

s _e ge; _s e g_e 3:13

where g 2 R3 3 > 0 is a diagonal gain matrix.At first the ideal system is expressed as Dq f _q; lij

sd as D_s Y _q; q; hh sd, where

Y _q; q; hh Dh f _q;lij; h qd g_e 3:14

where Y _q; q; h 2 R34 is a known matrix function,h(m,lij, J) 2 R

41 is an unknown parameter vector and

are formulated as

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Y _q; q; h

h1 rv 0 0 0

h2 ru 0 CA CA

0 h3 lrCA lrCA

264

375 3:15

h m;J; lbl;lbrT. 3:16

The actual system (3.11) is expressed as

D_s Yh ^Ks Kss sd 3:17

for which Kss can be further expressed as

Kss Yshs 3:18

where

hs lfr;lfl;1

R

T3:19

is the parameter vector, and

Ys

CsindA d CsindA d sr sl

CcosdA d CcosdA d 0

lfCsindA d lfCsindA d ltsr sl

264

375

3:20

is the known matrix function.Let hs be the estimate ofhs. Then, the dynamical system

(3.17) becomes

D_s Yh Yshs hs sd 3:21

For this system, the control law is designed as

sd Yh KDs 3:22

where KD KT

D > 0, KD 2 R33 is a gain matrix, h is the

estimate ofh, updated from an adaptive law. This controllaw could also be used for the ideal system D_s Yh sd.

With the control law (3.22), the closed-loop system is

D_s Yh h Yshs hs KDs 3:23

A Lyapunov-like positive-definite scalar function ischosen as

V1t 1

2sTDs

1

2~hTC11

~h 1

2~hTs C

12

~hs 3:24

where ~h h h, ~hs hs hs, C1 CT1 > 0 and C2

CT2 > 0.For the time t in each interval when the values ofh and

hs do not change, differentiating (3.24) with respect to thetime t and using (3.23) yields

_V1 ~hTC11

_~h YTs ~hTs C12

_~hs YTss s

TKDs 3:25

Choose the adaptive parameter update laws for_h and

_hs as

_h

_~h C1YTs 3:26

_hs

_~hs C2YTss 3:27

which are defined for all tP 0. Then, the derivative of theLyapunov-like function is found as

_V1 s

T

tKDst 6 0 3:28

for t in each interval when the values of h and hs do notchange.

The Lyapunov-like function V1(t) is positive-definiteand _V1 is negative semi-definite. Stability in the sense ofLyapunov has been shown in that the position trackingand velocity tracking errors are both bounded. When a slip

occurs at any of the wheels of the vehicle, a bounded jumpoccurs in the Lyapunov-like function V1(t), where its deriv-ative _V1 is undefined. The stability of the system is validfor every each time interval between the occurrence of eachjump. Due to possible persistent uncertain jumps, thetracking error may not converge to zero. However, if thereare only a finite number of jumps, the developed adaptivecontrol scheme ensures that the tracking error convergesto zero asymptotically. For the case when the parameterjumps are persistent, that is, the vehicle keeps entering asurface at one time and another surface at another time,practical tracking is achievable if the time interval ofswitching surface is large enough. In this case, to ensure

asymptotic tracking, a modified adaptive control schemecan be developed, using certain additional system informa-tion (see the known road condition case).

In order to ensure that ^Kst has a bounded inverse^K1s t, a projection algorithm is used in the design of theadaptive update law. The term ^Kss is expressed as

^Kss Ysd; sr; slhs 3:29

where hs is the estimate of the parameter vector andY

s(d,sr,sl) is the known signal vector.The adaptive update law for hs is given in Eq. (3.27) as

_

hs

_~

hs C2YT

s s 3:30and by applying a projection algorithm defined as

_hs Gt 3:31

Gt G1t; . . . ; G4tT

2 R4 3:32

where the size of G(t) is equal to the size ofhs which con-sists of four unknown parameters namely lfr, lfl, lbr andlbl.

Gkt

if hskt 2 hask; h

bsk or

Fkt if hskt hask and FktP 0 or

if^

hskt hb

sk and Fkt 6 00 otherwise

8>>>>>>>:

3:33

where F(t) is defined as Ft F1t; . . . ;F4tT C2Y

Tss

and hsk 2 ha

sk; hb

sk; k 1; . . . ; 4, hsk and hsk are the kth com-

ponents of hs and hs, respectively, hask; h

bsk; k 1; . . . ; 4, are

known such that for any hsk 2 ha

sk; hb

sk, where ha

sk is the

upper limit of lij, which is ldry and hb

sk is the lower limitoflij, which is lice, k= 1, . . . ,4, the resulting ^Kst is invert-ible, that is, ^K1s t is bounded, and n is the degree of thedifferential equations representing the dynamics of the sys-

tem, which is two in this study. It can be verified that the

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desired property (3.28) still holds, and in addition, it isensured that hsk 2 h

a

sk; hb

sk; k 1; . . . ; 4, so that^K1s t is

bounded.

3.3.2. Design with known road condition

It is possible to measure the road condition by using a

millimeter radiometer, a radar sensor [2426], in the senseit is known whether the vehicle is on a dry surface or onan icy surface, but the road friction coefficients areassumed to be unknown, that is, kijp(t) are known, but lijpare unknown.

Using (3.5) and (3.6), we express g _q; s;lij in (2.15) as

g _q; s;lij Ks1kij1 Ks2kij2 Kskkijks 3:34

where

Ksp

Csind A d lflp lfrp

d

1

R

1

RCcosd A d lflp lfrp

d0 0

lfCsind A d lflp lfrp

d

lt

R

lt

R

26666664

37777775

p 1; 2; . . . ; k

3:35

where A arctanvrlfurlt

.

Let ^Ksp be the estimate ofKsp, p = 1,2, . . . , k, and choosethe control law as

s ^K1s1 kij1t ^K1sp kijptsd 3:36

Then Eq. (3.11) becomes

Dq f _q;lij Ks1 ^Ks1kij1t Ksp ^Kspkijptssd.

3:37

Considering that sp = kijps and inserting this equationinto the previous equation results as

Dq f _q;lij Ks1 ^Ks1s1 Ksp

^Kspsk sd. 3:38

f _q;lij formulated as

f _q;lij Y _q; q; hh1kij1 h2kij2 hkkijk 3:39

By including Yp = Ykijp and Eq. (3.39) into Eq. (3.38),the actual system is expressed as

D_s Y1h1 Y2h2 Ykhk Ks1 ^Ks1s1

Ks2 ^Ks2s2 Ksk ^Ksksk sd 3:40

Considering that

Kss Ys1d; sr; slhs1 Ys2d; sr; slhs2

Yskd; sr; slhsk 3:41

^Kss Ys1d; sr; slhs1 Ys2d; sr; slhs2

Yskd; sr; slhsk 3:42

and by choosing the control law as

sd Y1h1 Y2h2 Ykhk KDs 3:43

the closed-loop system becomes

D_s Y1h1 h1 Y2h2 h2 Ykhk hk

Ys1hs1 hs1 Ys2hs2 hs2

Ysk^hsk hsk KDs 3:44

A Lyapunov-like positive-definite scalar function ischosen as

V2 1

2sTDs

1

2

Xkp1

~hTpC1

p~hp

1

2

Xkp1

~hTspC1s2

~hsp 3:45

For the time t in each interval when the values ofhp andhsp do not change, differentiating (3.45) with respect to the

time t and using (3.44) yields

_V2 Xk

p1

~hTpC1

p

_~hp YT

ps

Xkp1

~hTspC1sp

_~hsp YTsps s

TKDs 3:46

Choose the adaptive parameter update laws for_hp and

_hsp as

_hp

_~hp CpYT

ps 3:47

_hsp

_~hsp C2YTsps; p 1; 2; . . . ; k 3:48

which are defined for all tP 0, and the derivative of theLyapunov-like function is found as

_V2 sTKDst 6 0 3:49

for t in each interval when the values of hp and hsp do notchange, the Lyapunov-like function V2(t) is valid over thetime interval for each change in the road friction coefficientfor each wheel.

The fact that _V2 sTKDs 6 0 implies s 2 L

2 \ L1 andhp and hsp 2 L

1 for p = 1, . . . , k. Since s = _e + ge, it impliese and _e 2 L2 \ L1, and also q and _q 2 L1. From (3.22) itfollows that sd and _s 2 L

1 which results e 2 L1. By apply-ing the Barbalat Lemma [27] it is concluded that the posi-tion tracking error e and velocity tracking error _e go tozero as t goes to 1.

In summary, it is proven that with the control law (3.43)and adaptive update laws (3.47) and (3.48), when the roadcondition is measured, it is guaranteed that all closed-loopsignals are bounded and the tracking error e(t) and _e(t) goto zero as t goes to 1.

In order to ensure that ^Kst has a bounded inverse^K1s t, a projection algorithm is used in the design of theadaptive update law same as before.

4. Simulation results

The vehicle model formulated in Eqs. (2.15)(2.18) is

used in the simulations. Since, this study is more focused


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on design of the control algorithm, the assumptions men-tioned in Chapter 2 are made. Another assumption madein this research is also the fact that the drivers reactionis not important, since the purpose of this research is todemonstrate the design of the control algorithm. In reality,

the drivers reaction will be very important, and should beincluded in the field test of this algorithm on a model car.This is what we would like to achieve in the future research.

In this simulation study, the vehicle design shown inFig. 1 is driven on a straight road with a 10 m/s constant

0 2 4 6 8 10 12 14 16 18 20

0

2

4

X

0 2 4 6 8 10 12 14 16 18 201

0

0.1

0.2

0.3

Y

0 2 4 6 8 10 12 14 16 18 202

1

0

0.1

0.2

Time (sec)

Fig. 5. Position errors.

0 2 4 6 8 10 12 14 16 18 20

0

20

40

0 2 4 6 8 10 12 14 16 18 20

0

2000

4000

R

0 2 4 6 8 10 12 14 16 18 20

0

2000

4000

L

Time (sec)

Fig. 6. Control inputs.


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speed. At the fifth second the driver realizes the ice on theroad and maneuvers the vehicle to the left, but the rearright wheel drives on the ice and the effect of the changing

friction coefficient is shown in the following figures. Fig. 5

shows the position errors in the longitudinal, lateral andangular directions. Fig. 6 demonstrates the change in thecontrol inputs because of the changing road conditions.

The effect of the known road condition controller is shown

0 20 40 60 80 100 120 140 160 180 2001

0

1

2

3

4

Desired

0 20 40 60 80 100 120 140 160 180 2001

0

1

2

3

4

Withcontroller

1.5 1 0.5 0 0.5 1

x 109

5

0

5

10x 10

11

Lateral vs. longitudinal position

Without

controller

Fig. 7. Effect of the controller.

0 2 4 6 8 10 12 14 16 18 20

0

0.5

0.5

1

1.5AVSC vs. ESP

AVSCESP

0 2 4 6 8 10 12 14 16 18 20

0

1

1

2

3

Lateralchange(m)

0 2 4 6 8 10 12 14 16 18 200

50

100

150

200

Time (sec)

Longitudinalchange(m)

Fig. 8. Comparison of the AVSC system to an ESP system.


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in Fig. 7, where with the controller can achieve the desiredperformance, but without the controller the systembecomes unstable.

Fig. 8. shows the advantages of the proposed controlsystem Adaptive Vehicle Skid Control (AVSC) to an ESPsystem studied in [28], where the earlier simulation is

repeated for both AVSC and ESP systems. The result isthe vehicle with an AVSC system continues with the samespeed whereas the vehicle with an ESP system cannot keepits intended position in the longitudinal direction.

5. Concluding remarks

In this research, adaptive vehicle skid control concept isdeveloped. Dynamic equations of an electric vehicle are for-mulated and the effect of the road friction coefficient is stud-ied. When the road condition information is not available,the boundedness of the position and velocity error isproved. In the second part of the research, the performanceof the controller is examined when the road condition ismeasured (but the road friction coefficients are assumednot to be known). Simulation results show the effectivenessof the controller schemes. The effect of a flat tire and roll-over resistance of the vehicle, by selecting the steering anglesdfl and dfr differently, will also be studied for future research.

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