4param Transient Response (1)

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Four Parameter Evaluation Method of Lateral Transient Response

Tetsushi Mimuro, Masayoshi Ohsaki, Hiromichi Yasuna a, and Kohji Satoh {esearch Department

Passenger Car Engineering Center Mitsubishi Motors Corp.

Passenger Car Meeting and Exposition

Dearborn, Michigan September 17-20, 1990

Reprinted from SP-838 - Electronic and Non-Electronic Suspenslon

Systems and Steerlng Controls

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Four Parameter Evaluation Method of Lateral Transient Response

Tetsushi Mimuro, Masayoshi Ohsaki, Hiromichi Yasunaga, and Kohji Satoh Research Department

Passenger Car Engineering Center Mitsubishi Motors Corp.

There are many meaningfu l numer ical eva lua t ions o f v e h i c l e dynamic performance. However, e v a l u a t i o n methods us ing one o r two parameters have l i t t l e p r a c t i c a l use f o r development o f new ca rs o r f o r comparison o f s i m i l a r ca rs s ince d r i v e r s ' preferences vary w ide ly i n va r i ous hand l ing c h a r a c t e r i s t i c s .

We developed t h e Four Parameter Evalua- t i o n Method. The parameters are e x t r a c t e d from l a t e r a l t r a n s i e n t response da ta by curve f i t t i n g w i t h a two degree o f freedom model. I n t u i t i v e l y , we can recognize v e h i c l e p e r f o r - mance by a r rang ing these f o u r parameters i n a rhombus. Th is express ion i s s u i t a b l e f o r com- p lex hand l ing c h a r a c t e r i s t i c s . The r e s u l t i n g rhombus p a t t e r n r e f l e c t s a r a t h e r smal l d i f - ference between t e s t c a r s i n c l u d i n g 4WS.

HANDLING EVALUATION i s one o f t h e most a t t r a c t i v e themes i n automobi le engineer ing, hence much research has been made. But suspen- s ion and t i r e des igners want t o know how t h e i r design parameters c o n t r i b u t e t o hand l ing c h a r a c t e r i s t i c s , and t e s t d r i v e r s want t o know where t h e i r f e e l i n g s are de r i ved from. Th i s i s because hand1 i n g c h a r a c t e r i s t i c s are ra the r confusing.

We recognized t h a t t he re are severa l p re fe rab le hand l ing c h a r a c t e r i s t i c s f o r var ious d r i v e r s . Th is means, hand l ing evalua- t i o n should have many e v a l u a t i o n parameters, which e f f e c t each other . I t has been recog- n ized t h a t a v e h i c l e can be cha rac te r i zed by

the balance o f m u l t i p l e e v a l u a t i o n parameters and not o n l y by each e v a l u a t i o n parameter by i t s e l f ( 1 ) *.

P r a c t i c a l t e s t i n g methods should be ac- curate, easy t o execute, easy t o reproduce, and not dependent on a d r i v e r . A t r a n s i e n t response t e s t w i t h pu l se - i npu t i s t h e most popular t e s t and has good performance w i t h t h i s c r i t e r i a ( 3 and 4 ) . I n add i t i on , our r e l i - ab le da ta processing, which i s based on the curve f i t t i n g method w i t h a two degree o f freedom model, a l s o con t r i bu tes .

We have chosen th ree e v a l u a t i o n parameters from yaw v e l o c i t y response data. They are : steady s t a t e ga in " a l " , n a t u r a l frequency " fn" . and damping r a t i o " r" . As a f o u r t h parameter we have used phase de lay " 9 " a t 1Hz f rom l a t e r a l a c c e l e r a t i o n response data. Th i s combinat ion o f f o u r parameters i s p a r t i c u l a r l y important. Al though each parameter has proved i t s wor th iness i n - d i v i d u a l l y , combining them c rea tes a new and va luab le i n s i g h t .

To express these f o u r parameters i n a rhombus w i 11 he lp recognize v e h i c l e p e r f o r - mance i n t u i t i v e l y . The area o f t h e rhombus denotes v e h i c l e hand l i ng p o t e n t i a l , and t h e d i s t o r t i o n denotes t h e hand l i ng tendency. Th is expression i s s u i t a b l e f o r complex hand l ing c h a r a c t e r i s t i c s .

Th is method has been u t i l i z e d f o r severa l years i n t h e development o f new ca rs and i n t he ana lys i s o f many t e s t r e s u l t s f o r t h e study o f fundamental suspension

*Numbers i n parentheses designate References a t end o f paper.




characteristics, including 4WS and active suspension. The application of this evaluation has been reported previously ( 1 and 2). In this paper we have summarized the test method, data processing, measured data, and estimations using ADAMS full vehicle model.

In the above, each coefficient is determined with vehicle dimensions and speed. Where

ai : Steady state gain of each response fn : Natural angular frequency C : Damping ratio Tf : Lead time constant of yaw velocity

2 DOF MODELAND ITS TRANSFER FUNCTION Other coefficients are given in the APPENDIX.

Here, the classical 2 degree of freedom (DOF) model is reviewed. According to the notation and the definition in the APPENDIX, the 2 DOF model is written as:

Where

s = the Laplace transform operator @ = yaw velocity ay = lateral acceleration ,6 = side-slip angle bH= steering wheel angle

V = forward speed rn = vehicle mass Iz = yawing moment of inertia 1 = wheelbase, l=lf+lr If = distance from front axle to C. G. lr = distance from rear axle to C.G. Kf = equivalent front cornering power Kr = equivalent rear cornering power is = overall steering ratio

Then from Eq ( 1 ) , we get the transfer functions of yaw velocity response, side-slip angle response, and lateral acceleration response.

TRANSIENT TEST METHOD WITH PULSE INPUT

Of the many kinds of lateral transient tests (3-6), the steering pulse input method is the simplest. It doesn't require much ski1 1 of a driver. It also doesn't require a wide prov- ing ground since a test vehicle runs almost straight. Moreover, the result is available as a time history.

The following points should be noticed. First, the pulse width should be determined to cover sufficient frequency range by itself. From our experience. the pulse of 0. 4sec width has brought us good results, owing to the sufficient power spectrum level of the pulse between 0. 1Hz and 2Hz. Figs. 1 and 2 denote the time history and the power spectrum of a good steering pulse input, respectively.

Fig.1 - Time history of a pulse input

10 ,cl 2 3 5 7 2 1 5 HZ

Fig.2 - Power spectrum density of a pulse




Second, the pulse amplitude should be of passenger cars, the driver sits near the determined in every test to give the same in- center of gravity (C.G.) of the vehicle, so it tensity to the vehicle. This is because the would be suitable to adopt yaw velocity and characteristics of a real car reveal rather lateral acceleration at the C.G. to compare strong nonlinearity. We usually determine the with the driver's evaluation. amplitude to attain 0.4g peak lateral acceleration. 0.49 is within so-called linear characteristic range, although it is not linear from a precise point of view. Referring to a peak isn't always correct to get the same intensity, but it is a convenient method for daily use. Later, we will show the difference that comes from this intensity.

Last, the variation of the neutral steering angle before and after the pulse as diagrammed in Fig.3, makes a deflection of steady-state gain in a resulting transfer function. We can reduce the deflection to choose the pre-trigger time (preceding the pulse) adequate1 y , but not always. This problem is easily fixed by excluding very low frequency data from the curve fitting object as stated next.

Fig.3 - Neutral steering angle variation before and after the wulse

CURVE FITTING

The lines connecting symbols in Fig.4 and 5 show the measured transfer functions of a European: 4WD car at 100km/h. As the power spectrum of the steering pulse input have insufficient level over 2Hz, symbols scatter losing their coherence. Curve fittings with the 2 DOF model given in Eq(2) have executed between 0. 2Hz and 1. 9Hz. Their results are also shown in Figs.4 and 5 as smooth lines even over 2Hz.

Almost a1 1 transfer functions properly measured can get good curve fittings as these examples show. Besides yaw velocity and lateral acceleration, side-slip angle would normally be considered, but side-slip angle is very small in a hi-speed linear condition, so it is inconvenient to measure it. In the case

~1 * I l l I 0.1 0.2 0.5 1.0 2.0 3.0

F R E Q U E N C Y ( H z )

Fig.4 - Curve fitting of yaw velocity response (4-wheel-drive car)

0 160

c = I I I I l l - I I 1 , I 0.1 0.2 0.5 1.0 2.0 3.0

F R E Q U E N C Y (Hz)

Fig.5 - Curve fitting of lateral acceleration response - (4-wheel-drive car;)




It should be noticed, however, that successful curve fitting with the 2 DOF model doesn't mean that a real car would be expressed as the 2 DOF model we1 1. The parameters of the model such as wheelbase, vehicle mass, etc., need not be ones of the real car. In addition, the identified parameters from the yaw velocity response may contradict the ones from the lateral acceleration response. The usage of the 2 DOF model is only a technique to extract meaningful parameters from measured data.

FOUR PARAMETE RS

There are many proposals exclusively about numerical evaluation parameters. For example, TB value(7) which is the product of yaw velocity response time by steady state gain of

0. I side-slip angle, and equivalent time lag of

Although calculated values from these equations differ from measured ones, basic characteristics are the same. Some calculated results are shown in Fig.6. In these calcula- tions the vehicle dimensions applied are given in the APPENDIX.

stability factor

- 2 DOF model K = 3 . 8 6 ~ 1 0 - ~ (s2/n2)

- - - 2 DOF model K = 2 . 3 1 ~ 1 0 - ~ (sz/n2)

. . . . 2 DOF model K = 1 . 2 1 ~ 1 0 - ~ ( s 2 h 2 )

yaw velocity, are well known. Steady state gain, natural frequency, and damping of yaw velocity response are most commonly used(8 and 9). Phase delays of yaw velocity response and lateral acceleration response are also common (1 0) .

We have chosen the following four parameters:

a1:steady state gain of yaw velocity response,

fn:natural frequency of yaw velocity response,

:damping of yaw velocity response, d:phase delay at 1Hz of lateral ac-

celeration response, by curve fitting with the 2 DOF model. As representative values for the basic steering response characteristics, it would be adequate to use the test data of 100km/h and 0.49 peak intensity. In order to pick up the four parameters from many can- didates, we had analyzed our vast test data- 3ase. It was proven that these four parameters re stable enough to measure, sensitive to the hanges of vehicle characteristics, and dis- inct from each other.

These parameters are directly expressed n the 2 DOF model:

r , , , , ~ 80 100 120 140 160 180

SPEED I km/hl

I , , , , I 80 100 120 140 I60 180

SPEEO I km/h

Fig.6 - Parameter change vs. vehicle speed

It may be considered to use the stability factor or steady state gain of lateral acceleration response in stead of al. Although they have some common information as indicated in the 2 DOF model, a1 has the most fami 1 iar and direct meaning to the drivers.

It was important not to have adopted phase delay of yaw velocity but 4. From the 2 DOF model, we conclude that side-slip angle derivative defines the difference between these two phase delays. But actually. the phase delay of yaw velocity response is more insensitive than 4. In measured data, lateral acceleration includes the gravity component caused by vehicle roll motion because the ac- celerometer is fixed to the vehicle body. This makes influence on the steady state gain of lateral acceleration, but little on the delay.

PERFORMANCE EXPRESSION USING RHOMBUS

Expressing 4 parameters in a rhombus enables us to understand intuitively, steering response characteristics. Recent data of 20 compact cars at 100km/h are given in Fig. 7. Parameters at 100km/h of almost all normal




passenger c a r s are drawn w i t h i n t h i s char t . Axes have somewhat unusual sca les t o

express t h e average c h a r a c t e r i s t i c s i n a regu la r form. From t h e da ta o f t h e 20 cars, the average "AVE" o f each parameter l oca tes near t h e second g radua t i on f rom the center o f the rhombus and the s tandard d e v i a t i o n , "a " ,

takes one o r h a l f graduat ion.

f i

Fig .7 - Rhombuses o f recent 20 ca rs a t 100km/h

Al though t h e d i r e c t a s s o c i a t i o n o f a numer ical eva lua t i on w i t h a s u b j e c t i v e e v a l u a t i o n may be m is in te rp re ted , we s t a t e t h e f o l l o w i n g : The l e f t a x i s o f a1 denotes "heading easiness". The g r e a t e r ( t o t h e l e f t ) t h e b e t t e r . The upper a x i s o f f n denotes "heading responsiveness". The g rea te r (h igher) t he be t te r . The lower a x i s o f l " d i r e c t i o n a l damping". The greater ( lower) t h e b e t t e r . The r i g h t a x i s of q.5 denotes " f o l l o w i n g c o n t r o l l a b i l i t y " . The smal le r ( t o t h e r i g h t ) t h e b e t t e r .

I f C i s smal l , t h e d r i v e r w i l l n o t be s a t i s f i e d w i t h h i s c a r ' s course convergency over 120km/h. A t lower speed, smal l damping doesn ' t mat ter . But i t should be remembered. t h a t damping used t o become sma l l e r i f t he car speeds up. I f 4 i s smal l , t he emergency han- d l i n g performance w i l l be good, s ince t h e l a t e r a l a c c e l e r a t i o n generates respons ive ly . Small 4 makes a good course convergency w i t h c o r r e c t i o n a l hand1 ing, and l a r g e l makes a good d i r e c t i o n a l convergency w i t h o u t d r i v e r ' s c o r r e c t ion.

Formerly, we s t a t e d t h a t each parameter has a p r e f e r a b l e d i r e c t i o n (great o r sma l l ) , but o f course, excessive s e t t i n g s shou ld be avoided. I t may be n o t i c e d t h a t one parameter wouldn' t o f f e r any bad c h a r a c t e r i s t i c a t t he excessive s e t t i n g , bu t o t h e r parameters would

do some. The reason we d i d n ' t use any recom- mendatory range, i s t h a t t he rhombus expres- s i o n shows the parameters g e t t i n g b e t t e r as w e l l as o t h e r s g e t t i n g worse, s imu l taneous ly , so we can eva lua te i f i t i s excessive f rom t h e balance o f t he parameters c o n t r a d i c t i n g each o ther .

Accord ing ly t h e s i z e o f a rhombus denotes a b i l i t y , and the d i s t o r t i o n denotes tendency. These bas i c p a t t e r n changes a re immediate ly understood by the 2 DOF model. Using some c a l - c u l a t i o n r e s u l t s and measured data, b a s i c p a t - t e r n changes a re shown. Reference v e h i c l e speed i s 100km/h.

F igs .8 and 9 show t h e rhombus s i z e reduc- t i o n accord ing t o t h e i nc reas ing o f t h e v e h i c l e speed.

F i g . 8 - P a t t e r n changes vs. v e h i c l e speed (THe same model as F ig . 6)

2.01 f n ( H z )

f L0.6

F ig .9 - P a t t e r n changes vs. v e h i c l e speed (Front-wheel-dr ive car )

I f a v e h i c l e has a s t r o n g understeer(US) c h a r a c t e r i s t i c . t h e rhombus d e f l e c t s t o t h e r i g h t - h i g h e r d i r e c t i o n . I f a v e h i c l e has a week US c h a r a c t e r i s t i c , t h e rhombus d e f l e c t s t o t h e l e f t - l o w e r d i r e c t i o n . Fig. 10 shows the d i s t o r t i o n o f t h e 2 DOF model accord ing t o t h e




unders tee r -overs tee r (US.OS) change, w h i l e the t o t a l co rne r ing power i s constant . Figs. 1 1 and 12 show the d i s t o r t i o n s o f measured da ta ac- co rd ing t o the USsOS change by f r o n t o r rea r suspension r o l l s teer . I n t h e case o f f r o n t r o l l s teer , f n v a r i e s oppos i te t o t h e case o f the 2 DOF model. The reason i s t h a t t he t o t a l co rne r ing power decreases, w h i l e t h e r o l l unders teer i s n ' t e f f e c t i v e dynamical ly , because the v e h i c l e r o l l mot ion (hence t h e r o l l s teer ) delays t o t h e f r o n t t i r e s l i p angle.

-..,- .. -. . ~ = 1 . 2 1 ~ 1 0 - ~ (sZ/m2)

Fig. 10 - USaOS p a t t e r n changes (2 DOF model) 2 . 0 1 - f n (Hz)

.\.,-pi , , . . . . K=1. 31x10-3 ( s 2 / m z )

t"\,6.6 K = I . oax10-= (~2/.2)

Fig. 1 1 - P a t t e r n changes vs. f r o n t r o l l s tee r

FWD, 100km/h

K = 3 . 5 3 ~ 1 0 - ~ ( s 2 / m 2 )

- - - K = 2 . 8 4 ~ 1 0 - ~ ( s Z / m z )

1 .... . . K=2. 3 4 ~ 1 0 - ~ ( ~ 2 1 ~ 2 ) t 0 . 6

Fig. 12 - P a t t e r n changes vs. r e a r r o l l s t e e r

Fig. 13 shows t h e same d a t a as F ig . 12, p l o t t i n g t h e po les (see Eq(A12)) on the com- p l e x plane. We can see t h a t t h e yaw response loses i t s damping as t h e v e h i c l e speed b u i l d s up, and the s t ronges t US v e h i c l e i s unaccep- t a b l e i n t h e h i g h speed range. The o t h e r hand, F ig. 12 shows t h e damping r e d u c t i o n as w e l l as the f n and t h e d a r e g e t t i n g w e l l .

stability

3.53 x 10-

0 0

f a c t o r -3 SZ /ma

0 2.84 x

0 2.34 X lo-"

- 8 - 4

Fig .13 - Pole l o c a t i o n r e l a t i n g t o t h e r e a r r o l l s t e e r

RELIABILITY OF THE METHOD

To c e r t i f y t h e r e l i a b i l i t y o f t h i s f o u r parameter e v a l u a t i o n method, some exper imenta l r e s u l t s a re shown. The number o f samples was r a t h e r smal l t o assure t h e r e l i a b i l i t y f rom t h e s t a t i s t i c a l v iew po in t .

F i r s t . t h i s t r a n s i e n t response t e s t i s a so -ca l l ed open-loop t e s t , b u t i t shou ld be c e r t i f i e d t h a t t h e r e s u l t i s i n s e n s i t i v e t o t h e t e s t d r i v e r . Tes ts were a c t u a l l y executed by t h e same ca r ( f ron t -whee l -d r i ve ca r ) , on t h e same day, i n t h e same p r o v i n g ground by t h r e e d i s t i n c t d r i v e r s . The normal ized s tandard d e v i a t i o n s (d i v i ded by t h e average) o f t h e 4 parameters were w i t h i n 1. 7%. I f t h e s t e e r i n g i npu ts are w e l l c o n t r o l l e d i n i t s peak g a i n and pu lse width, d i f f e r e n c e s between t e s t d r i v e r s w i 11 be n e g l i g i b l e .

Second, t h e d i f f e r e n c e among c a r s i n t h e same s p e c i f i c a t i o n i s a l s o small . I n t h e case of t he th ree ca rs ( f ron t -whee l -d r i ve car ) produced i n t h e same product l i n e , t h e normal- i z e d standard d e v i a t i o n s were w i t h i n 1.6% i n a l l parameters. But t h e c a r produced as a p ro to t ype i n t h e same s p e c i f i c a t i o n has a b i t b igger d i f f e r e n c e .

Third, t he d i f f e r e n c e by the i n p u t pu l se i n t e n s i t y i s shown i n Fig. 14. f n and 4 are g e t t i n g worse as t h e i n p u t i n t e n s i t y i n -




creases. Th i s s t r o n g n o n l i n e a r i t y i s s t i l l un- der cons idera t ion . As mentioned before, we adopt t he da ta when t h e peak o f t h e l a t e r a l acce le ra t i on i s 0. 49.

2.01 * (Hz) FWD. 100km/h

- 0.25g - - - 0. 3 7 g

Fig.14 - D i f f e r e n c e by i n p u t pu lse i n t e n s i t y

APPLICATIONS

To demonstrate t h e e f f e c t i v e n e s s o f t h e method, two a p p l i c a t i o n s a re shown.

F i r s t , t he e f f e c t o f t h e d i f f e r e n c e s among s im i l a r t i r e s i s shown. The t i r e s . A, B, and C a re i n t he same s i z e and have the same t r e a d pa t te rn . Fig. 15 shows t h e t i r e ' s corner - i n g c h a r a c t e r i s t i c s on t h e t i r e t e s t stand. The d i f f e r e n c e s among t h e t i r e s are ve ry smal l and c lose t o t h e v a r i a t i o n l e v e l o f t h e products. F ig . 16 shows the r e s u l t i n g rhombuses us ing a f r on t -whee l -d r i ve c a r a t 100krn/h. The d i f f e rence o f t h e c o r n e r i n g power i s r e f l e c t e d t o t he rhombus s ize. The d i f f e r e n c e o f t h e phase de lay i s t o o sma l l t o d e t e c t on the car. The s u b j e c t i v e r a t i n g s by some p r o f e s s i o n a l d r i v e r s were 6-, 6, and 6t (10 p o i n t s method), respec t i ve l y .

100km/h

Load=2940N

+ ldeg

A: 175/10SR13 B: 115/70SR13 C: 175/701W13

FWD, 100km/h

Fig. 16 - D i f f e r e n c e by t i r e c o r n e r i n g charac- t e r i s t i c s

Second, t he a p p l i c a t i o n t o t h e development o f 4WS i s shown. The rhombuses i n F ig .17 correspond t o t he v a r i a t i o n o f t h e r e a r s t e e r c o e f f i c i e n t , kp ( the r a t i o o f r e a r s t e e r angle d i v i d e d by f r o n t s t e e r angle) . The t e s t car i s i n s t a l l e d an e l e c t r o - h y d r a u l i c 4WS system. I n t h i s case, t h e o v e r a l l s t e e r i n g r a t i o i s n ' t changed, so a1 i s decreasing as kp increases.

FWD. 100km/h

- - - kp=O. 15 \ / i /' . . . . . kp=O. 27

< wb.6 -.-.- kp=O. 4 0 Fig. 17 - P a t t e r n changes vs. r e a r s t e e r coef - f i c i e n t - (Front-wheel-dr ive car )

From t h e APPENDIX, we o b t a i n the 4WS t r a n s f e r f u n c t i o n s i f t h e r e a r wheel s t e e r f u n c t i o n i s assumed. I f t h e p r o p o r t i o n a l f unc- t i o n app l ies ,

8 r = k p . '

(9 f n and r are t h e same as t h e 2WS, because t h e c h a r a c t e r i s t i c po lynomia l doesn' t change (see Eq (A21)). I n t h e r e a l 4WS system, t h e r e wi 11

cornering force phase detay e x i s t some dynamics even i f n o t i n t e n t i o n a l l y

a t 1Hz (deg) caused by mechanical problems. These dynamics i n f l u e n c e the r e s u l t a n t performance o f t h e

F ig .15 - Corner ing c h a r a c t e r i s t i c s o f t he t r e e v e h i c l e c l e a r l y . For example, t h e open-loop t i r e s 4WS c o n t r o l method which ma in ta ins t h e s lue -




slip angle zero (see Eq (A22)), gives a excel- lent performance.

EST1 MAT10 N US1 NG FULL VEHICLE MODEL

This evaluation method is useful to designers as they have wanted to estimate the four parameters by calculation. To satisfy the designers needs, the simulation model should contain precise structure descriptions. For this objective, the full vehicle model written in ADAMS(8 and 9) is suitable. We had developed a pre-processing system to construct a large-scaled vehicle model named "Digital Vehicle System". Using this system, designers easily can get a precise model of their developing suspension system, and full- vehicle. Fig.18 shows one of our vehicle models which has about 80 degrees of freedom.

Fig. 18 Full-vehicle model

CONCLUSIONS

It is not an easy task evaluating vehicle handling characteristics. We adopted four parameters extracted from transient response data by curve fitting. Denoting the parameters in a rhombus, gives good information contain- ing basic and major vehicle characteristics.

The parameters are insufficient to evaluate the overall vehicle handling charac- terist ics. However, from our experiences over several years, the huge task of improving vehicle handling has turned into simple routine. The key point of this method is making it possible to see the parameters that contradict each other at a glance.

AC KN OWLE DG EM E NT

The application to the tire characteristics was made possible by support from Toyo Tire L Rubber Co., LTD. The authors wish to gratefully acknowledge the contributions of H. Yashima and S. Nishihata.

REFE RE NC ES

1. T. Mimuro and S. Takeuchi, "CAE for Vehicle Dynamics, " JSAE Review Vo1. 9, No. 3, Ju 1 y 1988.

2. T. Tanaka, et. al. "Active Control Technology for Passenger Car, " XXII FISITA. September 1988.

3. "Transient Response Test Procedure (Pulse Input) for Passenger Cars, " Japanese Automobi le Standard, 2-1 10, Mar. , 1983.

4. "Road Vehicles - Lateral Transient Response Test Methods," International Stand- ard, ISO/TC22, IS0 7401. 1988.

5. M. K. Verma and W. L. Shepard, "Comparison of Transient Response Test Procedures for Motor Vehicles," SAE Paper 81 0807.

6. R. L. Ni songer and P. S. Fancher. "Experimental Examination of Transient Direc- tional Response Tests, " SAE Paper 810808.

7. B. Richter. "Driving Simulator Studies: The Influence of Vehicle Parameters on Safety in Critical Situations", SAE Paper 741 105.

8. D. McRuer and R. Klein, "Effects of Automobile Steering Characteristics on Driver/Vehicle Performance for Regulation Tasks, " SAE Paper 0760778.

9. D. H. Wei r and R. J. DiMarc, "Correlation and Evaluation of Driver/Vehicle Directional Handling Data," SAE Paper 780010.

10. S. Sano et. al. "Effect of Vehicle Response Characteristics and Driver's Skill Level on Task Performance and Subjective Rating, " 8th International Conference on Ex- perimental Safety Vehicles, 1980.

SUMMARY OF THE 2 DOF MODEL

Here, the 2 DOF model of 4WS, which is more general than 2WS, is introduced. In the text, we used only 2WS, making the rear steer angle zero.

According to Fig. A1 and Table Al, and as- suming that 6f, 6r, af, ar, and are small, the differential equations of motion are:




Iz-$=lr Fyr-lr F y r

The side forces. Fyf and Fyr are:

The front and rear tire slip angles are:

From these equations, Eq (A7) is created through the Laplace transformation.

I 2 (11 Kc -1r Kr)

mV+ mV.s+2 ( K r + K r ) v

2 ( lr2Kr+lr2Kr) I Z - s + 2 (11 Kf -1r Kr

v r -8 ,- 7

Fig.Al - 2 DOF model

I $ I rad/s I Yaw velocity I I I B 1 rad I Side-slip angle B%v,/v, I I ( a, I ./s2 1 Lateral acceleratim I I I BH 1 rad I Steering wheel angle I I The state variables of Eq (A7) are 4 and B ,

and the two independent input variables are 6f and 6r. The determinant and the adjoint matrix of the system matrix are developed. These are:

I Br. 6, 1 rad ] Steer angle, f m v r e a r I I IX; Tire s l i p angle

Forrard/latcral velocity

Vehicle speed V=JvX2+ vV2 100 (k/h)

Vehicle MSS 1200 (k)

I 1, 1 kg.* 1 Yaw -t of inertia ( 1470 (Itp.6) (

1 I , 1 1 m I Distance fm each axle to C. C 1 1.0/1.6 (m) I

2 (lr2Kr+lr2Kr) 2 (Ir Kt - I r Kr) 1, -s+ -mV-

v

(A91

Kr. K,

i .

The characteristic polynomial and its poles, A 1 and 12, can be directly got from Eq(A8).

A p o l ( s ) = d e t ( s )

4 l K r K r In the above. K is the so-called stability factor defined in Table A2. Using Eq (A8 and A 9 ) , E q (A71 can be solved for its state variables.

N/rad

1

As the result, we can get yaw velocity response, side-slip angle response, and

Ewivalent cornering m r

Overall steering ratio

365W/56150(N/rad)

17.0 (1)




lateral acceleration response.

. A

Newly defined coefficients are also given in Table A2. Be careful that Tr is negative and Bf and rf turn to negative if the speed builds up. In the case of 2WS, using the relations;

8 , = 0 , G r = C i ~ / i . (A1 6)

we get the transfer functions:

In the case of the simple 4WS;

B r = k p . Q ~ (A201 where kp is a constant, the transfer function of yaw velocity is:

The open-loop 4WS control which maintain side- slip angle zero, is introduced from Eq(A14) ;

lateral acceleration responses have the same dynamics f rom Eq (A1 5).

Notation 1 Coefficient (hit) Definition

m (1, K,-lc Kt) K Stability factor (s2/#)

2 1'KrKr

Steadv state min of v

Steady state gain of 2Ie1Kr-IrmV2 21t1Kr+l , rnVz B t . B, side-siip angle response, -A , . A

f m w r e a r (1) 2 1 K , V 2 1 K r V

Lead time ccnstant of Tr, T, yaw velocity response. I I -

Zr. Zr

Wn

I Lead ti= cautant of side-slip angle r e s m e .

f m t / r e s r (s)

Natural angular frequency (raws)

( f . : Natural frequency) f .=0./2 z oh)

t *ma ( l /d 1 Kt+K, Ir2Kr+1,'Kr - (- + ) (C : Daaping ratio (1)) V m IZ

Using this control method, yaw velocity and




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4param Transient Response (1)