IEEE TRANSACTIONS ON MAN-MACHINE SYSTEMS, VOL. MMS-11, NO. 3, SEPTEMBER 1970

3) The pilots were able to execute better performancein minimizing the time outside the glide-slope errorlimits with the monitor display compared to the pro-jector display. However, for the lateral localizer per-formance, the error was smaller for the projectordisplay than for the monitor display.

4) Degradation of the monitor display resolutiontended to reduce the touchdown distance and slightlyincreased the corresponding standard deviation. Inaddition, the landings were made predominantly tothe right of the center line for the black/white moni-tor display and with nearly twice the standard de-viation as obtained for the color monitor display.

5) The relationship between standard deviations ofrate of descent and touchdown distance for all thetests with the projector and monitor displays sug-gests that a decrease in the rate-of-descent devia-tion will be offset by an increase in the touchdowndistance deviation.

6) The pilots showed more criticism of the black/whiteconfiguration and favored the use of color for bothdisplays. Their comments were also more favorabletoward the monitor display, because of better picturequality and depth perception than was evident inthe projector display.

7) In view of the sum of the performance measureresults, and particularly from the confirmation ofthe opinions by trained commercial pilots, a colli-mated monitor appears to be a more satisfying de-vice than the projector for this model visual simula-tion television system.

REFERENCES[1] T. F. Buddenhagen and M. P. Wolpin, "A study of visual

simulation techniques for astronautical flight training," WrightAir Dev. Div., Tech. Rept. 60-756, March 1961.

[2] T. F. Buddenhagen, A. B. Johnson, S. C. Stephan, and M. P.Wolpin, "Development of visual simulation techniques forastronautical flight training," AMRL-TDR-63-54, vol. 1, June1963.

[3] B. D. Armstrong, "Difficulties with the simulation of aircraftlandings," Royal Aircraft Establishment, Tech. Rept. TR68116, May 1968.

[4] C. T. Jackson, Jr., and C. T. Snyder, "Validation of a researchsimulator for investigating jet transport handling qualitiesand airworthiness criteria during takeoff," NASA TN D-3565,October 1966.

[5] J. S. Bendat and A. G. Piersol, Measurement and Analysis ofRandom Data. New York: Wiley, 1966.

[6] J. W. Stickle and N. S. Silsby, "An investigation of landing-contact conditions for a large turbojet transport during rou-tine daylight operations," NASA TN D-527, 1960.

[7] "Operational experiences of turbine-powered commercial trans-port airplanes," Langley Airworthiness Branch, NASA TND-1392, 1962.

[8] D. R. Geoffrian and V. M. Kibardin, "Statistical presentationof operational landing parameters for transport jet airplanes,"Federal Aviation Agency, Flight Standards Service Release470, August 8, 1962.

Variable Perception Time in Car Following and

Its Effect on Model Stability

LUCIEN DUCKSTEIN, ERNEST A. UNWIN, AND EUGENE T. BOYD

Abstract-The functional variation of perception time T incar following is studied for the purpose of introducing a variabletime delay into a previously proposed nonlinear model. The basichypothesis is that the ratio of a just noticeable difference in visualangle (AO) to the visual angle (0) is a constant. This hypothesisleads to a model in which perception time T is proportional torelative spacing and inversely proportional to relative speed.Experimental results agree very closely with the model for negativerelative speeds between 3 and 18 ft/s. A second-order approximationis used to explain behavior for relative speeds of less than 3 ft/s inabsolute value. This second-order approximation also agrees wellwith data taken previously with relative acceleration varyingbetween 1 and 4.7 ft/s2 and initial relative speed between -1.9 and

Manuscript received October 15, 1969; revised May 3, 1970.Parts of the results of this paper were presented at the 32ndORSA Natl. Meeting in Chicago, Ill., November 1967.L. Duckstein and E. T. Boyd are with the Department of Sys-

tems Engineering, the University of Arizona, Tucson, Ariz.E. A. Unwin was with the Department of Systems Engineering,

the University of Arizona, Tucson, Ariz. He is presently withSylvania Electronic Systems, Mountain View, Calif.

2.9 ft/s. A brief discussion of "ideal" following distance b is pre-sented to clarify the stability analysis. The introduction of a variabletime delay and ideal following distance into the proposed car-following model changes the size of the minimum asymptoticstability region but not the basic properties of the model. Theword "asymptotic" is used in the dynamic stability sense as in theclassical control theory literature, not in the sense of platoonstability.

INTRODUCTION

N a previous paper [10] it was proposed that a car-following model that exhibited asymptotic stabilitycharacteristics be written as:

a"(t + T1) = x"(t + T1) - a(t)+b (z'(t) + ez(t)) (1)where x,"(t) is the position of the nth vehicle of a platoonat time t in feet per second, T1 is a time lag in seconds,a is a sensitivity coefficient in feet per second, b is an

149

150

"ideal" following distance in feet, e is a stability char-acteristic parameter in second-', and

z(t) = xn(t) - x.r,,(t) - b.

The time lag T1 is composed of four elements:

IEEE TRANSACTIONS ON MAN-MACHINE SYSTEMS, SEPTEMBER 1970

DETECTION OF RELATIVE VELOCITY

Brown [2], in analyzing many experiments related tothe detection of angular motion, found that a Weber-likeratio seemed to hold for a certain range of angular ve-locities:

T=T + Td + Tr + T,,

where

T perception time of the following driver,Td decision time,Tr reaction time,T, mechanical delay of the following car.

Note that the perception time T is the time elapsed be-tween the instant a change of relative velocity (stimulus)is applied and the instant the following driver signals bypressing a button or a pedal that a change has been per-ceived, less a simple reaction time.With e-0, (1) is the reciprocal-spacing model [4],

which does not possess the asymptotic stability property.In this paper, functional forms for the perception time

T7 are derived and validated by experimental results. Sta-bility of the resulting model is then examined.

THRESHOLD FOR PERCEPTION OF CHANGE

Let 9 represent the visual angle at the following driv-er's position projected by the lead vehicle. If the widthof the lead car is B and the distance from the back ofthe lead car to the following driver is y, then

GBtan - = B.2 2yFor small0,,

^}~By

Now consider the rate of change of subtended visualangle:

d_dt

Thus,

B,co--2 Yy

y is the distance between the nth and the (n + 1) st ve-hicle; in terms of car-following variables:

y ' Xn - Xn+1;then approximately,

-B (x -nx+1 (2)(Xn Xn+ l)

This result is well known [8]. The intent now is to de-termine which of the stimuli 8 or o is the pertinent onein detection of relative change between lead and follow-ing car.

Aco-= 0.1co

0.002 <o <0.35,

where ,w is angular velocity in radians per second and Aois a just-noticeable difference (JND) in the angular ve-locity w. It should be emphasized that this result is basedon data in which the typical response is binary: differ-ence detected or not detected. This theory does not con-sider perception time of relative velocity, which is aparameter of primary interest in car-following theory.

Michaels and Cozan [7] determined that under care-fully controlled conditions, the lower bound for percep-tion of motion (an absolute threshold) is approximately0.0006 rad/s. Using this threshold and (2) (with a lead-car width of 6 feet), the relationship between minimaldetectable relative velocities V* as a function of follow-ing distance D is

V* = 0.0001 D2.

If Brown's results were combined with (2), detectiontime could be hypothesized, but this would not includerelative speeds smaller than those that give an angularvelocity of 0.002 rad/s.

In order to overcome this anomaly in the theory, ex-periments were performed [1], [6] to test the followinghypothesis on the perception mechanism.

Basic HypothesisOver some range of the visual angle 0, there exists a

Weber ratio

(3)0- h

where AG is a JND in visual angle. The advantage ofsuch a model would be its independence of the conceptof absolute threshold, since G can not tend to zero or thelead car would not be visible and car-following theorywould not be appropriate.

FUNCTIONAL VARIATION OF TIME DELAY

For a JND AG, assume

0(t + T) =0(t) + AO.Using (3), one obtains:

0(t + T) (1 + uh)0(t)with

u = 1

u = -1

For small AO,

AG > 0, i.e., y'(t) < 0,

AO <0, i.e., y(t) > 0.(4)

TABLE IMEAN PERCEPTION TIMES OF 12 SUBJECTS

Distance y (feet)

50 90 115

y y y

Relative Velocity Y Mean Y Mean Y Meany'(t) (ft/s) (second) (second) Variance (second) (second) Variance (second) (second) Variance

-1.47 34.0 5.51 7.20 61.2 7.12 8.69 78.2 10.6 15.0-2.20 22.7 4.18 7.02 40.9 5.33 16.2 52.3 6.98 9.18-2.99 16.7 1.20 0.197 30.1 1.72 0.164 38.5 2.61 0.304-4.40 11.4 0.738 0.113 20.4 1.35 0.267 26.2 1.77 0.121-5.87 8.52 0.627 0.0410 15.3 0.964 0.135 19.6 1.27 0.174-8.80 5.68 0.321 0.0252 10.2 0.693 0.0164 13.1 0.875 0.206-11.7 4.27 0.221 0.0289 7.70 0.408 0.0099 9.82 0.633 0.0179-17.6 2.84 0.155 0.0164 5.11 0.239 0.0089 6.55 0.463 0.0352

Takingyields

0(t) = By(t)

0(t + T) y(t + T)

the ratio of these two expressions and using (4)

y(t + T) = 1+ hy(t). (5)

If the relative velocity is constant over the time inter-val (t, t + T) (or if T is sufficiently small for a first-order Taylor series approximation to be reasonable, sayT < TM), then y (t + T) can be approximated as:

_ 10

8

0 60.

.2

1oO n

,WYy=50ty=0ty=115ft

- 2 -4 -6 -R -10 -12 -14 -16 -18Relative Velocity y' (ft/sec)

Fig. 1. Observed perception time T versus relative velocity y.

y(t + T) = y(t) + Ty'(t). (6)An approximate value of TM must be determined experi-mentally.

Equations (5) and (6) yield the following expressionfor the time delay T:

T_= uh y(t) = _k y(t)1 + uh y'(t) y'(t) (7)

with

uh- 1 -uh,>1 + uh

10.0

4.04-w

P 2flz0

1.0V

cc.waI.

w 0.4.ccCM 0A8 0.2

(8)

In order to verify (7), tests were conducted with mov-ing pictures as described in the Appendix. Only negativevelocities ranging from -1.47 to -17.6 ft/s, were usedin the analysis, with following distance values of 50, 90,and 115 feet. A plot of the observed perception times isshown in Fig. 1. The means and variances of the mea-sured times T and the ratios -y/y' are shown in Table I.A plot of T versus -y/y' is shown in Fig. 2. A linearregression of observed perception times T was fitted onthe ratio -y/y' for Iy'l > 2.99. The reason for excludingthe two lower velocities will be given later. The equationof the regression line is

0.1

lyl,' 2.99 ft/secox

/

I9* y=0f

IXXy9fI=|f

/

i

2 4 10 20-y/y' (sec)

40 100

Fig. 2. Regression of observed perception times on the ratio-Y/Y .

Note that the line shown in Fig. 2 does not exactlyrepresent this regression line, since a log-log scale wasused to obtain a better visibility of data points. However,the value of a is so close to zero that the log-log plotis linear. The calculated values of the coefficients are

T= -k-f+a.y k = 0.0660 a = -0.0202.

I~~~~~,LL-

DUCKSTEIN et al.: PERCEPTION TIME IN CAR FOLLOWING 151

1-

al

152

The 95 percent confidence limits are, respectively,

0.061 < k < 0.071,

-0.101 < a < 0.00604.The standard deviation about the fitted line is s = 0.0905second. The correlation coefficient between T and -y/y'is r = 0.99 with 16 degrees of freedom. Using (8), theWeber ratio constant is found to be k = 0.071.

CASE OF SMALL RELATIVE SPEED Y'Why was (7) such a poor predictor of observed percep-

tion times at relative velocities y'(t) of -1.47 and -2.20ft/s, whereas it seemed to give excellent results forly'(t) .> 2.99 ft/s? (See Fig. 2.) The reason will appearimmediately if the assumption under which (6) wasderived is recalled, namely, that y'(t) should be constantin the interval (t, t + T) or that T < Tm, so that a first-order Taylor series approximation be valid. It appearsfrom the data that (6) is valid for 3 < ly'(t) .< 18 ft/s,where the detection times T are small. In Fig. 1, one canobserve that the rate of change of T with y'(t) changessharply around y'(t) = 3 ft/s, which corresponds to avalue of T between 1.20 and 2.6 seconds.

This suggests that the upper limit TM of T be made afunction of the relative distance y; this function could betaken as TM = Cy. The constant C is determined byapplying this formula to the last point that fits the linearrelation in Fig. 2, e.g., T = 2.6 and y = 115. This yieldsTM = 0.022 y, for a value of y such that car-followingtheory remains adequate, for example, y < 120 feet.

For T > Tm, it may be that a long-term memoryprocess comes into play rather than a short-term memoryprocess. At any rate, the linear approximation for y (t +T) is not valid any longer. Then consider the second-order approximation

T2y(t + T) = y(t) + Ty'(t) + - y"l(t).

Using (5):

yQ( + T) = y1 + uh'

one obtains the second-order equation in T:

()T2 + y'T + uh Y=O. (10)

Consider the case y' < 0, so that u = 1 and

k = uh - 0.066.- ~~1+ uh

The sensitivity of T to a' very small relative accelerationis illustrated in Fig. 3, where the function

T= f(Y(t)

given by (10) with y' as a parameter is represented fory" > 0, y = 60 feet, y' = -0.66, -1.32, and -3.3 ft/s.

IEEE TRANSACTIONS ON MAN-MACHINE SYSTEMS, SEPTEMBER 1970

T sec.

10

5

_ _-

y= 60ft

/ y'= -.66 ft/sec

y'= -.32 ft/sec

y'= -3.3 ft/sec

0.1 0.2 0.4 0.8

2,y ftse2f ft/sec

1.6 3.2 6.4 128 25.6 512 102.4

Fig. 3. Theoretical perception time T versus relative accelerationy" and relative speed y'

Note that a semilogarithmic plot had to be used to give aclear picture of the dependence of T on y" and y'. Also,the right side of Fig. 3 ((y"/2) > 10 or 12) certainlydoes not apply to ordinary car-following situations.

Fig. 3 shows how a small relative acceleration makesthe difference between a large and a small perceptiontime T; hence, the variance in Table I for y' = -1.47and y' = -2.20.Another explanation, suggested by a reviewer of this

paper; is that the sense of personal risk to the driver isnot great when y' is small, so that more noise is intro-duced into the response.

CASE OF HIGH ACCELERATIONS

What is the functional relationship between y, y', y",and T in the case when the relative acceleration y" isthe dominant factor? In order to test if (10) remainsvalid in this case, data obtained earlier by Whitty andDuckstein [11] and shown in Table II were used to com-pute the expression

K - (0.5 y"T2 + y'T)y-'.

Before presenting the results, however, two importantdifferences between Whitty's [11] and Boyd's ([1], Ap-pendix A) studies should be pointed out.

1) Whitty's experiments were designed to investigatethe detection of acceleration by the following driver; asa result, the relative speed y' was kept as close as pos-sible to zero in the initial conditions and the accelerationy" was made as large as the lead car capabilities wouldallow; this operating procedure would hardly be adequateto test a concept of JND in visual angle.

2) In contrast to Boyd's laboratory tests, Whitty con-ducted tests on highways where environmental factorsare always much more difficult to control than in a labo-ratory. In particular, inconsistencies could be foundwithin subjects. For example, consider in Table II sub-ject 2 who detected an acceleration of 4.7 ft/s2 in 1.4seconds and a considerably smaller acceleration of 2.4ft/s2 in a shorter time of 1.2 seconds! The only way to

0 1f 1l

TABLE IIWHITTY'S EXPERIMENTAL DATA

Initial Lead car Relative Initial PerceptionRelative Speed y' Initial Speed x,' Acceleration Relative Distance Time T

Subject (ft/8) (ft/8) y" (ft/s2) y (feet) (seconds)

1 -0.88 35.0 3.7 59.5 1.62 -0.88 37.0 4.7 50.7 1.43 0.00 29.5 3.8 63.3 1.54 0.00 32.2 3.7 33.5 1.65 0.00 39.5 4.4 46.8 1.56 1.05 37.0 4.7 35.2 1.87 1.05 37.0 4.5 52.4 1.98 0.00 37.9 4.4 33.5 2.09 -0.88 37.4 3.2 60.7 1.911* 0.88 36.0 3.4 63.3 2.212 0.88 37.9 1.7 54.3 1.713 0.00 37.4 3.2 49.1 1.814 0.88 36.0 3.8 50.7 1.51 -0.88 37.4 1.6 54.3 3.72 0.00 34.1 2.4 41.9 1.23 2.93 33.2 1.6 64.7 1.84 -1.91 34.1 1.0 60.7 4.75 -1.91 39.9 2.7 50.7 2.86 -1.05 36.0 2.0 53.3 2.87 0.00 37.9 1.7 52.4 2.88 -1.05 40.6 1.7 45.3 3.39 1.05 37.0 2.0 75.7 2.8

11* 0.00 36.0 1.2 56.2 3.212 1.91 39.7 1.5 52.4 3.213 -0.88 37.4 1.5 50.7 2.614 0.88 38.7 1.0 56.2 4.7

* Information pertaining to subject 10 was partially lost due to a jammed recorder.

minimize the occurrence of such inconsistencies is to re-peat the road tests about ten times and take an averagevalue for each individual; clearly, this is an expensive aswell as a difficult proposition, because road tests are noteasily reproduced. Whitty's data shown in Table II areactual measurements; no averages have been computed.

Since the hypothesis here is that perception time isroughly proportional to K, and since perception time onthe road in nonemergency conditions is larger than in thelaboratory, the values of K computed from the data inTable II are expected to be larger than the values foundin this study.More precisely, (8) yields kc = 0.071, hence,

k = 0.066 y' < 0 (this study),kI = 0.076 y' > 0.

Thus K should be smaller for negative relative speedthan for positive ones so that one would have:

K > 0.066 y'<0

K > 0.076 y' > 0.With all due precautions taken, the computation of

K = (0.5 y"T2 + y'T) y'l yields the following results.

i relative speed mean value Ki of K sample variance s2

1 y' > 0 0.177 0.005012 y' = 0 0.112 0.002483 y' < 0 0.0828 0.00126

Using a t test with both variances unknown and notassumed equal, the test statistic for the null hypothesisH0: K3= K, is found to be t = 3.57 with 13 degrees offreedom. Since the rejection value at the 5 percent levelis t43,0.95 = 1.77, the null hypothesis is strongly rejectedand the difference between the values of K for y' > 0and y' < 0 is found highly significant. The 95 percentconfidence intervals on K3 and I., are, respectively,

0.126 <K, < 0.228

0.0572 <K3 < 0.1084

yI > 0

y' <0.

These results are certainly within an acceptable range ofvalues especially if the deviations from the assumptionleading to (10) are taken into consideration. In fact, themean value K3 = 0.083 obtained for y' < 0 is relativelyclose to the value k = 0.066 obtained in Boyd's experi-ments [1]; it may also be noted that the 0.95 confidenceinterval on K3 includes k.Thus it appears that (10) may give reasonable predic-

tions of T for a positive relative acceleration y" < 5 ft/s2and small relative velocities y' in the approximate interval(-2, +2) ft/s. Further experimental verification of thesepreliminary findings under carefully controlled conditionsare certainly in order.

In summary, the concept of a Weber ratio for the visualangle 0 explains in a reasonable fashion the variation of Twith y, y', and y". Furthermore, when y and y' are suchthat the influence of y" is negligible (Table I for ly'l >2.99 ft/s), an excellent fit is obtained between observedvalues of T and theoretical values given by (7).

DUCKSTEMet al.: .PERCEPTION TIME IN CAR FOLLOWING 153

154

"IDEAL" FOLLOWING DISTANCE

The car-following model under consideration includes aparameter b called the "ideal" following distance. Thereason for the introduction of this quantity and thechange of variable associated is to allow the formulationof stability criteria in the usual dynamic sense [9]. Sincethe parameter is inherent to the model, some discussionof model behavior related to previous hypotheses aboutfollowing distances (even though these hypotheses aresubject to considerable discussion) seems pertinent.The parameter b, for a given individual, seems to depend

mainly on the absolute speed x'(t) of the lead vehicle; insteady-state x'(t) is also the speed of the following vehicle.Let, as a first approximation,

b = bo + b1x'(t). (11)

Experimental evidence in support of such a relationshipwas found by Daou [3] who proposed the followingaverage values:

bo = 34 feet,

b== 1.4 second-'.

Experiments performed concurrently with the tests de-scribed earlier gave the average values [10]:

bo = 18 feet,

b, = 1.3 second-'.

The difference between the intercept values bo is prob-ably caused by environmental factors. Daou derived hisdata from measurements taken in the Lincoln Tunnel inNew York, whereas the second set of measurements wasmade in the laboratory using movies taken in a desertenvironment. On the other hand, the agreement betweenthe values of the slopes b1 is quite good. Note that ineither case the average ideal following distance (fromfront to front) is larger than the safe distance recom-mended by many state codes, namely, one car length(from back to front) per 10 mph, which is equivalentto 15 feet per 15 ft/s of speed or bo = 15, b1 = 1.The variation of b from driver to driver has yet to be

determined. A log-normal distribution of b can not berejected [5]; on the other hand, it is not impossible thatthe distribution of gaps in a platoon is bimodal. Thispoint requires further empirical study under various ex-perimental conditions. In the meantime, the mean valueof b as given by (11) can be used for stability studies.Varying b is an alternative to including a power of speedin the sensitivity coefficient, as suggested in the presenta-tion of a general car-following model [4].

STABILITY CONSIDERATION WITH VARYING TIME DELAYAND VARYING FOLLOWING DISTANCE

In [10] using the phase plane, it was found that aminimum region for asymptotic stability is given by theinequality

IEEE TRANSACTIONS ON MAN-MACHINE SYSTEMS, SEIPTMBM 1970

(12)with

Z=z(t), Z2 = z'(t)and

2 (b - aT)1 + T2 (13)

Using (7) or a simple transformation of (10) for T, and(11) for b, R can be calculated for any given drivingsituation. Strictly speaking, since both sides of the aboveinequality depend upon z (t) and z'(t), it makes no senseto vary R as a function of T (z, z', z") or b (z'). If how-ever, such a variation is studied in the sense of smalldisturbances, i.e., only a small portion of the curve isused at one time, useful information can be obtained.

In this spirit, consider the variation of R first versus Tthen versus b.

In the first case, let

b = 30 + 1.33 xl(t)

a = 20 ft/s

nx = 36, 48, and 72 ft/s.

The resulting variation of the region of convergence Rversus T and x' is shown in Fig. 4. The points whereT = TM, that is, the limits where T must be obtainedfrom (10) instead of (7) are also represented in Fig. 4.Below the curves, the model is asymptotically stable;above, no sufficient condition for stability has been found.

In the second case, assume that T < TM, so that (7) canbe used; also, that initial conditions correspond to theequilibrium point z = 0, or y = b, so that

T =yI

Substituting this expression into (13) yields

b(l _)ak

1+(y)

The curves representing this function are shown inFig. 5 with a = 20 ft/s, k = -066, y' = -3.3 and -6.6ft/s. As in Fig. 4, the model is asymptotically stable be-low the curves; it may be unstable above them.

Proceeding formally, the state trajectories of themodel can be computed from a previously derived result[11]

(z - Tz' + b)x"'- a(ez + (1 -ET)x')z - Tz' + b - aT (14)

To account for a variable T, the current value of Tis evaluated along the trajectories and can thus beobtained in a routine manner.

It is also possible to substitute (7) and (8) into (14)to obtain

DUCKSTEIN et a.l.: PERCEPTION TIME IN CAR FOLLOWING

R ft

120

100 x"= 72ft

I j/80/

x= 48ft

60 4

40->35fASYMPTOTICALLY

STABLE

20-

T sec

01 2 3 4 5 6

Fig. 4. Minimum radius R of stability region versus perceptiontime T and absolute speed x.

(1 + k)(z + b)z'x" - a[(ez + z')z' - (z + b)Ekz']zi" k)z' - ak'}(z + b)

T <0.022(z + b).

This method was tried and proved to be more inefficientcomputationally than using (14) with a step-by-stepevaluation of T along trajectories.The results found earlier with a constant time delay T

remain valid; a neighborhood near the origin whereasymptotic stability exists even in presence of persistentdisturbances can be found [9]. The introduction of a

variable ideal following distance into (14) should onlyenhance the stability property, since b increases withthe mean velocity of the lead car x'. However, such a

manipulation yields an unwieldly model for a relativelysmall gain.

- ~~~CONCLUSIONS1) The variation of the perception time with relative

distance, speed, and acceleration can be explained byhypothesizing a Weber ratio for the visual angle of a

following driver. Such a model does not depend on thethreshold concept.

2) Experimental data substantiate the model whereperception time T is obtained by a first-order Taylorapproximation of the relative distance at time t + Tfor relative speeds between -3 and -19 ft/s. The Weberratio constant is found to be approximately 0.07.

3) A second-order Taylor approximation of the rel-ative distance at time t + T gives a plausible explanationfor the large mean and variance of observed T obtainedfor ly'l < 3 ft/s; it also explains data taken with -1.91< y' < 2.93 ft/s and 1 < y" < 4.7 ft/s2.-4) For any given follow-the-leader driving situation,

a lower bound on the region of stability R of the model

R ft60

MAY BE UNSTABLE

50- y=-6.6 ft/sec, T- 0.6sec

40-

30

20/ y=- 3.3 ft/sec, T-1.2sec

10 // ~~ASYMPTOTICALLY STABLE

b ft0 20 40 60 80 100 120 140

Fig. 5. Minimum radius R of stability region versus ideal spac-ing b and relative speed y'.

in the phase plane can be computed. Graphs to be usedfor small disturbances only have been plotted to illus-trate how R varies as a function of the parameters andvariables of the model.

5) Variable time delay and variable ideal followingdistance do not appreciably change the asymptoticstability properties of the model.

APPENDIX

BOYD'S EXPERIMENTAL PROCEDURE [ 1]

Twelve subjects were shown a car-following film inthe laboratory. Their driving experience ranged from2 to 30 years with a mean of 10 years. The film hadbeen taken and edited especially for this experiment. Itshowed a series of shots of a lead car taken as thedriver of the following vehicle would see it from hisseat. Scene durations varied between 10 and 20 secondsso that subjects had ample time to detect any changein following distance. Scenes with relative speed positive,negative, and zero were randomly presented and sep-arated by 2-second intervals of blue sky projection.Care was taken to reproduce driving conditions as wellas possible; thus, the subjects could hear real noise, andmore important, the projection room was arranged sothat the lead car pictures would subtend the sameretinal angle as that car would on the road.By means of two pedals, simulating a gas and a brake

pedal, respectively, subjects would indicate whether thelead car on the screen was going faster, slower, or atthe same speed as the following car that they were sup-posedly driving. A slight depression of the gas pedalindicated the same speed; a strong depression, a fasterspeed; a change in pedal, a slower speed of the leadvehicle. Subjects could correct errors and were familiar-ized with the testing procedure with another film. Theywere told that all three types of responses were equallyimportant.

Basic reaction time of the subjects was measuredseparately and subtracted from the data in this paper.

155

156

ACKNOWLEDGMENT

The authors would like to thank the referees for severalclarifying comments.

REFERENCES

[1] T. E. Boyd, "Perceptual latency in car-following for a con-stant relative velocity," M.S. thesis, University of Arizona,Tucson, 1967.

[2] R. H. Brown, "Weber ratio for visual discrimination ofvelocity," Science, vol. 131, pp. 1809-1810, 1960.

[31 A. Daou, "On flow within platoons," Aust. Road Res. J., vol.2, no. 7, pp. 4-13, 1966.

[4] D. C. Gazis, R. Herman, and R^. W. Rothery, "Nonlinearfollow-the-leader models of traffic flow," Oper. Res., vol. 9,pp. 545-567, 1961.

IEEE TRANSACTIONS ON MAN-MACHINE SEYSTEMS, SEPTEMBER 1970

[51 I. Greenberg, "The log-normal distribution of headways.'Aust. Road Res. J. vol. 2, no. 7, pp. 14-18, 1966.

[6] E. F. Imler, "Motion pictures for velocity change detectiontesting in car-following," M.S. thesis, University of Arizona.Tucson, 1967.

[7] R. M. Michaels and L. W. Cozan, "Perceptual and fieldfactors causing lateral displacement," Highway Res. Rec., vol.25, pp. 1-13, 1963.

[8] L. A. Pipes, "Car-following models and the fundamental dia-gram of road traffic," Transport Res., vol. 1, no. 1, pp. 21-29,1967.

[9] E. A. Unwin, "Stability analysis of saturated traffic systems,"Ph.D. dissertation, University of Arizona, Tucson, 1967.

[10] E. A. Unwin and L. Duckstein, "Stability of reciprocal spac-ing type car-following models," Transport Sci. vol. 1, no. 2,pp. 95-108, 1967.

[111 W. Whitty and L. Duckstein, "Detection of acceleration incar-following," preprint of a paper presented at the 31stORSA Natl. Meeting, New York, N. Y., June 1967.

Communications

Computer-Assisted Fingerprint Encoding and Classifica-tion

Abstract-A proposed man-machine system for encoding finger-print ridge characteristics is described. The fundamental conceptunderlying the proposed system is to use an operator to recognizethe ridge characteristics and to impart to a computer the abilityto manipulate and compare the digitized locations and directionsof these characteristics for single-fingerprint classification.The proposed system and encoding schemes were simulated

using a RAND tablet and an IBM 1800 computer. Sample inputprints were encoded and stored on a magnetic tape. Experimentalresults on human factors and multiple-impression file searchesillustrate the feasibility of computer-assisted fingerprint encodingand classification.

I. INTRODUCTION

Fingerprints have been used systematically as a means ofestablishing personal identification for nearly a hundred years.The ten-finger file has been accepted by almost every countryto be the single most reliable and convenient way of generalpersonal identification [6], [7]. However as populations grow,the task of searching fingerprint files becomes increasingly moredifficult. In the United States, the Federal Bureau of Investi-gation has about 175 million ten-finger cards on file. In additionto the FBI file, almost all police departments maintain theirown extensive files. Since file-searching procedures are at pres-ent largely manual, obviously a great deal of manpower isinvolved in searching fingerprint files.

Recently there has been a growing interest in the possibilityof using a computer to automate the fingerprint-searchingprocess. Several feasibility studies have been made. Using theexisting gross typing as the basis of fingerprint description, ithas been demonstrated and generally concluded that com-

puterized file searches of ten fingerprints are reasonable. How-ever, for latent fingerprint searches where the gross typingprovides very little information, the minute ridge character-istics must be used. Several proposals have been made sug-

Manuscript received January 21, 1970; revised April 10, 1970.

gesting matching of minutiae patterns as a basis for auto-matic print identification [1]-[4], [9]. To date, experimentalverifications of these schemes have been sketchy.The chief obstacle to systematic investigation of com-

puterized single-fingerprint searches is the lack of a single-printfile comprised of single fingerprints described by their minutiae.Present-day manual methods for setting up such a single-printfile are so time consuming and expensive that even the settingup of a moderate size file for research in classification methodshas not been attempted [8].

In the joint study undertaken by the New York State Intel-ligence and Identification System and the System DevelopmentCorporation, a man-machine operation is proposed for the pur-pose of searching single prints [1]. The basic idea is to usean operator to locate and recognize the ridge characteristicsand to use a computer to perform data manipulations. Thesystem concept appears to be a sound one. Unfortunately, thetechnical evaluation of the system has not been as conclusiveas one would like to see because no experimental equipmentwas assembled in that study to investigate practical problemsof the system.The research described in this communication represents an

attempt to evaluate realistically the utility of a man-machinesystem for fingerprint encoding and classification. All experi-ments were performed on a system that could obviously beused as an operational system. The advantages of this typeof operational simulation, which uses personnel and equip-ment at operational facilities, should be obvious. The principalobjectives of the research are to examine the magnitude of thetask of using a computer-assisted method to set up a single-fingerprint file, to test the effectiveness of ridge characteristicsin computerized searching of single fingerprints, and to definea system of present-day hardware that may be assembled tohandle the single-print problem economically.

II. ENCODING OF RIDGE CHARACTERISTICS

A. Experimental FacilityThe elements in the experimental system are illustrated by

the block diagram in Fig. 1. The input device is a Bolt,Beranek, and Newman Grafacon 1OOA tablet, commonly