E.
R.
HOFFMANN
B.E., M.Eng .Sc., Ph .D., Department of Mechanical Engineering, University of Melbourne
DETECTION OF VEHICLE VELOCITY CHANGES IN
CAR-FOLLOWING
(Paper No . 443)
A survey is made of past work on the detection of vehicle velocity
changes in car-following. The hypothesis is made that detection
is dependent on changes of the visual angle sub tended by the lead
vehicle at the eye of a subject in the following vehicle, and also
on the time rate of change of this visual angle. Reanalysis of experimental data gives results consistent with this hypothesis. It
appears that when rate of change of visual angle is small, it has a
degrading effect on detection by changes in visual angle.
INTRODUCTION
1.
One of the basic problems in the theory of traffic flow is in the mathematical
modelling of a car-following situation. This is possibly the simplest of all traffic
flow situations, wherein a car follows a lead vehicle and must respond to the
changes in velocity of the lead vehicle. It is apparent that this model must include
both driver and vehicle characteristics. Driver characteristics which are likely to
be of importance are reaction and decision times, motor responses and ability to
detect changes in the velocity of the lead vehicle.
2.
The basic assumption in car-following theory is that each vehicle in a
single lane of traffic follows the one in front according to a form of stimulus
response law.
Response of vehicle at time (t + T) = sensitivity coefficient X stimulus at time t (1)
An equation which has been found by experiment to agree fairly well with reality
is (see, for example, Herman and Potts, Ref. 1)
d 2 x n(t + T)
[ dXn _ 1(t) - dXn(tJ]
- -d-e- - = ..1.0
dt
dt
xn _ 1(t)- xn (t)
(2)
3.
This equation states that the acceleration of the nth vehicle at a lag time T
after the change in the lead vehicle velocity is proportional to the difference in
velocity between that vehicle and the one leading it, and inversely proportional to
the distance between them. Much experimental and theoretical work h as been
done on this car-following model to determine stability of the motion, best fo rms
for the equ ation and effects of stimuli from other vehicles. A review of this work
is given by H erman (Ref. 2).
4.
The m otion and stability of the traffic flow is dependent on the way in
which a driver detects the differences in the distance between his own and the
leading vehicle, the way in which he converts this information into control reVolume 4, Part 1 (1968)
821
HOFFMANN -
VEHICLE VELOC ITY C HANGES IN CAR-FOLLOWING
sponses, the form of control on the vehicle and its requirements on the driver,
and also on the actual vehicle's response to the driver's control input.
The quantity T in eqn (2) is the driver vehicle delay time and is due to
S.
the driver's detection, reaction, decision and motor response times along with the
vehicle response time. Little systematic analysis of the way in which the driver
of the following car detects changes in velocity of the lead car appears to have
been made.
6.
The aim of this paper is to review the theoretical and experimental work
that has been done on the detection aspect of the problem and to determine
criteria for detection under different conditions of following and lead vehicle
motion.
MEANS OF DETECTING A CHANGE IN RELATIVE VELOCITY OR INTERVEHIClE
SPACING
7.
In the following, the relative motion of the two vehicles being considered
is that of motion in depth, that is, towards or away from the observer. In motion
of this form there is not necessarily any movement of the image of the lead vehicle
across the retin a of the observer's eye, rather there are changes in retin al image
size, contour, shape and also of patterns of li ght and shade on the lead vehicle.
8.
Detection of changes in spacing or relative velocity may occur as a result
of some or all of the following visual effects.
(a) By a change in the size of the retinal image. If this change occurs rapidly,
the rate of change of this image size may be detectable. This corresponds to
detection of changes in visual angle and rates of change of visual angle respectively.
(b) The driver may use information from the streaming of objects in the visual
field in order to infer changes of speed or spacing. Gordon (Ref. 3) has
considered the problem of this streaming visual field in relation to driving
and gives equations for the vector velocity fields.
(c) In (a) above only monocular cues are necess ary. Performance however is
enhanced by binocular vision due to the disparity of the images of the two
eyes. These cues are, however, only useful at short distances.
(d) Changes of distance are usually readily detected when the visual backgrou nd
is textured and the observer is station ary (Gibson, Ref. 4 ). However in this
case, both the observer: and the lead vehicle are in motion and hence the
background is constantly changing and so the texture effects are unlikely
to be of much use for detecting changes.
(e) There are also several other effects due to distance, these being the 'convergence' of the parallel sides of the roadway with increasing distance from
the observer and also the changing position of the lead vehicle relative to
the horizon with change of spacing.
9.
For the sake of simplifying the analysis and also due to the unknown
effects of some of the above cues, in the following only the effects of changes and
rates of change of visual angle will be considered. The hypothesis is m ade th at
822
A .R.R.B . PRO CEEDTNGS
HOFFMANN -
VEHICLE VELOCITY CHANGES IN CA R-FOLLOWING
-v
~
ARA
-----E:J
AR --r
----
-~
-
Fig . I -
B
~
I
L
Th e car-following situation . Car A
follows car B at an initial distance
l and equal velocity V. Car B
changes speed and car A attempts
to adjust speed so that spacing remains at l.
these two quantities are the main determinants of the detection of changes in the
relative velocity and spacing of the lead and following vehicles.
10.
Consider the two vehicles shown in Fig. 1, where the following vehicle is
at a distance L behind the lead vehicle and both vehicles are initially travelling at
a velocity V. In a typical car-following situation the following driver is to detect
changes in spacing between his and the leading vehicle and adjust his own speed
to attempt to maintain a constant spacing between the vehicles.
11.
The lead vehicle subtends an angle 0, say at the eye of the driver in the
following vehicle.
The angle 0 is given by
o=
(~)
2 artan
~
W
-
(3)
L
If now the lead vehicle accelerates or decelerates and changes the spacing to
(L + ~L) , the visual angle is changed to (0 + ~O) where
flO
o
artan
W
2(L
- artan (W12L)
flL)
artan W /2L
+
(4
For large val ues of L (i.e. small visual angles), this equation reduces to
-flL
flO
o
L
+
flL
(5)
12.
Michaels (R ef. 5) has suggested that at large, following distances, the
detection of changes in spacing is due entirely to detection of changes in visual
angle subtended by the lead vehicle. The driver at two instants of time estimates
the visual angle and if the change is sufficiently large over this time interval, he
can infer that the vehicle spacing has changed. At these large distances the driver
does not have any mechanism to determine the rate at which this visual angle is
changing. The change in size of the image on the retina is the psychological
correlate of distance perception in this case.
13 .
At smaller distances another cue is available to the following driver in the
rate at which the visual angle is changing. This rate of change of the size of the>
retinal image is given by
dO
dt
-
L2
WVre l.
+ W 2 j4
(6)
or approximately for the present purposes by
Volume 4, Part 1 (1968)
823
HOFFMANN -
VEHICLE VELOC ITY CHANGES I N CAR-FOLLOWING
de
dt
Here V r e I. is the relative velocity between the vehicles, Wand L are as defined
previously. At sufficiently large values of dO/ dt the driver is able to detect the
relative velocity by the movement of the image of the lead vehicle on the retina .
14.
H ence at large, following distances the following driver may rely on
changes of visual angle, and hence distance, for determining changes of velocity
of the lead vehicle and at small distances may use the cue of relative velocity from
movement of the image over the retina. At intermediate distances both cues may
be used.
AN IDEAL MODEL
15.
In the following, the simple situation of the lead vehicle accelerating or
decelerating at a uniform rate will be considered. At large, following distances
the angular velocity will be small an d detection is largely by changes in visual
angle subtended by the lead vehicle. Thus detection will occur when the change
in vehicle distance is such that the change in visual angle is equal to the just
noticeable difference (or some function of this) for visual angle. I t is likely that
there will be an equation for the just noticeable difference in visual angle similar
to Weber's law, i.e.
(Ile)
()
= constant
(8)
clet.
that is, the just noticeable change in visual angle at detection is proportional to
the initial visual angle.
16.
From eqn (5) it is seen that, for a given (t:.0/ 0), the spacing change for
an accel erating vehicle (t:.L positive) will be greater than that for a decelerating
vehicle (t:.L negative). This spacing change can be converted into a time lag T
for a uniform acceleration by use of the equation
ilL = t aT2
where 'a' is the lead vehicle acceleration.
(9)
17.
At smaller vehicle spacing, when the lead vehicle acceleration is sufficiently
large, the rate of change of visual angle may be large enough to exceed the
'threshold";' value and the followin g driver is likely to respond to this more sensitive
means of detection.
18.
Much work has been done on determining mean thresholds and Weber's
ratios for angular velocity in a plane normal to the line of vision (for example,
R ef. 6, 7 and 8); however th ere appears to be little information for motion along
the line 'of vision, that is motion in depth, as occurs in the car-following situation.
In a driving situation Michaels and Cozan (Ref. 9) found values of 3 to 10 X 10- 4
rad/sec. Smith (Ref. 10 ) carried out experiments on apparent motion in depth.
Baker and Steedman (Ref. 11 and 12) did similar experiments but with real
'There is still some doubt as to the existence of a 'threshold' . G reen and Swets (Ref. 18) in chapter 5,
co ncluded that 'there m ay be a sensory thresho ld . . . the e xistence o f a sensory th reshold has no t been
demons tr ated ' .
!Y24
A.R .R.B. PRO CEEDINGS
HOFFMANN -
VEHICLE VELOCITY CHANGES IN CAR-FOLLOWING
motion in depth of a luminous object in an otherwise stimulus-free surround. They
found that, under optimum viewing conditions, 75 per cent correct detection was
achieved when the visual angle increased or decreased by 2 per cent. Their results
show some effect of angular velocity but this is not analysed in detail.
19.
The distance change taking place when angular velocity is used as a
criterion, with a uniformly accelerating lead vehicle is then
(10)
or in terms of time to detection
WaT
(11)
20.
In Fig. 2 is shown the distance change ~L for detection alone, as a function of the initial spacing of the vehicles and the acceleration of the lead vehicle.
This is an ideal situation and the values are likely to be different for the vehicle
case, where the sensitivity of the detection processes may be decreased by the
visual 'noise' introduced by the vehicle motion and the effects of the visual environment. Also the spacing changes in the intermediate initial distances are likely to
be different, where the effects of visual angle and rate of change of visual angle
are both significant.
21.
Bierley (Ref. 13) points out that the total time for the driver to respond
depends on whether the lead vehicle accelerates or decelerates, the difference being
that in the case of deceleration the following driver must move his foot from the
accelerator to the brake pedal. He finds that for acceleration
total reaction time = detection time + 0.50 sec
and for deceleration
total reaction time = detection time + 0.69 sec
In each case, 0.50 sec is made up of reaction time plus decision time.
PAST WORK ON MEASUREMENT OF DETECTION OF VEHICLE VElOCITY CHANGES
22.
Measurements of driver responses to lead vehicle changes have been carried
out since the formulation of vehicle-following models and the corresponding
models of traffic flow. In all the experiments discussed in the following, the brake
+6L
Co
0
B
a-
(jet.
/1
0;
".8
ac:
D'>
'"
~~] = oonsta:-l
0
J:
U
ao
c:
'"
~
~ - [~= cons tant
(jet .
Accelerating
lead vehicle
Fig. 2 -
Init ial vehicle spacing
An ideal model fo r detection of
spacing changes as a function of
lead vehicle acceleration; 'a' a nd
initial spacing, l. At small vehicle
spacing and suffi cie ntly large ac·
celeration,
o
-6
Volume 4, Part 1 (1968)
detection
is
by
means
of rate of change of visual angle
and at large spacing is by means
of change of visual angle.
825
HOFFMANN -
VEHICLE VELOCITY CHANGES IN CAR-FOLLOWING
lights of the lead vehicle were disconnected so that this information could not be
used by the following driver in detecting velocity changes.
23.
Forbes et al. (Ref. 14) measured time and distance headway changes as
a result of an accelerating or decelerating lead vehicle· when driving under freeway
or tunnel conditions. However they do not give details of the accelerations of lead
vehicle and hence it is not possible to deduce anything from their data on the
effects of changes of visual angle or angular velocity. Bierley (Ref. 13) investigated the effects on car-following of a spacing display, giving information to the
following driver on the actual spacing or the relative velocity between the vehicles.
His data include information for the case of no display, which may be used in the
present work. Having determined the delay times for several accelerations and
decelerations at a fixed initial distance, he demonstrates that the mathematical
model of the situation given by a modified form of eqn (2) gives an excellent
approximation to the real situation, when these time values are used. He does not,
however, determine how these detection times are dependent on the visual variables. Braunstein and Laughery (Ref. '6) performed experiments somewhat similar
to those of Bierley but with no display information for the following driver. They
used two levels of acceleration and two of deceleration with two initial vehicle
spacings in each case. Braunstein and Laughery presented their data in the form
of distance change, latency and velocity change for each experimental condition.
They concluded that detection time increased with intervehicle separation and
decreased with increasing acceleration. The Fig. 2 of Braunstein and Laughery
is shown in the present Fig. 3.
24.
An analysis of variance showed that the effect of the magnitude of the
acceleration was significant at P < 0.05 for both accelerations and decelerations
of the lead vehicle and the vehicle separation effects were significant at P < 0 .05
only for the case of accelerating lead vehicle. No other analysis of these results
was presented.
25.
Hoffmann (Ref. 15) derived simple expressions for the time delay or distance change required by an observer in a moving vehicle to detect changes in
the velocity of a lead vehicle. The expressions were derived by the use of dimensional analysis and , after modification, were found to be in agreement with the
results of Braunstein and Laughery. The analysis also indicated that the detection
process was not significantly affected by the absolute speed of the vehicles, but
was dependent only on the relative speed.
u
~7
ClI
E
.~
6
>-
u
c
ClI
1Qs
C
tU
ClI
2: 4
826
=2501 I
~
~ I~
160
deceleration
I
I
f
0-
250
't)
I acceleration
~~--"""'"-L~=---~~-ft-I ~c~
Fig . 3 -
Th e ex perime ntal results of Braun stein and laugh ery, showing response latency as dependent on
vehicle spacing and lead vehicle
acceleratio n.
A.R.R.B. PROCE EDINGS
HOFFMANN -
VEHICLE VELOCITY CHANGES IN CAR-FOLLOWING
26.
This analysis showed that an equation similar to Webers law applied for
the noticeable change in distance, that is,
-~L =
1
const. ~ (12)
L
8
To allow for the effects of the direction of the acceleration of the lead vehicle,
Hoffmann modified this equation to
(13)
~L
L
~L
ea
1
L + t at
4
is for acceleration, and - is for deceleration of the lead vehicle.
- - -2 =+ -
or
where the
+
+
27.
Although not realized at the time, the equations are simply a statement
that the change in visual angle, flO, is proportional to the visual angle (Refer
eqn 5).
28.
In this work the effects of rate of change of visual angle were not considered. The results appeared to be reasonably well correlated by the above
equations except for one point which was far removed from the other data. (In
a later section this will be seen to be due to dO/ dt effects) .
29.
Torf and Duckstein (Ref. 16) carried out similar experiments but at a
much smaller following distance (approximately 80 ft) , so that rate of change
of visual angle effects are likely to be significant. The aim of their experiments
was to determine the feasibility of using motion picture testing in place of road
testing for determining perceptual latency in car-following and to find values for
this latency for acceleration and deceleration at fixed initial speed and following
distance.
30.
Their analysis of variance showed no significant difference between road
testing and photographic testing (P > 0.05). (However if they had considered
just the deceleration data it would have been found that the results are different
at P < 0.05.)
31.
They also found that the effects of magnitude of the rate of change of
velocity and the direction of this acceleration were significant at P < 0.05. They
found that the perception times for deceleration were longer than those for acceleration and explained their results in terms of the size of the image on the areas of the
eye most sensitive to motion. The explanation given does not appear, however,
to be in agreement with the physiological explanation for detection of motion given
by Motokowa (Ref. 17). These authors do not attempt to give any criteria for
detection of the velocity change.
These few references appear to be
32.
of velocity changes of a leading vehicle.
Bierley, Braunstein and Laughery and of
analysed to give a clearer picture of the way
Volume 4, Pa rt I (1968)
all that is available on the detection
In the following section the data of
Torf and Duckstein will be further
in which the detection process changes
827
HOFFMANN -
VEHICLE VELOCITY CHANGES IN CAR-FOLLOWING
with leap: vehicle spacing and accelerations and also to give some criteria and
equations which might be of use in the car-following modeL
ANALYSIS O F AVAilABLE DATA
33.
TABLE I gives a summary of the experimental data of Bierley, Braunstein
and L(!.ughery and of Torf and Duckstein, which is relevant to the present work.
DETECTION BY RATE O F CHANGE OF VIS UAL ANGLE
34.
As the data of Torf and Duckstein and of Bierley are at small vehicle
spacing and the accelerations are of reasonably large magnitude, it is likely that
both sets of data are in a region in which detection was largely on the basis of
rate of change of size of the visual image. This can be readily checked by graphTABLE I
SUMMARY OF EXPERIMENTAL DATA OF (a ) BRAUNSTEIN AND LAUGHERY (b) TORF AND DUCKSTEIN
AND (c) BIERLEY FOR TIME AND DISTANCE CHANGES FOR OF VelOCITY CHANGE OF LEAD VEHICLE
Source
Braunstein and Laughery
Vehicle
Spacing
(ft)
Initial
Velocity
(m.p.h.)
Lead Car
Accel.
159
54.9
1.74
21.9
(ft / sec
2
)
Distance
Change
L(ft)
Time*
Delay
(sec)
Velocity
Change
(V m.p .h.)
5 .1
6.1
and Laughery
159
54.9
2.21
25.5
4.7
7 .1
Braunstein and Laughery
256
55.0
1.76
28.0
5.9
7.1
Braunstein and Laughery
251
55.0
2.25
31.2
5.7
8.7
and Laughery
158
55. 1
-0.81
-15.0
6.6
- 3.7
Braunstein
Braunste in
Braunstein and Laughery
161
54.6
-1.43
-19.9
5 .2
-5.1
Braunstein and Laugherv
249
55.0
-0.78
-18.4
6.8
-3.6
Braunstein and Laughery
249
54.8
-1.40
-23.0
6.0
-5.7
43.7
2 .5
Torf and Duckstein (veh )
77.8
1.9
Torf and Duckstein (veh )
77.4
42.4
1.6
2.8
Torf and Duckstein (veh )
77.4
42 .8
-1.3
4 .7
Torf and Duckstein (photo)
77.8
43.7
2.5
1.6
Torf and Duckstein (photo)
77.4
42.4
1.6
2.6
Torf and Duckstein (photo)
77.4
42 .8
-1.3
3.2
Bierley
88.8
60.5
2.96
1.72
1.45
1.13
corrected
timet
Bierley
87.8
60.0
-3.06
1.59
Bierley
86.9
59.2
2 .85
1.68
1.41
Bierley
81.3
55.4
-2.98
1.68
1.22
* The data of Braunstein and Laughery and of Torf and Duckstein were obtained from an observer in the
following car and hence are a detection time and reaction time.
t The data of Bierley are from the driver of the se cond ca r and includes also a decision time and motor
response time . Hence subtract 0.27 sec from t he acceleration times and 0.46 sec from the d e celeration
times to give all results on a common basi s. The time (detection and reaction) is then a perception time
and is used in the following .
228
A.R.R .B. PROCE EDfNGS'
HOFFMANN -
VEHICLE VELOCITY CHANGES IN CAR-FOLLOWING
TABLE II
CALCULATED VALUES OF SPACING CHANGE AND ANGULAR VElOCITY FOR DATA OF T.QRF AND
OUCKSTEIN AND OF BIERLEY (DETECTION IN THESE CASES IS LARGELY BY ME.(>.NS OF RATE OF
CHANGE OF VISUAL ANGLE)
Source
Le a d Car
Accel.
(fl / sec')
Time
De lay
(sec)
~
L (ft)
(
d B ) (rad / se c)
dt
de t.
2 .5
1.9
4.52
-4.66 X 10-"
To rf and Duckslei n (veh)
1.6
2.8
6.27
-4.26 X 10-"
Torf and Duckslein (veh )
-1.3
4.7
- 14.35
10.24 X l 0-"
To rf a nd Ducksle in (veh )
Torf and Duckslei n (ph olo)
2.5
1.6
3.20
-4.06 X 10-"
Torf and Duckslein (pholo )
1.6
2.6
5.42
-4.04 X 10-"
Torf and Duckslein (p holo )
-1.3
3.2
- 6.65
5.54 X 10-"
3.11
-3 .04 X 10-"
2.96
Bierley (vehicle)
1.45
Bierley (vehicle)
2.85
1.41
2.84
-3.00 X 10-"
Bierley (vehicle)
-2.98
1.22
-2.22
3.48 X 10-"
Bierley (veh icle)
- 3.06
1.13
-1.95
2.82 X 10-"
a= - 0.1
-20
-1.0
-0.5
Decelerating lead vehicle
....§-10
0
.....
<l!
<l!
U
.8
0
{ f~.
<l!
OJ
c
1!u
<l!
u
c
2
10
. !!l
Accelerating
0
0.5
-1
3.0
<J
20
o
•
~
Fig. 4 -
Torf & Duckstein ( vehicle)
"
(photographic)
Bierley ( vehicle)
Vehicle spacing changes wilh de tection by rate of change of vis ual angle, for accelerati on and
deceleration of the lead vehicle. The experime ntal data of Torf and Duckstein an d of Bierl ey
are also shown .
Volume 4, Part 1 (1968)
829
HOFFMANN -
VEHICLE VELOCITY CHANGES IN CAR·FOLLOWING
ing (1l0/ 0) det. versus (dO/ dt) det. and seeing whether any relationship exists. In
this case (1l0/ 0) det. had no systematic variation with (dO/ dt) det. and as
(1l0/ 0) det. was approximately constant, it is apparent that the detection process
was as stated above.
35.
TABLE II shows the distance change to detection and the rate of change
of visual angle at detection calculated by eqn 11 for a vehicle width (W) of 6 ft.
The values of Bierley are seen to be lower than those of Torf and Duckstein and
one deceleration of Torf and Duckstein appears to be excessively large. Neglecting this value, the average for all accelerating trials is 3.84 X 10- 3 rad/sec and
for decelerating trials is 3.96 X 10-3 . As these are not significantly different the
overall mean value of 3.86 X 10- 3 rad/ sec will be used as a representative value
for the rate of change of visual angle for detection. Note that this is very much
larger than the values found by Michaels and Co zan in a vehicle situation. The
difference is possibly partly explained by the fact that the threshold value is that
which is detected on 50 per cent of occasions, whereas the experimental values in
the above are obtained in tests in which the vehicle speeds are changed until
detection takes place, that is, on 100 per cent of occasions.
36.
By substituting this value of (dO/ dt) det. into eqn (10), an expression for
the distance change occurring during the perception period can be obtained
L = - ~L + 46·8(a~L)t
This equation can be plotted for values of a and L to give Fig. 4 .
(14)
37.
The experimental data are also included on the figure and the fit is seen
to be reasonably good. In the above equation ilL and 'a' are positive for accelerations and negative for decelerations. This graph could be converted to the form
of perception time versus spacing and acceleration simply by using eqn (9) . At
the moment the· extent of spacing and accelerations to which this equation may
be applied is not known , but will be discussed later.
TABLE 1\1
RATE OF CHANGE OF VISUAL ANGLE AND THE WEBER FRACTION FOR VISUAL ANGLE CALCULATED
fOR THE DATA OF BRAUNSTEIN AND LAUGHERY (THE CALCULATIONS WERE MADE USING THE
£::, L VALUES GIVEN BY THE AUTHORS)
83 0
( ~e )
e
C~P)
det.
L (ft)
t:.
158
- 15.0
- 0.81
0.105
16 . 08 X 1~
161
- 19.9
- 1.43
0.141
25.24 X 10-'
249
-18.4
-0.78
0.080
6.70 X 10""
249
-2 3.0
-1.40
0.102
1 0.50 X 10-'
159
21.9
1.74
- 0.121
- 17.74 X 1~
L (fl)
. (fllsec')
del.
159
25.5
2.21
-0.138
-20.68 X 1~
256
28 .0
1.76
- 0.099
-8.2 0 X 1~
261
31.2
2.25
-0.111
-9 . 92 X 1~
A.R.R.B .
PROCE EDINGS
HOFFMANN -
f
VEHICLE VELOCITY CHANGES IN CAR-FOLLOWING
at detection
0.1
~tg at
Accelerat i ng
lead vehicle
4
detection x10
( rad /sec )
-0.1
Fig. 5 -
Experimental data of Braunstein and
Laughery, showing the locus of
visual angle and rate of change of
visual angle along which detection
takes place.
DETECTION BY VISUAL ANGLE CHANGES
38.
The analysis by Hoffman (Ref. 14) of the data of Braunstein and Laughery
suggested that the detection process was that of detecting changes in the visual
angle. This would be consistent with the fact that the vehicles were fairly widely
spaced and hence the rate of change of visual angle would be small. However a
certain amount of 'scatter' in the data prompted further investigation to see whether
any effects of angular velocity existed. TABLE III gives the calculated values of
(t..()/ ()) det. and (d()/ dt) det. for these data. Note that the values of d()/ dt are less
than the values found for the case when detection is by means of rate of change
of visual angle.
39.
The values of (t:..() / () versus (d()/ dt) at detection are graphed in Fig. 5
and it is seen that detection takes place along lines on which there is a fixed
relationship between these two dependent quantities.
tL
=-
1.0
-so
Decelerating lead vehicle
<lI
en
c
III
-5 -20
<lI
U
C
III
~-10
Fig . 6 0.1
°O~--~l00
~--~200
~---~~---aD
~---SOO~--~­
Initial vehicle spac ing ( f~
Volume 4, Part I (1968)
Graph of eqn (16) for a decelerating lea d vehicle showing the distance
chang e to detection when this
occurs mainly by means of changes
in vis ual angle . The data points of
Braun stei n and Laughery are also
included .
831
HOFFMANN -
VEHICLE VELOCITY CHANGES IN CAR-FOLLOWING
At detection the equation to these lines is given by
Lle)
(e
accelerating:
(~f})
decelerating :
-0'077 _ 2S '0(de)
dt det.
d e t.
det.
=
0·065
+
(15)
29'0(:~tcl.
Note that these equations are between two dependent variables and simply give
the locus of values along which detection takes place.
40_
By substituting expressions for these variables in terms of distances and
accelerations, the following equations are obtained
accelerations : L = 5-49LlL ± [42'2LlL 2 - 3425(aLlL)t ]t
decelerations : L = -S'70LlL ± [59-4LlL2 - 4200(aLlL)t ] t
(16)
41The eqn (16) for the case of deceleration of the lead vehicle is shown in
Fig_ 6 along with the data of Braunstein and Laughery. As the equation was
determined from these data, it is expected that the experimental points should be
a reasonably good fit.
42.
It is noted that the data point which appeared to be incorrect in the analysis
of Hoffmann (Ref. 14) now fits well in this scheme, apparently being accounted
for by the effects of the small ('sub-threshold') rates of change of visual angle
which are present. Even though the rate of change of visual angle is small in the
experimental conditions of Braunstein and Laughery, it may play a significant
part in the detection process. However, the effect it has is opposite to that which
might be expected, that is, its presence tends to degrade detection performance.
43.
Why this should be so the author cannot explain, but for both the accelerating . and decelerating cases of Braunstein and Laughery this effect is present.
Whether or not this effect vanishes with further increase of vehicle spacing is not
kriown but it is likely that detection purely by visual angle changes does exist at
large spacing.
0.2
[~8Jdetection
A
0.1
C
- 20
20
d8j x 104 ( rdd I sec)
[dt detection
E J - O•1
B
832
-0.2
Fig . 7 -
The
visual
locu s of
angle
rate
and
of change of
visual
angle
at
detection for both modes of detection , sh owing likely modification of
the locu s when the rate of change
of visual angle approaches threshold values.
A.R.R .B.
PROCEEDINGS
HOFFMANN -
VEHICLE VELOCITY CHANGES IN CAR ·FOLLOWING
/
-25
"
§-20
§
-0
~-1 5
"
g;.-10
c
-6
..
g -5
Dec eleral ing lead vehicle
Fig. 8 -
'"
11)
o
200
300 L (ftl
Distance changes to detection, cor·
responding to the locus of points
in Fig . 7. The figure is for deceleration of the lead vehicle.
THE DETECTION CONTINUUM
44.
The locus of detection points shown in Fig. 5 may be redrawn to include
the case of detection at high values of rate of change of visual angle (Fig. 7).
The corresponding spacing change taking place to detection is shown, for the case
of a decelerating lead vehicle, in Fig. 8.
45.
The equations for detection by the two different processes considered here,
intersect along the lines
~L = 0.23
L
~L
L
-0·15
for acceleration, and
for deceleration.
46.
As it is improbable that a sudden change from one form of detection to
the other is likely to occur (as shown at A and B in Fig. 7), some region in which
effects of both visual angle and rate of change of visual angle are of about the
same importance is likely. This would produce a spacing change shown by the
dotted lines in Fig. 8. If, at large distances , the detection process does become
entirely independent of the effects of dO j dt, the locus of points along which detection now occurs would be as shown by CD and EF on Fig. 7; that is, the detection
process is only dependent on changes of visual angle. (No data are available to
verify this.)
CONCLUSIONS
47.
A reanalysis of available experimental data gives results which are consistent with the hypothesis that there are several distinct processes for detection
of vehicle velocity changes; these are by means of detecting changes in visual angle
and by detecting rates of change of visual angle.
48.
In a region where rates of change of visual angle are smail, detection is
degraded by the presence of these angular velocities.
More experimental data are required for a further check on the hypothesis
49.
and to provide more accurate criteria for detection.
Volume 4, Part 1 (1968)
833
HOFFMANN -
VEHICLE VELOCITY CHANGES IN CAR-FOLLOWING
REFERENCES
1. Herman, R and Potts, R B., Single lane traffic theory and experiment, Theory
of Traffic Flow, Proc. of Symp., Elsevier Pub. (1961); R. Herman (ed.) .
2. Herman, R., Theoretical research and experimental studies in vehicular traffic,
Proc. 3rd Conf. ARR.B ., 3: 1,25 (Sydney 1966).
3. Gordon, D. A, Static and dynamic visual fields in human space perception,
J. Opt. Soc. Am., 55 : 1296 (1965).
4. Gibson, J. J., The Perception of The Visual World, Houghton Mifflin (Cambridge, Mass. 1950) .
5. Michaels, R M ., Perceptual factors in car following, Theory of Road Traffic
Flow, Sec. Intern. Symp., p. 44 (1963); Almond (ed.).
6. Braunstein, M. L. and Laughery, K. R , Detection of vehicle velocity changes
during expressway driving, Hum. Factor Lond., 6: 4, 327 (1964).
7. Brown, J. F., The thresholds for visual movement, Psychol. Forsch., 14, 249
(1931).
8. Brown, R H ., Weber ratio for visual discrimination of velocity, Science, 131,
1809 (1960) .
9. Michaels, R M. and Cozan, L. W. , Perceptual and field factors causing
lateral displacement, H .R.R 25, p. 1 (1963).
10. Smith, W. M., Effect of monocular and binocular vision, brightness and
apparent size on the sensitivity to apparent motion in depth, J. Exp. Psycho. ,
49, 357 (1955).
11. Baker, C A and Steedman, W. C , Perceived movement in depth as a fu nction of luminance and velocity, Hum. Factors Land. , 3, 166 (1961).
12. Steedman, W. C. and Baker, C A , Perceived movement in depth as a function of stimulus size, Hum. Factors Lond. , 4 : 6, 349 (1962) .
13. Bierley, R. L., Investigation of an inter-vehicle spacing display, Highw. Res.
Rec., 25, 58 (1963).
14. Forbes, T . W., Zagorski, H. J. , Holshouser, E. L. and Deterline, W. A ,
Measurement of driv er reactions to tunnel conditions, H.RB . Proc. , 37, 345
(1958) .
15. Hoffmann, E. R, Note on detection of vehicle velocity changes, Hum. Factors
Land., 8: 2, 139 (1966) .
16. Torf, A S. and Duckstein , L. , A methodolgy for the determination of driver
perceptual latency in car following, Hum . Factors Lond ., 8 : 6,441 (1966) .
17. Motokowa, K. , R etinal traces and visual perception of movement, J. Exp.
Psychol. , 45, 369 (1953).
18. Green, D. M. and Swets, S. A, Signal Detection Theory and Psychophysics,
Wiley (New York 1966).
ADDENDUM
50.
Some fur ther explanation of how the detection locus of Fig. 7 was arrived
at is contained in the following. All the experimental data have been obtained
with constant acceleration of the lead vehicle; here a second condition with
constant velocity difference is also considered.
CONSTANT ACCELERATION OF LEAD VEHICLE
51.
834
H ere we have the relationship
A.R.R .B . P ROCEEDINGS
HOFFMANN -
VEHICLE VELOCITY CHANGES IN CAR-FOLLOWING
lI 8
8
Fig. 9 -
d8/dt
gives the direction of increasing 'a '
l=O
d8
(d t
)2
and decreasing l.
= - 2aW 21l8 (
L3
The detection locus with a uniformly
accelerating lead vehicle, showing
changes in visual variables with in ·
crease of time. The dashed arrow
7i
I
1l8)3
+ 7i
and as shown in Fig. 9, both il8/ 8 and dO/ dt increase with time along the lines
shown (for varying L and a) until detection takes place when the detection locus
is reached.
CONSTANT V REL BETWEEN VEHICLES
52.
It might be reasonable to assume that the same detection locus exists for
other methods of changing il8/ O and dO/ dt. The simplest means is with a
constant V r e l · and with the subject commencing to view the lead car when t = 0,
at which time the vehicle spacing is Lo. Then
dO _ dt -
WVreJ. (
q
1
+
fl(J) 2
0
The time plot of this relationship is shown in Fig. 10.
53 .
This case needs to be tested by experiments. If the detection locus is
similar to that for acceleration of the lead vehicle, it is likely that other forms of
distance change can be plotted on this graph to determine detection conditions.
54.
Rockwell and Snider (Ref. 20) have carried out experiments on the ability
of drivers to detect changes in headway of a leading vehicle. They give linear
regressions of ilL as a function of both Land a. The mean regressions over the
eight subjects are
acceleration
ilL = 0.93 + 0.069L + 1.40a
deceleration
ilL = 2.1 2 + 0.053L -0.76a
55 .
These equations compare reasonably well with those presented in this
paper, however the effects of the acceleration are not as great as found in the data
detection points.
Fig . 10 -
d8/dt
Volume 4, Part I (1968)
Conjectured detection locus with
constant velocity difference
between vehicles. The dashed arrow
indicates the direction of increasing
relative velocity and decreasing
initia l spacing .
835
HOFFMANN - VEHICLE VELOCiTY CHANGES IN CAR-FOLLOWING
(DISCUSSIONS BY ANDREASSEND AND CLARK/ AUTHOR'S CLOSURE)
of Bramstein and Laughery. It is noted also that these equations show that the
presence of a rate of change of visual angle decreases the sensitivity of detection
by changes in visual angle - as reported in the paper.
DISCUSSIONS
D. C. AND REA SSE N D , B.Sc., M.Eng.Sc., Chief Engineer, Traffic Commission of
Victoria
56.
Does the author have any knowledge of studies made of the detection of
vehicle velocity changes during hours of darkness? If not, does he intend extending
his work to cover this period as the incidence of rear end accidents is proportionally a lot higher at night than it is during the day.
57 .
It would be interesting to see figures comparing the use of brake lights
with the figures quoted by the au thor, which were made with brake lights disconnected. Perhaps the author in his future studies could compare vehicles where
the brake lights were operated by a hydraulic switch (when the brakes are
actually operating) against vehicles where the brake lights were operated by a
mechanical switch (coming on when the foot touched the brake pedal and before
the brakes were actually operating). The night studies would be of particular
value to any analysis of night accidents.
B. A. J .
Melbourne
C L ARK, M.App.Sc., B.Sc., Dip.Mech.E. , Defence Standards Laboratories,
58.
Howard (Ref. 21) found that for a viewing distance of 6 m., a least
depth distance discriminated on the basis of image size is about twenty times as
great as one discriminated on the basis of stereoscopi.c vision. Since the ratio of
stereoscopic to monocular depth discrimination decreases linearly with distance, I
should expect that stereoscopic depth discrimination would be more important
than depth discrimination based on image size for all distances shorter than 120
m, at least for static situations. In view of this, what is the basis of para. 8 (c) ,
and what effects could be expected if the drivers had one eye occluded?
AUTHOR ' S CLOSURE
To C. CAM E RON, Australian Road Resea.rch Board (see Introductory Remarks to
this sessio n)
59.
The present work was aimed at finding some useful criterion for detection
of changes in spacing and the variables used (changes and rates of change of
visual angle) may not be the only visual cues being used by the driver or necessarily even the correct ones . The discussion leader is quite correct in his statement
that it is 'as if' the driver operates on these variables, even though he has no
direct perception of them. Dr. Michael Cook of AN.V. has pointed out to the
author that changes and rates of change of declination angle will produce precisely
the same results in the analysis.
836
A.R.R.B .
PROCEEDINGS
HOFFMANN -
VEHICLE VELOCITY CHANGES IN CAR-FOLLOWING
(AUTHOR'S CLOSURE)
60.
In perception phenomena, it is usually a combination of cues which
produce a given result and often the same result can be obtained even when some
of the apparently important cues are removed. It is thus difficult to separate out
the effects of the various cues. As the results of a number of experimenters
seem to fit into the proposed scheme, it could be that the variables used, or some
closely related, may be of importance.
61.
The discussion leader has pointed out that, in many situations, visual
angle has been found to be very insensitive as a measure of changes in distance,
due to size constancy effects. The difference between these experiments and the
car-following situation is that here we have movement of the object and observer.
An increase in sensitivity can be achieved in some of the constancy experiments
if the subject 'cheats' by providing a relative velocity by movement of his head .
To D . C. AND REA SSE N D , Traffic Commission of Victoria
62.
The author knows of only one study of detection of changes in spacing
between vehicles under night conditions (Parker, Gilbert and Dillon, Ref. 19).
However this work is of little use since these authors presented contradictory cues
to the subjects.
63.
Experimental data with brake lights fitted do not appear to be available,
however, one could expect the delay time to be reduced as the perceptual process
of detecting braking with a stoplight is less complex than detecting changes in
visual angle. Bierley (Ref. 13) tested several types of displays for the following
driver and these displays were found to decrease the response time.
To B. A. J. C L ARK , Defence Standards Laboratories, Melbourne
64.
The experiments of Howard referred to in the comment used the HowardDolman apparatus, consisting of two vertical rods, one fixed and the other
moveable. Howard found the just noticeable difference for change in depth using
the method of constant stimuli. However the results of Howard are not applicable
to the present work for two reasons:
(a) in the present case there is no 'fixed' object which is used in detecting a
change, and
(b) the observer and lead vehicle are in relative motion whereas Howard studied
a static situation.
It is predicted that the present results would not be significantly changed
with one eye occluded.
REFERENCES
19. Parker, J. F., Gilbert, R. R. and Dillon, R. F. , Effectiveness O'f three visual
cues in the detection of rate O'f closure at night, Biotechnology Inc. Rept. No.
64-1 (1964).
20. Rockwell, T. H . and Snider, J. N., An investigation of variability of driving
performance on the highway, Ohio State University, Systems Res. Group,
Project RF1450 (1965).
21. Howard, Am . J . Ophthal., 2, 656 (1919).
Volume 4, Part I (1968)
83 7