I Perception, 1993, volume 22, pages 1227-1244
Misperception of time-to-collision by drivers in pedestrian
accidents
Douglas Stewart, Christopher J Cudworth, J R Lishman
Department of Engineering, University of Aberdeen, Aberdeen AB9 2UE, Scotland, UK
Received 30 June 1992, in revised form 14 April 1993
Abstract. The approach of an object can be monitored from its optic flow. More specifically, it
has been postulated that time-to-collision at constant velocity is perceived by relating visual
angle 6 to its rate of change 0, time-to-collision being 6/6.
This hypothesis is reappraised, and an alternative based on the parameters 6 and angular
acceleration 6 is proposed. The expression 2 6/6 also specifies time-to-collision, with the
benefits that it removes reliance on 6 and permits time-to-collision to be determined from even
momentary perception of an approaching point. This is supported by tests in which subjects
responded to computer simulations of approaching objects. A further benefit is that if the
object is accelerating rather than at constant velocity, time-to-collision is adjusted by 2 6/6,
but not by 6/6.
As time-to-collision increases, however, its cognitive derivation should transfer from optic
flow to separate perceptions of distance and speed. It is proposed that when drivers of road
vehicles are in potential collision with pedestrians their perception of distance is based primarily on familiar size, resulting in overestimation of size and therefore of time-to-collision with
child pedestrians. This is supported by further computer simulations and is corroborated by
predicting the effect of that overestimation on certain types of accidents, then testing from
national accident statistics.
These analyses indicate that drivers' misperception of time-to-collision has a dramatic effect
on the accident rate of child pedestrians. It is proposed that this could be greatly reduced by
the provision of remedial measures.
1 Introduction
A basic principle of road design is that road users should see potential hazards in
time to avoid accidents. This paper has evolved from studies of situations which have
not met that criterion (Stewart 1988; Stewart and Cudworth 1990). These studies
suggested that perceptual error by drivers is a common cause of road accidents, but is
so subtle that it is seldom detected. Even overt perceptual problems, such as guard
rails which mask children, had previously been ignored in road safety research.
Investigation of masking hazards suggested that other forms of driver perceptual error
are prevalent and could explain why child pedestrians are so much at risk. T h e focus
for this paper is therefore to investigate that problem, with a view to reducing child
accidents.
T h e perceptual task of driving is complex. Drivers have to perceive and respond
to the relative movement of other road users and of the road environment, at speeds
many times greater than the pedestrian speeds of previous human history. T h a t may
be why it takes about 1 0 0 0 0 0 km of driving experience to become 'safe' ( O E C D
1986). A vital element of this learning process must be the perception of time-tocollision, the time it would take to reach a collision point. Cavallo and Laurent
(1988) have demonstrated significant differences in the ability of novice and experienced drivers to make that assessment.
Two principal cognitive strategies for perceiving time-to-collision have been postulated. These will be considered for the standard condition of observer and object in
colinear convergence at constant velocity, V, which is relevant to the driving situation
of specific interest in this paper. O n e strategy requires evaluation of the relationship
D Stewart, C J Cudworth, J R Lishman
1228
d/V from separate perceptions of distance d and relative velocity V. The other strategy
is computation from optic flow, the apparent dilation of the approaching object from
a distant focus of expansion. Lee (1976) has demonstrated that time-to-collision may
be obtained directly from the optic flow relationship 6/6 if 6 is small, as it is in
head-on collision. 6 is the angular position of any approaching point relative to the
focus of expansion and 6 is its angular velocity (figure 1). Lee designates 6/6 as r,
but in this paper time-to-collision will always be denoted as Tc to avoid confusion
from alternative derivations of r.
Development of these two hypotheses in relation to vehicle control may be
followed through Schiff and Detwiler (1979), McLeod and Ross (1983), and Cavallo
and Laurent (1988), but there is still no consensus on their validity (Groeger and
Brown 1988). Time-to-collision should be computed more rapidly from optic flow
parameters than from d/V (Tresilian 1990), making it the dominant method if timeto-collision is short. This preference should reverse, however, as time-to-collision
increases and optic flow approaches its perceptual threshold. Although this transfer
is evident from the literature (eg Cavallo and Laurent 1988), there has been little
consideration of its implications for road safety.
Our main purpose in this paper is to examine these factors, particularly in relation
to child pedestrian accidents. During initial computer simulations of driver view,
however, it became apparent that the estimation of time-to-collision from optic flow
may be based on an alternative relationship to 6/6. This will first be discussed,
because it clarifies why optic flow can be an effective source of time-to-collision.
P ^
I
I
I
I
I
I
d
o
z
Figure 1. Plane motion kinematics of point P moving with velocity v and acceleration a in any
direction relative to O, showing velocity and acceleration vectors.
2 Derivation of time-to-collision from optic flow
The elegant simplicity with which 6/6 could define time-to-collision Tc without any
need for additional information about size, speed, or distance, is so attractive that its
limitations may have been neglected. The main one is the problem of judging 6. It is
impractical to do so from the focus of expansion, both because that can be perceived
with an accuracy only in the order of 1° (Warren and Hannon 1990) and because it
must shift with any deviation of a vehicle from a straight path. The alternative
proposed by Lee (1976) of using 6 between pairs of points equidistant from the
observer, such as the tail lights of a vehicle, would be ineffective if the object was, for
example, very small or irregular and tumbling. Yet such objects, say a table-tennis
ball or rugby ball, can be responded to without difficulty.
These doubts about the importance of 6 are reinforced by research (Savelsbergh
1991) in which 6 and 6 were grossly distorted by deflating an approaching ball.
Although timing of grasp was perturbed slightly in the predicted direction, subjects
were successful in catching the ball, which suggests that some neural computation of
optic flow parameters other than 6/6 had been used. A clue was provided by the
pioneer work of Gibson (1950), who referred to the explosion of optic flow when
1229
Misperception of time-to-collision by drivers in pedestrian accidents
collision was imminent. The dynamic characteristic of explosion is not speed but
acceleration. It therefore seemed possible that acceleration of optic flow, denoted
as 0, was a factor in perceiving time-to-collision. That hypothesis was first explored
theoretically.
2.1 Kinematics of a moving point
Consider the plane motion kinematics of a point P at distance r from observer O, OP
being at angle 0 to an initial line OZ, as in figure 1. In the general case P is moving in
any direction relative to O, at an instantaneous velocity v and acceleration a, which
can therefore be expressed in polar coordinates as vr, ve, ar, and ad. The standard
kinetic relationships for P are:
vr = f;
ve = rd ;
ar = r-rd2;
ae = r0 + 2f0 .
That general case is adapted in figure 2 to the perceptual situation of observer O
looking along initial line OZ to the focus of expansion of point P, which is approaching him parallel to OZ at constant velocity V. Polar coordinates are therefore as
shown. For this case v = V = constant; a = 0; d = rcos 0. Hence
vr = -Vcosd
d
V
T =
;
ve = Fsin0;
ar = -acosO
= 0;
a0 = asinO = 0 .
rcosd
>
V
vd = Vsind = r0 =
YlA.
cos 6
so that
sin 6 cos 6
sin 20
6
" 20 '
T =
If 6 is small, sin2 0 -- 2 0, and
Tn
-*c
26
6
— =—
26
6'
-•
This ratio of optic variables is, of course, the same as that which Lee derived (1976)
from the Cartesian geometry of the optic flow field. Another optic ratio, however,
can also be shown to specify Tc. For a = 0 we have
ae = asind = 0 = r0 + 2r6 ,
whence
20
0
=
_r_
r'
V = const. <J—
Figure 2. Plane motion kinematics of point P moving parallel to OZ at constant velocity V,
showing velocity vectors. Acceleration vectors are zero.
1230
D Stewart, C J Cudworth, J R Lishman
But vr = r = - Fcos 6, so that
20
r
=
6
VcosO
=
Tc
cos2 (9'
whence
2 0cos 2 0
T =
-.
6
For small 0, cos 6 -* 1, and
2(9
rc -*• 0—•^
Hence time-to-collision can be predicted from 0 and 0 as directly as from 0 and 0,
but without the need for any additional information about location or size of object.
This suggests that perception of imminent collision is far more sophisticated than
previously understood. Even a brief view of an object could provide the necessary 0
and 0 to judge time-to-collision. A further benefit is that if the object is accelerating
rather than at constant velocity, time-to-collision is compensated by 2 6/0, as discussed later.
The above calculation could equally relate to a static driver and moving object, to
a moving driver and static object, or to both driver and object in motion. In the latter
situation the directions of motion need not be parallel. Rather, the requirement for
20/0 to approximate closely to time-to-collision is that the angle between relative
velocity and visual axis should be small.
The first experiment was designed to test the hypothesis that 6 and 6 can provide
sufficient visual information to obtain time-to-collision, and that 6 is therefore
unnecessary.
3 Experiment 1
3.1 Subjects
Twelve male volunteers participated in the experiment. All were students aged from
17 to 31 years, who had current driving licences and normal vision.
3.2 Method
A standard method of testing perception of time-to-collision is to occlude an object
approaching at constant velocity, and to ask the subject to judge when it would have
reached him (eg Schiff and Detwiler 1979; McLeod and Ross 1983; Cavallo and
Laurent 1988). He therefore has to respond not to a visible object, but to a memory
of the object. That interval between perception and response differs so much from
reality that it was considered essential to devise a test in which the subject would
respond directly to an approaching object.
The new test was analogous to a braking manoeuvre. When the subject responded
to the approaching object, he initiated a constant rate of deceleration which stopped
the object in a specified period. The task was to minimise the distance between that
terminal point and a target a short distance ahead of him. Final distance from object
to target could then be divided by initial velocity to obtain the subject's error in
estimating time-to-collision.
3.3 Computer simulation
A computer graphics program was written to create the scenario of an object
approaching the observer head on, at constant velocity. There was no background
detail, to eliminate other visual cues, so the simulation could equally be visualised as
a stationary object being approached by a moving observer.
Misperception of time-to-collision by drivers in pedestrian accidents
1231
The monochrome monitor screen was the picture plane on which the view
appeared in true perspective when viewed from a distance of 1.3 screen widths. A
fixed horizontal line was located near the bottom of the screen to provide the target,
specified as if it were a line on the road, ahead of and below the subject (figure 3). To
relate the experiment to the task of avoiding collision with pedestrians the line could
be envisaged as the bonnet of the vehicle whose windscreen was the monitor screen
and whose driver was the subject.
Several geometric and dynamic parameters could be selected for the simulation.
To correspond to the braking task, object velocity was set at an urban traffic speed of
50 km h _ 1 (13.9 m s"1). Stopping time was 1.0 s, which represented emergency braking
with a high coefficient of friction and provided values of 6 and 0 sufficiently large to be
readily perceived. The resulting stopping distance at constant deceleration was 7.0 m.
A side view of the simulation is depicted in figure 4. The target line was located
7 m from the subject, to be midway between him and the point at which braking had
to be initiated to stop at the target. Time-to-collision (at constant speed, as defined) to
the subject from the braking point was then the same as the stopping time because for
uniform deceleration
stopping time = 2 x
stopping distance
initial velocity
The perceptual task of the subject in predicting when to start braking was therefore
the same as it would have been if instead he had been asked to respond when the
object was one second from him, and it had continued at constant velocity.
The approaching object had two alternative forms, as shown in figure 3, for both of
which the eye level of the driver above the road ('driver height') was 1.8 m. One was
\
><
s
J
Figure 3. Computer display for experiment 1, showing target line being approached by square
(left) and pixel (right). The crosses indicate foci of expansion, and the arrows represent optic
flow, but only the white shapes are actually displayed.
stopping distance = 7.0 m
7.0 m
3.6 m cube
I 13.9 ms
I
target line ->
(transverse) \
error e (m)
-J L
I
invisible road
point (pixel)
Figure 4. Side view of simulation in figure 3. Cube and point are alternative objects approaching the subject head-on at constant velocity V. After response by the subject they decelerate
uniformly to a stop in 1.0 s. Terminal error e provides time-to-collision error from e/V.
1232
D Stewart, C J Cudworth, J R Lishman
a square of side 3.6 m, depicted in figure 4 as the face of a cube, whose focus of
expansion was therefore its centre. Its approach provided 0, 0, and 0 from its size
and dilation, so the subject could derive Tc from either 0/0 or 2 0/0. The other was a
single point (a pixel) in the same position as the mid-point of the bottom of the
square. Because the pixel did not dilate and its focus of expansion was not immediately evident, it provided no basis for quickly determining 0 or distance. The subject
could therefore derive Tc only from 0 and 0, both being available as the pixel
accelerated down the screen.
3.4 Procedure
The subject was seated at the correct viewing distance from the screen. He was told that:
• The experiment was a driving simulation, in which he was the driver.
• He would see an object ahead of him, either a square box or a point like a cats-eye,
and a target line representing the bonnet of his vehicle.
• The initial distance to the object would vary (so that he could not use it as a
datum for time or 0).
• The object would always approach at the same speed, and would decelerate
uniformly to a stop in one second after the space bar was pressed.
• The requirement was not to stop before the target, but as near to it as possible.
Variation in initial distance was provided by viewing times of 2 s, 4 s, and 8 s
before the point at which the 1 s braking should be initiated. That gave a total of 6
variants for the square and the pixel together.
The subject was first presented with each of these 6 variants, in random order, to
familiarise himself with the 1 s braking time. They were followed by 30 randomised
trials, comprising 5 of each variant. The success of each trial was immediately
apparent to the subject by noting the terminal position of the object relative to the
target, but no other feedback was given.
3.5 Results and discussion
For each trial, the distance from terminal position to target was converted to the
corresponding travel time at 5 0 k m h - 1 , to give the error in predicting time-tocollision. These were combined for each variant to give the mean error and the
standard error of that mean, as shown in figure 5.
The results confirm that the optic flow of a point is an effective source of time-tocollision. Time-to-collision was on average more accurately perceived for the pixel
than for the square, despite its lack of information about 0, as is apparent from
0.2
r
Viewing time/s
Figure 5. Errors in judging time-to-collision of 1.0 s for square and pixel in experiment 1, as
function of viewing time. Vertical bars represent standard errors, which are standard deviations divided by ^/12.
Misperception of time-to-collision by drivers in pedestrian accidents
1233
figure 5. For each viewing time, the difference between the errors in mean Tc for the
square and the pixel was tested by making a pooled estimate of their variance, then
calculating Student's t. This confirmed that in every case the difference between the
means was highly significant (p < 0.001).
The results also suggested that subjects could respond more consistently to the
pixel than to the square because the standard errors, as plotted in figure 5, are greater
for the square than for the pixel. Statistical testing confirms that this difference also is
highly significant (p < 0.001). The dilation of the square may therefore have been
more of a distraction than an aid to perceiving time-to-collision, a view expressed by
some of the subjects in subsequent discussion.
Both curves also show that, as viewing time increased, subjects responded earlier.
This is probably a regression effect. That is, the subject would tend to respond too
early when viewing time is longer than average, and the converse when it is shorter
than average.
The success of the subjects in responding to the pixel has interesting implications
for understanding interceptive or avoiding action. For instance, the ability to respond
rapidly and accurately in dynamic situations which seem impossible in terms of the
6/6 or d/V hypotheses, such as catching a ball with only 0.1 s view time (Whiting
et al 1970), becomes more plausible.
Another complication for the 6/6 hypothesis is an accelerating object, because
time-to-collision is bound to be less than 6/6, yet such an object can be accurately
responded to (eg Runeson 1975; Lee etal 1983). The difficulty of doing so would
be reduced, however, by using 2 6/6 rather than 6/6 because acceleration changes
the value of 6, but not 6 or 6. That is, acceleration toward the observer increases 6
which therefore reduces perceived Tc. This is confirmed in the Appendix.
It must be asked, however, how even 2 6/6 could provide time-to-collision for a
featureless, irregular object, or for Savelsbergh's shrinking ball. The answer might be
that 6 and 6 are based on average or 'whole body' values of these parameters, so that
the reference point is in effect the centroid of the object. The results of experiment 2
will support this hypothesis, and suggest that 'whole body' Tc is the preferred perceptual strategy for any approaching object, regardless of its visual form.
Use of 'whole body' Tc will fail only if the object is moving directly towards the
observer's eye, making 6 and 6 zero. For that condition, perception must presumably
revert to alternative means of estimating Tc, which may be why it has been found
(Schiff and Oldak 1990) that prediction of time-to-collision is less accurate if a
trajectory is head-on rather than by-pass.
Loss of accuracy should also result if a trajectory is curved rather than straight.
That would distort 6, hence leading to overestimation or underestimation of time-tocollision, depending on whether the curve is convex or concave relative to the
observer as it passes him. In the road safety context, this source of time-to-collision
error could be a cause of accidents where drivers have to accept gaps in traffic on a
curved path, as at a T-junction on a bend.
Both acceleration and curvature, however, occur in most natural motions so our
perceptual process would be expected to operate in such a way as to minimise the
resulting errors in time-to-collision. The most plausible method is continuous monitoring of an approaching object, as has been postulated from previous research (eg
Lee etal 1983; Savelsbergh etal 1991), so that time-to-collision can be continually
updated. Such monitoring must be easier if Tc is derived from 2 6/6 rather than
6/6, because of the facility of perceiving 2 6/6 and because errors due to an accelerating
object are much less, as shown in the Appendix.
1234
D Stewart, C J Cudworth, J R Lishman
4 Derivation of time-to-collision from familiar size
As proposed earlier, direct perception from optic flow may not be an adequate source
of time-to-collision between drivers and pedestrians in all circumstances. It was
postulated (Stewart 1991) that a cause of child pedestrian accidents might be difficulty in interpreting optic flow from a moving figure, which could encourage the use
of familiar size as an additional cue for estimating time-to-collision from distance d
divided by velocity V.
Subsequent testing with a variant of the computer simulation has supported that
hypothesis, although the perceptual difficulty appears to be due to reduction in optic
flow with increase in distance d, rather than to the movement of the pedestrian. The
effect of that reduction will be more severe if time-to-collision is derived from 20/0
than from 0/6, because 6 is proportional to d~3, whereas 0 and 6 are proportional to
d~l and d~2 respectively. Difficulty of perceiving optic flow information may therefore
increase so rapidly with distance that it is useful only if time-to-collision is brief.
This thesis is reinforced by considering the effect on 2 0/0 of an approach which is
accelerating rather than at constant speed, as is done in the Appendix.
That limitation may, however, have been unimportant before the advent of highspeed transport. The primal or 'ecological' purpose of perceiving time-to-collision is
to initiate self-action for interception or avoidance. For these tasks time-to-collision
is most critical at less than 1 s, as in catching (eg Alderson et al 1974) and jumping
(Lee et al 1982), so optic flow should be an adequate source of time-to-collision. It
would not be, however, for driving a vehicle. Simply to start braking can take over
1 s (Road Research Laboratory 1963), and stopping time at an urban speed of
5 0 k m h _ 1 is about 3 s, even in good conditions. In adverse conditions it can be
considerably more—far beyond 'ecological' requirements for time-to-collision.
It is not surprising, therefore, that increase in time-to-collision beyond its ecological
range increases error (eg McLeod and Ross 1983; Schiff 1979). That is probably
why drivers resort to indirect estimation of time-to-collision from d/V well before
reaching the perceptual threshold for 0 (DeLucia 1991). We must therefore ask how
these separate perceptions of velocity and distance are made by a driver when
approaching a pedestrian.
4.1 Estimation of velocity
Relative velocity between driver and pedestrian is largely dictated by the speed of the
vehicle. Unlike time-to-collision, however, velocity cannot be directly perceived from
optic flow unless the location of the point generating the flow is known. It is
therefore apparent that a driver's perception of velocity should be most reliably
provided by the optic flow of the road surface as it is occluded by the bonnet of the
vehicle. That point on the road is at constant distance and orientation to the driver,
so the angular velocity of its optic flow is directly proportional to vehicle velocity,
which is therefore constantly available to the driver. This is probably why driver
error in determining the velocity of his vehicle tends to be small (Spurr 1969).
In both experiments described in this paper, however, road surface texture was
omitted. But subjects were told that every trial was at the same velocity, so that was
not a variable which they had to monitor. To derive Tc from d/V, therefore, only
distance d from driver to 'pedestrian' had to be estimated.
4.2 Estimation of distance
Determination of distance is more difficult. On a level road the distance d to an
approaching point on the road surface could, in theory, be determined from its angle
0p below the horizontal. Distance would then be obtained from the relationship
d « H/0p, where H is driver height. Such a computation would, however, be unreliable because of the large error resulting from even slight vertical curvature of the road.
Mis perception of time-to-collision by drivers in pedestrian accidents
1235
It has also been demonstrated (Sedgwick 1983) that angle from the horizontal cannot
be judged with the necessary accuracy.
Many other cues for distance have been postulated. The most informative is
familiar size (Kunnapas 1968), that is the actual size as recollected from previous
experience, which is the principal distance cue for driving (Hills 1980). Unlike other
cues, the scaling of retinal size against familiar size permits distance to be reliably
estimated, provided that familiar size is correct, because a unique value for distance
is obtained from the relationship
distance =
familiar size „
,
;
;— x focal length of eye
retinal size
4.3 Application to pedestrians
To appreciate why scaling from familiar size is liable to endanger child pedestrians
consider the perceptual sequence necessary to estimate distance from the familiar
size of an object:
(i) The object must be recognised.
(ii) The familiar size of the object must be recollected.
(iii) Familiar size must be compared to retinal size to give distance.
Objects of uncertain or 'nonfamiliar' (Gogel 1964) size would therefore be expected
to create distance errors, especially in visually impoverished conditions such as fog.
This situation has been examined by Ross (1967), who found distance estimates twice
as great as reality. But every estimate from familiar size must be liable to error, even
if the observer is unaware of it. The influence of these errors on road safety has been
generally ignored, although it has been proposed (Eiberts and MacMillan 1985) as a
cause of accidents involving small cars.
Eibert's hypothesis is that a small car tends to be misperceived as a larger car at a
greater distance, a scaling error which will be termed 'regression to mean size'. He
suggests that this would lead to delay in responding to impending collision with the
car, causing small cars to be over-represented in accident statistics.
Similar logic suggests that regression to mean size would influence pedestrian
accidents. A small pedestrian would tend to be misperceived as a larger pedestrian at
a greater distance but subtending the same visual angle at the observer (figure 6). The
smaller the child, the greater the error, which may help to explain why, for instance,
boys have 6 times the risk of accident per vehicle encounter at age 5 as at age 10
(Howarthetal 1974).
This argument may be elucidated by considering more closely the perceptual
sequence required for a driver to estimate distance from height of pedestrian. In the
absence of proximate cues for size, as is common on a road, the driver must deduce
height from other characteristics, such as age, shape, and gait, all of which are
correlated with height. But the complexity of such a judgement suggests that when
velocity V
adult
Figure 6. If retinal image is wrongly scaled against familiar size of adult rather than of child, the
resulting distance error e produces time-to-collision error e/V.
1236
D Stewart, G J Cudworth, J R Lishman
collision is imminent it would be preceded by a simpler one. Initially an average or
mean size of pedestrian would be assumed as the familiar size, and would then be
refined as the situation permits. To examine whether an error due to regression to
mean size occurs, another computer simulation was developed.
5 Experiment 2
5.1 Subjects
Ten male volunteers participated in the experiment. All were students aged from 21
to 43, who had current driving licences and normal vision.
5.2 Method
As in experiment 1, an object approaching at a constant velocity of 50 km h _ 1 had to
be stopped as near as possible to a target line 7 m ahead of the subject. To accentuate the difficulty of identifying familiar size, an object of nonfamiliar size was used,
a simple rectangle whose actual size could be estimated by subjects only from
experience during testing. Its height to width ratio was 5:1, and it had two variants, a
1.8 m 'adult' and 1.0 m 'child', which always stood upright on the horizontal plane of
the 'road'. There were also two variants of driver height, 1.0 m representing a car,
and 1.8 m representing a large vehicle, which dictated the perspective of the simulation.
These variants are depicted in figure 7.
As before, road or other background detail was absent from the display, to minimise visual cues. Earlier tests had shown, for instance, that inclusion of a horizon
encouraged subjects to use horizon scaling, a method of estimating size which has
been elucidated by Sedgwick (1983). Care was taken that the screen had no marks or
reflections which could provide datums for scaling. The view was seen in true
perspective when picture distance was 1.3 times the width of the screen, as in
experiment 1.
In each trial the object appeared about 7 s before the subject had to brake, slight
variation in that time being applied randomly by the program to preclude its use as a
temporal cue. These and other parameters of the experiment were selected to test
several predictions:
(i) With increase in braking time, time-to-collision error would increase, as it becomes
more difficult to derive Tc from optic flow. Four braking times were therefore used,
0 s, 2 s, 4 s, and 6 s, a range covering most emergency braking situations at 50 km h" 1 .
(ii) Subjects would respond too late for 'child' and too early for 'adult', because of
regression to familiar size, the mean size of 'pedestrian'. That mean size would be
weighted according to the ratio of 'children' to 'adults'. In each set of trials equal
numbers of 'adults' and 'children' were provided, so that mean height would be 1.4 m.
Figure 7. Computer displays of 1.0 m 'child' (left) and 1.8 m 'adult' (right) used in experiment 2.
The crosses indicate foci of expansion for driver heights of 1.0 m (lower) and 1.8 m (upper), and
the arrows represent optic flow at the base of the rectangles. Only the white shapes are
actually displayed.
Misperception of time-to-collision by drivers in pedestrian accidents
1237
(iii) Tc error would decrease with increase in driver height. Angular velocity 6 of a
point on the road surface ahead of a driver is approximately proportional to his eye
level, so increase in driver height will facilitate his judgement of 6 and 6, and hence
of 20/0. That should postpone the transfer from 2 0/0 to d/V as time-to-collision
increases, and thus reduce scaling error.
5.3 Procedure
The subject was seated in front of the screen, at approximately the correct picture
distance. He was told that:
• The experiment was a driving simulation, in which he was the driver.
• He would see a rectangular object ahead of him, which could vary in size and
initial distance, but not in shape.
• The object would always approach at the same speed, and would decelerate
uniformly to a stop X seconds after the space bar was pressed. The tester would
nominate X for each group of trials. (It had been established that 4 familiarisation
trials were adequate to accustom a subject to X).
• The requirement was not to stop before the target line, but with the base of the
rectangle as near to it as possible.
The first set of trials had driver height 1.0 m and braking time 0 s. The 4 familiarisation trials were followed by 10 actual trials. Subsequent sets of trials had braking
times of 2 s, 4 s, and 6 s with driver height kept at 1.0 m. As in experiment 1, the
subject had visual feedback from the terminal position of the rectangle, but was
given no other information.
After these 56 trials, a further 56 followed the same sequence, but with driver
height increased to 1.8 m. Finally, the trials for 1.0 m driver height and 2 s braking
time were repeated to check for consistency. On completion, details of age, driving
experience and road accident record were noted.
5.4 Results and discussion
For each trial, the terminal distance between object and target was converted to the
equivalent travel time at 5 0 k m h _ 1 , to find the temporal error when braking was
initiated. If there was no error, time-to-collision at the braking point would, from
section 3.3, have been half the braking time plus 0.5 s, the time from target to subject
at 50 km h" 1 . That is, the braking times of 0 s, 2 s, 4 s, and 6 s correspond to timesto-collision of 0.5 s, 1.5 s, 2.5 s, and 3.5 s, respectively.
For each of the 8 combinations of driver height and braking time, the mean and
standard deviation of all trials were calculated. Note that the task at Tc = 0.5 s was a
judgement of coincidence, rather than of time-to-collision.
Misjudgements of Tc for 'child' and for 'adult' are plotted in figure 8 for 1.0 m
driver height, and in figure 9 for 1.8 m driver height. The standard error is also
plotted, to indicate the reliability of the curves. On comparing them by /-test, as in
experiment 1, it was found that the only differences between 'child' and 'adult' which
were nonsignificant (p > 0.05) were those in which the standard error bars overlapped.
It is evident that all three predictions are supported.
(i) Tc error increased as Tc increased.
(ii) Tc error was an overestimate for 'child' and an underestimate for 'adult' in almost
all trials,
(iii)The difference in Tc error between 'child' and 'adult' decreased, by about 50%,
when driver height was increased from 1.0 m to 1.8 m.
To facilitate interpretation of the results, lines have been added to show a hypothetical
Tc error derived solely from d/V, with V correct but d based on a familiar height of
1.4 m, the mean of the 'child' and 'adult' heights. These 'regression to mean size'
D Stewart, C J Cudworth, J R Lishman
1238
lines therefore represent:
T error = T x | m e a n h e i g h t _
\ actual height
1
As would be expected, the actual errors for 'child' and 'adult' tend to be less than
these because the subjects could estimate Tc from optic flow as well as from d/V. An
exception is the 'adult' 1.0 m driver height combination, where errors are close to the
regression line, suggesting that optic flow was not used. If so, the reason may be that
Tc is normally interpreted from 'whole-body' 6 and 0, as postulated in section 3.5.
The centroid of the 1.8 m 'adult' being only 0.1 m below the focus of expansion when
driver height is 1.0 m, 'whole body' 6 would be very small, so d/V would become the
prime source of Tc. 'Regression to mean size' errors would therefore approach the
predicted maxima.
The perceptual strategy suggested by these results therefore is that subjects strive
to use optic flow even if 6 is very slow. But as 0 approaches its perceptual threshold
they have to place increasing reliance on d/V.
Two of the subjects, however, responded differently from the other ten. Both had
almost identical error curves, showing no regression to mean size until time-tocollision exceeded 2.5 s, which suggested that optic flow was not being supplemented
by d/V. It may therefore be significant that these two subjects were the only ones to
have had several accidents as drivers.
Figure 8. Errors in judging time-to-collision for experiment 2, with driver height 1.0 m. Vertical
bars represent standard errors, which are standard deviations divided by ^10.
r 'adult'
Figure 9. Errors in judging time-to-collision for experiment 2, with driver height 1.8 m. Vertical
bars represent standard errors, which are standard deviations divided by ^10.
Misperception of time-to-collision by drivers in pedestrian accidents
1239
In this experiment, V was constant so it should not have been a source of bias, but
d could be misjudged because of regression to mean size of 'pedestrian'. Therefore
the greater the reliance on d/V rather than on optic flow, the closer Tc error
approaches the regression line. That conclusion was supported by additional trials
with 'pedestrians' of constant height, to remove regression-to-mean-size error. Mean
Tc errors then remained close to zero as Tc increased.
6 Comparison with actual child pedestrian accidents
The question whether differences in response to 'child' and 'adult' rectangles were
reflected in real accidents with actual pedestrians is a vital one.
With driver height 1.0 m the subjects stopped on average 0.6 s later for 'child' than
for 'adult', a difference in stopping distance of about 8 m. To investigate how this
might be related to the accidents which each year kill or injure 21000 children
crossing roads in Britain it is necessary to ask how real Tc errors might differ from
the experimental ones.
Real errors might be smaller because familiar size should more readily be identified
for a real pedestrian than for a test rectangle. Also, there will be the additional cue of
carriageway width, which should facilitate distance perception just as runway width
does for landing an aircraft (Chappelow and Smart 1982).
On the other hand, actual Tc errors might be greater than in the experiment.
Whereas the subjects encountered equal numbers of 'children' and 'adults', in reality
adult pedestrians are more frequent than children. That would weight the mean
height towards adults, and therefore increase Tc error and risk for child pedestrians.
It could also be argued that whereas the subjects could concentrate on the experimental task, a driver has other tasks and distractions which would increase his Tc
error.
It is impractical to evaluate these opposing influences, but the need to do so can be
obviated by a different approach. The 'scientific method' of formulating a hypothesis,
then testing and corroborating it, is regrettably rare in road safety research, particularly in view of the massive data base which is available—the 260000 injury accidents
documented each year by the police as STATS 19 records. In 60000 of these
accidents pedestrians are killed or injured.
6.1 Casualties on pedestrian crossings
The height of a pedestrian crossing a road should be an important distance cue
because his immediate surroundings tend to lack other sources of familiar size.
Where these are available, however, Tc error due to regression to mean size should
decrease.
The most obvious source of familiar size near pedestrians is the black and white
stripes of a zebra crossing—in effect a standard scale across the road, as seen in
figure 10. A driver approaching a pedestrian on the crossing should be able to judge
distance more reliably from the retinal size of these conspicuous markings than from
that of the pedestrian, although he will be unaware that he is doing so. The risk
differential between children and adults should therefore decrease. It can therefore
Figure 10. Zebra markings and high-visibility guard rail to standardised dimensions provide
familiar size cues, which counter 'regression to mean size' error for pedestrians.
1240
D Stewart, C J Cudworth, J R Lishman
be predicted that the proportion of pedestrian casualties who are children (the 'child
casualty percentage') should be lower on zebra crossings than elsewhere.
That prediction was checked from published statistics (Department of Transport
1991). Away from crossings, the child casualty percentage was 36%, but on zebra
crossings only 24%. Pelican crossings, however, are marked only with inconspicuous
studs, so the child casualty percentage should be higher. This prediction was also
confirmed, the child casualty percentage on pelican crossings being 29%. Other
factors could influence these figures, of course, so more robust tests were applied.
6.2 Pedestrian casualties and driver height
The results of experiment 2 supported the hypothesis that increase in driver height
reduces Tc error because it accelerates optic flow and therefore reduces reliance on
d/V. Children should therefore comprise a higher proportion of casualties for cars
than for lorries. Accordingly, Department of Transport accident statistics were analysed
and plotted as curves in figure 11.
A dramatic difference between cars and lorries was found. Children between 5 and
15 years old exhibit a pronounced casualty peak for cars, which disappears completely for lorries. If this is due to driver height, then the intermediate height for vans
should produce an intermediate curve. This also was confirmed, as seen in figure 11.
It may be argued that these differences simply reflect exposure—that lorries tend to
predominate on roads where children are less common. But when the curves are
redrawn for individual classes of road—A, B, and minor—the pattern of the curves
remains similar. The implication, therefore, is that if time-to-collision error by car
drivers could be reduced to the level of lorry drivers, there would be a massive reduction in child pedestrian accidents.
15
8 10 f
0
10
20
40
60
80
Age of pedestrian casualty/years
Figure 11. Age distribution of pedestrian casualties in Great Britain, 1988, by vehicle type.
6.3 Location of pedestrian casualties
That conclusion is so remarkable that corroborative evidence was essential. The tests
dealt with situations where perceptual information had been enhanced, first by zebra
markings, then by increasing driver height. For corroboration, a completely different
method of testing for regression-to-mean-size error was desirable; an accident situation
which instead of facilitating perception of time-to-collision, removed the need for it.
Tc error would then be irrelevant, so it would no longer influence casualty statistics.
Such accidents should be typical of footways. Before mounting a footway a vehicle
is usually out of control, so misjudgement of Tc to pedestrians should be much less
Misperception of time-to-collision by drivers in pedestrian accidents
1241
important than on the carriageway. That difference should reveal itself in two ways.
First, the child casualty percentage should be far smaller on the footpath than on the
carriageway. This was confirmed: for all types of vehicle combined, child casualty
percentage was 36% on carriageways but only 18% on footways. Second, cars and
lorries should again show different results. If increase in driver height reduces Tc
error, as postulated, that benefit should be much less in footway collisions, where
estimation of Tc is unimportant. Thus the reduction in child casualty percentage from
carriageways to footways should be smaller for lorries than for cars.
That prediction also was confirmed. For cars the child casualty percentage decreased
from 42% to 19%, but for lorries only about half as much, from 19% to 13%. Both
tests indicate, therefore, that misjudgement of time-to-collision to children is prevalent
on the carriageway, especially for drivers of cars.
6.4 Summary of tests
In each of the three accident situations examined—type of crossing, height of driver,
and footways—the predicted statistical anomalies were found to exist. x2 tests were
applied to check whether the anomalies could simply have occurred by chance, and in
each case the probability was found to be negligible (p < 0.0001).
These corroborations provide convincing evidence, therefore, that misjudgement of
time-to-collision by drivers causes a large proportion of child pedestrian casualties.
That proportion cannot be reliably estimated from the first test, however, because of
nonperceptual differences between zebra and pelican crossings.
The tests based on driver height and on footway/carriageway comparison suggest,
however, that about half of all child pedestrian casualties, or about 30 per day in
Britain, would not occur if car drivers could judge time-to-collision as reliably as
lorry drivers. It follows that if time-to-collision error by drivers of all types of vehicle
were eliminated, the improvement would be even greater.
7 Conclusions
It has been demonstrated that perceptual error by drivers is the main reason that
children have a much higher pedestrian accident rate than adults. The innate difficulty
of judging time-to-collision from optic flow when driving requires that judgement to
be supplemented by separate perceptions of distance and speed. Distance to a
pedestrian is primarily based on apparent size, so if a child is misperceived as a larger
person at a greater distance, time-to-collision will be overestimated. Analysis of
accident data indicates that this error causes over half of Britain's daily toll of about
60 child pedestrian casualties.
It must be asked why this has only now become apparent. One reason is that
research into child pedestrian accidents usually starts from the premise that the
inexperience and carelessness of children is the main causation. As a result, a standard conclusion is that safety education must be improved, despite failure to show
that this is effective in reducing accidents (eg Ampofo-Boateng and Thomson 1989).
Another reason that driver error has seldom been implicated in child pedestrian
accidents is that drivers are unaware that they are misjudging time-to-collision. The
perceptual process is so automatic that its inerrancy is taken for granted, both by
drivers and by safety professionals. Hence time-to-collision error by drivers is
unlikely to feature in any accident report.
A radical reappraisal of the safety of child pedestrians is therefore necessary.
Their high level of risk may have been tolerable if they themselves could be blamed
for their accidents, but not if drivers have prime responsibility.
What must also be reviewed is why Britain has the worst child pedestrian accident
record in Europe. If driver perceptual error is so critical, are there features of
1242
D Stewart, C J Cudworth, J R Lishman
Britain's roads which exacerbate it? Suspicion must fall, for instance, on pedestrian
crossings.
Countries comparable to Britain make far wider use of zebra crossings, whose
white stripes should facilitate perception of distance and time-to-collision. In Britain
the number of zebra crossings has greatly declined as they are replaced by pelican
crossings, despite evidence that this change increases accidents (eg Lalani 1975).
Other sources of familiar size have been proposed, such as high-visibility guard
rails (Stewart 1988) and delineator posts (Landwehr 1991) of standard height, to give
a rich scaling environment such as shown in figure 10. Stewart (1988) proposed that
this was why accident reduction after erection of high-visibility guard rails was
higher than predicted.
Risk due to time-to-collision error should also be greatly reduced by lower traffic
speeds, which would accelerate optic flow at stopping distance. Research to develop
and validate such ideas is necessary, if significant reductions in pedestrian casualties
are to be achieved.
The more fundamental conclusion that when time-to-collision is obtained from
optic flow it is computed from speed 6 and acceleration 0 has wider implications. In
particular, the facility of using 'whole body' optic flow to derive time-to-collision from
2 0/0 clarifies how it is possible to respond effectively even in difficult situations, such
as a momentary view of an accelerating object. Not only do these developments
indicate a need to reappraise previous research, but they promise a new understanding of the perception of motion.
Acknowledgement. The assistance of the Rees Jeffreys Road Fund in the initial stages of this
research is gratefully acknowledged.
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APPENDIX
Time-to-collision with an accelerating object
From the standard kinematic equations in section 2.1, for a head-on approach with
constant acceleration A we have
/
rd
or sin0 = —
I
v
ve = rd = vsin0
ae = A s i n 0 = rB + lfO .
From these two expressions we obtain
= rO + lrB
or rd = 0
(T- •If
V
whence
20
6
2rv
Ar-2rv'
For small 0, r -+ d, r -»• - v, and
20
2dv
I
2
0 ' Ad + 2v ~ I 2v
d
'
1244
D Stewart, C J Cudworth, J R Lishman
The last expression corresponds to the reality that an object accelerating towards an
observer has a shorter time-to-collision than if it maintains a constant velocity. That
is, if 2 6/6 is being monitored, then it automatically reduces time-to-collision as
acceleration increases. How reliably it does so can be determined by comparison with
the actual time-to-collision, which from the Newtonian equations of motion, is
Z =
{v2 +
2Adf/2-v
If this comparison is applied to 'ecological' situations such as a falling rock, it is
found, on ignoring wind resistance, that 2 6/6 slightly underestimates time-to-collision,
whereas 6/6 greatly overestimates it. This is illustrated in figure A l , in which the
rock is falling from an initial height D of 20 m with gravitational acceleration
g = 9.81ms" 2 .
The generating equations for the three curves are developed from the optic and
dynamic relationships derived for 6/6, 2 6/6, and Tc. It is then found that
2 6/6
Tc
=
2k{l-k)l/2
[l-(l-fc)1/2](4-3fc)'
where k = d/D. This demonstrates that 26/6 underestimates time-to-collision by a
factor which is dependent only on k. At mid-trajectory, for instance, 2 6/6 will always
underestimate time-to-collision by 3.4%, whereas the 2 1 % overestimate for 6/6 in
figure A l is true only for D = 20 m. Both the safety margin implied by that slight
underestimate and its constancy reinforce the case for 2 6/6 being the preferred
cognitive algorithm. But 2 6/6 is much less than time-to-collision at the beginning of
the trajectory. That apparent deficiency may have 'ecological' value by promoting the
startle response when you see something begin to fall towards you. Having activated
the observer by falsely indicating instant collision, 2 6/6 would almost immediately
provide a reliable source of time-to-collision for avoiding or interceptive action.
20
/*r c
e
1/
//
15
1
1
1
/
\
y'^-e
Curve generators
ynn
'
/*
1 '
10h
r
r
'
„---
HT)1
/'
e
e "
/
5h
d
[2g(D-d)]l/2
2{D-d)
2d
4D-3d [
8
2d
6
i_
— J
0
1
1/2
2
3
4
rc/s
Figure Al. Time-to-collision with an accelerating object.
© 1993 a Pion publication printed in Great Britain