RESEARCH ARTICLE
Intercepting moving targets: a little foresight helps a lot
Gabriel Jacob Diaz Æ Flip Phillips Æ Brett R. Fajen
Received: 14 July 2008 / Accepted: 31 March 2009 / Published online: 26 April 2009

Springer-Verlag 2009
Abstract Behavioral studies suggest that humans inter-
cept moving targets by maintaining a constant bearing
angle (CBA). The purely feedback-driven CBA strategy
has been contrasted with the strategy of predicting the
eventual time and location of the future interception point.
This study considers an intermediate anticipatory strategy
of moving so as to produce a CBA a short duration into the
future. Subjects controlled their speed of self-motion along
a linear path through a simulated environment to intercept
a moving target. When targets changed speed midway
through the trial in Experiment 1, subjects abandoned an
ineffective CBA strategy in favor of a strategy of antic-
ipating the most likely change in target speed. In Experi-
ment 2, targets followed paths of varying curvature.
Behavior was inconsistent with both the CBA and the
purely predictive strategy. To investigate the intermediate
anticipatory strategy, human performance was compared
with a model of interceptive behavior that, at each time-
step t, produced the velocity adjustment that would mini-
mize the change in bearing angle at time t ? Dt, taking into
account the target’s behavior during that interval. Values of
Dt at which the model best fit the human data for practiced
subjects varied between 0.5 and 3.5 s, suggesting that
actors adopt an anticipatory strategy to keep the bearing
angle constant a short time into the future.
Keywords Interception
Locomotion
Visual control

Anticipation
Introduction
Running to intercept a target moving across the ground
plane is one of the basic locomotor tasks performed by
creatures ranging from predators in the wild to humans
on the playing field. The study of locomotor interception
has provided a window onto the nature of the informa-
tion–movement coupling that characterizes many forms
of visually guided behavior. However, most of the efforts
(both theoretical and empirical) to understand locomotor
interception have been directed at the interception of
constant velocity targets. In only a small number of
studies is it acknowledged that moving targets change
speeds and directions, oftentimes in ways that cannot be
perfectly predicted. In this study, we begin by consid-
ering the usefulness of previous models designed for
constant velocity targets within a somewhat more real-
istic context that involves targets that change speed or
direction. We then present two experiments designed to
investigate human behavior in this situation. In Experi-
ment 1, subjects were tested for the ability to accurately
anticipate probable changes in a target’s speed. In
Experiment 2, subjects were presented with targets that
approached along either a rectilinear path or one of
several possible curved trajectories. Subjects’ behavior
was compared to that of an ideal pursuer that accurately
positions itself in anticipation of the expected target
position and velocity a brief period into the future. The
degree of similarity between subjects’ behavior and the
behavior of the ideal provided an indication of subjects’
ability to act so as to bring about a desired future state
G. J. Diaz (&)
B. R. Fajen
Department of Cognitive Science, Rensselaer Polytechnic
Institute, 110 8th Street, Troy, NY 12180-3590, USA
e-mail: diazg2@rpi.edu
F. Phillips
Department of Psychology/Neuroscience, Skidmore College,
815 North Broadway, Saratoga Springs, NY 12866, USA
123
Exp Brain Res (2009) 195:345–360
DOI 10.1007/s00221-009-1794-5

that is dependent upon accurate anticipation of the tar-
get’s dynamics.
Situations involving moving targets that change velocity
can often be observed in a game of American football. In
football, the defenseman’s task of intercepting the ball
carrier is complicated by unpredictable changes in the
speed and direction of the ball carrier as he evades pursuit.
The experienced defenseman orients his approach in
anticipation of these changes and in such a way that limits
the ball carrier’s chances of evasion. In other words, an
effective control strategy accounts not just for current tar-
get behavior, but also for the possible changes in behavior.
The constant bearing angle model
The most widely accepted model of locomotor interception
is based on a strategy that has been used for centuries by
sailors to avoid collisions with nearby vessels (Le Brun
2002). Known amongst scientists as the CBA model of
interception, the strategy proposes that by maintaining a
CBA the observer is guaranteed to intercept the moving
target. The bearing angle (W) is the direction of the target
with respect to an exocentric reference line (see Fig. 1).1
An increasing bearing angle suggests that the observer will
pass in front of the target and that it is necessary to
decelerate or turn toward the target for a successful inter-
ception. A decreasing bearing angle suggests that the
observer will pass behind the target, and that it is necessary
to accelerate or turn ahead of the target for a successful
interception. As would be expected based on the CBA
hypothesis, subjects perceive an impending collision when
their self-motion is accompanied by the movement of a
target whose bearing angle remains constant (Cutting et al.
1995). In addition, the active interception strategies of both
humans and of several non-human animals including
dragonflies (Olberg et al. 2000) and teleost fish (Lanchester
and Mark 1975) conform to the predictions of the CBA
model. The CBA model of interception has also been
suggested as a potential control law for guiding lateral
movements when running to catch a fly ball (Chapman
1968).
The first two studies to test the CBA strategy in humans
were reported by Lenoir et al. (1999a, 1999b). Subjects
were instructed to hit a target moving along a track by
controlling their approach speed on a tricycle with heading
fixed. Although results were consistent with the CBA
model, the task was somewhat oversimplified due to
technical limitations. The target’s movement was restricted
to one approach angle, and the target moved at one of only
two possible speeds. Each speed was presented ten times
for a total of twenty trials of limited variety. This simplicity
of design may have allowed for behavior stereotyping in
response to the observed speed. Furthermore, target motion
started only after the observer was a short fixed distance
from the interception point, limiting the analysis to the final
seconds of approach. Thus, although the data were sug-
gestive of a CBA strategy, the findings were far from
conclusive due to methodological limitations.
In a subsequent study (Lenoir et al. 2002), more con-
clusive findings were provided by showing that behavior
was still consistent with the CBA model even when there
was more variety in initial conditions and the task and
environment were more natural. Chardenon et al. (2005)
found that subjects walking on a treadmill adjust locomotor
speed in accordance with the predictions of the CBA
model, making more pronounced velocity adjustments
when the target approaches from a greater (i.e., more
orthogonal) angle. Whereas locomotor heading was fixed
in the aforementioned studies, subjects in Fajen and War-
ren (2004) intercepted moving targets while walking in a
large area virtual environment that also permitted changes
in walking direction. They found that subjects turn while
walking until they lead the target by a bearing angle that
can be maintained until interception.
Compelling support for the CBA strategy was provided
by Bastin et al. (2006) using targets that approached along
1 Fajen and Warren (2007) presented simulations to show that the
bearing angle whose change must be nulled is defined in an exocentric
reference frame. When simulated agents keep the target at a constant
egocentric (rather than exocentric) direction, the resulting trajectory
spirals behind the moving target for some initial conditions. By
comparison, human subjects follow a straight path ahead of the target
(Fajen and Warren 2004).
α
Ψ
Ex
oc
en
tri
c
Re
fe
re
nc
e
Li
ne
Fig. 1 An overhead view of an exemplary interception situation. The
pursuer (triangle) and target (circle) approach the invisible intercep-
tion point (diamond). W denotes the exocentric direction of the target
(bearing angle) and a denotes the target’s approach angle
346 Exp Brain Res (2009) 195:345–360
123

curvilinear paths. The experiment was set up in such a way
that, on a subset of trials, subjects could successfully
intercept the target without changing speed. No speed
adjustments were necessary on such trials because when
the target first appeared, subjects were already moving at
the speed that would eventually bring them to the inter-
ception point at the same time as the target. Although it
was not necessary for a successful interception, subjects
regulated their velocity on the basis of changes in bearing
angle that resulted from the target’s curvilinear approach.
These velocity regulations fit qualitative predictions of the
direction and magnitude of the subjects’ initial and sub-
sequent velocity adjustments. Furthermore, regression
analyses based upon the CBA strategy were able to explain
an average of 56% of the total variance at the trial level,
and 75% of the total variance at the group level, after the
data has been averaged across trials and subjects.
To summarize, numerous previous studies on locomotor
interception suggest that actors adjust locomotor speed
and/or direction to keep the target at a CBA. Additional
past research on this problem was aimed at uncovering the
informational basis for perception of the change in bearing
angle, and specifically, the contributions of visual, propri-
oceptive, vestibular, and podokinetic information (Fajen
and Warren 2004; Bastin and Montagne 2005; Chardenon
et al. 2005; Fajen and Warren 2007).
Interception strategies in the presence of variability
Unlike the experimental conditions in which the CBA model
has been tested, real world conditions are subject to sources
of variability that may complicate behavior. In an intercep-
tion task variability may arise from unpredictable changes in
target velocity. Consider the (admittedly unrealistic and
predictable) situation in which the target moves along the
same trajectory at the same initial speed as on every other
trial until a certain moment at which it accelerates to some
new speed, which is also the same on every trial. It will also
be assumed that the pursuer can adjust speed but not heading.
(In Experiment 1, subjects are presented with a more realistic
situation in which targets change speeds by an amount that
randomly varies from trial to trial, but with some statistical
regularity.) One would expect that, contrary to the predic-
tions of the CBA model of interception, a human pursuer
would eventually learn to anticipate the target’s predictable
increase in speed. A pursuer with accurate expectations of
the target’s acceleration could make anticipatory speed
adjustments before the change in target speed that would
bring about the desired state (a CBA) immediately following
the change in target speed.
By how much should the pursuer change her own speed
in anticipation of the change in target speed to maximize
her chances of intercepting the target? Because both target
heading and subject heading are fixed, the point of inter-
ception is defined in advance. We define the target’s first-
order time-to-contact (TTC) as the amount of time it would
take for the target to reach the interception point assuming
target speed does not change. Before the pursuer has
enough experience to realize that the target always accel-
erates, it is expected that she will adjust speed in such a
way that her first-order TTC will equal the target’s first-
order TTC. Such behavior would result from maintaining a
CBA.
If the target accelerates, then the target’s actual TTC with
the interception point would be less than the first-order
TTC. If the pursuer can perfectly anticipate the increase in
target speed, then she should adjust her speed even before
the target accelerates so that her TTC with the interception
point equals the target’s actual TTC with the interception
point. Once the pursuer’s TTC is equal to the target’s actual
TTC, the pursuer can simply maintain speed to intercept the
target. Note that this is not intended to be a possible control
strategy used by human actors, as it assumes knowledge that
humans are not likely to have when intercepting targets.
Rather, our objective here is simply to explain how the data
will be analyzed to test the hypothesis that actors can
anticipate changes in target speed. If subjects can perfectly
anticipate changes in target speed, then their TTC at the
moment that the target changes speed should equal the
target’s actual TTC at that moment. On the other hand, if
subjects cannot anticipate the change in target speed, then
their TTC will be biased toward the first-order TTC.
Experiment 1: anticipating changes in target speed
In the previous example, the simplifying assumption was
made that the target always accelerated by the same
amount at the same time. In Experiment 1, we investigated
whether subjects can anticipate changes in target speed in a
more realistic situation in which the target changes speed
by an amount that varies from trial to trial. A spherical
target approached the interception point from one of three
angles at one of three initial speeds. Between 2.5 and
3.25 s after the trial began, target speed changed to a final
speed that was randomly selected from a normal distribu-
tion. Because the mean of the final speed distribution was
greater than all three initial speeds, the target usually
accelerated. However, the variance of the distribution was
large enough that the target occasionally decelerated to its
final speed.
When the change in target speed is unpredictable as in
Experiment 1, the pursuer cannot perfectly anticipate the
change in speed on every trial. The best the pursuer can
possibly do is to anticipate the most likely change in speed;
Exp Brain Res (2009) 195:345–360 347
123

that is, adjust speed so that TTC is equal to the actual TTC of
a target that changes speeds to the most probable final speed
at the most probable time. So the goal of Experiment 1 was to
determine if subjects can learn to anticipate the timing and
magnitude of the most likely change in target speed.
Methods
Participants
Twelve undergraduate students from Rensselaer Polytech-
nic Institute participated in the experiment. Each had
normal or corrected-to-normal vision.
Displays and apparatus
A Dell Precision 530 Workstation with a 1.7 GHz Intel
Xeon processor and an nVidia Quadro2 Pro graphics card
generated the experimental stimuli and recorded the posi-
tion and velocity of the subject and target at 60 Hz. The
stimuli were rear-projected via a Barco Cine 8 projector at
a resolution of 1280 9 1024 onto a 1.8 9 1.2 m screen at
60 Hz. To reduce the salience of the screen frame, black
felt covered the border of the frame and the surrounding
walls. Participants viewed the stimuli from a distance of
approximately 1 m using unrestricted binocular vision of
the monocular display.
Displays simulated the observer’s movement at 1.1 m
over a ground plane along a fixed path. The ground texture
resembled a green grass covered field that is free of dis-
tinguishing landmarks, and the simulated sky was light
blue. At the beginning of each trial, a moving target with a
radius of 0.35 m approached the observer’s path of motion
from the right side (see Fig. 2).
The simulated target approached the unmarked inter-
ception point from an initial distance of 45 m, along an
initial approach angle of 135, 140, or 145 from the
subject’s path of motion (a in Fig. 1). The target traveled
along a fixed path at one of three initial speeds 11.25, 9.47,
8.18 m/s, which correspond to initial first-order time-to-
contact values of 4, 4.75, and 5.5 s. At a randomly selected
time between 2.5 and 3.25 s after the start of the trial the
target gradually changed speed to a new value that was
selected from Gaussian distribution. Final target speed was
independent of initial target speed. The mean of the final
target speed distribution was 15 m/s, and the standard
deviation was 5 m/s. The range of the distribution was
truncated to exclude speeds that lay farther than one stan-
dard deviation from the mean. The target changed speed at
a constant rate over a duration of 0.5 s.
Subjects controlled their simulated speed using an ECCI
Trackstar 6000 spring-loaded foot pedal. To begin each
trial, subjects completely released the foot pedal and
pressed a button. Initial distance from the unmarked
interception point was randomized between 25 and 30 m.
Subjects’ velocity had a lagged first-order relationship with
the pedal position, defined by the equation:
_V ¼ K  Vp  Vs


ð1Þ
where Vp is the speed specified by the current pedal posi-
tion, Vs is the subject’s current velocity, and K is a constant
lag coefficient. The lag coefficient was set to 0.017 because
it produces a smooth relationship between the pedal
movements and the resultant velocity changes, while still
allowing the responsiveness needed for a successful inter-
ception. The range of possible speeds extended from 0 to
14 m/s. To successfully intercept the target, the subject had
to pass within a distance of 0.35 m from the center of the
target.
Procedure
Upon arriving at the lab area, each subject provided written
consent and was asked to read a set of instructions. The
subject was then brought into the experimentation room
and seated in a chair approximately 1 m from the projec-
tion screen. Eyeheight was approximately 1.1 m, equal to
the simulated eyeheight used in the stimuli. The location of
the foot pedal unit was manually adjusted to ensure com-
fort during participation.
Prior to the experiment, subjects completed a short
practice session, during which each possible combination
of initial approach angle and initial target speed was pre-
sented once, producing a total of nine practice trials.
During the experimental session, each combination was
presented ten times within each of four blocks. The
Fig. 2 A sample frame of the experimental stimulus
348 Exp Brain Res (2009) 195:345–360
123

3 (initial approach angles) 9 3 (initial target speeds) 9 10
(repetitions) 9 4 (block) design produced a total of 360
trials during the experimental session.
Results and discussion
Task performance
The mean hit rate across all subjects and all four blocks
was 47% (SD = 11.31). However, one subject’s perfor-
mance was particularly poor. Because his mean hit rate of
20% was 2.39 standard deviations below the group mean,
he was treated as an outlier and his data were excluded
from further analysis.
The 11 remaining subjects had a hit rate of 39.1% in
block 1, but improved over blocks up to 54.9% in the final
block (Fig. 3). This steadily improving hit rate was mir-
rored by a steady decrease in the percentage of misses with
the target passing in front. The percentage of misses with
the target passing behind the subject was fairly constant
and relatively low. This may be due to the fact that the
target usually increased (rather than decreased) speed, but
may also reflect a bias to keep the target within the *80
field of view provided by the projection screen for as long
as possible. A Chi-squared test confirmed the significant
effect of block on the distribution of trial outcomes X2(6,
N = 3,960) = 57.00, p B 0.001.
Did subjects use a constant bearing angle strategy?
The remaining analyses were based on measurements taken
on the frame immediately before the onset of the target’s
acceleration from its initial speed to its final speed (here-
after referred to simply as the ‘‘onset’’). The reason for
taking measurements at onset is twofold: (1) subjects do
not yet have visual information about the magnitude of the
impending change in target speed, and (2) if subjects are
going to modulate their approach speed in anticipation of
the target’s likely change in speed, they will have had
sufficient time to do so in the 2.5–3.25 s before the target
changes speed.
If subjects used a CBA strategy, then the instantaneous
change in bearing angle at onset should be close to zero.
t tests revealed that the rate of change of bearing angle at
onset was statistically different from zero on all four blocks
t(10) = 7.22, 9.47, 9.91, and 13.54, respectively, with
p B 0.001 for each individual t test (Fig. 4). Although the
change in bearing angle did not significantly differ from
zero in some conditions, there was a consistent trend for
the rate of change in bearing angle to grow with approach
angle.
Furthermore, if subjects were using a CBA strategy,
then subjects’ TTC should match the target’s first-order
TTC at onset. This was clearly not the case (Fig. 5), as
subject TTC was consistently less than the target’s first-
order TTC. This was confirmed by calculating the differ-
ence between the subject’s TTC and the target’s first-order
TTC on each trial, and then running Bonferroni tests for
each of the 36 conditions (3 initial speeds 9 3 approach
angles 9 4 blocks). Of the 36 tests used to test for a sta-
tistically significant difference between the subject’s TTC
and the target’s first-order TTC, 28 show a significant
difference from zero.
To summarize, there was no evidence that subjects were
trying to maintain a constant bearing angle during the first
1 2 3 4
0
10
20
30
40
50
60
Block
Pe
rc
en
t
Hits
In Front
Behind
Fig. 3 Percentage of trials in
which the target was
successfully hit, passed in front
of the observer, or passed
behind the observer in
Experiment 1
Exp Brain Res (2009) 195:345–360 349
123

part of the trial. The change in bearing angle was consis-
tently greater than zero and subject TTC was consistently
less than the target’s first-order TTC, suggesting that sub-
jects anticipated the likely increase in target speed.
Did subjects use a predictive strategy?
If subjects modulated speed in anticipation of the most
likely change in target speed, then the subject TTC should
match the target’s mean actual TTC. Visual inspection of
Fig. 5 suggests that, for the majority of initial conditions,
the subjects’ TTC was more similar to the target’s mean
actual TTC than the first-order target TTC. This observa-
tion is strengthened when one compares the frequency with
which the subject TTC is significantly different from first-
order TTC (28 of 36 Bonferroni tests), and mean actual
TTC (18 of 36 t tests).2
Differences in subject TTC and measures of target
TTC appear to have varied with the target’s approach
angle and speed: subject TTC was sometimes greater
than the first-order TTC when the target was approaching
from a less head-on angle at a high speed (the top-right
of Fig. 5), and lower than mean actual TTC values when
a more slowly moving target approached from more
head-on trajectory (the lower-left of Fig. 5). A three-way
repeated measures ANOVA revealed main effects of
initial target speed F(2,20) = 112.85, p B 0.001,
approach angle F(1.33,13.1) = 95.26, p B 0.001, and
block F(3,30) = 11.30, p B 0.001. In addition, there was
an interaction of initial target speed and angle on the
subject’s TTC F(1.55,15.54) = 4.17, p = 0.044. The
main effect of initial speed simply reflects the effect of
initial target speed on the overall time it takes for the
target to reach the interception point. Simply put, the
subject must increase his or her speed in order to catch
faster moving targets. Figure 5 shows that the effect of
initial speed on subject TTC was more pronounced when
the target’s approach angle was smaller (corresponding
to a more orthogonal trajectory), giving rise to the sig-
nificant initial speed 9 approach angle interaction. The
interaction could reflect difficulty detecting differences in
initial speed when the target approached along a more
head-on trajectory. When the target approaches along a
more orthogonal trajectory, differences in initial speed
are easy to detect because they are accompanied by
salient differences in the rate at which the bearing angle
changes. In contrast, when the target approaches along a
more head-on trajectory, differences in the rate of change
135 140 145
0
1
2
3
4
Block #1
Approach Angle ( ° )
Ch
an
ge
in
B
ea
rin
g
A
ng
le
( d
eg
ree
s/s
ec
)
135 140 145
0
1
2
3
4
Block #2
Approach Angle ( ° )
Ch
an
ge
in
B
ea
rin
g
A
ng
le
( d
eg
ree
s/s
ec
)
135 140 145
0
1
2
3
4
Block #3
Approach Angle ( ° )
Ch
an
ge
in
B
ea
rin
g
A
ng
le
( d
eg
ree
s/s
ec
)
135 140 145
0
1
2
3
4
Block #4
Approach Angle ( ° )
Ch
an
ge
in
B
ea
rin
g
A
ng
le
( d
eg
ree
s/s
ec
)
8.18 m/s
9.47 m/s
11.25 m/s
Fig. 4 Change in bearing angle
at onset by initial target speed,
approach angle, and block
2 Although it is common to apply Bonferroni tests and thus to
decrease alpha to compensate for the greater family-wise probability
of a type 1 error when performing multiple t tests, our strategy of
maintaining an alpha of 0.05 was the more conservative approach in
that it increased the odds of rejecting the null hypothesis when we
should not have rejected it.
350 Exp Brain Res (2009) 195:345–360
123

in bearing angle across initial speeds are small. So
detecting differences in initial speed requires subjects to
discriminate among small differences in the optical
expansion of the target. This would explain why the
degree to which subject TTC varies with initial speed is
less when approach angle is greater. It would also
account for the largest deviation between subject TTC
and actual target TTC, which occurred when initial speed
was slow and approach angle was large (bottom left of
Fig. 5). When subjects had difficulty perceiving initial
target speed, they may have increased their speed to
avoid missing the target when it accelerated. Such
behavior would result in a subject TTC that was con-
sistently smaller than actual target TTC when initial
speed was slow, as in the bottom left of Fig. 5. Thus, the
few situations in which subject TTC deviated from
actual target TTC may be attributable to an inability to
accurately anticipate the change in target speed under
certain extreme conditions.
The preceding analyses indicate that subjects adopted
a strategy that takes into account the behavior of the
target on previous trials. In this regard, our findings are
similar to those of de Lussanet et al. (2001, 2002), who
found an effect of the target’s speed on the previous trial
on hand movements in a rapid pointing task. To further
investigate this issue, we ran a follow-up analysis to
determine whether subjects’ strategy was based on target
behavior on the most recent trial, an average of the last
few trials, or an average of all trials between the first
trial and the most recent trial. More specifically, this
analysis provides an estimate of the number of previous
trials that were taken into account when anticipating the
change in target speed. For each trial, we calculated an
estimate of target TTC at onset based on the assumption
that the target would change to a speed equal to the
mean of the final speeds on the last n trials. Hereafter,
this estimate is called ‘‘windowed target TTC.’’ Five
values of n were tested: 1, 2, 4, 8, and 16. In addition,
we also generated estimates based on the average of all
trials that were completed up to the most recent trial.
The absolute value of the difference between subject
TTC at onset and each estimate of windowed target TTC
was then calculated. If subjects took the last n trials into
account, then the mean absolute difference between
subject TTC and windowed target TTC should be least
for the estimate that uses the last n trials.
1 2 3 4
0
1
2
3
Approach Angle: 135°
Target Speed: 8.18 m/s
Block
TT
C
Subject
First-Order Target
Mean Actual Target
1 2 3 4
0
1
2
3
Approach Angle: 135°
Target Speed: 9.47 m/s
Block
TT
C
1 2 3 4
0
1
2
3
Approach Angle: 135°
Target Speed: 11.25 m/s
Block
TT
C
1 2 3 4
0
1
2
3
Approach Angle: 140°
Target Speed: 8.18 m/s
Block
TT
C
1 2 3 4
0
1
2
3
Approach Angle: 140°
Target Speed: 9.47 m/s
Block
TT
C
1 2 3 4
0
1
2
3
Approach Angle: 140°
Target Speed: 11.25 m/s
Block
TT
C
1 2 3 4
0
1
2
3
Approach Angle: 145°
Target Speed: 8.18 m/s
Block
TT
C
1 2 3 4
0
1
2
3
Approach Angle: 145°
Target Speed: 9.47 m/s
Block
TT
C
1 2 3 4
0
1
2
3
Approach Angle: 145°
Target Speed: 11.25 m/s
Block
TT
C
Fig. 5 Mean subject TTC, actual TTC, and first-order TTC at the onset of the change in target speed
Exp Brain Res (2009) 195:345–360 351
123

The results of this analysis are summarized in Fig. 6a,
which shows that the mean |subject TTC—windowed tar-
get TTC| decreased as n increased up to n = 4, beyond
which point it leveled off. This suggests that subjects took
into account the behavior of the target on the last four or
more trials. Because the difference did not increase for
larger values of n, we cannot rule out the possibility that
subjects took into account more than the last four trials.
However, the fact that the difference leveled off after
n = 4 suggests that there was no advantage in taking into
account more than the last four trials. Thus, subjects were
able to use previous target behavior to improve perfor-
mance without having to rely on memory of target behavior
in the distant past.
Given the length of the error bars in Fig. 6a, one might
wonder whether this result was consistent across subjects.
A closer look at the individual subject data reveals large
overall differences in mean |subject TTC—windowed tar-
get TTC| (see Fig. 6b, which shows individual subject data
from block 4.). However, mean |subject TTC—windowed
target TTC| consistently decreased as n increased up to
n & 4, confirming that the pattern was consistent across
subjects.
Did subjects use a task-specific heuristic?
One might wonder if subjects adopted a simple task-
specific heuristic that worked within the range of con-
ditions experienced in this experiment, but would not
work when conditions vary across a wider range. Of
course, there are an infinite number of possible heuris-
tics. In this section, two such heuristics will be consid-
ered and ruled out.
One possibility is that subjects developed a stereotyped
pattern of velocity adjustments that could be applied
independently of the initial conditions to get within the
ballpark. However, our analysis of the mean subject TTC at
onset indicates that behavior was influenced by the initial
condition (Fig. 5), ruling out the possibility of a stereo-
typed approach.
n = 1 n = 2 n = 4 n = 8 n = 16
0.25
0.3
0.35
0.4
0.45
n = all previous
trials
|TT
C


- T
TC




|
su
bje
ct
w
in
do
w
ed
|TT
C


- T
TC




|
su
bje
ct
w
in
do
w
ed
Block 1 Block 2 Block 3 Block 4
0.3
0.35
0.4
0.45
n = 1
n = 2
n = 4
n = 8
n = 16
n = all previous trials
A
B
Fig. 6 a Mean |subject TTC—
windowed TTC| for each value
of n and each block. b Mean
|subject TTC—windowed TTC|
for each value of n and each
subject in block 4
352 Exp Brain Res (2009) 195:345–360
123

The second possible heuristic that was considered was
that subjects tried to maintain a constant non-zero rate of
change in bearing angle. To investigate this possibility, a
three-way ANOVA was performed to test the effects of
approach angle, initial target speed, and block on the rate of
change in bearing angle at onset (Fig. 4). The rate of change
in bearing angle increased with both the target’s approach
angle F(2,20) = 66.48, p B 0.001, initial target speed
F(1.33,13.30) = 33.42, p B 0.001, and block F(3,30) =
9.47, p B 0.01. There was also an interaction between angle
and initial target speed F(2.05,20.50) = 7.96, p B 0.003,
and a marginally significant interaction between initial target
speed and block F(6,60) = 2.25, p B 0.05. Such variation in
the rate of change in bearing angle across initial conditions
rules out the possibility that subject simply tried to maintain a
constant non-zero rate of change in bearing angle at onset.
Summary
The results of Experiment 1 suggest that subjects did not
use the constant bearing angle strategy to intercept targets
that change speeds. Instead, values of subject TTC at onset
more closely matched predictions that were based on
accurate anticipation of the most likely change in target
speed given the trial’s initial conditions, as well as target
behavior on the most recent four (or so) trials.
Experiment 2
Experiment 1 demonstrated that subjects were able to
accurately anticipate changes in target speed. Experiment 2
tests whether these findings will generalize to a new situ-
ation, in which the target approaches along a curvilinear
path with either concave curvature that bends away from
the subject, or convex curvature that bends towards the
subject (Fig. 7).
The situation in Experiment 2 is similar to that in
Experiment 1 in that use of a pure constant bearing angle
strategy will often result in a failure to intercept the target.
This is especially true on trials with large convex curva-
ture: if a subject maintains the bearing angle early in the
trial when the target’s heading is roughly parallel to the
subject’s path of motion, then the subject will be forced to
continually increase his or her velocity as the target’s
heading moves in the direction that is roughly perpendic-
ular to the subject’s path of motion (the dark dotted line in
Fig. 7). If the subject reaches maximum speed before
interception, he or she will be unable to maintain a constant
bearing angle, and unable to intercept the target.
To avoid this situation subjects might make adjustments
early in the trial in anticipation of the expected change in
target heading. Bastin et al. (2006) showed that actors
cannot predict the eventual time and location of the future
interception point for curvilinear targets. However, even if
actors do not rely on such long-term predictions, they may
still be able to gain an advantage by anticipating the change
in target direction a brief period into the future. The
problem then becomes one of learning the correct pedal
adjustment at time t that will produce a constant bearing
angle at time t ? Dt. Because the bearing angle is a
function of both the target’s and the subject’s behavior,
solving this problem requires that subjects take into
account the expected target dynamics. These expectations
may be encoded in terms of a learned mapping from a
desired future state with a constant bearing angle at time
t ? Dt to the velocity adjustment at time t that is necessary
to bring about this desired future state.
Interestingly, this simple characterization of anticipatory
behavior might be used to explain a spectrum of possible
human behavior by simply varying the value of Dt: if Dt is
close to zero, the modeled velocity adjustments will
resemble those of a subject using a pure constant bearing
angle strategy. Greater values of Dt will produce behavior
that suggests perfect anticipation of the future time and
location of the target’s passage over the subject’s path of
motion. By comparing subject velocity adjustments with
those of ideal pursuers with varying values of Dt, we are
able to estimate the temporal distance across which sub-
jects are able to anticipate future target dynamics.
Co
nv
ex
Co
nc
av
e
Fig. 7 An overhead view of the task in Experiment 2. The pursuer is
marked by a triangle, and the target by a circle
Exp Brain Res (2009) 195:345–360 353
123

Methods
Participants
Twelve undergraduates from the Rensselaer Polytechnic
Institute participated in the experiment. Each had normal or
corrected-to-normal vision.
Displays and apparatus
The stimuli were identical to those of Experiment 1, with
the following exceptions. The initial position of the target
was 40 m from the interception point and 135 or 145
from the subject’s path of motion. The target moved at an
initial tangential speed of 10 or 8.89 m/s, corresponding to
initial time-to-contact values 4 and 4.5 s, along one of five
trajectories, four of which were curvilinear, and one of
which was rectilinear. The radii of the curvilinear paths
were 35 or 60 m, and the direction of a path was either
concave in that it bent away from the pursuer, or convex in
that it bent toward the pursuer. The 2 (initial target
angles) 9 2 (initial target speeds) 9 5 target path curva-
ture 9 5 (repetitions) 9 3 (block) design produced a total
of 300 trials per subject during the experimental session.
As in Experiment 1, subjects had lagged first-order
control of their speed defined by Eq. 1. However, the lag
coefficient was adjusted to 0.03 to accommodate the dif-
ferent conditions in Experiment 2. The change in lag
coefficient had the effect of allowing the actual speed to
more closely follow the speed defined by the position of the
pedal. The minimum speed was 0 m/s and the maximum
was 15 m/s.
The model
This section describes the model, and how it was used to
generate predictions for different values of Dt between zero
and the remaining movement time of the target. To antic-
ipate, the model selected (at each time-step) the speed
adjustment that nulled the change in bearing angle at some
future time t ? Dt, taking into account the dynamics of the
controlled system and the behavior of the target up until
t ? Dt.
Initial conditions, target behavior, and controller
dynamics
The initial conditions and target behavior were identical to
those used in the actual experiment. In order to make the
model as realistic as possible, it was also necessary to
incorporate into the model the various sources of controller
lag that subjects experienced in the actual experiment. Due
to the inertia of the subject’s foot and the foot pedal sys-
tem, all pedal adjustments in the actual experiment were
smooth and continuous. In addition, recall that there was a
first-order lag between the position of the foot pedal and
the simulated speed. These two sources of lag were com-
bined in the model by adding a second-order lag between
the intended speed selected by the model and the current
speed. More specifically, the agent’s actual speed was
treated as an over-damped harmonic oscillator about the
intended speed. Calculations were made using the Matlab
function ODE45 that, at each time-step t, solved the
equation:
€v ¼ b  _v  x20  v  vð Þ ð2Þ
where v is the current speed, _v is acceleration, €v is jerk, and
v* is the intended speed. This introduced two additional
free parameters: the damping term b, and the (undamped)
natural harmonic frequency x0
2. Speed was recovered by
taking the double integral of jerk.
Assumptions
The intended speed selected by the model at each time-step
was based on perfect knowledge of both the controller
dynamics and the target’s behavior from t to t ? Dt.
Because the controller dynamics were fixed, and subjects
practiced the task before the experiment began, it is rea-
sonable to assume that they were familiar with the con-
troller dynamics. Of course, one cannot assume that
subjects actually knew the future behavior of the target on
each trial. Nonetheless, it might be possible to learn a
control strategy that allows one to take advantage of reg-
ularities in the target’s behavior, without actually having
explicit knowledge of such behavior. [This is analogous to
the way in which outfielders learn a control strategy that
allows them to move into position to catch a fly ball
without having explicit knowledge of the dynamics of
projectiles (e.g., McLeod et al. 2006)]. Our aim is not to
address the issue of how such a control strategy could be
learned, but rather to determine if such a model could
account for human behavior. If it can, then a logical next
step would be to explain how the control strategy is
learned.
Updating speed
The model updated intended speed at 60 Hz, equal to the
frame rate of the display used in Experiment 2. To reflect
the fact that the human subjects needed time to react after
the onset of the display, no speed adjustments were made
for the first 330 ms of each trial. This duration corresponds
to the average trial time (calculated across all trials and all
subjects) at which speed adjustments were initiated.
354 Exp Brain Res (2009) 195:345–360
123

At each time-step after 330 ms, the intended speed that
would null the change in bearing angle at time t ? Dt was
found via brute-force search of each possible speed from 0
to the maximum speed of 15 m/s, with a search resolution
of 0.25 m/s. For each possible intended speed, the change
in position and speed from t to t ? Dt was calculated using
the universal oscillator equation (see above). The position
and speed at time t ? Dt was then used together with the
position and speed of the target at time t ? Dt to calculate
the rate of change in bearing angle, which was then stored
in an array. Once all possible intended speeds had been
tested, the intended speed that produced the minimum
change in bearing angle at time t ? Dt was selected for that
time-step. Because there is no advantage to predicting
beyond the point of interception, Dt was adjusted on each
frame after target time-to-contact dropped below Dt such
that Dt equaled target TTC.
Fitting the parameters
The model was used to find the set of parameters (Dt, b,
and x0) that best fit the human data. The three parameters
were always fixed within each simulated trial. Because
some trajectories may be more predictable than others, the
parameters were allowed to vary across target radius and
direction, but were fixed across initial target speed and
approach angle.
The model was fit to the mean speed profile from the
human data. The speed profiles from each trial in the actual
experiment were averaged across repetitions within a
condition, and then across subjects. This resulted in 48
mean speed profiles (4 radius/direction pairs 9 2 initial
target speeds 9 2 target approach angles 9 3 blocks).3
The RMSE between the speed profiles produced by the
model and by human subjects was then calculated. The
parameters b, and x0 were fixed at the values that produced
the least total RMSE across the four radius/direction pairs,
while the parameter Dt was allowed to vary across radius/
direction pairs, but not across speeds or angles within each
radius/direction pair.
The search space
The process tested each value of x0 between 2.5 and 4.5 in
increments of 0.25. The parameter b was defined as a scalar
multiple of x0, where a scalar of 2 produces a critically
damped oscillator, values greater than 2 produce an over-
damped oscillator, and values less than 2 produce an under-
damped oscillator. The scalars used to define b ranged from 2
to 4.5 in increments of 0.25. The range of Dt values explored
extended from 0.25 to 3.5 in increments of 0.25. The lower
value of 0.03 was also included in the fitting process.
Results and discussion
Task performance
The average hit rate for all subjects improved from 38.8%
in block 1, to 51.5% in block 2, to 56.3% in block 3
F(2,22) = 27.05, p B 0.01. Repeated measures contrasts
across blocks indicate that performance on block 2 was
significantly different than performance on block 1
F(1,11) = 38.18, p B 0.001, and that performance on
block 3 was significantly different than performance on
block 2 F(1,11) = 21.11, p B 0.01. Performance also dif-
fered by radius/direction pair that defined the targets tra-
jectory F(2.08,22.95) = 33.40, p B 0.01. Visual inspection
of Fig. 8 suggests that performance was worst on largely
convex trials, better on less convex trials, and best for
rectilinear and concave curvatures. A significant interaction
was found between block and curvature F(8,88) = 3.11,
p B 0.01, possibly because improvement across blocks
appears to have been greatest on conditions in which initial
performance was poorest.
Speed profiles: did subjects use a predictive strategy?
Subject speed profiles will be analyzed with respect to the
pure predictive strategy first, followed by the CBA and
intermediate anticipatory strategies. A five-way repeated
measures ANOVA was used to investigate variations of
subject speed in response to changes in radius/direction pair,
target approach angle, initial target speed, block, and time.
Time was incorporated into the ANOVA by sampling subject
speed at 1 s intervals from 0.5 s until 3.5 s into the trial.
If subjects relied on a pure prediction strategy, then
subject behavior should be similar across changes in target
trajectories (within each initial target speed condition).
However, as one would expect based on the findings of
Bastin et al. (2006), the evolution of subject velocity dif-
fered by radius/direction pair F(12, 132) = 90.72,
p B 0.01 (Fig. 9). This is most apparent on concave trials,
in which subjects accelerated in the first part of the trial
before decelerating. If subjects were able to predict the
future time and location of the interception point, then one
would expect an initial increase in speed followed by a
plateau regardless of target trajectory. Thus, the findings of
Experiment 2 allow us to rule out the possibility that
subjects adjusted speed based on an accurate prediction of
the time and location of the interception point.
3 For each combination of radius/direction, initial target speed, and
target approach angle, the model was then used to produce one
simulated speed profile for each possible combination of Dt, b, and
x0.
Exp Brain Res (2009) 195:345–360 355
123

The findings do not allow us to rule out the possibility
that subjects relied upon an inaccurate prediction of the
interception point. Similar velocity profiles might result
from initially relying on an inaccurate prediction, and
gradually refining the prediction during approach.
Speed profiles: did subjects use a CBA strategy?
The rise and fall of subject speed on concave trials is
qualitatively consistent with the CBA strategy, as main-
taining a CBA on such trials would require one to accel-
erate early on when the target’s heading was roughly
perpendicular to the subject’s path of motion, and subse-
quently decelerate as the target’s trajectory aligns with the
subject’s direction of motion. However, the anticipatory
strategy with small values of Dt could also account for such
behavior. The model proves useful in distinguishing
between these two hypotheses. Table 1 shows the best-
fitting Dt values, along with the RMSE and r2 values, for
each radius/direction pair in blocks 1 and 3, and Fig. 10
compares the simulated speed profiles (using the best-fit-
ting parameters) with the human data for block 3. On
concave trials, the best-fitting Dt values were 0.75 and
0.03 s in block 1 and 0.50 and 1.00 s in block 3, suggesting
that practiced subjects adjusted speed to null the change in
bearing angle 0.5–1.0 s into the future. In other words,
although the rise and fall of subject speed on concave trials
is qualitatively consistent with the CBA strategy, speed
adjustments were better fit by the intermediate strategy
with a little bit of anticipation. The good fit between the
model and human data (i.e., slow RMSE values, high r2
values) suggests that the best-fitting Dt values can be
trusted as an estimate of the degree of anticipation. For the
purposes of comparison, Table 1 also includes RMSE and
r2 values for both a CBA strategy (Dt = 0) and a purely
predictive strategy (Dt = 3.5), neither of which fit the data
as closely as an intermediate strategy (0 \ Dt B target
TTC).
When the target follows a convex trajectory, maintain-
ing a CBA would require one to increase speed gradually at
first, and then more rapidly as the target’s direction shifts
from roughly parallel to the subject’s path to roughly
perpendicular. Although subject speed increased through-
out the trial, the majority of acceleration took place early in
the trial rather than later, and velocity changed little
towards the end of the trial when the change in bearing
angle was greatest (see Fig. 9). This suggests that subjects
learned to anticipate the predictable evolution of the tar-
get’s trajectory. Not surprisingly, the best-fitting values of
Dt were quite high (3.5 s), and when Dt was set to 0.03,
performance was quite poor (see Table 1). This might be
interpreted as evidence that subjects were able to anticipate
further into the future on convex trials. However, it is not
clear why subjects would be able to anticipate further into
the future on convex trials compared to concave trials.
Furthermore, the poorer quality of fit on convex trials (see
Table 1, Fig. 10) raises some concerns about the reliability
of the high Dt value as an estimate of the degree of
anticipation. In particular, Fig. 10 suggests that the model
with Dt = 3.5 s captured speed adjustments during the first
2 s, but not during the latter part of the approach. After the
first 2 s, the model’s speed consistently exceeded subject
speed, suggesting that subjects did not look ahead as far as
Dt = 3.5 suggests. One alternative explanation for the
inflated Dt values is that subjects accelerated more than
necessary during the first part of the trial to avoid the
1 2 3
0
10
20
30
40
50
60
70
80
90
100
Block
Pe
rc
en
t S
uc
ce
ss
fu
l
35m convex
60m convex
Rectilinear
60m concave
35m concave
Fig. 8 Percentage of successful
interceptions in Experiment 2
by block and curvature
356 Exp Brain Res (2009) 195:345–360
123

situation in which the speed needed to intercept the target
exceeded the maximum possible speed. Because the best-
fitting parameters were chosen based on the entire trajec-
tory, the estimate of Dt may have been inflated by behavior
during these first 2 s. Whether the observed values of Dt
accurately reflect the temporal distance of anticipation, or a
difficulty for the model to account for adjustments early in
the trial, behavior was clearly inconsistent with the CBA
strategy on both convex and concave trials.
One might wonder whether different results would be
obtained if the procedure used to estimate Dt was based on
successful trials only. The concern is that what appears to
be a strategy of anticipating target behavior a brief period
into the future might actually result from anticipating far-
ther into the future and occasionally being wrong about the
target’s behavior. Assuming that subjects were less likely
to intercept the target when they inaccurately anticipated
target behavior, excluding unsuccessful trials might yield a
different estimate of Dt. To investigate this possibility, we
isolated successful trials and re-fit the model. Successful
trials from blocks 2 and 3 were pooled to increase the
number of data points. Only minor differences were found
between the best-fitting parameters for successful trials and
those for both successful and unsuccessful trials. The best-
fitting values of frequency (3.75) and damping (12.19)
were only slightly different for those when the model is fit
to all trials (see Table 1 for values for all trials). Values of
Dt were identical with the exception of the 60 m concave
trajectory, in which Dt was 1.25 compared to 0.5 s for the
analysis with all trials. Thus, we can rule out the possibility
that the evidence for an intermediate anticipatory strategy
resulted from including both successful and unsuccessful
trials.
Speed profiles: effects of target speed and approach
angle
Subject speed profiles within each radius/direction pair were
also grouped according to initial target speed, and unaffected
by approach angle (see Fig. 9). An interaction of sample and
initial target speed was found F(3,33) = 107.84, p B 0.01,
whereas none was found between sample and approach angle
F(3,33) = 1.61, p = 0.21. The significant sample 9 initial
target speed interaction simply captures the fact that subjects
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
0
5
10
15
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
0
5
10
15
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
0
5
10
15
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
0
5
10
15
Target Angle:
Target Speed:
135˚
10 m/s
135˚
8.9 m/s
145˚
8.9 m/s
145˚
10 m/s
Ve
lo
ci
ty
Time
Fig. 9 Mean subject velocity
over time for block 3 of
Experiment 2. Each panel
corresponds to a different target
trajectory. Within each panel,
velocity profiles are broken
down by angle and speed
Exp Brain Res (2009) 195:345–360 357
123

must travel faster to intercept faster targets. The non-sig-
nificant sample x approach angle interaction is also adaptive,
as the target reaches the interception point at the same time
regardless of approach angle.
The model was able to capture the grouping of velocities
according to the initial target speed within each radius/
direction pair, with higher simulated velocities accompa-
nying higher values of initial target speed (Fig. 10). Like
the subject data, the simulated velocity profiles were
unaffected by the initial approach angle.
General discussion
The purpose of this study was to examine human control
strategies for the interception of moving targets that change
velocity. Experiment 1 tested for the ability to accurately
anticipate probable changes in the target’s speed. Shortly
after the start of each trial, the target accelerated to a new
speed that was randomly selected from a Gaussian
distribution. The question was whether subjects could learn
how to modulate their speed during the first part of the trial
in anticipation of the change in target speed that was most
likely based on past experience. In most conditions, subject
behavior matched predictions that take into account the
most probable change in target speed given past experience
and the initial conditions of that trial. The implication is
that, given the insufficiency of a CBA strategy to deal with
changes in target speeds, subjects adopted a strategy that
exploited regularities in the target’s behavior to anticipate
the most probable change in target speed. Importantly, our
findings are inconsistent with the conclusions of other
papers on interception (e.g., Chardenon et al. 2005),
according to which actors continue to use the CBA strategy
even when target speed and/or direction changes.
Whereas, in Experiment 1, the change in the target’s
trajectory occurred over a short duration (0.5 s), targets in
Experiment 2 approached along curvilinear paths, causing
a continuous change in trajectory throughout the trial.
Although these simple and continuous changes in target
trajectory would seem ideal for a predictive model of
interception, Bastin et al. (2006) found that subject
behavior was more consistent with a CBA strategy of
interception than a pure prediction of the future time and
location of the target’s passage over the interception point.
We extend this research by considering intermediate
strategies, by which subjects anticipate the change in target
behavior and adapt their speed accordingly to null the
change in bearing angle a short time into the future.
As in Bastin et al. (2006), the trajectory of the target
significantly affected behavior, ruling out the predictive
strategy. Although speed profiles were affected by target
trajectory, the differences were not as great as one would
expect based on a CBA strategy. To further investigate
intermediate strategies involving anticipation of target
behavior a brief period into the future, human behavior was
compared to a model that, at each time t, makes an
adjustment that nulls the change in bearing angle at the
future time t ? Dt. When the target followed a concave
trajectory, the model best fit the human data using inter-
mediate values of Dt. For convex trajectories, higher values
of Dt best fit the human data. However, these values may
have been inflated by the initial speed adjustment. Taken
together, behavior was consistent with neither a pure pre-
dictive strategy nor a CBA strategy, and was best captured
by an intermediate strategy that allows for some degree of
anticipation.
From feedback to anticipation
Our conclusions are consistent with certain aspects of
information-based control (Warren 1998), according to
Table 1
Block 1: x0 = 3.25 b = 10.56
Radius/direction RMSE r2 Dt (s)
35 m, convex 1.55 0.93 3.5
60 m, convex 1.22 0.95 3.5
60 m, concave 0.61 0.98 0.75
35 m, concave 0.88 0.96 0.03
Block 3: x0 = 3.75 b = 12.19
Radius/direction RMSE r2 Dt (s)
35 m, convex 1.12 0.94 3.5
60 m, convex 1.2 0.91 3.5
60 m, concave 0.5 0.98 0.5
35 m, concave 0.89 0.95 1
Block 3: x0 = 3.75 b = 12.19 (fixed Dt)
Radius/direction RMSE r2 Dt (s)
35 m, convex 3.37 0.65 0.03
60 m, convex 2.28 0.69 0.03
60 m, concave 0.66 0.97 0.03
35 m, concave 1.09 0.95 0.03
Block 3: x0 = 3.75 b = 12.19 (fixed Dt)
Radius/direction RMSE r2 Dt (s)
35 m, convex 1.12 0.94 3.5
60 m, convex 1.20 0.91 3.5
60 m, concave 1.92 0.75 3.5
35 m, concave 1.87 0.77 3.5
358 Exp Brain Res (2009) 195:345–360
123

which action is coupled to information in optic flow
according to a law of control. Consistent with Warren’s
definition, our control law is task-specific, and operates on
the basis of task-specific control information (e.g., the rate
of change in bearing angle). However, the model differs
from most information-based models in that movements
are made to bring about a desired optical consequence
(e.g., a constant bearing angle) at some point in the near
future. This difference, though subtle, necessitates at least
one form of learning that is not typically considered in
theories of information based control: the actor must learn
how to choose an appropriate action that will bring about
the desired change in the relevant optical variable(s).
Similar inverse problems where a desired outcome pre-
cipitates the causal action are commonplace in motor
control literature resulting in a variety of frameworks with
which one might model the learning of solutions (Jordan
and Wolpert 1999). Whatever framework is chosen, the
results of Experiments 1 and 2 suggest that the solution
must take into account the likely target behavior based on
both past experience and currently available information.
However, additional research is required to address the
issues of how and which aspects of past experience are
factored in. One approach would be to build upon what was
done in the present study by developing more sophisticated
models that allow one to generate predictions by changing
the way in which past experience is taken into account. For
example, to the degree that actors optimally weight past
experience based on the consistency of target behavior and
currently available information based on its reliability, a
Bayesian approach may be best suited to capture behavior.
Alternatively, if past experience and current information
are not optimally integrated, then simpler models may be
sufficient.
The proposed framework also brings new insight to the
dichotomized models of prospective control, that link
action to temporally proximal events as they unfold, and
predictive control, in which a control plan is chosen on the
basis of a temporally distant desired outcome (Montagne
2005). The model straddles this dichotomy by linking
action in a prospective manner to a temporally distant
desired outcome that changes as new information is
received. This new characterization adds additional
explanatory power by, for example, accounting for how
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
0
5
10
15
* Radius: 35 convex * RMSE: 1.12 * r2: 0.94 * ∆t: 3.50s
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
0
5
10
15
* Radius: 60 convex * RMSE: 1.20 * r2: 0.91 * ∆t: 3.50s
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
0
5
10
15
* Radius: 60 concave * RMSE: 0.50 * r2: 0.98 * ∆t: 0.50s
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
0
5
10
15
* Radius: 35 concave * RMSE: 0.89 * r2: 0.95 * ∆t: 1.00s
Target Angle:
Target Speed:
135˚
10 m/s
135˚
8.9 m/s
145˚
8.9 m/s
145˚
10 m/s
Time
Ve
lo
ci
ty
Fig. 10 Mean simulated
velocities (lines and circles) for
block 3 of Experiment 2
overlaid upon subject velocity
(shading). Each panel
corresponds to a different target
trajectory. Within each panel,
simulated velocity profiles are
broken down by angle and
speed
Exp Brain Res (2009) 195:345–360 359
123

movements are guided during brief periods of interrupted
visual feedback. If the actor knows how to move so as to
bring about a desired change in optic flow, then movement
can still be guided (albeit not as effectively) even when
vision is temporarily occluded.
Acknowledgment This research was supported by grants from the
National Science Foundation (BCS 0545141 and BCS 0236734)
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