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Behavior of College Baseball Players in a Virtual Batting Task
Rob Gray
Arizona State University East
A baseball batting simulation was used to investigate the information used to hit a baseball. Measures of
spatial and temporal swing accuracy were used to test whether batters (a) use speed to estimate pitch
height, (b) initiate a constant swing duration at a fixed time to contact, (c) are influenced by the history
of previous pitches and pitch count, and (d) use rotation direction. Batters were experienced college
players. Pitch speed variance led to predictable spatial errors, and spatial accuracy was worse than
temporal accuracy. Swing duration was generally variable. The history of the previous 3 pitches and the
pitch count had significant effects on accuracy, and performance improved when rotation cues were
added. There were significant effects of expertise on hitting strategy.
From the time the first game of baseball was played, players,
coaches, and researchers have sought to understand the act of
hitting a pitched ball. Whereas batting coaches and players have
explored countless ingenious ways to evaluate and improve batting
average and power, researchers have faced the difficult challenge
of explaining how this incredible feat of perceptual–motor control
is achieved.
1
With margins for error of only a few milliseconds and
a fraction of an inch and with processing times of less than half a
second, baseball batting truly pushes the limits of human
performance.
Despite the Herculean effort that has been put into understand-
ing the act of hitting, relatively little is known about which sources
of perceptual information hitters actually learn and how this in-
formation is used to control the swing. The major weaknesses of
much of the previous research on hitting have been methodolog-
ical. In the typical laboratory experiments used to study hitting,
participants make judgments about stimuli presented on a monitor
without actually swinging a bat. There is increasing evidence that
the cognitive and perceptual processes involved in these passive
judgment tasks may be quite different from those used during
goal-directed actions (e.g., Milner & Goodale, 1995). Furthermore,
passive judgment experiments cannot show how the information is
used to control the motor response. The other major approach
researchers have taken is to study real batting (e.g., during batting
practice or during use of a pitching machine). These active tasks
have the advantage that both perception and action are involved.
However, it is often difficult to have fine control over the stimulus
parameters and to isolate different sources of information in these
real-world settings.
In the present study, I used a novel virtual baseball batting
simulation to overcome some of the limitations of previous re-
search. This simulation had the advantage that active motor re-
sponses could be combined with fine control over the visual
stimulus. Subsequently, I review the previous research on baseball
batting to give a background for the specific issues that were
addressed in this study.
Perceptual Information Available for Hitting a Baseball
The informational support for hitting a baseball has been ex-
plored in some detail (reviewed in Regan, 1997). It has been
proposed that the perceptual component of the act of hitting can be
reduced to the judgments of where and when; a batter needs only
to know the position of the ball when it crosses the plate and the
instant in time that it will be there (Bahill & Karnavas, 1993).
There are two primary sources of visual information about when
the ball will cross the plate (the time to contact; TTC). TTC
information is provided by the change in angular size of the ball’s
retinal image,
(Hoyle, 1957), that is, when an approaching object
is moving at a constant speed directly toward the observer’s eye
2
:
TTC ⬇
共d
/dt兲
. (1)
1
Some of the more interesting methods developed to improve hitting
performance include writing numbers on tennis balls and then trying to
swing only at odd-numbered balls (used by Barry Bonds; Savage, 1997),
trying to hit a Wiffle ball off a tee so that it travels with no spin (used by
Tony Gwynn; Gwynn, 1990), and swinging a bat at a telephone pole to
notice if one is rolling his or her wrists (used by Ted Williams; Williams
& Underwood, 1970).
2
Of course, the direct approach assumption is not true in the game of
baseball, as the point of contact is a considerable distance from the batter’s
eye. However, Equation 1 does provide a TTC estimate within the temporal
margin for error in this situation. If one assumes that the batter makes his
or her estimate when the actual TTC is 0.3 s and assumes that the point of
contact is 1 m from the eye, the estimation error would be approximately
3.5 ms for a 70-mph (31.3-m/s) pitch and 2.1 ms for a 90-mph (40.2-m/s)
pitch (see Tresilian, 1991, for derivation). Both of these errors are well
within the ⫾9-ms temporal margin for error.
I thank Kristen Macuga for her invaluable assistance in recruiting
participants and collecting data for some of these experiments. In addition,
I thank Mike McBeath for his many insightful comments on a draft of this
article.
Correspondence concerning this article should be addressed to Rob
Gray, Department of Applied Psychology, Arizona State University East,
7001 East Williams Field Road, Building 20, Mesa, Arizona 85212.
E-mail: robgray@asu.edu
Journal of Experimental Psychology: Copyright 2002 by the American Psychological Association, Inc.
Human Perception and Performance
2002, Vol. 28, No. 5, 1131–1148
0096-1523/02/$5.00 DOI: 10.1037//0096-1523.28.5.1131
1131
It has recently been demonstrated that accurate TTC information
is also provided by the change in the binocular disparity (
␦
)ofthe
approaching ball (Gray & Regan, 1998):
TTC ⬇
I
D共d
␦
/dt兲
, (2)
where D is the distance of the ball from the eye, and I is the
interpupillary separation.
The use of the information expressed in Equation 1, commonly
called tau (
) after Lee (1976), has been studied for a wide range
of actions (reviewed by Regan & Gray, 2000). For the act of
hitting a baseball, Bahill and Karnavas (1993) calculated the value
of d
/dt to be roughly 30 times above the discrimination threshold
at the moment the ball is released, leading to the conclusion that
“from the instant the ball leaves the pitcher’s hand, the batter’s
retinal image contains accurate cues for time to contact” (p. 6). As
evidence for this claim, they note that batters rarely make purely
temporal errors that result in line drives hit into foul territory.
However, the ability to make fine discriminations of an informa-
tion source is necessary but not sufficient for accurate estimation
of an absolute value.
3
In contrast to Equation 1, the approaching ball’s distance enters
into Equation 2. This may limit its effectiveness, because there are
no reliable retinal image correlates of the ball’s absolute distance
in the hitting situation (Bahill & Karnavas, 1993; Regan, 1997).
However, because a baseball is always the same physical size,
Equation 2 can be rewritten as
TTC ⬇
I
B共d
␦
/dt兲
, (3)
where B is the diameter of the ball.
For the question of whether a hitter can use visual information
to make estimates of absolute TTC with the accuracy required to
hit a baseball, one should consider the psychophysical findings of
Gray and Regan (1998). In that study, the accuracy of observers’
estimates of absolute TTC for a simulated approaching ball was
measured over a range of TTC values (from 1.8 to 3.2 s). They
reported 2.0%–12.0% errors for judgments based on Equation 1
alone, 2.5%–10.0% errors based on Equation 2 alone, and 1.3%–
3.0% errors when both sources of information were available. If
one assumes that these percentage values can be generalized from
a TTC value of 1.8 s (a 23-mph [10.3-m/s] pitch) to the 0.4–0.6 s
TTC range involved in hitting, then a 1.3% estimation error
corresponds to a temporal error of approximately 5 ms. This value
is well within the ⫾9-ms error margin calculated by Watts and
Bahill (1990). The findings of Gray and Regan suggested that a
hitter could estimate the TTC of an approaching ball more accu-
rately if both changing size and changing disparity information are
used (although the best estimation performance for either cue
alone is also within the required margin for error).
Bahill and Karnavas (1993) have proposed that the more diffi-
cult judgment for the batter (and the one with the smaller margin
for error; Watts & Bahill, 1990) is estimating where the ball will
be when it crosses the plate.
4
The aspect of this judgment that is
particularly difficult is predicting how far the ball will drop in
height. Although batters are exquisitely sensitive to the angular
drop speed of the ball (Regan & Kaushal, 1994), and although this
information is well above threshold from the instant the ball leaves
the pitcher’s hand (Bahill & Karnavas, 1993), the angular drop
speed is insufficient for judging height, because the relationship
between the angular drop speed and the physical drop speed
depends on the ball’s distance. In the absence of cues to the ball’s
absolute distance, two possible means of scaling the angular drop
speed with distance to get an accurate estimate of height have been
identified. Bahill and Karnavas have proposed that hitters use the
pitch speed in lieu of distance information to estimate the height of
the ball. In particular, the height of the ball when it crosses the
plate (Z
P
) is given by
Z
P
⬇ 共D
M
⫺ tS兲共d
/dt兲
, (4)
where D
M
is the distance to the mound, t is the time since pitch
release, S is the estimated pitch speed, and d
/dt is the angular
drop speed.
Alternatively, Bootsma and Peper (1992) have suggested that
batters could take advantage of the fact that the ball is always the
same physical size. In particular, the ball height is given by
Z
P
⬇
B共d
/dt兲
d
/dt
, (5)
where B is the diameter of the ball (for similar derivations, see
Regan & Kaushal, 1994; Todd, 1981).
What evidence is there for the use of these information sources
in hitting? Bahill and Karnavas (1993; following McBeath, 1990)
argue that the use of Equation 4 to estimate height is evidenced by
a perceptual illusion that is occasionally experienced by batters:
the rising fastball. If a batter underestimates the speed of a pitch,
the height estimate based on Equation 4 will be too low. Therefore,
at the point of contact, the ball will appear to jump over the hitter’s
bat. However, Profitt and Kaiser (1995) have argued that hitting
strategies requiring an estimate of pitch speed would not produce
the temporal precision exhibited by professional hitters.
As support for the use of Equation 5 in estimating height,
Bootsma and Peper (1992) cited evidence that introducing balls of
different sizes alters the judged spatial position of objects in the
horizontal plane. Equation 5 can be used to estimate the lateral
distance at which an approaching object will pass the midpoint (by
using the angular speed in the horizontal meridian). Using real
approaching objects, Bootsma and Peper found that the passing
3
For example, consider the case of a batter judging the TTC of two
pitches, a 95-mph (42.5-m/s) fastball with a TTC of 0.43 s and an 85-mph
(38.0-m/s) curveball with a TTC of 0.48 s. If the batter judged that the
95-mph pitch would arrive in 0.6 s and that the 85-mph pitch would arrive
in 0.65 s, discrimination of relative TTC would be precise (i.e., the hitter
would correctly judge that the 95-mph pitch would arrive sooner, with only
a 12% difference in TTC between the pitches), but estimation of absolute
TTC would be quite inaccurate (an error of 0.17 s is twice the temporal
accuracy required).
4
Watts and Bahill (1990) calculated the temporal margin for error to be
⫾9 ms and the spatial margin for error in the vertical dimension to be ⫾0.5
in. (1.27 cm). Relative to a 95-mph (42.5-m/s) pitch, a 90-mph (40.2-m/s)
pitch arrives 21 ms later (i.e., 2.3 times the temporal margin for error) and
crosses the plate 2.8 in. (7.11 cm) higher (i.e., 5.6 times the spatial margin
for error).
1132
GRAY
distance at which participants judged an approaching ball to be
reachable increased with ball size, as predicted by Equation 5. To
my knowledge, this study has not be replicated for judgments of
height.
In addition, there have been no investigations of observers’
ability to use the relation specified by Equations 4 or 5 to estimate
the absolute height of an approaching object. However, Regan and
Kaushal (1994) have shown that the discrimination threshold for
judging direction on the basis of Equation 5 can be as small as
0.03°–0.12°.
In summary, there are multiple sources of perceptual informa-
tion available for a batter when estimating where a pitch will be
and when it will arrive. The question of which particular sources
are used by baseball batters and in what combination remains
largely unresolved. In the present study, I directly tested whether
batters use speed to estimate pitch height and evaluated whether
Equation 1 (i.e., tau) alone is sufficient to control the timing of a
swing. I next review the literature on the motor responses involved
in hitting.
Biomechanics of Hitting a Baseball
Investigations of the coordinated movements involved in a
baseball swing have revealed that hitting involves a complex chain
of muscle activity (Shaffer, Jobe, Pink, & Perry, 1993; Welch,
Banks, Cook, & Draovitch, 1995). Figure 1 illustrates the phases
of a baseball swing. Hitters generate bat speed by a coiling process
that involves a rotation of the arms, shoulders, and hips away from
the oncoming pitch (the windup). This coiling begins when the
batter’s weight is shifted toward the back leg by lifting the front
foot off the ground. In generating an effective swing, it is impor-
tant that the hip rotation leads the shoulder rotation, which in turn
leads the arm rotation, forming a kinetic link. As the hitter drives
forward out of the coil (toward the oncoming ball), his or her foot
returns contact with the ground (the preswing). The momentum of
the uncoiling of the hips and shoulders is then transferred to the
arms (by decelerating each of these components in turn) to create
a maximum bat speed (up to 70 mph [31.3 m/s], measured from the
end of the bat) at the bottom of the swing (midswing). The
preswing begins 175 ms prior to bat–ball contact, and the maxi-
mum bat speed occurs approximately 15 ms before contact. It is
clear that generating a powerful swing involves much more than
just using the arms, as “skilled baseball batting relies on a coor-
dinated transfer of muscle activity from the lower extremities to
the trunk, and finally to the upper extremity” (Welch et al., 1995,
p. 293). In the present study, I explored how perceptual informa-
tion is used to modulate some of these swing components.
Another interesting aspect of the motor act of hitting a baseball
is how batters use eye and head movements to track the ball.
Hubbard and Seng (1954) were the first to study the visual tracking
used by baseball batters. In their study, 35-mm films of profes-
sional batters were visually inspected to determine at what inter-
vals during the ball’s flight gross eye and head movements oc-
curred. The major finding was that the hitters’ smooth pursuit eye
movements do not continue until the point of contact: No move-
ments were observed within roughly 150–200 ms prior to contact.
This finding is not surprising given that major league pitches can
travel at rates up to 1,000 degrees/s, whereas the fastest pursuit eye
movements recorded in humans are only about 90 degrees/s (Watts
& Bahill, 1990). Hubbard and Seng also somewhat surprisingly
found that batters do not seem to reduce this large discrepancy by
moving their heads with the flight of the ball.
A more fine-grained analysis, using modern eye- and head-
movement recording techniques, was conducted by Bahill and
LaRitz (1984). This study compared movements made by Brian
Harper, a major league player, with those of novice hitters. The
main finding confirmed the observations of Hubbard and Seng
(1954): Major league hitters cannot track from release until con-
tact. Harper could track the ball very well until it was roughly 5.5
ft (1.68 m) from the plate, at which point it was no longer foveated.
As might be expected, novice hitters lost foveation of the ball
considerably earlier, when it was an average of 9 ft from the plate.
Harper also used a superior strategy for following the ball: He used
a combination of head and eye movements to follow the ball,
whereas amateurs tended to predominately move one or the other.
In summary, a baseball swing comprises a complex series of
muscle activations involving several different muscle groups. The
response complexity involved in hitting may bring into question
how well laboratory research on judgments of TTC extends to
realistic game situations. For example, in the experiments demon-
strating that observers can accurately estimate TTC on the basis of
Equation 1 alone, participants viewed the approaching object while
seated with their chins in a headrest (Gray & Regan, 1998). Can
people still use this information source (Equation 1) effectively
when they are moving their heads, eyes, and limbs? In addition,
when trying to hit a ball, there are severe demands on when the
estimate of the TTC can be made. In the present study, I evaluate
the use of Equation 1 in a more realistic active paradigm.
Previous research has also shown that batters’ eye movements
are not fast enough to keep the ball on the fovea from the point of
release to bat–ball contact and that there appears to be some clear
expert–novice difference in hitting strategy. The results of the
present study provide evidence for further expertise effects in
hitting.
Figure 1. Phases of a baseball swing. Values are proportions of the total
swing time. From “Baseball Batting: An Electromyographic Study,” by B.
Shaffer, F. W. Jobe, M. Pink, and J. Perry, 1993, Clinical Orthopaedics
and Related Research, 292, p. 286. Copyright 1993 by Lippincott Williams
& Wilkins. Reprinted with permission.
1133
VIRTUAL BASEBALL BATTING
Mutual Constraints of Perception and Action
Unfortunately, very few studies have investigated the specifics
of how the perceptual information is used to control the various
motor responses involved in hitting. It is clear that there is more
involved in hitting than judging where and when; the batter needs
to use this information to modify the complex stages of swinging
a bat. In addition, the motor responses (in particular, the limitations
of eye movements) may constrain the type of information that can
be used.
Hubbard and Seng (1954) were the first to note the necessity of
this combined analysis: “The sensory and response aspects may be
separated for analysis, but in the batting situation the batter’s step
and wind-up for the swing must occur while the sensory–
perceptual process is under way” (p. 42). Along with the important
findings concerning eye movements, this study identified some
important relationships between the movements of the swing and
the TTC of the pitch. Figure 2A shows the preswing initiation time
(defined as the point in time when the batter’s front foot broke
contact with the ground) as a function of pitch speed. It is clear
from this figure that the timing of the swing initiation varied with
pitch speed. Figure 2B plots the TTC at which the preswing was
initiated (i.e., the ball’s TTC when the batter lifted his front foot
off the ground) as a function of speed. It is evident that the
preswing occurred at roughly a constant TTC for all pitch speeds,
resulting in a constant swing duration (time between preswing and
contact). This finding has important implications for understanding
how batters use visual information during hitting. As suggested by
Fitch and Turvey (1978), the timing of constant swing duration
could be easily controlled by gearing initiation to a critical value of
TTC specified by Equation 1 and/or Equation 2. However, Profitt
and Kaiser (1995) have argued that a constant swing duration
could also simply be a consequence of batters attempting to
achieve maximum bat speed (i.e., swinging the bat as hard as
possible over the same distance would always produce roughly the
same duration). The use of this control strategy is tested formally
in the present study.
Bahill and LaRitz (1984) explored the relationship between the
perceptual information available for hitting and the use of eye and
head movements. They found that hitters appear to use two dif-
ferent tracking strategies. Figures 3A and 3B show records of the
position of the ball and the position of a professional baseball
player’s eyes for two swings. The first tracking strategy (shown in
Figure 3A, top) is to use smooth pursuit eye movements in com-
bination with head movements to foveate the ball for as much of
its flight as possible. The consequence of this strategy is that the
ball will be off the fovea (by up to 35°) for the last 5–6 ft (1.5–1.8
m) of its flight. The alternative strategy (shown in Figure 3B) is to
follow the ball with eye movements for the first part of its flight
(until it is roughly 25 ft [7.6 m] from the plate) and then make a
quick saccade to a point in space that is predicted to be just ahead
of the ball. After the saccade is finished, the ball is foveated and
tracked with smooth pursuit eye movements until contact is made.
There are advantages to each of these strategies. Bahill and LaRitz
termed the strategy shown in Figure 3A the optimal hitting strat-
Figure 2. A: Preswing initiation time as a function of pitch speed. B: Preswing time to contact (TTC) as a
function of pitch speed. (Data from Table 1 of Hubbard & Seng, 1954, were used to calculate these variables.)
1134
GRAY
egy, because it gives the hitter more time to judge the height and
TTC of the pitch. The main advantage of the second strategy is that
the batter can evaluate the accuracy of his or her height and TTC
estimates when the ball is foveated just before it crosses the plate.
For this reason, it has been called the optimal learning strategy,
because it will probably give the batter the best chance of getting
hits in future times at bat.
Another interesting problem that arises when considering the
mutual constraints of perception and action is that the accuracy of
the perceptual information for hitting depends on when it is used.
A batter can predict the height of the pitch on the basis of Equation
4 or 5. However, both of these estimates are constant-velocity
approximations (i.e., they give the height of the ball at contact
assuming it continues to fall at the current drop speed), whereas a
baseball accelerates due to gravity. This approximation has the
important consequence that the accuracy of the batter’s height
estimate depends on the point along the ball’s trajectory at which
the estimate is made. The open circles in Figure 4 plot the height
estimate (i.e., the ball’s height when it reaches the contact point)
provided by Equation 4 as a function of the time since pitch release
(bottom axis) and the distance from the tip of the plate (top axis).
When the ball is farther than roughly 15 ft (4.6 m), Equation 4
overestimates the height at contact (solid line), and when the ball
is nearer than roughly 10 ft (3.0 m), Equation 4 underestimates the
Figure 3. Eye and head movement recordings during a professional baseball player’s swing. A: Optimal hitting
strategy. The batter follows the ball with his eyes until it is roughly 5–6 ft (1.5–1.8 m) in front of the plate. B:
Optimal learning strategy. The batter follows the ball until it is roughly 25 ft (7.6 m) in front of the plate and
then makes a saccade so that the ball is on the fovea at the point of contact. From “Why Can’t Batters Keep Their
Eyes on the Ball?” by A. T. Bahill and T. LaRitz, 1984, American Scientist, 72, p. 251. Copyright 1984 by Sigma
Xi, The Scientific Research Society. Reprinted with permission.
1135
VIRTUAL BASEBALL BATTING
height. The results of the present study provide evidence that the
accuracy of a baseball swing is influenced by this particular
estimation error.
Cognitive and Indirect Information for Hitting
Ted Williams (a career .344 hitter considered by many to be the
greatest hitter ever) once said “Proper thinking is 50 per cent of
effective hitting” (Williams & Underwood, 1970, p. 29). There is
abundant anecdotal evidence that hitters use variables such as the
history of previous pitches and the pitch count (i.e., the number of
balls and strikes) to predict the location and speed of the upcoming
pitch. However, to my knowledge, the use of this cognitive infor-
mation during baseball batting has not been explored in detail.
5
An indirect perceptual cue that baseball batters could use to
distinguish between fast and slow pitches is the ball’s rotation
direction. Because of the biomechanics and physics of each pitch,
a fastball travels with an underspin (or backspin; i.e., from ground
to sky), whereas a curveball travels with an overspin. In laboratory
judgment experiments, it has been demonstrated that college base-
ball players can distinguish between a fastball and a curveball from
a 200-ms video of the ball’s flight at a 90% accuracy rate (Bur-
roughs, 1984). In the present study, I test whether batters can use
this information source to improve hitting performance.
Aims of the Present Study
The aims of the study reported here were to address some of the
outstanding issues from previous research on baseball batting and
to evaluate the effectiveness of the virtual batting task as a research
tool. Measures of the temporal and spatial swing accuracy were
used to test the following specific hypotheses:
1. If batters use pitch speed to estimate height, introducing a
large variance in speed should cause large spatial errors in the
swing.
2. Batters can effectively control the timing of their swing when
only the TTC information expressed in Equation 1 is available.
3. Batters use a control strategy of initiating a constant swing
duration when the ball reaches a critical TTC.
4. Swing accuracy is related to the history of previous pitch
types.
5. Swing accuracy is related to the pitch count.
6. Batters can use the direction of ball rotation to improve their
swing accuracy.
General Method
Participants
A total of 6 experienced college-level baseball players from the Boston
area participated in this study. At the time of these experiments, 2 of the
batters (1 and 2) were playing in Division 1A (the highest college level),
2 were playing in Division 3 (3 and 4), and 2 were playing in competitive
recreational leagues (5 and 6). The mean number of years of playing
experience was 15. All were given 30 min of simulated batting practice
prior to beginning the experiments.
Apparatus
Participants swung a baseball bat at a simulation of an approaching
baseball. The simulated ball, generated using OpenGL (Silicon Graphics,
Inc., Mountain View, CA), was an off-white sphere texture mapped with
red laces. The background was black. Simulated lighting from above
5
The term cognitive information is used in this article to refer to sources
of information that do not provide direct information about the flight of the
ball (and that are not given by the motion of the ball) but that are related
to characteristics of the ball’s flight. For example, a pitch count of 3–0
provides cognitive information that the next pitch will be fast, whereas a
small value of Equation 1 provides direct perceptual information that the
pitch is fast.
Figure 4. Estimate of final pitch height derived from Equation 4 as a function of time since release and distance
from the tip of the plate. The solid horizontal line shows the actual final height for a 90-mph (40.2-m/s) pitch.
1ft⫽ 0.3048 m.
1136
GRAY
produced shadows on the bottom of the ball. The simulated ball was
displayed on a 28-cm (vertical) ⫻ 36-cm (horizontal) SVGA monitor
(Viewsonic model PT795) that ran at 120 Hz. The monitor was viewed
from a distance of 3.5 m in a dimly lit room. A sensation of motion toward
the batter was created by increasing the angular size of the ball. The
vertical position of the ball on the monitor was changed to simulate the
drop of the ball as it approached the batter. Unless otherwise stated, the ball
rotated at 200 rpm with simulated underspin (i.e., from ground to sky).
6
All
participants reported a compelling impression of motion in depth.
Mounted on the end of the bat (a Louisville Slugger Tee Ball bat; 25 in.
[63.5 cm] long) was a sensor from a FASTRAK (Polhemus, Colchester,
VT) position tracker.
7
The x, y, z position of the end of the bat was recorded
at a rate of 120 Hz.
Pitch Simulations
The pitch simulation was based on that used by Bahill and Karnavas
(1993). Balls were launched horizontally (i.e., 0°) from a simulated dis-
tance of 59.5 ft (18.5 m; i.e., the pitcher released the ball 1 ft in front of the
rubber). The only force affecting the flight of the ball was gravity. (The
effects of air resistance and spin on the ball’s flight were ignored.) The
height of the simulated pitch, Z(t), was changed according to
Z共t兲 ⫽ ⫺0.5g共t
2
兲, (6)
where g is the acceleration of gravity (32 ft/s [9.8 m/s]). In the present
study, I used simulated pitch speeds ranging from 60 mph (26.8 m/s) to 87
mph (38.9 m/s). Unless otherwise stated, the ball was released from a
height of 6 ft (1.83 m) and traveled across the center of the plate (i.e., the
lateral position was not varied).
Procedure
Batters attempted to hit the simulated approaching ball. No instructions
were given as to how hard to swing the bat. A progress bar that indicated
the time until pitch release was presented on the monitor 10 s before the
ball appeared on the screen. This was used to give a crude simulation of the
pitcher’s windup. The progress bar disappeared 0.5 s before the pitch was
released.
Visual feedback indicating the success of the batter’s swing was given as
follows. The x, y, and z coordinates of the bat and ball as a function of time
were used to estimate the point of contact on the bat (if there was contact)
and bat speed. These variables were then used to approximate the trajectory
and speed of the ball leaving the bat. An image of a baseball diamond was
presented on the monitor 5 s after the virtual ball crossed the plate. A red
line extending from home plate, indicating the trajectory and distance of
the ball, was drawn on the diamond. If no contact was made, a large, red
X was presented to indicate a strike. Text messages were also displayed to
indicate foul balls and home runs (i.e., balls estimated to travel further than
300 ft [91.4 m]). In preliminary experiments, some hitters had difficulty
determining why their swings were unsuccessful (e.g., they were too late or
too high). Therefore, text messages were displayed in the simulation to
indicate large temporal and spatial errors in the swing (e.g., “way late” and
“get the bat down”).
The time interval between successive pitches was 20 s. Each block of
trials consisted of 20 pitches. At the end of each block, a statistical
summary showing the numbers of contacts, fair balls, and home runs was
displayed on the monitor. Batting performance was measured in four
different experimental conditions. The order of experiments was random-
ized across the 6 participants to reduce any practice effects.
Data Analysis
Figure 5 shows a typical recording of bat and ball heights as a function
of time. To explore the perceptual–motor strategies used in the simulated
hitting task, I calculated the following variables for each swing: the
minimum bat height, which was the minimum Z value of the bat during the
swing (A in Figure 5); and the onset of the downward motion of the bat,
which was the point in time that the bat began moving downward from the
batter’s shoulder (B in Figure 5). The criterion for the latter variable was
five consecutive height samples that were lower than the previous sample;
this variable is equivalent to the preswing in Figure 1.
When hitting a baseball, the spatial and temporal components of the
swing are tightly coupled; for example, for the ball to be contacted earlier
in its flight, the bat must be swung higher. To dissociate the spatial and
temporal components of the swing, I made the following simplifying
assumptions: (a) the batter uses visual information about the height of the
ball to control the minimum bat height, (b) the batter uses visual informa-
tion about the TTC to control the point in time when minimum bat height
is reached, and (c) the batter attempts to make contact with the ball when
it is 0.9 m in front of the plate (Bahilll & Karnavas, 1993). From these
assumptions, the spatial swing error can be expressed as
Z_ball
y⫽ 0.9
⫺ min共Z_bat兲, (7)
where Z_ball
y⫽0.9
is the height of the ball when it was 0.9 m in front of the
plate, and min(Z_bat) is the minimum bat height. The temporal swing error
is given by
t_ball
y⫽ 0.9
⫺ t_bat
min共Z兲
, (8)
where t_ball
y⫽0.9
is the time (measured from the point of release) when the
ball was 0.9 m in front of the plate, and t_bat
min(Z)
is the point in time when
the minimum bat height occurred.
Experiment 1: Batting for a Wide Range of Pitch Speeds
The purpose of Experiment 1 was threefold. First, I sought to
evaluate batting behavior in a virtual simulation. Given that accu-
rate information about TTC and pitch height was available in the
simulation, I predicted that the batters would be able to control the
dynamics of their swings appropriately for the simulated pitches.
Second, I sought to test the hypothesis that large variances in pitch
speed lead to spatial errors in the swing (Bahill & Karnavas, 1993;
McBeath, 1990). Third, I sought to test the proposal that batters
use a motor control strategy of initiating a constant swing duration
at a critical TTC (Fitch & Turvey, 1978).
Method
In Experiment 1, I examined batting performance when pitch speed was
chosen randomly on each trial from a wide range of speeds (63–80 mph
[28.2–35.8 m/s]). Each participant completed three blocks of 20 swings.
There was a 5-min rest period between each block.
6
The ball spin rate of 200 rpm was chosen so that it would be percep
-
tible to the batters given the 120-Hz frame rate of the display and to avoid
any aliasing. This rate is well below the rotation rate of a major league
fastball (roughly 1,600 rpm; Watts & Bahill, 1990) and is closer to the
range of a knuckleball (25–50 rpm).
7
All batters reported that the position tracker in no way hindered their
swing and that swinging the bat felt “easy and natural.” It should be noted,
however, that the bat used in the present study was lighter and shorter than
bats typically used in college hardball. Given that bat size can have a
dramatic effect on the dynamics of a swing (Watts & Bahill, 2000, p. 109),
it will be interesting for future research to address whether bat size also
affects the hitting strategy.
1137
VIRTUAL BASEBALL BATTING
Results
Overall performance. The most basic requirement for per-
forming the simulated hitting task was to swing the bat at different
heights and at different times depending on the pitch speed. Figure
6A shows the minimum bat height [min(Z_bat)] as a function of
pitch speed for Batter 3.
As can be seen from the figure, the minimum bat height signif-
icantly correlated with pitch speed (R ⫽ .60, p ⬍ .001). However,
the variation in swing height (slope ⫽ 0.03 m) was much less than
the actual variation in pitch height (thin line; slope ⫽ 0.10 m).
Variability was also quite large for this batter. Similar results were
obtained for the other 5 batters. The slope and correlation values
for minimum bat height as a function of pitch speed for the 6
batters are shown on the left side of Table 1. Speed and minimum
bat height were significantly correlated for all 6 batters, and the
slope values ranged from 26%–57% of the actual pitch height
slope.
Figure 6B plots the point in time since release when the mini-
mum bat height [t_bat
min(Z)
] occurred as a function of pitch speed
for Batter 3. There was a significant negative correlation between
speed and t_bat
min(Z)
for this batter (R ⫽⫺.43, p ⬍ .001). The
slope of the line of best fit (⫺11.25 ms; solid line) was much closer
to the predicted slope (⫺17.90 ms; thin line) in this component
than it was in the case of the spatial component (Figure 6A),
indicating that this batter was much better at controlling the timing
of his swing than the swing height. The slope and correlation
values for time of the minimum bat height as a function of pitch
speed for the 6 batters are shown on the right side of Table 1. There
was a significant negative correlation between speed and t_bat
min(Z)
for all 6 batters, and the slope values ranged from 48%–118% of the
actual arrival time slope.
For the 60 swings shown in Figure 6, 10 resulted in some
simulated contact with the ball. Of the swings that resulted in
Figure 5. Example data record for one swing. Open circles represent the bat height as a function of time since
release, and solid circles represent the ball height. Two main variables were calculated for each plot: minimum
bat height (A) and onset of the downward motion of the bat (B).
Figure 6. Minimum bat height (A) and time of minimum bat height (B)
as a function of speed for Batter 3 in Experiment 1 (random speed
simulation). Thick lines are the lines of best fit. The thin line in A is the
actual variation in height, and the thin line in B is the time of arrival.
1138
GRAY
contacts, there were 3 fair balls (none of which were home runs)
for this batter. Overall, the success rate of the 6 batters was quite
poor; the mean number of hits was 1.7 (SE ⫽ 0.3), for a batting
average of .030. The mean number of swings that were within the
temporal margin for error (M ⫽ 12.5, SE ⫽ 0.6) was significantly
greater than the mean number of swings within the spatial margin
for error (M ⫽ 5.0, SE ⫽ 0.7), t(5) ⫽⫺3.3, p ⬍ .05. At this point,
it should be noted that the hitting task in Experiment 1 was in
many ways more difficult than that faced by a major league player;
from the comments made by professional hitters (e.g., Williams &
Underwood, 1970), it would seem unlikely that a pitcher can
randomly vary the pitch speed by 17 mph (7.6 m/s) from pitch to
pitch without giving other cues (e.g., arm motion or ball rotation).
In Experiments 2–4, I examined more realistic game situations.
Subsequently, I present a more detailed analysis of the Experiment
1 data.
Spatial accuracy. Figure 7A shows the data from Figure 6A
replotted as spatial error (i.e., Equation 7) versus pitch speed for
Batter 3. Negative errors indicate that the bat was above the
ball. It is clear from Figure 7A that spatial accuracy was
strongly related to pitch speed (R ⫽ .89, p ⬍ .001); this batter
swung over the ball at slow speeds and under the ball at fast
speeds. What might cause a batter to exhibit this pattern of
errors? As shown in Figure 4, the accuracy of the perceptual
correlates that can be used to estimate pitch height depends on
when the estimate is taken. The dashed line in Figure 7A shows
the errors that would be predicted if the batter always estimated
pitch height a constant time after the pitcher released the ball.
This constant time fit (using a value of 440 ms) gave a reason-
ably good fit to the data (R
2
⫽ .49). As shown in Table 2,
similar results were obtained for the other 5 batters. The mean
absolute spatial error for the 6 batters was 0.2 m (SE ⬍ 0.1), 16
times the spatial margin for error.
Temporal accuracy. Figure 7B shows temporal error (i.e.,
Equation 8) as a function of pitch speed for Batter 3. Negative
errors indicate that the swing was late. The temporal error was not
significantly correlated with pitch speed for Batter 3 (R ⫽⫺.25, p
⬎ .05). Slope and correlation values for all 6 batters are shown in
Table 3. For 3 of the 6 batters, there was a significant negative
correlation between temporal error and pitch speed. The mean
temporal error for the 6 batters was 58.7 ms (SE ⫽ 9.8), 6.5 times
the margin for error.
Swing initiation and duration. Figure 8A plots swing duration
as a function of pitch speed for Batter 3. For this batter, swing
duration was significantly shorter at faster pitch speeds (slope ⫽
⫺7.60 ms; R ⫽⫺.35, p ⬍ .01). This pattern is consistent with the
findings of Hubbard and Seng (1954; see Figure 2), although the
effect of speed on swing duration was much larger in the present
study; the mean slope for the 6 batters was ⫺10.77 ms. Slope and
correlation values for all 6 batters are shown in Table 4. Pitch
speed and swing duration were significantly correlated for all 6
Figure 7. A: Height error as a function of pitch speed for Batter 3 in
Experiment 1 (random speed simulation). The dashed line shows values
that are predicted if the batter estimated height using a constant-velocity
approximation 440 ms after the pitch was released. B: Timing error as a
function of pitch speed for Batter 3. The solid line is the line of best fit.
Table 1
Spatial and Temporal Swing Data for Experiment 1
Batter min(Z_bat) slope % actual slope
a
Rpt_bat
min(Z)
slope
% actual slope Rp
1 0.0570 57.0 .54 ⬍ .001 ⫺14.10 78.60 ⫺.44 ⬍ .001
2 0.0550 55.0 .39 ⬍ .01 ⫺21.29 118.60 ⫺.55 ⬍ .001
3 0.0260 26.0 .60 ⬍ .001 ⫺11.25 62.70 ⫺.43 ⬍ .001
4 0.0511 51.1 .63 ⬍ .001 ⫺8.77 48.80 ⫺.38 ⬍ .01
5 0.0385 38.5 .45 ⬍ .001 ⫺10.60 59.08 ⫺.26 ⬍ .05
6 0.0287 28.7 .32 ⬍ .01 ⫺8.66 48.26 ⫺.39 ⬍ .01
Note. min(Z_bat) ⫽ minimum bat height; t_bat
min(Z)
⫽ point in time when the minimum bat height occurred.
a
Derived from the height of the simulated ball when it was 0.9 m in front of the plate.
1139
VIRTUAL BASEBALL BATTING
batters. The mean swing duration for the 6 batters was 148 ms
(SE ⫽ 13). This value is similar to that found in the study by
Welch et al. (1995), which used electromyography.
Figure 8B plots TTC at swing onset. There was a strong ten-
dency to start the swing at a shorter TTC for faster pitches
(slope ⫽⫺14.30; R ⫽⫺.49, p ⬍ .001). Again, the effect of speed,
although consistent with the general pattern of Hubbard and
Seng’s (1954) data, was much larger in the present study; the mean
slope for the 6 batters was ⫺17.30 ms. Slope and correlation
values for all 6 batters are shown in Table 5. There was a signif-
icant negative correlation between speed and TTC at swing initi-
ation for all 6 batters. The pattern of data in Figure 8B is consistent
with the batter initiating the swing at a roughly constant time after
pitch release. For all 6 batters, pitch speed and swing initia-
tion time were not significantly correlated (see the right side of
Table 5).
Discussion
Ted Williams (Williams & Underwood, 1970) said that “hitting
a baseball . . . is the single most difficult thing to do in sport” (p.
7), and the results of Experiment 1 certainly provide support for his
claim. The simulation used in the present study contained theoret-
ically accurate perceptual information about where and when the
ball would cross the plate, yet all 6 experienced college players
could barely make contact with the ball. It is clear that successful
batting is nearly impossible in the situation in which pitch speed is
random and in which no auxiliary cues (e.g., pitcher’s arm motion
or pitch count) are available to the batter.
Relative to the margins for error, batters in the present experi-
ment were significantly better at controlling the temporal compo-
nent of the swing than the spatial component of the swing. The
mean number of swings within the temporal margin for error was
significantly greater than the mean number of swings within the
spatial margin for error. This experimental finding is consistent
with the theoretical analysis by Bahill and Karnavas (1993). This
may occur because there is no direct perceptual correlate of ball
height; instead, batters must estimate height indirectly using ab-
solute speed or absolute distance. Conversely, TTC can be directly
estimated using the ball’s rate of expansion. Another problem
associated with estimating pitch height is when to take the esti-
mate. The curve fit in Figure 7A suggests that a major source of
error in the spatial component of the swing is the batter estimating
pitch height at a constant time after the pitch is released.
In general, the results from Experiment 1 are roughly consistent
with the following perceptual–motor control strategy: (a) Initiate
the swing at a constant time after the pitch is released (Figure 8B),
(b) estimate pitch height using a constant-velocity approximation
at the point in time that the swing is initiated (Figure 7A), and (c)
Figure 8. A: Swing duration as a function of pitch speed for Batter 3 in
Experiment 1 (random speed simulation). B: Time to contact (TTC) at
swing onset as a function of pitch speed for Batter 3. Solid lines are the
lines of best fit.
Table 2
Spatial Swing Error Slopes and Curve Fits for Experiment 1
Batter Spatial error slope R
Constant t value
(curve fit) R
2
1 0.042 .43 0.540 .21
2 0.060 .48 0.436 .19
3 0.070 .89 0.440 .48
4 0.051 .61 0.435 .36
5 0.060 .61 0.617 .25
6 0.075 .65 0.494 .28
Note. For all Rs, p ⬍ .001.
Table 3
Temporal Swing Errors for Experiment 1
Batter Temporal error slope Rp
1 ⫺9.1 ⫺.47 ⬍ .001
2 4.2 .12 ns
3 ⫺6.7 ⫺.25 ns
4 ⫺8.6 ⫺.51 ⬍ .001
5 ⫺6.1 ⫺.15 ns
6 ⫺9.5 ⫺.42 ⬍ .01
1140
GRAY
adjust the swing duration based on an estimate of TTC or pitch
speed. This control strategy is very different from that proposed by
Fitch and Turvey (1978) and is discussed in more detail below.
Experiment 2: Simulation of a Two-Pitch Pitcher
It is a very unusual situation in which a baseball pitcher can
randomly vary pitch speed over a 17-mph (7.6-m/s) range. It is
more common for a pitcher to have two or three pitches (e.g., a
fastball, a curveball, and a change-up) with distinct ranges of
speeds. In Experiment 2, I examined the more natural situation in
which the simulated pitcher could throw only two pitches (a
fastball and a slow ball [i.e., a change-up]), and I measured the
effect of pitch type on swing error. I hypothesized that batting
performance would be significantly improved in this more natural
two-pitch scenario.
Abundant anecdotal evidence exists in support of batters using
the sequence of previous pitches to control their swing (e.g.,
Williams & Underwood, 1970). For example, after seeing several
off-speed (i.e., slow) pitches, batters often gear up for a fastball.
Thus, I hypothesized that the temporal and spatial errors in Ex-
periment 2 would be related to the pitch type of the previous few
pitches.
Method
Slow pitches were simulated to travel at 70 ⫾ 1.5 mph (31.3 ⫾ 0.67
m/s), and fast pitches were simulated to travel at 85 ⫾ 1.5 mph (38.0 ⫾
0.67 m/s). Fast or slow speed was chosen randomly on each trial. As in
Experiment 1, all pitches were strikes and traveled down the center of the
plate. Each batter completed three blocks of 20 pitches, with rest intervals
between each block.
Results
Overall performance. Figure 9A plots the minimum bat height
[min(Z_bat)] as a function of pitch speed for Batter 3. Compared
with the results from Experiment 1 (Figure 6A), this batter was
substantially better at controlling the spatial component of the
swing in Experiment 2. There was a significant correlation be-
tween minimum bat height and pitch speed (R ⫽ .62, p ⬍ .001),
and the slope of the line of best fit (0.08) was much closer to the
actual pitch height slope. Similar results were obtained for the
other 5 batters. For all 6 batters, there was a significant positive
correlation between min(Z_bat) and pitch speed (Rs ranged from
.42 to .71). Slope values for the 6 batters ranged between 67% and
90% of the actual pitch height.
In Figure 9A, it can also seen that the distribution of min(Z_bat)
values was bimodal for both the slow and the fast pitches. For
about seven slow pitches (as indicated by the solid arrow in Figure
9A), Batter 3 swung the bat at a height that would be appropriate
for a fast pitch (at roughly ⫺0.5 m), and for about eight fast pitches
(as indicated by the dashed arrow in Figure 9A), he swung the bat
at a height that would be appropriate for a slow pitch (at roughly
⫺1.0 m).
Figure 9B shows the point in time that the minimum bat height
occurred [t_bat
min(Z)
] as a function of pitch speed for Batter 3. As
was the case in Experiment 1 (Figure 6B), this batter was relatively
good at controlling the temporal component of the swing. Mini-
mum bat height was significantly negatively correlated with pitch
speed (R ⫽⫺.48, p ⬍ .001), and the slope of the line of best fit
was 53% of the actual arrival time slope. Similar to the spatial
swing component, the distribution of t_bat
min(Z)
values was
roughly bimodal for both the slow and fast pitches. Similar results
were obtained for the other 5 batters. For all 6 batters, there was a
significant negative correlation between t_bat
min(Z)
and pitch
speed (Rs ranged from ⫺.39 to ⫺.66). Slope values for the 6
batters ranged between 51% and 99% of the actual pitch height.
For the 60 swings shown in Figure 9, 20 resulted in some
contact with the ball. There were 8 fair balls (2 of which were
home runs) and 12 foul balls. The mean number of hits for the 6
batters was 7.1 (SE ⫽ 0.2), for a batting average of .120. A paired
t test revealed that the number of hits was significantly greater in
Experiment 2 than in Experiment 1, t(5) ⫽ 3.9, p ⬍ .05. In
Experiment 2, the mean number of swings that were within the
temporal margin for error (M ⫽ 24.1, SE ⫽ 0.5) was not signifi-
Table 4
Swing Duration Slopes for Experiment 1
Batter Duration slope Rp
1 ⫺9.71 ⫺.35 ⬍ .01
2 ⫺16.63 ⫺.38 ⬍ .01
3 ⫺7.62 ⫺.35 ⬍ .01
4 ⫺5.80 ⫺.41 ⬍ .01
5 ⫺18.95 ⫺.54 ⬍ .001
6 ⫺5.91 ⫺.26 ⬍ .05
Table 5
Swing Initiation Slopes for Experiment 1
Batter TTC at initiation slope RpInitiation time slope Rp
1 ⫺18.879 ⫺.53 ⬍ .001 ⫺1.2 .03 ns
2 ⫺12.469 ⫺.25 ⬍ .05 ⫺5.3 ⫺.10 ns
3 ⫺14.320 ⫺.49 ⬍ .001 ⫺3.6 ⫺.13 ns
4 ⫺16.503 ⫺.69 ⬍ .001 ⫺1.6 ⫺.08 ns
5 ⫺26.047 ⫺.48 ⬍ .001 ⫺8.4 .16 ns
6 ⫺15.490 ⫺.52 ⬍ .001 ⫺2.7 ⫺.10 ns
Note. TTC ⫽ time to contact.
1141
VIRTUAL BASEBALL BATTING
cantly different from the mean number of swings within the spatial
margin for error (M ⫽ 22.3, SE ⫽ 0.7).
Spatial and temporal errors. In Experiment 1, there was a
strong relationship between spatial errors and pitch speed, and the
pattern of errors was consistent with the batter estimating pitch
height at a constant time after release. In Experiment 2, the
constant time strategy curve could not be fit to the data because of
the limited range of pitch speeds; however, there was a significant
positive correlation between spatial error and pitch speed for
Batter 3 (R ⫽ .29, p ⬍ .05). This relationship appeared to be due
to Batter 3 being fooled on some pitches: He swung too high for
some slow pitches and too low for some fast pitches (see the
arrows in Figure 9A). These expectancy effects are discussed in
detail below. Similar results were obtained for the other 5 batters.
For all 6 batters, there was a significant positive correlation be-
tween spatial error and pitch speed (Rs ranged between .21 and
.42).
In Experiment 1, the relationship between temporal error and
pitch speed was not consistent across the 6 batters. This was not
the case in Experiment 2. For all batters, there was a significant
negative correlation between speed and temporal error (Rs ranged
between ⫺.29 and ⫺.61). Again, this appeared to be due to an
expectancy effect, as batters swung early for some slow pitches
and late for some fast pitches (see Figure 9B).
Expectancy effects. The bimodal distributions in Figures 9A
and 9B suggest that for at least some pitches, Batter 3 generated a
swing that was appropriate for the opposite pitch type. Figure 10
plots spatial swing error versus temporal swing error for Batter 3.
The relationship in Figure 10 is what would be predicted if the
batter’s expectations about speed were incorrect on some pitches.
For example, if the batter was expecting a fast pitch when the pitch
was slow, he should swing too high and early. On the other hand,
if he incorrectly expected a slow pitch, he should swing too low
and late. There was a strong negative correlation (R ⫽⫺.87, p ⬍
.001) between spatial and temporal errors for Batter 3. Similar
results were obtained for the other 5 batters (Rs ranged from ⫺.67
to ⫺.95).
What might batters be using to develop these expectations about
pitch speed? One likely possibility is the pitch sequence. For
example, if a batter is thrown several pitches of the same speed in
a row, he or she might come to expect that the next pitch will also
be the same speed. If the batter is using this strategy, then his or
her swing error should be much larger when a fast pitch is
preceded by a series of slow pitches (i.e., incorrect expectation)
than when a series of consecutive fast pitches is followed by a fast
pitch (i.e., correct expectation). Figure 11 shows that this was
indeed the case. For all 6 batters, the mean spatial (Figure 11A)
and temporal (Figure 11B) errors for fast pitches that were pre-
Figure 9. Minimum bat height (A) and time of minimum bat height (B)
as a function of speed for Batter 3 in Experiment 2 (two-pitch pitcher
simulation). Thick lines are the lines of best fit. The thin line in A is the
actual variation in height, and the thin line in B is the time of arrival.
Arrows show pitches for which the batter’s swing was appropriate for the
opposite pitch type: The solid arrow indicates slow pitches for which Batter
3 swung the bat at a height appropriate for a fast pitch, and the dashed
arrow indicates fast pitches for which Batter 3 swung the bat at a height
appropriate for a slow pitch.
Figure 10. Relationship between temporal errors and spatial errors in
Experiment 2 (two-pitch pitcher simulation). Data are for Batter 3. The
solid line is the line of best fit.
1142
GRAY
ceded by three consecutive fast pitches (solid bars) were substan-
tially smaller than mean errors for fast pitches that were preceded
by three consecutive slow pitches (open bars).
8
Paired t tests
revealed that the mean errors were significantly different in these
two conditions: spatial errors, t(6) ⫽ 2.6, p ⬍ .05; temporal errors,
t(6) ⫽ 3.1, p ⬍ .05. For comparison, the mean errors from
Experiment 1 (striped bars) are also plotted in Figure 11. For 5 of
the 6 batters, errors were smaller for the random speeds in Exper-
iment 1 than for the unexpected fast pitch in Experiment 2.
In a companion article (Gray, 2002), I used a two-state Markov
model to predict expectancy effects for all possible pitch se-
quences. This mathematical model uses a simple set of transition
rules to determine the probability that the batter will expect a given
pitch on the basis of the type of the previous three pitches. This
probability can then be used to predict the accuracy of the swing
(i.e., if there is a high probability that the batter is expecting a fast
pitch, he or she will be more accurate when the pitch actually is
fast). This model provided a good fit to the swing error data in this
article (R
2
values for the 6 batters ranged from .51 to .96).
Discussion
As predicted, hitting performance was considerably better when
the simulated pitcher was limited to throwing only fastballs and
change-ups. In comparison with Experiment 1, the batters made
significantly more contacts with the ball, and control of the spatial
component of the swing significantly improved. The dramatic
improvement in hitting performance for this two-pitch scenario
emphasizes why it is important for baseball pitchers to learn to
throw at least three different types of pitches (Ryan & House,
1991).
Why is it easier to hit against a two-pitch pitcher? The results of
Experiment 2 indicate that it is because batters can anticipate the
speed of the upcoming pitch on the basis of the prior sequence of
pitches and use this cognitive information to control their swing.
For the 6 batters, the spatial swing error was up to five times larger
and the temporal swing error was up to eight times larger when a
fast pitch was preceded by three consecutive slow pitches than
when it was preceded by three consecutive fast pitches. This
dramatic difference in performance occurred even though the
perceptual information was identical in the two situations. It is
clear that, as I hypothesized, the batters were supplementing the
visual information about the current pitch with expectations about
speed on the basis of the history of previous pitches.
Despite the improvement in hitting performance in Experiment
2, the batters had very large swing errors on some pitches, and the
batting average (.120) was still low relative to real-game perfor-
mance. The strong relationship between errors and pitch speed
shown in Figure 10 suggests that the major source of error in
Experiment 2 was incorrect expectations about pitch type. When
batters did make a large error, they made it because their swing
was appropriate for the opposite pitch speed. Bahill and Karnavas
(1993) have proposed that batters need to use supplemental infor-
mation about pitch speed to estimate pitch height (see Equation 4).
Further, these authors predicted that if batters are using Equation
4, then they should swing too low for unexpectedly fast pitches
(which give the batter the illusion that the ball has risen) and too
high for unexpectedly slow pitches (which give the batter the
illusion of a hard-breaking pitch). The pattern of errors shown in
Figures 9 and 10 provides strong experimental support for their
predictions.
Experiment 3: Effect of Pitch Count
In Experiment 2, I found that swing accuracy was related to the
history of previous pitches. It is well-known that baseball batters
also generate expectations about the upcoming pitch on the basis
of the pitch count: “Certainly the pitch you anticipate when the
count is 0 and 2 (a curve ball probably, if the pitcher has one) is
not the pitch you anticipate when the count is 2 and 0 (fastball,
almost without exception)” (Williams & Underwood, 1970, p. 30).
In Experiment 3, I examined the effect of pitch count on hitting
performance for the two-pitch scenario used in Experiment 2.
From the narratives from experienced players, I predicted that
batters would expect fast pitches when they were ahead in the
count (i.e., more balls than strikes) and that they would expect
slow pitches when they were behind in the count. The rationale for
these predictions is as follows. In general, for slower pitches such
8
Similar results were obtained when swing errors for slow pitches were
analyzed. The mean temporal (and spatial) errors for a slow pitch preceded
by three consecutive slow pitches were significantly lower than those for a
slow pitch preceded by three consecutive fast pitches.
Figure 11. Mean absolute spatial errors (A) and mean absolute temporal
errors (B) for the 6 batters in Experiment 2 (two-pitch pitcher simulation).
Solid bars are means for fast pitches that were preceded by three consec-
utive fast pitches (F, F, F, F), open bars are means for fast pitches that were
preceded by three consecutive slow pitches (S, S, S, F), and striped bars are
means for Experiment 1 (random speed condition). EXPT. ⫽ Experiment.
1143
VIRTUAL BASEBALL BATTING
as a change-up or curveball, it is more difficult for a pitcher to
control the location. When the pitcher falls further behind in the
count, it is more likely that the next pitch thrown will be fastball
to ensure a strike (Williams & Underwood, 1970, p. 30). On the
other hand, as the hitter falls behind in the count, it becomes more
likely that the pitcher will attempt to force the batter to chase a
pitch such as a slow curveball or change-up (Williams & Under-
wood, 1970, p. 30).
Method
In Experiment 3, the apparatus and procedure used were identical to
those in Experiment 2, except that the horizontal locations of the simulated
pitches were varied such that some of the pitches were strikes and some of
the pitches were balls. For strikes, the horizontal location of the simulated
baseball when it crossed the plate was 0.0 ⫾ 1 in. (0 ⫾ 2.54 cm), for which
0.0 was the center of the plate. For balls, the horizontal location of the pitch
when it crossed the plate was either ⫺12 ⫾ 1 in. (⫺30.5 ⫾ 2.54 cm) or
⫹12 ⫾ 1 in. (30.5 ⫾ 2.54 cm). Pitch location was chosen randomly on
each trial to be either a ball or strike. Pitch speed on each trial depended on
the pitch count (balls to strikes). For counts of 0–0, 1–0, 0–1, 1–1, 2–1,
2–2, and 3–2, fast pitches and slow pitches occurred with equal probability.
For counts of 0–2 and 1–2, slow pitches occurred with a probability of .65,
and fast pitches occurred with a probability of .35. For counts of 2–0, 3–0,
and 3–1, fast pitches occurred with a probability of .65, and slow pitches
occurred with a probability of .35. Thus, as is generally the case in real
baseball, fast pitches in the simulation occurred with a higher probability
when the hitter was ahead in the count, and slow pitches in the simulation
occurred with a higher probability when the hitter was behind in the count.
If the batter swung the bat at a ball and the bat crossed the front of the plate,
the call for that pitch was a strike. Visual feedback was given for the pitch
call, total pitch count, walks, and strikeouts.
Results and Discussion
Figure 12A shows mean spatial swing errors for two different
pitch counts: when the batter was ahead in the count at 2–0 (open
bars) and when the batter was behind in the count at 0–2 (solid
bars). The swing errors shown are only for fast pitches.
9
If, as
predicted, batters expected fast pitches when they were ahead in
the count and slow pitches when they were behind in the count,
then swing errors for fast pitches should be much smaller for a
count of 2–0 than for a count of 0–2. This was indeed the case. For
all 6 batters, the mean spatial swing error for fast pitches was lower
for the 2–0 count than for the 0–2 count. A paired t test revealed
that errors for the two pitch counts were significantly different,
t(5) ⫽ 2.5, p ⬍ .05.
Similar results were obtained for the temporal swing errors for
fast pitches, as shown in Figure 12B. The mean temporal swing
error was significantly smaller for the 2–0 count than for the 0–2
count, t(5) ⫽ 2.9, p ⬍ .05. When these pitch-count transition rules
(i.e., expect a fast pitch when ahead in the count, and expect a slow
pitch when behind in the count) were incorporated into the two-
state Markov model described above (Gray, 2002), the model
provided a good fit to the swing error data for all possible pitch
counts (R
2
values for the 6 batters ranged from .59 to .73).
Experiment 4: Effect of Rotation Cues
The purpose of Experiment 4 was to investigate whether rota-
tion direction cues influence batting performance. Because expe-
rienced batters appear to be quite sensitive to this cue (Burroughs,
1984), I predicted that swing errors would be significantly smaller
when the rotation direction cue was added to the batting
simulation.
Method
The procedure was identical to that described for Experiment 2, except
that for fast pitches, underspin was simulated, whereas for slow pitches,
overspin was simulated. The spin direction did not otherwise change the
trajectory of the simulated pitch; it simply added an additional source of
information related to pitch speed. The rotation rate was 200 rpm. All
pitches traveled down the center of the plate. Each batter completed three
blocks of 20 pitches with rest intervals between each block.
Results
The open bars in Figures 13A and 13B show, respectively, the
mean spatial errors and the mean temporal swing errors in Exper-
iment 4. In comparison with the identical two-pitch scenario with-
out rotation cues (Experiment 2; replotted with solid bars), errors
were substantially smaller for the majority of the batters. For
9
Again, the opposite pattern of results was obtained when errors for
slow pitches were analyzed: Swing errors (both spatial and temporal) for
slow pitches were significantly larger for a count of 2–0 than for a count
of 0–2.
Figure 12. Mean absolute spatial errors (A) and mean absolute temporal
errors (B) for the 6 batters in Experiment 3 (pitch-count simulation). Solid
bars are means for a count of 0–2 (i.e., no balls and two strikes), and open
bars are means for a count of 2–0 (i.e., two balls and no strikes). Error bars
represent standard errors.
1144
GRAY
Batters 1, 2, 3, and 4, spatial errors were significantly smaller
when rotation cues were present. Results of t tests were as follows:
Batter 1, t(118) ⫽ 2.0, p ⬍ .05; Batter 2, t(118) ⫽ 2.4, p ⬍ .01;
Batter 3, t(118) ⫽ 2.5, p ⬍ .01; Batter 4, t(118) ⫽ 1.8, p ⬍ .05. For
Batters 1, 2, and 3, temporal errors were significantly smaller when
rotation cues were present. Results of t tests were as follows:
Batter 1, t(118) ⫽ 1.7, p ⬍ .05; Batter 2, t(118) ⫽ 2.5, p ⬍ .01;
Batter 3, t(118) ⫽ 5.1, p ⬍ .001.
Discussion
The results of Experiment 4 indicate that for some batters,
rotation direction cues improve hitting performance. This behav-
ioral finding is consistent with psychophysical discrimination find-
ings (Burroughs, 1984). The ability to use rotation cues appears to
depend on the hitter’s ability to detect the movement of the seams
on the ball. Hyllegard (1991) compared hitters’ ability to discrim-
inate between video clips of fastballs and curveballs for a ball that
was painted white so that it had no seams, for a regulation baseball,
and for a ball for which the visibility of the seams was enhanced
with thick red stripes. College players achieved 74% correct for the
ball with no seams, 81% correct for the regulation ball, and 85%
correct for the ball with enhanced seams.
One limitation of the present finding is that the display frame
rate necessitated the use of a rotation rate that was well below that
which occurs for real fastballs. Whether batters can still use the
rotation direction cue when the spin rates are set at their normal
level needs to be addressed in future experiments.
Supplemental Analysis: Effect of Playing Level
Method
The experimental results showed that the simulated batting task requires
some of same skills involved in playing baseball. I therefore predicted that
several of the experimental variables would correlate with playing level. To
test this prediction, I grouped batters into one of three categories according
to their current playing level: high (College Division 1), medium (College
Division 3), or low (college recreational leagues). In the present study,
there were two batters in each category (see the General Method section).
I performed correlational analyses on the following variables: Experiment
1, spatial and temporal accuracy and total number of ball contacts; Exper-
iment 2, differences in absolute errors for the conditions in which a fastball
followed three fast pitches and a fastball followed three slow pitches;
Experiment 3, differences in absolute errors when the batter was ahead in
the count (2–0) and when the batter was behind in the count (0–2); and
Experiment 4, decrease in absolute errors produced by the addition of
rotation cues.
Results and Discussion
Table 6 shows the correlation values for the different experi-
mental variables and playing level. As expected, measures of
Figure 13. Mean absolute spatial errors (A) and mean absolute temporal
errors (B) for the 6 batters in Experiment 4 (rotation cue condition). Solid
bars are the means for Experiment 2, in which both slow and fast pitches
had the same rotation direction. Open bars are the means for Experiment 4,
in which slow pitches had overspin and fast pitches had underspin. Error
bars represent standard errors. *p ⬍ .05. **p ⬍ .01. ***p ⬍ .001. (Bars
without asterisks show nonsignificant results.) EXPT. ⫽ Experiment.
Table 6
Correlations Between Playing Level and Experimental Variables
Experiment Variable Description Rp
1 Spatial accuracy The min(Z_bat) slope compared with the actual pitch height slope .74 ⬍ .05
Temporal accuracy The t_bat
min(Z)
slope compared with the actual arrival time slope
.76 ⬍ .05
Total number of contacts Fair balls plus foul balls .73 ⬍ .05
2 Expectation effect size Difference between mean absolute temporal error when a fastball was
preceded by three fast pitches and mean absolute temporal error when
a fastball was preceded by three slow pitches
⫺.77 ⬍ .05
3 Pitch count effect size Difference between mean absolute temporal error for a pitch count of 2–0
and mean absolute temporal error for a pitch count of 0–2
.07 ns
4 Rotation cue effect size Difference between mean absolute temporal error for Experiment 2 (no
rotation cues) and mean absolute temporal error for Experiment 4
(rotation cues)
.86 ⬍ .05
1145
VIRTUAL BASEBALL BATTING
hitting performance in Experiment 1 were highly correlated with
playing level. Batters playing at a higher level had a greater
number of ball contacts, and the variation in their swing height and
timing was significantly closer to the actual variation in pitch
height and time of arrival. For Experiment 2, higher-level players
showed less of an effect of pitch sequence than did lower-level
players. Finally, the addition of rotation cues improved perfor-
mance more for higher-level players than for lower-level players.
This result is consistent with the finding that college baseball
players are significantly better than novices at discriminating be-
tween fastballs and curveballs presented in short video clips (Bur-
roughs, 1984).
General Discussion
Virtual Baseball Batting Task
For many years, researchers interested in visually guided action
have struggled with the problem of combining fine control over
stimulus parameters with realistic, active motor responses. The
baseball batting simulation used in the present study may be a step
in the right direction toward addressing this issue. In comparison
with hitting real baseballs, virtual batting has the advantage that
different information sources can be isolated and dissociated (see
Rushton & Wann, 1999, for a comparable catching simulation).
10
The participants in this study reliably adjusted the dynamics of
their swing based on the parameters of the simulation and, while
doing so, exhibited some of the well-known behaviors observed in
real baseball. The fine control over parameters such as pitch speed
in the simulation allowed direct testing of some questions that
were untenable to judgment experiments or measurements of real
batting. Finally, the strong correlations between the dependent
measures and playing level in the present study indicate that virtual
batting may be a useful diagnostic tool for comparing individual
hitters and for studying the nature of expertise.
Perceptual–Motor Control Strategies for Hitting a
Baseball
It has been repeatedly emphasized that hitting a baseball places
severe demands on the perceptual–motor system. A 100-mph
(44.7-m/s) major league fastball travels the distance between the
mound and the plate in 410 ms. Combining this with the batter’s
limited ability to track the ball with his or her eyes (Figure 3) and
the movement time required to execute the stages of a swing
(Figure 1), it is clear that a baseball batter must predict the future
location of the ball from visual information provided in the first
150–200 ms of the ball’s flight.
Despite the severity of this task, it should be emphasized that
accurate predictive visual information is available through direct
perceptual variables. The TTC can be predicted from the ball’s rate
of expansion (Equation 1), and the ball’s height can be predicted
from ball diameter, rate of expansion, and angular drop speed
(Equation 5). Therefore, as proposed by Bootsma and Peper
(1992), it is possible to hit successfully entirely on the basis of
perceptual information picked up during the ball’s flight. How-
ever, the results of Experiment 1 in the present study do not
support this direct pick-up proposal. When pitch speed was varied
randomly from trial to trial, the batters could not consistently make
contact with the ball, even though the information provided by
Equations 1 and 5 was available. Furthermore, the large improve-
ment in hitting performance in the two-pitch scenario (Experiment
2) and with the addition of rotation cues (Experiment 4) is not
compatible with this proposal, as neither of these manipulations
affect the values of Equation 1 or 5.
Instead, these results provide support for the indirect perceptual
model proposed by Bahill and Karnavas (1993). These authors
have proposed that the height of the ball when it crosses the plate
is predicted indirectly from an estimate of pitch speed (Equation
4). Further, the authors argue that “the speed estimator probably
uses memory and other sensory inputs: some visual, such as the
motion of the pitcher’s arms and body” (p. 8). In the present study,
the pattern of swing errors in Experiments 2 and 3 was consistent
with the batter generating expectations about pitch speed on the
basis of pitch sequence and pitch count. As predicted by Bahill and
Karnavas, when the batters in this study attempted to hit an
unexpectedly fast pitch (e.g., when a fast pitch followed a series of
slow pitches), they swung the bat too low. The improvement in
batting performance resulting from the addition of rotation cues in
Experiment 4 also provides support for this indirect model. Rota-
tion direction does not provide any information to the batter about
the future location of the ball, so it presumably aids the batter by
influencing his or her estimate of pitch speed.
How does the batter actually use this perceptual information to
control the complex motor responses involved in swinging a bat?
It is surprising that there have been very few models put forth to
explain this essential component of “America’s Game.” One no-
table exception is the simple control model proposed by Fitch and
Turvey (1978). These authors argued that batters could easily
control the timing of their swing by initiating a constant-duration
ballistic swing at a critical value of TTC. The results of the present
study are not consistent with this proposal, as the batters appeared
to initiate a variable-duration swing at a constant time after the
pitch was released. Further, the results of Experiment 1 suggest
that the batters were adjusting their swing duration on the basis of
an estimate of pitch speed or an estimate of TTC. It is clear that
further research is needed to understand how a baseball swing is
controlled. Some possible future experiments are discussed below.
Cognitive Information for Baseball Batting
Anecdotal evidence from players and coaches (e.g., Williams &
Underwood, 1970) indicates that cognitive processing (e.g., ex-
pectations about the upcoming pitch) plays an important role in
successful baseball batting, yet this aspect of hitting has not been
investigated in detail. The present study provides the first quanti-
tative experimental evidence that the history of previous pitches
and the pitch count significantly influence the spatial and temporal
components of a baseball swing. The pattern of errors produced by
manipulations of these variables is consistent with the notion that
batters generate expectations about the type of the upcoming pitch.
Furthermore, the findings reported here indicate that in some
10
Simulation also has the advantage that impossible pitches (such as
rising fastballs or sharp-breaking curveballs) can be used.
1146
GRAY
instances, these expectations may play a larger role in swing
execution than does visual perception. For example, consider the
data from Batter 2 in Figure 11B. On average, this batter swung
137 ms later for a fast pitch that was preceded by three slow
pitches than for a fast pitch that was preceded by three fast pitches.
Given that the actual mean difference in time of arrival between
the fast and slow pitches in Experiment 2 was only 105 ms, it
would appear that in this instance, the batter completely ignored
the perceptual information and controlled his swing entirely on the
basis of his expectation about the speed. This finding may come as
no surprise to many baseball fans given the regularity with which
professional hitters are fooled by slow pitches. Finally, the results
of the present study suggest that picking up of indirect perceptual
cues, such as ball rotation direction, is also very important for
successful batting.
The hitters in this study seemed to use very simple rules for
generating expectations about pitch speed. The results of Experi-
ment 2 suggest that hitters use a simple strategy of expecting the
pitcher to throw the ball at the same speed as on the previous two
or three pitches in the sequence. The results of Experiment 3 are
consistent with the batter expecting fast pitches when they are
ahead in the count and slow pitches when they are behind in the
count. The emphasis placed by the batter on regularities in the
pitch sequence emphasizes why it is so important for a pitcher to
vary the speed and location of pitches (Ryan & House, 1991). In
a companion article, I provided a detailed mathematical model of
the effect of expectations on baseball batting (Gray, 2002).
Effects of Skill Level
Comparisons between athletes of different skill levels are one of
the most effective techniques for understanding the specific skills
that make great athletes. If an elite player has some ability that a
novice does not, then it is presumably important to his or her sport.
In the present study, I was able to compare the data for players
competing at three different levels. The correlational analyses in
Table 6 point to some interesting differences that may be related to
hitting success. The finding that less experienced batters are more
influenced by the prior sequence of pitches may imply that the
degree to which expectations are combined with perceptual infor-
mation is related to skill level (i.e., good hitters are better at
tempering their expectations with perceptual cues). Combining this
with the finding that more experienced batters appear to make
more use of rotation cues, one can make the general conclusion
that experienced players use more sources of information when
batting. Finally, the finding that basic performance measures (e.g.,
temporal and spatial swing errors) were strongly correlated with
playing level implies that the simulated hitting task used here
required some the same skills used in real baseball.
Limitations and Future Research
It is clear that the task used in the present study was a very
simplified simulation of a real baseball game and lacks some
information sources that may have a significant effect on hitting
performance. One such information source may be the pitcher’s
delivery. It is well-known that a pitcher’s body language can be
used to anticipate the upcoming pitch. For example, consider this
description by elite major league hitter Tony Gywnn: “His arm is
less extended than usual when he throws one kind of pitch, his grip
on the ball is too visible on another” (Will, 1990, p. 33). Another
important limitation of this simulation is that there was no binoc-
ular information available. Binocular cues are important for accu-
rate estimates of TTC (Gray & Regan, 1998). In the optimal hitting
situation in the present study (Experiment 4), the mean batting
average was only .220, suggesting that these missing information
sources are indeed important. It should be emphasized that this
simulation also lacked some variables that may make the hitting
task more difficult. For example, in the present study all pitches
were launched at the same angle, such that slower pitches arrived
at a lower height than did fast pitches. In real baseball, this is
clearly not the case, as pitchers can (and do) throw high change-
ups and low fastballs. I also did not simulate the complex effects
of air resistance and spin (Adair, 1990) on the ball flight that would
make the trajectory of the ball even less predictable from the
perceptual information. Finally, simulations of pitches such as
curveballs, knuckleballs, and sliders were not used. The simulation
parameters used in the present study were chosen to permit for
reasonable success in the hitting task while allowing particular
information sources to be isolated and tested and to allow for direct
comparison with a previous model of hitting (Bahill & Karnavas,
1993). Of course, to fully understand hitting, it will be necessary
for the effects of all these variables to be considered. I plan to
pursue this systematically in future experiments.
Implications for Visually Guided Actions
Why were the batters using cognitive and indirect perceptual
information when direct visual correlates of height and TTC were
available? Is it because hitting is an unusually demanding task that
is beyond the limits of the perceptual–motor system? Or is this a
general, multipurpose strategy for the control of action? The role
of cognitive processing in the control of visually guided action is
often overlooked. In fact, most experiments in this area are delib-
erately designed to remove expectancy and memory effects
through randomization of conditions and through the use of unfa-
miliar objects presented out of context. These highly controlled
laboratory conditions remove many of the regularities that are
available when people perform actions in the real world. For
example, in many situations in which people are required to
intercept an approaching object (e.g., when playing baseball or
football), the object has a familiar size and travels at a predictable
speed. Yet in psychophysical experiments on TTC, it is necessary
to vary object speed and object size over a large range (e.g., Gray
& Regan, 1998). The present findings suggest that it is important
to augment previous psychophysical studies with more realistic
conditions to determine if these other sources of information are
used (see also Tresilian, 1999).
Finally, a review of earlier research on hitting (see the intro-
duction) suggested that baseball batting could be controlled en-
tirely on the basis of direct perceptual variables. Psychophysical
experiments have demonstrated that observers are sensitive to
optical variables such as rate of expansion and angular drop speed,
and theoretical analyses have shown that these variables can be
combined to predict height and time of arrival. However, the
present findings indicate that a baseball swing is not controlled in
1147
VIRTUAL BASEBALL BATTING
this manner. This discrepancy further emphasizes why it is neces-
sary to study perception and action together.
Summary
This study effectively implemented a novel simulation of base-
ball batting to study the perceptual and cognitive information used
during hitting. It was shown that varying the speed from pitch to
pitch leads to large errors in the height of the swing and that batters
use the history of previous pitches, knowledge of the pitch count,
and ball rotation direction to control their swing. These findings
are consistent with the proposal that batters use an estimate of
speed to predict pitch height (Bahill & Karnavas, 1993) and that
this estimate is largely based on expectations generated before the
ball is released. Indeed, Ted Williams seems to be correct: Think-
ing appears to be a major part of effective hitting. Combining
perception and action together in this manner facilitated one of the
first studies to address the important question of how these infor-
mation sources are used to control the swing. Contrary to previous
findings (Hubbard & Seng, 1954), these batters seemed to vary the
duration of their swing from trial to trial, suggesting that this may
be a swing parameter that is controlled on the basis of visual
information.
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Received March 30, 2001
Revision received August 30, 2001
Accepted February 19, 2002 䡲
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