Journal of Experimental Psychology:Human Perception and Performance2001, Vol. 27, No. 2, 395-410

Copyright 2001 by the American Psychological Association, Inc.OO96-1523/Ol/$5.OO DOI: 10.1037//0096-1523.27.2.395

Monocular Optical Constraints on Collision Control

Matthew R. H. Smith, John M. Flach, Scott M. Dittman, and Terry StanardWright State University

A simulated ball-hitting task was used to explore the optical basis for collision control. Ball speed andsize were manipulated in Experiments 1 and 2. Results showed a tendency for participants to respondearlier to slower and larger balls. Early in practice, participants would consistently miss the slowest andlargest balls. Experiments 3 and 4 examined performance as a function of the range of speeds.Performance for identical speeds differed depending on whether the speeds were fastest or slowest withina range. Asymmetric transfer between the 2 ranges of speeds showed that those trained with slow speedswere very successful when tested with a faster range of speeds. Those trained with fast speeds did notdo as well when tested on slower speeds. The pattern of results across 4 experiments suggests thatparticipants were using optical angle and expansion rate as separate degrees of freedom for solving thecollision task.

Humans and other animals are able to perform precisely syn-chronized actions over a range of speeds and with objects ofvarying sizes (e.g., plummeting gannets [Lee & Reddish, 1981],table tennis [Bootsma & van Wieringen, 1990], volleyball [Lee,Young, Reddish, Lough, & Clayton, 1983], cricket [Regan, 1992],and driving a car [Lee, 1976]). In some cases, the precision ofresponse has been estimated to be on the order of ±2 ms (Regan& Vincent, 1995). The tau hypothesis provides one explanation forthis ability. Hoyle (1957) observed that the ratio of the instanta-neous visual angle and the angular expansion rate created by anapproaching object provided a convenient estimate for the time toarrival of that object. Lee (1974, 1976, 1980) introduced the termtau (to indicate time). He proposed that this ratio of optical angleand optical expansion rate was an invariant that allowed direct

Matthew R. H. Smith, John M. Flach, Scott M. Dittman, and TerryStanard, Department of Psychology, Wright State University.

This research was made possible by support from the Air Force Officeof Scientific Research Grant F49620-97-1-0213 and the state of Ohio.M H. Smith (University of Auckland) made several contributions regard-ing the mathematics and physics of the simulated events used in theseexperiments. One reason that we are excited about this research is thedifferent ways that we have discovered for representing the data. The ideaof graphing performance against an explicit representation of the actionconstraints (e.g., the white regions in Figure 9) evolved from discussionswith several other researchers. In particular, discussions with P. J. Stappers(Delft University) stimulated our ideas on data representation. The idea ofan optical state space is a simple generalization of state space representa-tions used by control engineers. However, the extrinsic dimensions (posi-tion and velocity) were replaced by their optical correlates (optical angleand expansion rate). This representation was critical in helping us tovisualize optical margins different from tau or critical expansion rate.Discussions with R. J. Jagacinski (Ohio State University) about the simi-larities of our results to data from manual control studies with quickeneddisplays inspired us to examine our data in this state space.

Correspondence concerning this article should be addressed to John M.Flach, Department of Psychology, 345 Fawcett Hall, Wright State Uni-versity, Dayton, Ohio 45435. Electronic mail may be sent tojohn.flach@wright.edu.

perception of time to contact without intermediate judgments ofobject speed and distance.

Regan and Hamstra (1993) have shown that humans are capableof making perceptual discriminations based on tau (ratio of angleto expansion rate) independent of the expansion rate alone. Theseresults led them to conclude that the human visual pathway con-tains a specialized mechanism that is sensitive to the ratio of angleto expansion rate. They also pointed to work by Wang and Frost(1992) that "found neurons in the pigeon's brain that gave theirmaximum firing response at a constant time before contact with asimulated approaching object even when the size of the stimulus orits velocity was varied widely" (p. 459). Also, Wagner (1982)found that the initiation of deceleration in approach to landing forhouseflies was initiated at a constant tau margin from contact.

Despite the evidence in favor of the tau hypothesis, numerousstudies of human performance show errors that are inconsistentwith a tau strategy. Errors have been shown to be associated withboth the speed of closure and the size of objects. This is inconsis-tent with a tau strategy, because the value of optical tau is invariantover changes in size and speed. Studies that have varied speed ofclosure have shown that participants tend to respond early and thatthey consistently respond earlier to slower events than to fasterevents (Li & Laurent, 1995; McLeod & Ross, 1983; Schiff, Oldak,& Shah, 1992; Sidaway, Fairweather, Sekiya, & McNitt-Gray,1996). Studies that have varied the size of objects have shown thatparticipants respond earlier to larger objects than to smaller objects(Caird & Hancock, 1994; DeLucia, 1991; DeLucia & Warren,1994; Van der Kamp, Savelsbergh, & Smeets, 1997).

The effects of speed and size on judgments about collisions haveled researchers to consider alternate optical variables as the per-ceptual basis for performance. Flach, Smith, and Stanard (in press)found that the patterns in their data could be predicted by assumingthat observers responded on the basis of a critical expansion rate.They showed that results from timing errors in numerous previousstudies (DeLucia & Warren, 1994; Li & Laurent, 1995; Sidaway etal., 1996; Stanard, 1998; Stanard, Flach, Smith, & Warren, 1997;Stanard & Smith, 1998) are consistent with a strategy of respond-ing to the optical expansion rate rather than to tau.

395

396 SMITH, FLACH, DITTMAN, AND STANARD

Recently, Kerzel, Hecht, and Kim (1998) also found evidenceconsistent with the use of expansion rate in judgments aboutcollisions. They evaluated judgments of time to passage as afunction of the angular position between an object (that a movingobserver is passing) and the observer's heading. This angle be-tween object and focus of expansion has been referred to as globaltau (Kaiser & Mowafy, 1993; Tresilian, 1991). Kerzel et al. (1998)found that when global tau was brought into conflict with expan-sion rate by varying the lateral offset of objects, observers showeda strong tendency to rely on expansion rate. Note that, geometri-cally, the manipulation of lateral offset of the object has similareffects for global tau as changing the size of an object in a head-oncollision event. Michaels, Zeinstra, and Oudejans (1997) reportedsimilar results for a ball-punching task. They found that perfor-mance in a monocular viewing condition suggested that partici-pants were responding on the basis of a critical expansion rate(they used the term looming).

In the same series of studies in which Regan and Hamstra(1993) found evidence of sensitivity to tau independent of expan-sion rate, they also demonstrated that participants could makediscriminations based on expansion rate alone, independent ofvariations in time to contact. They concluded that "the humanvisual system contains a neural mechanism that encodes the rate ofangular expansion, and that, over the range of parameters studied,this mechanism is independent of time to contact" (Regan &Hamstra, 1993, p. 456). Regan and Hamstra also found that whenboth tau (angle-expansion rate) and expansion rate were simulta-neously available as discrimination cues, both cues were used withdiscrimination thresholds near what would be expected for prob-ability summation.

In summary, there are numerous observations of errors that areinconsistent with a tau strategy. In many cases, these errors areconsistent with the use of optical expansion rate as the primarymonocular source of information. It is also clear that there aremany situations in which the estimation of time to arrival providedby a tau strategy is inadequate for solving the collision controlproblem (e.g., Tresilian, 1997; Wann, 1996). The fact that peopleare successful in solving these control problems suggests thatsources of information other than tau are available. A possibleexample is binocular information such as changing disparity(Cumming & Parker, 1994; Gray & Regan, 1998; Harris, McKee,& Watamaniuk, 1998; Harris & Watamaniuk, 1995; Kohly &Regan, 1999; Regan & Beverley, 1979; Rushton & Wann, 1999).Van der Kamp et al. (1997) reported that size effects on accuracyof ball catching observed early in practice under monocular view-ing conditions did not occur under binocular conditions, in whichtiming was more accurate. With practice, performance in themonocular conditions conVerged to be equivalent to performancein the binocular conditions.

Thus, although Lee's (1976) identification of tau as a potentialoptical invariant stimulated much of the interest in collision con-trol, the early optimism that tau might be the single solution to thecollision problem is suspect. It seems that other sources of infor-mation may be involved, and some have questioned the role of tauin collision judgments (e.g., Wann, 1996). In the experimentsreported here, we manipulated speed and size to determine whetherthe timing errors that have been reported in previous studies(responding too early to slower and larger events, respectively)could be replicated. The'unique feature of these experiments is that

the speed and size manipulations were tested in identical taskcontexts. This common dynamic context may help to determinewhether there is a common source for these errors. In particular,our hypothesis was that these errors reflect the use of expansionrate to control collisions.

The experiments reported here explored the optical constraintson collision control. Figure 1 illustrates the experimental task. Aball approached an observer along a head-on trajectory at a con-stant velocity. This produced a symmetric expansion in the ob-server's field of view. The observer was asked to release a pen-dulum so that it would make contact with the object at a fixed pointon the trajectory. The task was implemented as a computer simu-lation with a monocular display. This task was designed to satisfythe criterion that Tresilian (1997) outlined for situations in whicha tau strategy should be successful.

Experiment 1

Experiment 1 focused on the effects of speed variation onhuman performance in a collision task. Figure 1 shows the virtualtask apparatus. A simulated ball approached the participant on astraight-line collision course at a constant speed. Speed was con-stant within a trial but varied from trial to trial. Speed was animportant independent variable in Experiment 1. A pendulumrested above the observer's eye position, ready to swing away fromthe observer. The participant's task was to release the pendulum atthe appropriate time to create a collision between the sphere on theend of the pendulum and the approaching ball. Once released, thevirtual pendulum swung down through the trajectory of the ball.

The timing goal for this task was to release the pendulum so thatit made contact when it was in a vertical position. We refer to thisas the ideal hitting position, in that it resulted in the ball beingdeflected in the desired direction. Tau specifies the time at whichthe ball will contact the eye (TEYE). To accomplish an ideal hit,one must release the pendulum when the TEYE of the ball is equalto the sum of the time for the pendulum to reach the ideal hittingposition plus the time it would take the ball to travel from the idealhitting position to the eye (see Figure 1). This sum determines the

InitialPosition

FinalPosition

Ideal HittingPosition

Ball at IdealRelease^

-TpENDULUM-

lEYE-

Figure 1. A schematic representation of the experimental task. Theobjective of the task was to release the pendulum so that it collided with anapproaching ball when it (the pendulum) reached its vertical position (idealhitting position). The ideal time to release the pendulum was when TEYE

was equal to the time it took for the pendulum to reach the ideal hittingposition plus the time that it would take the ball to travel from the idealhitting position to the eye. TPFNDUl UM = pendulum time; T E Y E = time atwhich the ball contacts the eye.

OPTICAL CONSTRAINTS FOR COLLISION CONTROL 397

appropriate time for initiating a response. In this experiment, thedistance from the ideal hitting position to the eye was constant.However, ball speed varied from trial to trial. This variation ofspeed meant that the time from the ideal hitting position to the eyealso varied. Thus, correct timing of the pendulum swing requiredknowledge of the pendulum dynamics, and there was a dependenceon ball speed (e.g., see Tresilian, 1991, for a discussion of theimplications of interceptions that do not occur at the eye for the tauhypothesis). These types of dynamic constraints are representativeof many natural collision tasks in which correct timing must reflectthe action dynamics (e.g., neuromuscular lags or braking dynam-ics) and contact is made at a position other than the retina (e.g., atthe extended hand or at the front end of a vehicle). To test for anyinteractions between the perceptual constraints of the event and thetiming requirements arising from the action constraints, we in-cluded two different pendulum dynamics as a second independentvariable. One group used a slow pendulum (400 ms to ideal hittingposition), and the other group used a fast pendulum (200 ms toideal hitting position).

A third independent variable was the texture of the approachingball. A flat black ball was used for half of the trials, and a bit-maptextured ball was used for the other half. Texture was includedbecause it might have implications for the optical information thatwas available to the observer. A flat shaded object (which has onlyone local optical expansion at the edge) may provide less infor-mation about TEYE

t n a n a textured object that shows an entiredistribution of expansion rates. Vincent and Regan (1997) inde-pendently varied the expansion of an object and the texture ele-ments on its surface. Expansion rate of the object was consistentwith TEYE, and expansion of the texture elements varied from zeroto twice the expansion rate of the object. Judgment of TEYE wasvery accurate when the expansion of object and texture elementswere identical (both consistent with actual time to collision).However, slower or faster texture expansion rates led to significanterrors in directions consistent with the texture expansion rate. Thatis, participants overestimated TEYE with the slower texture expan-sion rates and underestimated TEYE with faster texture expansionrates. Thus, texture was included as an independent variable to testwhether performance might be more accurate with textured objectsthan with flat shaded objects.

A final independent variable was session. As discussed earlier,the synchronization of the pendulum with the approach of the ballrequired a relative timing judgment. Thus, it is reasonable toexpect that individuals must become attuned to the action dynam-ics to do the task correctly (see Tresilian, 1997). Also, as the Vander Kamp et al. (1997) study showed, performance under monoc-ular viewing conditions can improve with practice. Thus, partici-pants may need to tune to the appropriate perceptual information.Participants were tested over five sessions. Our analyses focusedon the first and last sessions. The first session is important becauseit should reflect the initial orientation of participants, that is, howparticipants' past experiences with collision events generalize tothe pendulum task (assimilation). The fifth session should showsome degree of accommodation or attunement to the particulardemands of the pendulum task.

In summary, the independent variables for Experiment 1 wereapproach speed, pendulum dynamics, ball surface texture, andsession. Ball size was constant. The dependent variable was theTEYE at which the pendulum was released. Figure 2 illustrates the

predictions for a control strategy based on tau and a controlstrategy based on expansion rate. Figure 2A illustrates the strate-gies in terms of the optical primitives. The coordinates for thisspace are the visual angle and the visual expansion rate. Thecurved trajectories represent the optical changes associated withthe different approach speeds. The outer (shallower) curve repre-sents the slowest approach, and the inner (steeper) curve representsthe fastest approach. The arrows along the trajectories show equalincrements in time. Figure 2B illustrates the strategies in terms ofthe primary independent (speed) and dependent (time of release)variables. The solid gray line within the white regions of bothgraphs represents the ideal release point. In Figure 2A, this linerepresents different combinations of angles and expansion rates. InFigure 2B, this line represents a nearly constant TKYH. Surroundingthese lines (67 ms before and after) are white regions that represent"hits" for each of the two dynamics. The shaded regions representmisses. The curve of the hitting region in Figure 2B reflects thedependence of the travel time from the ideal hitting position to the

A

^%r5^__^ac"

B

| 500:

-? 400j

2(X)

2M) ms

-

5 7.5

— '2 - - -.

10 12.5 15Ball Speed (m/s)

17.5

Tau

CER

Tau

CER

20

Figure 2. A: An optical state space that illustrates the stimulus events forExperiment 1 as a function of optical angle and optical expansion rate. Thecurved trajectories with arrows represent the paths for balls moving atseven different speeds. The arrows are spaced at 100 ms along the flightpath. The white regions represent hit zones for the two pendulums. If thependulum is released when the ball is in this zone, the pendulum will makecontact with the ball. The dark curves within the hit zones represent perfecthits. If the pendulum is released when the ball trajectory intersects thesecurves, a perfect hit will result. B: Hit zones as a function of the indepen-dent variable (ball speed) and the dependent variable (TnYF_). The small-dashed lines in both the top and bottom sections represent an appropriatetau margin for hitting the balls. Medium-dashed lines represent a criticalexpansion rate (CER) that would be appropriate for the range of speeds.TEYE = time at which the ball contacts the eye. deg = degrees.

398 SMITH, FLACH, DITTMAN, AND STANARD

eye on the sp>eed of the ball. This time is larger for slow balls. Notethat the relation between position in the spaces and timing of swing(too early or too late) is different in Figure 2A and Figure 2B. InFigure 2A, the shaded regions below the hit regions representreleases that were too early (i.e., the TEYE at release was largerthan the ideal TEYE). Shaded regions above the hit region representlate releases. In Figure 2B, the shaded regions above the hitregions represent releases that were too early. Shaded regionsbelow the hit region represent releases that were too late.

The small-dashed lines represent predictions for a tau strategy.In the optical state space (Figure 2A), the tau strategy predicts adiagonal line with slope equal to the inverse of the critical taumargin and a zero intercept (e.g., see Wagner, 1982):

6(1)

The medium-dashed lines represent the predictions for an ex-pansion rate strategy. The expansion rate strategy predicts a hor-izontal line (constant expansion rate) in optical state space (Figure

2A):

0 = k = Critical Expansion Rate. (2)

Both strategies (tau and expansion rate) predict linear margins inoptical state space. Tau predicts a line with zero intercept, andexpansion rate predicts a line with zero slope. Thus, an importanttest of these strategies is whether the slopes or intercepts aredifferent from zero. The critical values for the margins shown inFigure 2A were selected to provide predictions that matched theideal release time for the median value of speed used in thisexperiment (12.5 m/s). Note that a critical value strategy is invitedbecause of the discrete nature of the control (button press to releasependulum).

Figure 2B shows the projections of the linear margins onto theindependent (speed) and dependent (TEYE) variables. Note that thetau strategy predicts responses that are late for slow balls and earlyfor fast balls but generally within the hit region for the full rangeof speeds. Thus, a critical tau strategy would be a good (but notperfect) solution for this task. The expansion rate strategy, how-ever, predicts responses that are much too early for the slowerspeeds (missing the slowest speeds for the slow pendulum) and toolate for fast speeds. Thus, an expansion rate strategy is not such agood solution to this task. The bias of the task in favor of a taustrategy was intentional. Because the research was motivated byskepticism about tau, this was a conservative strategy for testingour hypothesis about expansion rate as an optical primitive.

Method

Participants. Eight volunteers were recruited from graduate and un-dergraduate psychology classes. The minimum corrected vision require-ment in each eye was 20/30. Four participants were assigned to eachpendulum dynamic.

Apparatus. We generated real time displays that simulated a ballapproaching a pendulum using a Silicon Graphics Deskside Onyx WithReality Engine and Open GL and Borland C + + software. Displays werepresented on a 48-bit color Silicon monitor with a refresh rate of 60 Hz.Displays had a fixed resolution of 1,280 (horizontal) X 1,024 (vertical) andwere presented at a rate of 60 frames per second. The screen subtended a

visual angle of approximately 48° horizontal X 39° vertical when viewedmonocularly from a distance (fixed by a chin rest) of 40 cm.

Virtual task layout. A pendulum with a sphere (radius: 5 cm) wassimulated to be 50 cm from the center of the pendulum swing (see Figure1). The center of the pendulum was located 50 cm above and 60 cm in frontof the participant's eye position. When the space bar was pressed, thependulum swung away from the viewer's eye position (its angle followinga purely sinusoidal function of time). The pendulum accelerated from theinitial position and reached peak velocity at its lowest point in the swing.From there, the pendulum began to decelerate, reaching zero speed at thetop of the swing (on the opposite side of the initial position). It remainedin that position for the duration of the trial. Participant eye height (150 cm)was fixed at the virtual horizon, which was the same height as the centerof an oncoming spherical ball and the center of the pendulum sphere whenthe pendulum was in the ideal hitting position. The pendulum and ball werecontained inside a shaded-gray virtual room 3 m tall X 4 m wide X 100 mlong, with a black line on the back wall at the height of the horizon. Ballswere 5 cm in diameter, beginning 25 m away from the eye position. Thiscreated an initial visual angle of 0.114°.

Procedure. Participants were tested in five hourly sessions. A sessionconsisted of 10 blocks. A block contained 70 trials that reflected fiverepetitions of the factorial crossing of speed (seven levels) and texture (twolevels). Blocks 2-10 were averaged over a session for each individualparticipant. The first blocks of these sessions were considered to representan attunement period allowing the participants to become familiar with thependulum response dynamics.

The trials were self-paced. The participant initiated a trial by pressing theenter key. The ball began a constant speed approach in a straight line. Thependulum could be released by pressing the space bar. Participants wereinstructed that the goal was to release the pendulum so that its spherewould collide with the oncoming ball directly below the center of thependulum, propelling the ball back toward its point of origin. If thependulum completely missed the ball (i.e., it was released too early or toolate), the ball continued toward the eye until it filled the screen. If thependulum hit the ball, it could deflect straight back toward the horizon(perfect hit) or deflect downward if the swing was early or late withinbounds (timing error < 67 ms). Table 1 illustrates the deflection logic.Early and late hits of the same timing error magnitude produced identicalresults. The angle of deflection provided immediate qualitative feedbackthat participants could use to adapt to the task demands and improveperformance. On completion of each block, participants were providedwith statistics summarizing their performance on the block, includingnumber of hits, percentage of hits, and root mean square (RMS) timingerror.

Design. A 2 X 2 X 7 X 2 mixed factorial design was used in eachexperiment. Pendulum time, the amount of time it took for the pendulumto reach the ideal hitting position, was a between-subjects variable (seeFigure 1). Pendulum times were 200 and 400 ms. Texture was a within-subject variable. The approaching spherical balls had two types of surface

Table 1Deflection Angle as a Function of Absolute Valueof Timing Error

Deflection angle(degrees)

180166153139126

0

Frames

0111|2||3||4|

>|4|

Timing error

Milliseconds

0

>

16.7|33.350.066.766.7

OPTICAL CONSTRAINTS FOR COLLISION CONTROL 399

texture: a homogeneous black surface (black) and a texture-mapped surface(texture). Two levels of texture were crossed with seven levels of ballspeed (20, 17.5, 15, 12.5, 10, 7.5, and 5 m/s). The order of texture type andspeed was random across trials. Finally, data were analyzed for twosessions: early (Session 1) and late (Session 5) in practice. The TEYE ofeach response was recorded as the dependent measure.

Results

The data for the two different pendulum dynamics were ana-lyzed separately in 2 X 7 X 2 (Texture X Speed X Session)repeated measures analyses of variance (ANOVAs). The reasonfor the separate analyses was that we did not believe it would bemeaningful to average data over the two different dynamic taskconstraints (slow and fast pendulums).

Slow pendulum dynamic (400 ms). Figure 3 (top) shows thesignificant interaction between speed and session, F(6, 18)= 18.22, p < .0001, TJ2 = .098. Each point in the graph is anaverage over the 4 participants. The error bars show plus-minusone standard error of the mean. Note that the pattern seen in the toppanel of Figure 3 is consistent with the pattern for each individualparticipant (it is not an artifact of averaging). Performance on thefirst session (open circles) showed good agreement with the pre-dictions of a critical expansion rate strategy (medium-dashed

Expansion Rate

Session ]Session 5

400 msT

5 7.5 10 12.5 15 17.5 20Ball Speed (m/s)

Figure 3. Top: The significant interaction between speed and session forthe 400-ms pendulum from Experiment 1. Bottom: The significant inter-action between speed and session for the 200-ms pendulum. The small-dashed lines illustrate the best fit for a tau strategy. The medium-dashedlines illustrate the best fit for an expansion rate strategy. The solid linesillustrate the best fit for a linear combination of angle and expansion rate(linear margin in optical state space). TEYE

= time at which the ballcontacts the eye; TPENDUI UM = pendulum time. Error bars indicate stan-dard errors of th'e mean.

lines). Participants tended to swing early on the slow trials, as thismodel predicted. The slowest speeds were consistently missed.

Performance on the fifth session (solid circles) showed thatparticipants had improved with practice and now were consistentlyhitting the ball at all speeds. Performance in this session was notin complete agreement with either the tau or expansion rate strat-egy. There remained a trend in which participants continued torespond earlier to slower balls than faster balls (inconsistent witha tau strategy). However, participants were not swinging so earlythat they missed the ball on slow trials, as would be expected in anexpansion rate strategy.

The data for the fifth session could be fit with a linear margin inoptical state space:

= a + bd, (3)

where a and b correspond to the intercept and slope of the opticalmargin. Note that a would be zero for a tau strategy (Equation 1)and that b would be zero for an expansion rate strategy (Equation2). Figure 4 illustrates linear fits to data from the slow pendulum.The linear fits for Session 5 data are illustrated in Figure 3 as solidcurves. In Session 1, the data were best fit with a near horizontalline, consistent with an expansion rate strategy (i.e., zero slope).However, in Session 5, the best fitting line had a positive slope andintercept. The slopes and intercepts for linear fits to the data inoptical state space are given in Table 2. For Session 1, there wasa significant intercept and a very low correlation between angleand expansion rate (see Table 2). This is consistent with anexpansion rate strategy and is inconsistent with a tau strategy,which would predict a high correlation between expansion rate andangle. For Session 5, there was a high correlation between expan-sion rate and angle, indicating that performance was a joint func-tion of these two optical primitives. However, the intercept wasstill significantly different from zero. This suggests that the com-bination of angle and expansion rate was different than what wouldbe expected from a tau strategy. In both Session 1 and Session 5,the data seemed to conform to a linear threshold in optical statespace (see RMS errors in Table 2 and the solid curves in Figure 3).

There was a main effect of ball speed, accounting for a largeamount of the variance, F(6, 18) = 152.65, p < .0001, rf = .724.A tendency to respond too early to slower balls is clearly evidentin the top section of Figure 3. This is consistent with the hypothesisthat expansion rate was being used to initiate pendulum release.

There was a significant interaction of texture and ball speed,F(6, 18) = 2.76, p < .05, rf = .007. However, the pattern did notsupport our hypothesis that performance with textured balls wouldbe more accurate. The trend to respond too early to slow balls wasslightly more pronounced for the textured balls. Also, there was atendency for participants to respond late to fast, textured balls. Thethree-way interaction among texture, speed, and session was notsignificant, F(6, 18) = 1.09, p > .05, 172 = .003.

Fast pendulum dynamic (200 ms). Figure 3 (bottom) showsthe significant interaction between ball speed and session, F(6,18) = 4.37, p < .01, T)2 = .032. Again, performance on the firstsession (open circles) showed good agreement with the predictionsof a critical expansion rate strategy. Participants tended to swingtoo early on the slow trials, as this model predicted.

Performance on the fifth session (solid circles) showed thatparticipants had improved with practice and now were consistentlyhitting the ball. However, there remained a trend in which partic-

400 SMITH. FLACH, DITTMAN, AND STANARD

r 0 Session[ • Session

i

, . . .

/

/ / /

/ / /V/:

/ / Sm/s /

400 ms T I' PENDDtyM J

Figure 4. The data for Sessions I (open circles) and 5 (solid circles) forthe slow pendulum in Experiment 1, plotted as a function of angle andexpansion rate. The small-dashed line illustrates the best fit for a taustrategy. The medium-dashed line illustrates the best fit for an expansionrate strategy. The solid line illustrates the best fit for a linear combinationof angle and expansion rate (linear margin in optical state space).TP i l;M = pendulum time, deg = degrees.

ipants continued to respond earlier to slower balls than fast balls.The trend was not so large that the slow balls were missed, as theexpansion rate strategy would predict. Performance in Session 5could be modeled as a linear margin in optical state space reflect-

ing a combination of visual angle and expansion rate. The darksolid line in the bottom section of Figure 3 represents this opticalmargin. Table 2 shows the intercepts and slopes for linear fits tothe data in the optical state space. Again, data for Session 1showed a significant intercept and a low correlation betweenexpansion rate and angle. This suggests that participants wereresponding on the basis of expansion rate independently of angle,as predicted by an expansion rate strategy. In Session 5, however,there was a strong correlation between expansion rate and angle.This suggests that participants were responding as a joint functionof both optical primitives. However, the intercept was significantlydifferent from zero, indicating that the combination of opticalprimitives was not the combination predicted by the tau strategy.Modeling the data using linear critical thresholds in optical statespace resulted in low RMS errors. The goodness of this fit isevident by the solid curves in Figure 3.

There was a main effect of ball speed, accounting for 829f of thevariance, F(6, 18) = 106.96,/; < .0001, rf = .820. As can be seenin the bottom panel of Figure 3, participants tended to swing earlierto slower balls than to faster balls. This trend was consistent withan expansion rate strategy.

Again, there was a significant interaction effect of texture andball speed, F(6, 18) = 7.80, p < .0005, rf = .025. Participantstended to respond later to the faster balls when the balls weretextured. This trend was again not consistent with our hypothesis

Table 2Slopes, Intercepts, R2 Values, and Root Mean Square (RMS) Error forthe Best Fitting Optical Margin

Experiment

Experiment 1 (speed)400 ms

Session 1Session 5

200 msSession 1Session 5

Experiment 2 (size)400 ms

Session 1Session 5

200 msSession 1Session 5

Experiment 3 (speed)Slow range

Session 1Session 5

Fast rangeSession 1Session 5

Experiment 4 (transfer)Slow range: transferFast range: transfer

Slope

M

0.0621.271

0.5321.599

0.8561.714

0.8592.496

1.5231.829

9.928-1.257

0.9002.102

Linear

SE

0.4990.145"

0.5970.237h

0.279"0.076"

0.5730.137"

0.102"0.024b

1.0061'1.398

0.168b

0.063"

parameter

Intercept

M

0.9590.381

2.6111.726

0.6870.218

2.4211.053

0.2760.218

-2.8421.276

0.5410.085

SE

0.262h

0.084"

0.562b

0.234"

0.164"0.048"

0.553".0145"

0.074"0.020"

0.392"0.512

0.108"0.025"

R2

.003

.939

.137

.901

.653

.990

.310

.985

.978

.998

.951

.139

.852

.996

RMS

error(ms)

32.7321.54

13.548.27

24.5415.97

14.847.05

20.916.01

27.5320.87

23.436.19

Note. The R2 column describes the correlation between the expansion rate and the angle at the moment ofrelease. The RMS error column evaluates the goodness of fit for a linear margin in optical state space." Significant in a two-tailed t test with an alpha of .05. " Significant in a two-tailed t test with an alpha of .01.

OPTICAL CONSTRAINTS FOR COLLISION CONTROL 401

about texture. The three-way interaction among speed, texture, andsession was not significant, F(6, 18) = 0.42, p > .05, TJ2 = .001.

Discussion

The data from Experiment 1 show a very clear and reliabletendency for participants to respond earlier (at a larger TEYE) forslower balls. This pattern is consistent with previous research (Li& Laurent, 1995; McLeod & Ross, 1983; Schiff et al., 1992;Sidaway et al., 1996). Although average data are presented in thefigures, the trends for each individual participant were consistentwith the patterns seen in Figures 3 and 4.

The tendency to respond early to the slow balls was particularlystriking for Session 1 (Figure 3). In fact, participants consistentlymissed the slow balls. The pattern seen in Session 1 was consistentwith the predictions of an expansion rate strategy. According tothis strategy, participants should release the pendulum when theoptical expansion rate for the ball reaches a critical margin. Theparticular critical value was dictated by the timing constraintsarising from the pendulum swing time and the range of ball speeds.

The tendency to respond too early for slower balls remainedclearly evident in the data from Session 5. However, the magnitudeof this tendency was much lower than would be predicted by asimple expansion rate strategy. By Session 5, participants wereconsistently making contact with the slow balls, and the variabilityin responding to these balls was reduced. Clearly, participantswere learning to better synchronize their actions with the demandsof the task. This ability is consistent with the predictions of a taustrategy and is consistent with previous studies that have shownprecise control of timing. However, there are many alternativehypotheses that could account for this ability. Information obtainedin participant debriefings suggests one possible alternative. Anumber of participants acknowledged the difficulty of hitting theslow balls. These participants reported that they had to learn to"hold back" or "wait" on the slow balls. Such comments suggestthat they continued to use a fixed critical expansion rate to triggerthe pendulum release but that they compensated for the mismatchbetween the optics and dynamic constraints by adding a delay tothe response dynamics for the "slow" balls (i.e., waiting). This alsoimplies that participants were able to discriminate "slow" ballsfrom "fast" balls. This suggests that they were using other sourcesof information in combination with expansion rate.

To test the delay hypothesis, we examined the residuals betweenthe response times and the times predicted by an expansion ratestrategy. If there was a constant time difference on the slow trials,this might be evidence for a simple response delay strategy, that is,a strategy in which the participants delayed their response for afixed amount of time beyond where expansion rate reached acritical value. Table 3 shows the mean differences between theresiduals for the two slowest trials. The results show that theresiduals were significantly (p < .05) different in every case.Thus, the data do not support a simple delay strategy. Of course,more complicated delay strategies (e.g., strategies in which thedelay is proportional to speed) cannot be ruled out, but again thiswould imply that information about ball speed (other than expan-sion rate) was being used.

The data could be fit very well with a linear margin in opticalstate space (illustrated by the solid curves in Figure 3). That is,participants appear to have used a combination of optical angle and

Table 3Differences Between Residuals From an Expansion Rate Marginfor the Two Slowest or Largest Balls

Experiment

Experiment 1 (speed)Session 1

200 ms400 ms

MSession 5

200 ms400 ms

MExperiment 2 (size)

Session 1200 ms400 ms

MSession 5

200 ms400 ms

M

M

-28.70-87.74-58.22

-47.70-89.63-68.66

-40.87-77.94-59.41

-56.88-105.00

-80.94

SE

13.9019.9435.34

9.1710.8524.26

25.5923.9930.33

5.4516.1128.02

t

-4.13-8.80-4.66

-10.41-16.53

-8.00

-3.19-6.50-5.54

-20.86-13.03

-8.17

df

337

337

337

337

p"

.05

.01

.01

.002

.002

.002

.05

.01

.002

.002

.002

.002

a A two-tailed t test was used in determining p values.

optical expansion rate. Figure 4 illustrates the "fit" of the linearmargin to data from the slow pendulum. In Session 1, the datawere best fit with a near horizontal margin consistent with anexpansion rate strategy. However, in Session 5 the best fitting linehad a positive slope and intercept. The rotation from horizontaltoward a diagonal could reflect a strategy in which increasedattention was given to the angle as a critical source of information.In other words, the rotation may reflect perceptual learning inwhich the perceptual system adapts from a simple (expansion rate)strategy to a more complex strategy that is sensitive to covariationsin both angle and expansion rate. This learning is made possible asa result of the feedback from hits, partial hits, and misses onprevious trials. Note that such learning suggests that participantsare moving toward a tau strategy. Although even in Session 5 theintercepts for the optical threshold were significantly differentfrom zero, the thresholds were moving toward a diagonal line withzero intercept, as the tau strategy would predict. It is certainlypossible that, if more practice were involved, the data would movecloser and closer to the predictions of the tau strategy.

The effects of texture were small but reliable. There was aconsistent tendency to respond later to fast textured balls than tofast black balls. This pattern was not consistent with our hypoth-esis that the distribution of expansion rates available with texturedsurfaces would lead to more precise judgments of TEyE . Onepossible explanation is that the less accurate performance with thetextured balls, particularly fast textured balls, reflected a differencein contrast. The contrast between the background and the blackballs was much higher. Participants reported that the textured ballstended to blend with the background when they were distant(optically small).

Experiment 2

Experiment 2 focused on the effect of object size on perfor-mance in a collision control task. The task used in Experiment 2

402 SMITH, FLACH, DITTMAN, AND STANARD

was identical to that used in Experiment 1. Participants attemptedto hit an approaching ball by releasing a pendulum. In Experi-ment 1, speed varied and ball size remained constant. In Experi-ment 2, this was reversed, with size varying and speed remainingconstant. The two experiments were designed to be as similar aspossible to each other. Some changes, however, were necessary. InExperiment 1, the initial distance was fixed. All objects were of thesame size, maintaining a fixed initial visual angle of 0.114°. InExperiment 2, ball size changed. If the initial distance were fixed,the initial visual angle would systematically vary with ball size,providing information to participants about which task they werecurrently doing. To avoid this, we varied the initial distance so thatthe initial angle was fixed at 0.114°. Larger balls would beginfurther away, whereas smaller balls would begin closer to the eyeposition.

To make meaningful comparisons between the two experiments,it would be useful to have one dimension that quantifies both speedchange and size change, capturing important qualities of each.Smith (1998) suggested that radial travel time, the amount of timeit takes a ball to travel its own radius, captures the consequencesof both ball size and speed for the optic flow of the ball's expand-ing image. To understand how radial travel time determines opticflow, consider the optics of an approaching sphere:

(4)

To rewrite Equation 4 so that 6 is a function of TEYE, o n e c a n

divide the numerator and the denominator of the interior expres-sion by the velocity of approach:

,i r /

0= sin"1 - ; - —rl-x + xl-x

(5)

where xl—x is TEYE and rl—x is radial travel time. (Note thatbecause ball size and speed were constant on a given trial, radialtravel time was a constant; that is, it was not dependent on time.)Equation 5 can be written as

— „; „ - 1 TRADI us

-t- T\ n ii i s ^ 1 R

(6)

where TRAD1US is radial travel time. Differentiating this equationwith respect to time yields

(T RADIUS 'E) \ 'TEYE(2TRADIUS ~*~ TEYE)(7)

used exactly the same seven values of TRADIUS and the same initial6 at the onset of each trial. Even though Experiment 1 varied speedbut not radius and Experiment 2 varied radius but not speed, trialsfor specific radial travel times in Experiment 1 appeared to be

Note that both 6 and 6 can be written as a function of onlyTRADIUS anc l TEYE- This implies that when the radial travel timesof two balls are equivalent, the monocular optic flow of the ballswill be identical, despite differing sizes and approach speeds (seeFigure 5). The three-dimensional problem (time, size, and speed)is redefined as a two-dimensional problem (radial travel time and

TEYE)-

Equations 6 and 7 demonstrate that radial travel time determines Method6 and d as a function of TEYE. Experiment 1 and Experiment 2

Figure 5. Reciprocal relation between size and speed in determining theoptical trajectory of an approaching ball. The larger ball (A) has twice theradius of the smaller ball (B) and twice the speed. Thus, radial travel timeis equivalent for the two balls, and the optical trajectories are identical.That is, the optical angles and expansion rates are identical for every T,.Y1..TEYF = time at which the ball contacts the eye.

identical to trials for equal radial travel times in Experiment 2 (seeTable 4 and Figure 6). The differences between the two types oftrials were discernible only on the basis of what occurred once thependulum was released. This can be seen in Figure 6A. Note thatthe trajectories through optical state space for the balls of differentsizes are identical to the trajectories for the balls in Experiment 1(Figure 2A). However, the hit zones in Figure 2A and Figure 6Aare different. This is because the hit zone is determined by the ballspeed. The balls represented in Figure 6A are all moving at thesame speed.

In addition to size of the ball, Experiment 2 also includedpendulum dynamics, texture, and session as independent variables.These manipulations were identical to Experiment 1. As withExperiment 1, the dependent variable was the TFYli at which thependulum was released. Figure 6 shows the predictions for a taustrategy and critical expansion strategy in regard to the effects ofthe size manipulation on TEYE

a t release, both in terms of theoptical state space and in terms of the independent (size) anddependent (time of release) variables. The logic of this graph is thesame as for earlier graphs. The white regions represent the hitzones for the two pendulum dynamics. Note that in Figure 6B thereis no curvature to these regions as in Experiment 1. This is becausespeed was constant across size conditions. The small-dashed linerepresents the predictions of the tau strategy. Note that this strategypredicts accurate performance for all object sizes. The medium-dashed line represents the predictions for an expansion rate strat-egy. This strategy predicts that participants will respond too earlyfor the larger objects.

Participants. Eight volunteers were recruited from graduate and un-dergraduate psychology classes. The minimum corrected vision require-ment in each eye was 20/30. Four participants were assigned to eachpendulum dynamic.

Apparatus. The apparatus was the same as in Experiment 1.

OPTICAL CONSTRAINTS FOR COLLISION CONTROL 403

Table 4Radii and Speed Levels Dictating an Equivalent Set of Radial

Travel Times in Experiments 1 and 2

Radial traveltime (ms)

10.006.675.004.003.332.862.50

Radius

5555555

Experiment 1

(cm) Speed (m/s)

5.07.5

10.012.515.017.520.0

Experiment 2

Radius (cm)

12.508.346.255.004.163.583.13

Speed (m/s)

12.512.512.512.512.512.512.5

Virtual task layout. The task was identical to the task used in Exper-iment 1. Speed was fixed at the median level of Experiment 1, 12.5 m/s.Also, the initial visual angle was fixed at the same level as in Experiment 1(0.114°). With a fixed speed and initial visual angle, initial TEYE wasdictated by ball size. The seven ball sizes were selected to create the same

A

I 1.5Angle (deg)

CERKR '

400 ms

••

2(K)ms

7 8 9 10 i i "

-

Tau

CER

Tau

12

«KS

Size (cm)

a

Figure 6. A: Optical state space illustrating the stimulus events forExperiment 2 as a function of optical angle and optical expansion rate. Thecurved trajectories with arrows represent the paths for balls of sevendifferent sizes. The arrows are spaced at 100 ms along the flight path. Thewhite regions represent hit zones for the two pendulums. If the pendulumis released when the ball is in this zone, the pendulum will make contactwith the ball. The dark curves within the hit zones represent perfect hits. Ifthe pendulum is released when the ball trajectory intersects these curves, aperfect hit will result. B: Hit zones as a function of the independent variable(ball size) and the dependent variable (TEYE). Small-dashed lines representan appropriate tau margin for hitting the balls. Medium-dashed linesrepresent a critical expansion rate (CER) that would be appropriate for therange of speeds. TEYE = time at which the ball contacts the eye. deg =degrees.

seven initial TEYE values used in Experiment 1. This created ball sizesof 12.5, 8.33, 6.25, 5, 4.17, 3.57, and 3.13 cm.

Procedure. The procedure was identical to that of Experiment 1.Design. A 2 X 2 X 7 X 2 mixed factorial design was used in this

experiment. Pendulum time was a between-subjects variable. The pendu-lum times were 200 and 400 ms. The approaching spherical balls had twotypes of surface texture: a homogeneous black surface (black) and atexture-mapped surface (texture). Two levels of texture were crossed withseven levels of ball size (12.5, 8.33, 6.25, 5, 4.17, 3.57, and 3.13 cm).Finally, data were analyzed for two sessions (early and late in practice).The TEYE of each response was recorded.

Results

The data for the two different pendulum dynamics were ana-lyzed separately in 2 X 7 X 2 repeated measures ANOVAs(Texture X Size X Session).

Slow pendulum dynamic (400 ms). Figure 7 (top) shows thesignificant interaction between ball size and session, F(6,18) = 26.78, p < .0001, rf = .113. In Session 1 (open circles),performance was somewhat consistent with an expansion ratestrategy. Participants consistently responded too early to the largerballs, as this model predicts, although the magnitude of the errorswas not as extreme as predicted by the model.

Performance on the fifth session (solid circles) showed thatparticipants had improved with practice, although they still con-

700 r

>- 500

• Session 5Expansion Rate

Taji

400 ms T

Ball Size (cm)

Figure 7. Top: The significant interaction between size and session forthe 400-ms pendulum from Experiment 2. Bottom: The significant inter-action between size and session for the 200-ms pendulum from Experi-ment 2. Small-dashed lines illustrate the best fit for a tau strategy. Medium-dashed lines illustrate the best fit for an expansion rate strategy. Solid linesillustrate the best fit for a linear combination of angle and expansion rate(linear margin in optical state space). TEYE = time at which the ballcontacts the eye; TPENDULUM = pendulum time. Error bars indicate stan-dard errors of the mean.

404 SMITH, FLACH, DITTMAN, AND STANARD

sistently missed the largest ball and the overall trend was torespond earlier to larger balls. However, the errors for the largesttrials were not as extreme as an expansion rate strategy wouldpredict.

The data from Sessions 1 and 5 were fit with a linear margin instate space (fits are shown as solid curves in Figure 7). The slopesand intercepts for the linear fits are shown in Table 2. Note that,even in Session 1, there was a reasonably strong correlation (R2 =.653) between angle and expansion rate. This suggests that partic-ipants were responding on the basis of a joint function of angle andexpansion rate in this session. The fact that the intercept wassignificantly different from zero is inconsistent with a tau strategy.

There was a main effect of ball size, F(6, 18) = 21.46, p <.0001, T)2 = .657. This effect accounted for a large proportion ofthe variance. As can be seen in Figure 7 (top), participants re-sponded earlier to the larger balls.

There was also a significant interaction between texture and ballsize, F(6, 18) = 13.50, p < .0001, r\2 = .033. Participants tendedto respond later to small textured balls than to small black balls andearlier to large textured balls than to large black balls.

Fast pendulum dynamic (200 ms). Figure 7 (bottom) showsthe significant interaction between ball size and session, F(6,18) = 8.80, p < .0001, T)2 = .070. Performance on the first session(open circles) showed good agreement with the predictions of anexpansion rate strategy. Participants tended to respond too early tothe larger balls, as this strategy predicts.

Performance on the fifth session (solid circles) showed thatparticipants had improved with practice and now were consistentlyhitting the ball. Participants were more successful than the simpleexpansion rate strategy would predict. Yet, the predicted trend torespond earlier to larger balls remained clearly evident. The datacould be fit with a linear margin in optical state space (solid curvesin Figure 7). Table 2 shows the slopes and intercepts for linearfunctions fit to the data in the optical state space. For Session 1, thecorrelation between expansion rate and angle was low, and theintercept was significantly different from zero. This suggests anexpansion rate strategy. For Session 5, the slopes and interceptswere significantly different from zero, and the correlation betweenexpansion rate and angle was quite high. Again, this suggests thatresponses were being initiated on the basis of a joint function ofexpansion rate and angle.

There was a main effect of ball size, accounting for 57% of thevariance, F(6, 18) = 19.84, p < .0001, rf = .569. The data in thebottom section of Figure 7 show a reliable tendency for partici-pants to respond earlier to larger balls. The tau strategy does notexplain this pattern. However, the pattern is consistent with anexpansion rate strategy.

There was a significant interaction of texture and ball size, F(6,18) = 8.26, p < .0005, T/2 = .040. Small textured balls wereresponded to later than were small black balls, and large texturedballs were responded to earlier than large black balls.

Discussion

The data from Experiment 2 show a reliable trend for partici-pants to respond earlier to larger objects. This trend is consistentwith data from previous studies (Caird & Hancock, 1994; DeLu-cia, 1991; DeLucia & Warren, 1994; Michaels et al., 1997; Vander Kamp et al., 1997). For Session 1, the magnitudes of the errors

were consistent with the predictions of an expansion rate strategy.However, as in Experiment 1, participants improved with practiceto a point at which they were performing better than a simpleexpansion rate strategy would predict in Session 5. Again, consis-tent with the tau strategy, participants learned to consistently hitthe balls, regardless of size. However, other strategies (not depen-dent on tau) could account for this success.

Phenomenological reports collected in debriefing participantsindicated that all of the participants were unaware that ball sizewas changing; their impression was that speed was varying. Thus,they talked about "slow" balls and "fast" balls. Note that the largeballs did take longer to arrive at the hitting zone, because theybegan their approach at a greater distance (initial optical angle wasconstant). Again, they reported that they had to learn to "wait" onthe slow balls. That is, their natural impulses resulted in swingsthat were too early. Again, we tested this waiting strategy bycomparing the residuals from a critical expansion rate for the two"slowest" (actually largest and farthest initial position) balls. For asimple delay strategy, the residuals should be the same. As inExperiment 1, the data did not support this hypothesis.

The fact that participants had the impression of different speedsis not surprising, because optically there was no difference in thetrajectories for Experiments 1 and 2. However, it is important tonote that despite their phenomenological reports, the participantslearned to release the pendulum at the appropriate time. Thus,there was dissociation between the phenomenological reports andparticipants' actions. Because the actions were inconsistent withthe phenomenological reports, this might indicate that the actionswere "automatic" or "directly" controlled rather than "controlled"or mediated by higher cognitive processes. As in Experiment 1, thepattern of responses could be modeled with a linear margin inoptical state space (illustrated as the solid curves in Figure 7). Thissuggested that participants attended to both angle and expansionrate.

Figure 8 shows the data from Experiments 1 and 2 plotted as afunction of radial travel time and TEYE. This representation allowsa direct comparison of the effects of speed and the effects of sizeusing common metrics. Note that the patterns are strikingly sim-ilar. The trends due to radial travel time are similar and are closeto the predictions of the expansion rate strategy (shown as a dashedline). Also, the interactions with pendulum dynamic and withsession are comparable for the two experiments even though thetask constraints were different (e.g., the different hitting zones).The similarities for the two experiments suggest that the perfor-mance patterns resulted from a common cause and that this com-mon cause might have been a dominance of critical optical expan-sion rate in determining the pendulum release for Session 1. Alsoof note is the similarity of the performance shift with practice. Thissuggests that the "learning" reflected in both experiments resultedfrom a common attunement process. The shift seems to reflectincreased attention to the covariation of expansion rate and visualangle.

As in Experiment 1, the effects due to texture do not support thehypothesis that "global" properties of a texture distribution mayprovide better information than the single optical margin createdby the edge of the black ball. If anything, performance was betterwith the black balls than with the textured balls. We suspect thatthis was largely due to increased contrast with the background forthe black balls.

OPTICAL CONSTRAINTS FOR COLLISION CONTROL 405

720- Session 1

640-

Figure 8. TEYE measured in Experiment 1 (as a consequence of speedvariations) and Experiment 2 (as a consequence of size variations), plottedas a function of radial travel time (TRAQ^S). Top: Data obtained inSession 1. Bottom: Data obtained in Session 5. The lines represent fits foran expansion rate strategy to the data for speed (dashed lines) and a linearoptical margin strategy to the data for size (solid lines). TEYE = time atwhich the ball contacts the eye.

Experiment 3

Performance in Experiments 1 and 2 suggests that participantswere tuning to critical optical margins that reflected the dynamicconstraints of the pendulum and the range of speeds or sizes of theballs used. For example, performance on the first sessions sug-gested that participants responded at a critical expansion rate thatled to success for the middle and upper speeds (middle and smallersizes) but led to swings that were early for the slowest speeds(bigger balls). Even though the slow balls were missed, the marginchosen appears to have been a good choice given the range ofspeeds. Thus, consistent with Tresilian's (1997) claim that partic-ipants use "any available cues" (p. 1274), the performance indi-cates that participants were using information about the "range" ofspeeds.

Experiment 3 tested the dependence of performance on therange of speeds presented to the participants. Range effects arecommonly observed in motor control studies (e.g., Poulton, 1974).For example, in target acquisition experiments, participants tend toovershoot near targets and undershoot far targets. In other words,the range of target distances presented to participants can biasperformance. Identical targets may yield different responses if therange of targets in which they are imbedded is different. If partic-ipants are flexibly tuning to the constraints of the hitting task, asTresilian (e.g., 1997) has argued, then it is not unreasonable to

expect a range effect with this task. The participants should choosethe optical margin that is most effective for the range of speedspresented. Experiment 3 compared performance for two groups ofparticipants. Each group was presented with a different range ofspeeds such that the fastest speeds for one group were the slowestspeeds for the other group. The hypothesis was that the responsesto the overlapping speeds would be different for the two groups.For the group in which the overlapping speeds are fast relative tothe other speeds, performance should be good, with participantsconsistently hitting the balls. For the group in which the overlap-ping speeds are slow relative to the other speeds, performanceshould be poorer, with participants responding too early to the"slow" balls.

Method

Participants. Eight volunteers were recruited from undergraduate andgraduate psychology courses. The minimum corrected vision requirementfor participation in the study was 20/30. Four of the volunteers wereassigned to each range of ball speed.

Apparatus. The apparatus was the same as in Experiments 1 and 2.Virtual task layout. The task was identical to that used in

Experiment 1.Procedure. The procedure was identical to that of Experiment 1.Design. A 7 X 2 X 5 mixed factorial design was used. Eleven levels

of ball speed were divided into two ranges of seven speeds per range. Thus,there were two conditions. The slow range consisted of balls with speedsof 4, 6, 8, 10, 12, 14, and 16 m/s. The fast range consisted of balls withspeeds of 12, 14, 16, 18, 20, 22, and 24 m/s. The two range conditions hadthree ball speeds (12, 14, and 16 m/s) in common. Both groups used the400-ms pendulum from Experiment 1. Finally, the data were averaged overBlocks 2-10 of each session. Participants were tested over five sessions.The TEYE of each response was the dependent variable.

Results and Discussion

Figure 9 shows TEYE as a function of ball speed for Sessions 1and 5. Circles represent performance with the slow range of speedsand squares represent performance with the fast range. Consistentwith the hypothesis, the two groups responded differently to theoverlapping speeds. For the group in which the overlapping speedswere the slowest of the range, there was a tendency to swing early(at a larger TEYE) and consistently miss the balls. For the group inwhich the overlapping speeds were the fastest in the range, theballs were hit consistently. A 2 (range) X 3 (speed) X 2 (session)mixed-design ANOVA was performed on the overlapping speedsfor Sessions 1 and 5. There was a significant effect of range, F(l,6) = 73.19, p < .001, Tj2 = .528. The fast range group consistentlyresponded at a longer TEYE than the slow range group. Thedirection of this effect is consistent with the assumption that theparticipants were tuning to a critical expansion rate that reflectedthe range of speeds. Thus, the data support our hypothesis thatparticipants use an optical margin strategy that is tuned to theparticular dynamics (in this case, the range of speeds) of a task.Furthermore, this suggests that "errors" with slow approaches donot reflect an absolute constraint (e.g., threshold) of the perceptualmechanism. Rather, these errors seem to reflect attunement tospecific characteristics of the task (i.e., the range of speeds).

There was also a main effect of speed, F(2, 12) = 8.57, p < .01,T}2 = .068. In general, TEYE was larger for the slower speeds.However, this trend was much stronger for the fast group, as

406 SMITH, FLACH, DITTMAN, AND STANARD

Figure 9. TEYE data from Experiment 3, plotted as a function of ballspeed and session. The squares represent data for the group that receivedthe fast range of speeds, and the circles represent data for the group thatreceived the slow range of speeds. TEYE = time at which the ball contactsthe eye. Error bars indicate standard errors of the mean.

reflected in an interaction between group and speed, F(2,12) = 8.76, p < .01, rf = .070. This interaction is illustrated inFigure 9. There was also a main effect for session, F(l, 6) = 39.66,p < .001, rf = .116. Although it appears that much of theimprovement was for the fast group, there were no significantinteractions between session and the other variables.

The solid lines in Figure 9 show fits for a linear margin inoptical state space. Table 2 shows the slopes and intercepts for thecritical margin in optical state space that best fit the data. The slowrange group exhibited significant slopes and intercepts for bothearly and late sessions. Note also the large R2 values indicatingthat, even in Session 1, participants were attending to the covaria-tion of expansion rate and angle. Figure 10 shows the data inoptical state space. The data were consistent with an optical marginthat reflects a linear weighting of expansion rate and angle.

Performance of the fast range group was not fit so easily. Notethat, for the fast range group (squares), the intercept on Session 1was negative, and the slope for Session 2 was negative. Thispattern is not consistent with the performance of the other partic-ipants (from earlier experiments). Note that although the speedsused for the two groups in this experiment were of equal ranges (interms of ball speed), the ranges in optical state space were notequal. The fast speeds resulted in a very narrow range of anglesand expansion rates, especially in comparison with the slowspeeds. It appears that the participants in the fast speed conditionmay have used a pure angle strategy, especially in Session 1.Because the data were so tightly clustered in Session 5, the

parameters of the best fitting line are unreliable for guiding infer-ences about the optical strategy that was being used. It appears thata constant angle strategy was being used.

Experiment 4

Perhaps the tight clustering of the optical primitives for the fastspeed condition in Experiment 3 made it difficult for participantsto relate the covariations of visual angle and expansion rate to thetask of hitting the ball. To test this interpretation, we measuredperformance in a transfer condition. The same participants fromExperiment 3 were tested for one session. The range of speeds wasswitched between the two groups. The participants who had thefast range in Experiment 3 received the slow range in Experi-ment 4, and vice versa. The hypothesis was that there would beasymmetric transfer. The strategy of attending to covariation ofexpansion rate and angle that was apparently adopted by the slowgroup (Figure 10) should lead these participants to perform veryeffectively when tested with the fast range in Experiment 4. On theother hand, the pure angle strategy adopted by the fast group(Figure 10) should not be effective for the slower range of speeds.Thus, the fast group from Experiment 3 should have difficulty withthe slow speeds. This is somewhat counterintuitive. One mightexpect that training in the more difficult task (fast range of speeds)would make hitting the slower balls easier, particularly if perfor-mance was based on time. However, this counterintuitive predic-tion follows logically from the assumption that participants weretuning to a margin in optical state space. Practice with the slow

Session 5

Figure 10. Data from Experiment 3, plotted as a function of optical angleand expansion rate. The squares represent data for the group that receivedthe fast range of speeds, and the circles represent data for the group thatreceived the slow range of speeds, deg = degrees.

OPTICAL CONSTRAINTS FOR COLLISION CONTROL 407

speeds provided a broad range of experience in optical state space,allowing participants to discover a satisfactory optical margin.However, practice with the fast speeds provided such a narrowvariance on both angle and expansion rate that it may have beendifficult for participants to discover an appropriate margin.

Method

Participants. The same 8 volunteers who participated in Experiment 3took part in this experiment.

Apparatus. The apparatus was the same as in Experiments 1 and 2.Virtual task layout. The task was identical to that used in

Experiment 1.Procedure. The procedure was identical to that of Experiment 1.Design. A 7 X 2 mixed factorial design was used. Eleven levels of ball

speed were divided into two ranges of seven speeds per range, as inExperiment 3. The slow range consisted of balls with speeds of 4, 6, 8,10, 12, 14, and 16 m/s. The fast range consisted of balls with speeds of 12,14, 16, 18, 20, 22, and 24 m/s. The two range conditions had three ballspeeds (12, 14, and 16 m/s) in common. The participants who were testedwith the fast range in Experiment 3 were tested with the slow range inExperiment 4, and vice versa. Both groups used the 400-ms pendulum fromExperiment 1. Finally, the data were averaged over Blocks 2-10 of thesingle session. TEYE of each response was recorded for these analyses.

Results and Discussion

Figure 11 shows data for the transfer conditions as a function ofthe optical variables (Figure 11 A) and as a function of the inde-pendent variables (Figure 11B). The circles represent data for thegroup that was trained with the slow range of speeds. Note thatwhen tested with the fast range of speeds, this group exhibitedexcellent performance; the data are aligned with the perfect hitline. Performance in this group for the first exposure to the fastspeeds was superior to performance of the participants who hadfive sessions of practice with that range of speeds (see Figure 9).A 2 X 7 mixed ANOVA compared performance between theSession 5 data for Experiment 3 and Experiment 4 with the fastrange of speeds. There was a main effect for speed, F(6,36) = 34.71, Tj2 = .412. This was the typical trend; slower speedswere responded to earlier (larger TEYE) than faster speeds. Also,there was a significant interaction between group (training groupvs. transfer group) and speed (seven speeds), F(6, 36) = 28.38,p < .001, Tj2 = .337. The group that was trained with the fastspeed (Figure 9) consistently missed the slower balls (by swingingtoo early) and the faster balls (by swinging too late). Even afterfive sessions of practice, this group was still swinging early for theslower speeds in the range. However, the group that was trainedwith the slow spee(ds consistently hit all speeds when tested withthe fast range of speeds in the transfer condition (even though thiswas their first exposure to that range of speeds).

It seems that practice with the slow range of speeds was betterpreparation for hitting the fast range than practice with the fastrange of speeds. This is a very nonintuitive result. Generally, themore similar the transfer and training conditions, the greater thepositive transfer. Our hypothesis to explain this result is that theoptical margin that participants learned for the slow range ofspeeds was effective for hitting the fast range. This can be seenmost clearly in Figure 10. Note that the optical margins that werefit to the data from the slow group (circles) pass through thetrajectory for the fast speeds very near to the perfect hit line. Thus,

-'/ (A)

0.2

Figure 11. Data from Experiment 4, plotted as a function ot optical angleand expansion rate (A) and as a function of the independent variable speed(B). The circles represent data from participants who trained with the slowspeeds and were tested with the fast speeds. The squares represent datafrom participants who trained with the fast speeds and were tested with theslower speeds. TEYE = time at which the ball contacts the eye. deg =degrees.

if participants respond at the same optical margin that they learnedfor the slow speeds, they should be successful in hitting the fasterspeeds. The group that was trained with the fast speeds appearedto have difficulty discovering an appropriate optical strategy be-cause of the narrow range of stimuli in optical state space (squaresin Figure 10).

The squares in Figure 11 show the data for the group trainedwith the fast range of speeds and tested with the slow range. Firstnote that the data show a fairly well-defined margin in optical statespace. Table 2 reports the parameters for this optical margin. Therewas a high correlation between optical angle and expansion rateindicating that participants were considering covariations of thetwo optical primitives. Note that this was not true in training(Figure 10). Although Session 1 of training showed a high corre-lation, the negative intercept and the steepness of the slope lead usto believe that participants were primarily considering angle inde-pendently of expansion rate. This belief is supported by the datafor Session 5 showing a negative slope and a weak associationbetween angle and expansion rate. However, exposure to thebroader range of primitives associated with the slow speeds in thetransfer condition seemed to help this group discover an opticalmargin that resulted in a more universally successful strategy.Note, however, that training experience with the fast speeds re-sulted in negative transfer. That is, this group was not performing

408 SMITH, FLACH, DITTMAN, AND STANARD

as well with the slow range as the group that was trained with thisrange. A 2 X 7 mixed-design ANOVA compared performance ofthe transfer group and the training group (Session 1) for the sevenslow speeds. As with the fast range, there was a significantinteraction of group and speed, F(6, 36) = 11.57, p < .05, TJ2 =.131. The group that was trained with the slow speeds consistentlyhit balls at every speed. The group that was trained with the fastrange of speeds consistently missed the slower balls when testedwith the slower range of speeds. In this case, training with the fastspeeds inhibited the ability to successfully hit the slow speeds.Again, there was the usual main effect of speed in which partici-pants responded earlier to slower speeds, F(6, 36) = 11.57, p <.001, -rf = .456.

In summary, the pattern of asymmetric transfer produced inExperiment 4 is consistent with the assumption that participantswere tuning to a linear margin in optical state space (i.e., aparticular combination of angle and expansion rate). The positivetransfer from training with slow speeds to a test condition involv-ing a fast range of speeds reflects the fact that the optical margin,which was discovered with the slow speeds (i.e., reflecting co-variations of angle and expansion rate), was a very effectivemargin with the fast speeds. On the other hand, the negativetransfer from training with fast speeds to a test condition involvinga slow range of speeds reflects the fact that the optical margin (i.e.,an emphasis on optical angle), which was discovered with the fastspeeds, was not appropriate for the slower range of speeds. Thetransfer and range effects illustrate how participants adapt or tuneto task demands by using event constraints. This tuning allowsgeneralization to similar task environments (positive transfer), butit can cause difficulties when the environment constraints changeso that they are no longer consistent with the control strategy(negative transfer).

In Experiments 3 and 4, the participants were given equal rangeswith respect to speed. However, this did not result in equal rangeswithin the optical state space. Does the asymmetric transfer reflectthe speeds used (slow speeds are better for training), or does theasymmetric transfer reflect the differential variances in opticalstate space (narrow distributions in state space make it difficult todiscover the appropriate optical margins)? Our current hypothesisis that it is the ranges in optical state space that are critical for theattunement process. Using the language of nonlinear dynamics, itis tempting to conclude that the participants trained with the fastspeeds were trapped in a local minimum. We are currently design-ing experiments in which the ranges used in training will bematched with respect to the distributions in optical state space totest this hypothesis.

General Discussion and Conclusions

The first important result is that the errors in which peoplerespond early to slower approaches and larger objects were repli-cated. The data early in practice were consistent with a strategy ofresponding based on expansion rate. With practice, however, theseerrors were greatly reduced, and participants learned to consis-tently make contact with the ball. However, there remained a smalltendency to respond early to slower approaches and larger objects.

The data could be modeled by assuming that participants re-sponded at a linear margin in optical state space. Note that both thetau and expansion rate strategies predict specific linear margins. A

tau strategy predicts a margin with zero intercept, and an expan-sion rate strategy predicts a margin with zero slope. However, ascan be seen in Table 2, linear fits to the data showed significantslopes and intercepts (particularly for later sessions). Thus, al-though tau and expansion rate predicted linear margins with one ofthe parameters constrained (slope or intercept equal to zero), thedata were best fit by margins with two parameters (slope andintercept). This suggests that angle and expansion rate are func-tionally independent degrees of freedom with respect to the colli-sion control task.

It appears that, with practice, participants learned to use amargin that was a joint function of angle and expansion rate. Thisis a tau-like strategy (in that both angles and expansion rates wereinvolved). However, the patterns suggest that tau is not a singleoptical primitive (invariant). Rather, optical angles and expansionrates appear to be separate degrees of freedom that can be usedalone or in combination to solve the collision problem. The tau-like strategy appears to reflect the ability of people to tune to theseindependent degrees of freedom as required by the task. If precisetiming is demanded by the task (as in an interception task), then itis expected that the attunement to angle and expansion rate wouldcorrespond very closely to what would be expected by a taustrategy.

We have long appreciated the benefits (and difficulties) associ-ated with multiple degrees of freedom on the action side of theequation (e.g., Bernstein, 1967; Fowler & Turvey, 1978). How-ever, constructs such as tau (and, more generally, optical invariant)have led us to assume that the perceptual side of the equation ismore constrained. Perhaps this is a reaction to earlier theories thatposited internal computational degrees of freedom (i.e., interme-diate inferences about extrinsic dimensions such as size, distance,speed, and time) that may not be necessary for solving the collisionproblem. The hypothesis of a flexible attunement to angle andexpansion rate offers a solution that is direct (i.e., based in theoptical array without the need to introduce intermediate computa-tional variables) but is also flexible (i.e., allows multiple degrees offreedom). Such a solution might be characterized as "smart" in thesense of Runeson's (1977) "smart mechanism." This flexibilityhelps to explain both successes and failures in solving the collisionproblem and also leads to interesting hypotheses about the devel-opment of perceptual skill (e.g., differences between novices andexperts). Such flexibility is consistent with repeated observationsthat Tresilian has made about collision control and about the logicof experimental inferences with regard to tau (Tresilian, 1990,1991, 1993, 1994, 1997). It is also consistent with recent researchon the kinematic specification of dynamics showing a similarattunement process (Michaels & de Vries, 1998; Runeson, Juslin,& Olsson, 2000).

This flexibility is most evident in Experiments 3 and 4. Thedifferential response to identical speeds as a function of the asso-ciated range of speeds in Experiment 3 suggests that the errorswith slow speeds do not reflect a hard constraint of the perceptualsystem (e.g., a threshold). Rather, it suggests that these errorsreflect a flexible attunement to specific properties of the taskenvironment. The asymmetric transfer in Experiment 4 shows thatthis attunement can result in a skill that generalizes or in a bias thatinterferes with performance in another context. These results sug-gest that the range of speeds should be considered when comparingresults from different collision studies. It also suggests that it is

OPTICAL CONSTRAINTS FOR COLLISION CONTROL 409

possible to bias performance toward different optical primitives(angle, expansion rate, or tau) by selecting a particular range ofspeeds. Research to test this prediction is currently under way.

Recent research on neuronal firing patterns in the visual system(nucleus rotundus) of pigeons also shows some evidence for flex-ibility in responding to a looming event. Sun and Frost (1998)found neurons that were selectively tuned to three different pa-rameters of the looming event: tau, expansion rate, and a thirdparameter that was tuned to a different combination of expansionrate and optical angle (these optical parameters were labeled T, p,and Tj, respectively). The selectivity of these neurons seems tan-talizingly similar to the different "strategic" solutions that werefound in our experiments. Note that r\, like tau, is a joint functionof angle and expansion rate. However, the 17 function is consistentwith an early response to larger objects (consistent with the resultsof Experiment 3). This function was originally reported by Hat-sopoulos, Gabbiani, and Laurent (1995) to model the loomingdetector responses of locusts. Laurent and Gabbiani (1998) com-mented on the importance of the Sun and Frost (1998) study:

The results indicate that the brain reconstructs object approach usingseveral parallel (and possibly serial) computations. Each one providesa different piece of information about the state of the environment,and the animal thus presumably makes an informed decision on thebasis of these different inputs, (p. 262)

In generalizing from the experiments reported in this article toother collision control problems, it is critical to understand that thediscrete pendulum task as implemented here invited (if not de-manded) a critical margin strategy. This was a conscious experi-mental choice to minimize the variance that might result fromdifferent motor control strategies. A consequence of this choice isthat it would be erroneous to conclude, on the basis of thesestudies, that the critical margin strategy was an invariant propertyof how people control collisions. It is expected that, in less con-strained control tasks, different styles of control might beobserved.

By constraining the degrees of freedom on the response, weexcluded many smart solutions to the collision problem. For ex-ample, previous research has shown that the precision of control ina collision or timing task is greater when the actor is allowed freecontrol of the swing than when the action is constrained to a singlebutton press, as in this study (e.g., Bootsma, 1989). Thus, thepresent experiments probably underestimate the ability of theunconstrained perception-action system. We expect that an experttable tennis player will use degrees of freedom of action as well asperceptual degrees of freedom to discover a solution to the task ofhitting the ball. For example, altering the backswing so that theswing takes slightly longer when hitting slow lobs may be a smartsolution to the fact that some optical margins (expansion rate) tendto bias the system toward swinging too early for slow balls. Thiswould be consistent with the fact that the initiation of the back-swing and other aspects of the action are more variable than thecontact point of the swing (e.g., Bootsma & van Wieringen, 1990).Some of this variation in the trajectory may reflect a complemen-tary relation between degrees of freedom on perception and actionthat helps to ensure precise timing at the point of contact. Inessence, variation on the action side (e.g., a slower backswing)might complement variation on the perception side (e.g., underes-timating time to contact for the slow lob; e.g., Smeets & Brenner,

1995). The key point is that the range of possible solutions to thecontrol problem is a joint function of perception and action con-straints (Bootsma, Fayt, Zaal, & Laurent, 1997).

Although it is important not to mistake the action constraintsimposed by the experimental task as reflecting invariant con-straints of the human perception-action system, we do believe thatit is possible to cautiously generalize from our discrete experimen-tal task to less constrained tasks in more natural situations. That is,we would argue that, in continuous control situations (e.g., hittinga baseball), it is unlikely that there would be fewer perceptualdegrees of freedom than in our discrete task. That is, we expectthat careful analysis will show that angle and expansion rate areindependent degrees of freedom and that the combination of thesetwo optical primitives will reflect an ability to adaptively tune tospecific task dynamics (including the range of speeds). Thus, wewould predict that both successes (e.g., hitting a fastball) anderrors (e.g., swinging out in front of a change-up) might reflectattunements to linear margins within the joint optical state space inwhich angle and expansion rate are independent dimensions.

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Received August 16, 1999Revision received February 24, 2000

Accepted July 3, 2000 •