Exp Brain Res (1998) 122467±474 Springer-Verlag 1998 RESEARCH ARTICLE

Exp Brain Res (1998)122:467±474 Springer-Verlag 1998

R E S E A R C H A R T I C L E

Eli Brenner ′Jeroen B.J.Smeets Marc H.E.de Lussanet

Hitting moving targets

Continuous control of the acceleration of the hand on the basis of the target s velocity

Received:5January 1998/Accepted:12May 1998

E.Brenner ()

)′J.B.J.Smeets ′M.H.E.de Lussanet Vakgroep Fysiologie,Erasmus Universiteit Rotterdam,Postbus 1738,3000DR Rotterdam,The Netherlands

e-mail:BRENNER@FYS1.FGG.EUR.NL,Fax:+31-10-4367594

Abstract Previous studies on how we hit moving targets have revealed that the direction in which we move our hand is continuously adjusted on the basis of the target's perceived position,with a delay of about 110ms.In the present study we show that the acceleration of the hand is also under such continuous control.Subjects were in-structed to hit moving targets (running spiders)as quickly as possible with a rod.We found that changing the veloc-ity of the target influenced the speed with which the rod was moved.The influence was noticeable about 200ms after the target's velocity changed.The extent of the in-fluence was consistent with a direct dependence of the ac-celeration of the hand on the target's velocity.We con-clude that the acceleration of the hand is continuously ad-justed on the basis of the speed of the target,with a delay of about 200ms.

Key words Motor control ′Velocity ′Acceleration Arm movement ′Vision ′Human

Introduction

Subjects have a tendency to move their hand more quick-ly towards fast targets than towards slow ones (Bairstow 1987;Bootsma and van Wieringen 1990;Carnahan and McFadyen 1996;van Donkelaar et al.1992;van Don-kelaar and Lee 1994;Li 1996;Savelsbergh et al.1992;Wallace et al.1992).In the only case we know of in which they failed to do so,this can be explained by the hand having moved less far to intercept the faster targets (Chieffi et al.1992).Subjects even move their hand more quickly towards fast targets when explicitly instructed to hit the targets as quickly as possible (Smeets and Brenner 1995).Our explanation is that subjects do not use all the available visual information to predict when and where they will hit the target.Instead,they independently con-trol the direction in which the hand moves and its accel-eration (as suggested for direction and extent by Ghez et al.1997).The direction in which subjects move their hand is based on the perceived position of the target.The hand's acceleration ±and thereby the movement time ±is based on the target's perceived velocity.This separa-tion limits the subjects'options concerning the movement time,because their success in hitting the targets relies on the combination of the two influences (Brenner and Smeets 1996).

The suggestion that subjects do not use all the avail-able visual information when dealing with moving targets is not new.For instance,information about the target's acceleration is either ignored altogether (Lee et al.1983)or at least not fully utilised (Lee et al.1997).Even the perceived velocity appears to be ignored when deter-mining the initial direction in which the hand will move (Bairstow 1987;van Donkelaar et al.1992;Smeets and Brenner 1995).

Subjects may occasionally respond before the visual information has been fully interpreted and modify their movements as the interpretation proceeds (van Donkelaar et al.1992).However,information about target velocity must usually already have been interpreted by the time the hand starts to move,because the perceived target ve-locity influences the velocity of the hand from the start (Bairstow 1987;Smeets and Brenner 1995).It also influ-ences the reaction time (van Donkelaar et al.1992;Port et al.1997;Smeets and Brenner 1995).

We have proposed that visual information about the target's position and velocity are not combined into a sin-gle prediction of when and where the target will be hit.Instead,the perceived position is used to determine the di-rection in which the hand will move and the perceived ve-locity to determine how fast it will move (Brenner and Smeets 1996).These two (simultaneous)mechanisms need not interact until the stage at which actual com-mands for the muscles are generated.

The advantage of separating the visual control of the hand in the proposed manner is that it simplifies the link between the visual information and the controlled aspect

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of the movement.Simplicity is an advantage not only in terms of the required neuronal connectivity but also be-cause computations that need fewer steps,or that involve fewer parameters,can presumably be done faster and more accurately.Speed of computation is important,be-cause short visuo-motor delays can be beneficial when dealing with the unpredictable movements of everyday targets.An emphasis on quick processing of visual infor-mation,however,is only useful if the movement of the hand is under continuous visual control.

The direction in which the hand moves is known to be under the continuous control of the perceived position of the target,with a visuo-motor delay of about110ms (Brenner and Smeets1997;Goodale et al.1986;Prablanc and Martin1992).This continuous control helps compen-sate for errors that arise when the target's displacement is not anticipated correctly(Bairstow1987;Smeets and Brenner1995).If variations in movement time±due to differences in hand acceleration±are also essential for

hitting moving targets(Brenner and Smeets1996),the ac-celeration of the hand should change when the target's ve-locity changes.In the present study we show that the ac-celeration of the hand is indeed under continuous visual control.

Materials and methods

The study consists of a single experiment in which the subjects'task was to hit moving targets with a rod(Fig.1).They were explicitly instructed to do so as quickly as possible.The target always ap-peared on the left side of the screen and was always moving to the right.We manipulated the velocity of the target's rightward mo-tion.

Targets and background

The targets and background were presented on a computer monitor (38 28cm;815 611pixels)that was tilted backwards at an angle of 11 .Images were presented at a rate of120Hz.Liquid-crystal shut-ter spectacles were used to present alternate images to the left and right eyes,in order to make the visible background appear to coin-cide with a transparent,protective screen.The latter was tilted back-wards at an angle of28 to make the movement more comfortable. The distance between the centre of the monitor and the protective screen was about8cm.Predominantly red stimuli were used be-cause the shutter spectacles work best at long wavelengths.

The target was a simulated spider.Its body consisted of three segments with a total length of0.85cm.Eight1.5-cm legsaat-tachedoto the middle segment moved in accordance with the spi-der's simulated velocity.The background was a plane consisting of500lines.The lines were distributed at random within15cm of the centre of the protective screen.Outside of the central20cm (diameter),their intensity faded gradually with their distance from the centre of the screen.The lines were4cm long and were oriented at random within the simulated plane.

The combination of presenting the target and background in front of the computer screen with the aid of shutter spectacles, and not restricting head movements,meant that structures'images had to change their positions on the computer monitor±if they were to appear to remain at the same position on the protective screen±when subjects moved their heads.The positions of the subjects'eyes were therefore taken into account when rendering the images(note that we account for the positions of the eyes in space,not their ori-entations).Measuring the subjects'movements

The positions of the subject's head and hand were recorded at250Hz by a movement-analysis system based on active infrared markers (Optotrak3010;Northern Digital).The markers for measuring move-ments of the hand were attached to a Perspex rod(22cm long,0.9cm radius)with which the subject was to hit the targets.Subjects held the rod between their fingers and thumb as they would hold a pen.The wires attached to the markers were long,thin and flexible so as not to restrain the subject's movements.We describe our data in terms of the position and movement of the hand,although strictly speaking we will always be reporting on the position and movement of the tip of the rod.This position was determined by extrapolation from the measured positions of two markers on the rod's central axis.

Four more markers were attached to the right ear-piece of the shutter spectacles for measuring movements of the head.We deter-mined each eye's position from the positions of these markers and the distance between the subject's eyes.The delay in adapting the visual image to changes in the positions of the eyes was21 3ms (mean and standard deviation).

Synchronisation

Theablueocomponent of the image generated by the computer was used to synchronise the information about the position of the rod(mea-sured at250Hz)with the appearance and change in velocity of the tar-get on the screen(presented at120Hz).This signal never reached the monitor,but was filtered(low-pass;125Hz)and fed to an analogue input channel of the movement-analysis system.In this manner,we were able to determine the moment that the target actually appeared on the screen(blue signal on)and the moment that the change oc-curred(blue signal off)with the4-ms resolution with which the posi-tions of hand and head were determined(Brenner and Smeets1997).

Procedure

The target only appeared if the tip of the rod was less than5cm from theastarting positiono:40cm away from a point20cm below the centre of the protective screen.Instructions on the screen helped subjects place the tip of the rod within the required region.Some time after this was accomplished,the target appeared.The target al-ways appeared at the same position relative to the tip of the rod (20cm above and8cm to the left of its orthogonal projection on the protective screen).Consequently,it did not always appear at the same position on the screen.We imposed no restrictions on how subjects should sit or move during the experiment,except

that Fig.1Schematic view of a subject hitting a running spider with a rod.By presenting different images to the two eyes,with the aid of liquid-crystal shutter spectacles,we could make the spider appear to be running on a screen that protects the computer monitor from the impact of the rod

469 they had to hold the rod at the starting position,without occluding

the markers,to start each trial.

The initial velocity of the spider was either4.5cm/s or7.5cm/s.

At some time within400ms of its appearing,this velocity changed.

The change was either a3cm/s increase or a3cm/s decrease in ve-

locity.Thus,there were four conditions:two initial target velocities,

each combined with either an increase or a decrease in velocity.The

condition and the moment at which the change occurred were deter-

mined at random for each trial.A few trials(less than1%)were dis-

carded because of errors in synchronisation,because the movement

of the hand stopped before hitting the screen,because the subject

missed the target by more than10cm,or because the subject did

not react within750ms of the moment the target appeared.

Subjects received feedback on their performance.The spider was

asquashedoif we considered it to have been hit.This was so if the

centre of the rod was within1.8cm of theacentreoof the spider.

If subjects hit to the left of the spider,the latter ran away to the right.

If they hit to the right,it ran away to the left.If they hit above it,it

ran downwards.If they hit below it,it ran upwards.Subjects could

vaguely see their hand's contour occluding part of the image when

the hand was close to the screen.

Subjects and instructions

Seven subjects took part in the experiments,including the authors. The only special instructions subjects received was that they should hit the targets as fast as they could.Each subject tried to hit between 800and1200spiders,during two or three sessions.Two(non-au-thor)subjects'data were excluded after preliminary analysis,be-cause the final velocity of their hand did not increase systematically with target speed(on the trials in which the target velocity changed within25ms of the target appearing:3±15trials per velocity;see section Comparing the model with the data).We observed system-atic shifts in these subjects'movement time during the sessions, which may have masked the expected influence of target velocity. Alternatively,they may have used a different strategy that does not involve changes in movement time.In either case their data can-not be used to determine whether the acceleration of the hand is un-der continuous control on the basis of target velocity.

Analysis

Figure2shows the velocity of one subject's hand on an arbitrary tri-al.The velocity is shown from the moment the target appeared and was computed by dividing the distance between two consecutive po-sitions of the tip of the rod by the4-ms interval that separates the measurements.To obtain the velocity at the moment that the posi-tion of the tip of the rod was determined,we used the mean of the velocities during the intervals before and after that moment.

As in our previous studies(Smeets and Brenner1995;Brenner and Smeets1997),the rod moved towards the screen with an almost constant acceleration.The final velocity of the hand was rather ar-bitrarily defined as the velocity40ms before the hand stopped on the screen.The graphical representation in Fig.2is not quite correct, because the figure only shows the component of the velocity of the hand that is orthogonal to the screen.This was used to determine the moment the hand stopped on the screen and the reaction time (threshold of0.2m/s),but the values that were used for the further analysis were the final tangential velocity of the hand and the hand's tangential acceleration.

To determine the delay with which changes in target velocity in-fluence the acceleration of the hand,the rod's acceleration was com-puted by dividing the difference between the velocities during the intervals before and after a given moment by the4-ms interval that separates them.The resulting acceleration traces were synchronised with respect to the moment at which the target velocity changed,and then averaged(first within and then across subjects).In order to de-tect a change in the acceleration of the hand,the hand must be mov-ing when the change occurs.If it takes200ms for a change in target velocity to result in a change in the acceleration of the hand(our ini-tial estimate based on Fig.6;see Results),a change in the acceler-ation of the hand can only be detected if the hand is moving200ms after the target's velocity changes.Trials were therefore selected in which the change in target velocity occurred between200and50ms before the reaction time.This ensured that the rod will have started moving but will not yet have reached the screen by the time the re-sponse was expected.The high spatial resolution and large number of trials allowed us to do without any additional filtering,which could influence the time at which a response appears to occur.To examine the mean pattern of the acceleration during a hit,we also averaged the same acceleration traces after synchronising them with respect to the reaction time.

A simple model

A very simple model for the control of the acceleration of the hand was formulated to help interpret the measured final hand velocities. To keep the model simple,all variability was ignored.The only pa-rameters involved are the distance to the screen,the mean reaction time,the visuo-motor delay,and the mean measured value of the fi-nal velocity of the hand for each target velocity on its own(based on trials in which the change occurred within25ms of the target ap-pearing).The model is based on the assumption that the acceleration of the hand is constant from the moment the hand starts to move,and that the acceleration is directly controlled by the velocity of the tar-get(with the visuo-motor delay that we determined in the manner described in the preceding paragraph).

Figure3shows a schematic representation of this model.Two trials are shown;during one trial the target moves at a constant ve-locity(dashed line),during the other it increases its velocity some time before the reaction time(solid line).In the former case,the ac-celeration of the hand is constant from the moment it starts to move. In the latter it increases abruptly some time after the change in target velocity.We will henceforth substitute a value of200ms for this visuo-motor delay,in anticipation of the results.An increased accel-eration obviously results in a steeper slope of the velocity and in a faster change of position.As the distance to the screen is constant, a larger acceleration therefore results in a shorter movement time and a higher final velocity.

For each target velocity(i),the value of the acceleration of the hand(a i)was determined from the mean final velocity of the hand (v i)on trials during which the target only moved at that velocity, and the distance to the screen(d).Assuming that the velocity in-creases linearly with time(i.e.constant acceleration),

v i a i?MT and d 12a i?MT2

where MT is the movement time,so that

a i v2i a2

d

Fig.2Velocity of the tip of the rod during a single trial.The veloc-ity is shown from the moment the target appeared(left)until just af-ter the rod hit the screen(right).The final velocity of the hand is de-fined as the velocity40ms before the rod changed its direction with respect to the screen.Subject J.B.

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Having determined the acceleration for each target velocity,and the delay between a change in target velocity and the change in the ac-celeration of the hand,we can predict the final velocity of the hand for any interval between the moment the target appears and the mo-ment its velocity changes.

If the change in target velocity occurs more than 200ms before the reaction time,the final velocity of the hand will be the same as it would have been if the target had immediately been moving at the second velocity.If the change occurs later,then the acceleration of the hand may change during the movement (as shown by the solid lines in Fig.3).This will be so unless the hand reaches the target before its acceleration can change,i.e.unless the change in target velocity takes place less than 200ms before the hand hits the screen.If the acceleration of the hand does change,the final velocity of the hand will depend on how long the hand undergoes each magnitude of acceleration.These durations are determined by the moment of the change (plus the 200-ms delay)and by the distance that the hand has to move to reach the screen (which determines when the move-ment ends;see the hand velocity and position traces in Fig.3).Comparing the model with the data

In order to determine whether the measured data are consistent with this simple model for the control of the acceleration of the hand,we first averaged the five subjects'data.We did not want more weight to be given to subjects with a larger final velocity of the hand or to subjects who responded more vigorously to the differences in target velocity.Each measured final velocity of the hand was therefore first transposed to a normalised value (Smeets and Brenner 1995).This value,the equivalent target velocity ,was determined as fol-lows:First,the mean final velocity of each subject's hand was deter-

mined for each target velocity in isolation.This was achieved by se-lecting the trials in which the change in target velocity occurred less than 25ms after the target appeared.One subject's mean values are shown as circles in Fig.4.Next,a line was fit to these four points.This line is an estimate of the relationship between the target's ve-locity and the final velocity of the subject's hand.Such an estimate can be used to find the target velocity for which any given final ve-locity of the hand would be expected,as shown by the dotted line in Fig.4.We call this target velocity the equivalent target velocity,be-cause it is the single velocity for which we would expect the subject in question to have the same final velocity of the hand as for the pre-sented combination of two target velocities.

Each measured final velocity of the hand was atranslatedointo an equivalent target velocity.The equivalent target velocities for each condition,and each delay between the target appearing and its velocity changing,were then averaged across all five subjects.The delays between the target appearing and its velocity changing were random multiples of about 8ms (due to the 120-Hz frame rate of the screen),but have been grouped into 50-ms bins for clarity of presentation.Each bin includes all values lying between 25ms be-fore and 25ms after the indicated value.Statistics

The systematic influence of target velocity on movement time is small in comparison with the variability due to other causes.We therefore confirmed the effects by subjecting the calculated equiva-lent target velocities to an analysis of variance with the factors Sub-ject,Initial target velocity (4.5cm/s or 7.5cm/s),Change in velocity (increase or decrease)and Time of change in velocity (the 9bins).To evaluate whether our model can help understand the data,we also calculated the difference between each equivalent target veloc-ity and the value predicted by the model.These aresidualsowere subjected to an analysis of variance with the same

factors.

Fig.3Schematic representation of the model.Time is from left to right,starting just before the target appears.The dashed lines are for a target moving at a constant velocity.The solid lines are for a target that increases in velocity some time before the reaction time.The different parts show (from top to bottom )the velocity of the tar-get;and the predicted acceleration,velocity and position of the hand.The vertical dotted lines indicate the moments at which (from left to right ):the target appears;its velocity changes;the hand starts to move;the hand responds to the increase in target velocity;and the hand reaches the screen (two lines because this occurs sooner if tar-get velocity

increases)

Fig.4Determining the equivalent target velocity.The circles show the mean final velocity of the subject's hand (and standard errors)when the target moved at a single velocity.The thick line is a fit to these four points.The relationship between target velocity and fi-nal velocity of the hand ±as defined by this line ±is used to translate the measured final velocity of the hand on each individual trial into an equivalent target velocity (dotted lines ).Data for subject J.B.

471

Results

Figure 5shows the distribution of the horizontal positions at which the rod hit the screen.The position is shown both relative to the horizontal position of the rod when the tar-get appeared (Fig.5A)and relative to the spider's position when the rod hit the screen (Fig.5B).It is evident from Fig.5A that the spiders were hit long before they reached

the edge of the screen and that the distance to the screen hardly depended on where the target was hit.It is evident from B that subjects hit most of the spiders:the thick black line below the histogram indicates the range for which the spider is considered to have been hit.

Figure 6shows the final velocity of one subject's hand for the targets that initially moved at 4.5cm/s.The hori-zontal axis shows the time at which the change in target velocity took place.At this time the target either de-creased its velocity to 1.5cm/s (open symbols)or in-creased its velocity to 7.5cm/s (solid symbols).The data in the first bins are for targets that were moving at the new velocity (1.5cm/s or 7.5cm/s)within 25ms of their ap-pearing on the screen.They can therefore be considered to represent targets moving at 1.5cm/s and 7.5cm/s.The difference between the open and solid symbols confirms that this subject hit slower targets more gently.

If the change in target velocity occurs just before the end of the trial,the subject is unable to modify the veloc-ity of his hand,so that the direction of the change is irrel-evant.This appears to be the case in the last bin (at 400ms).In that case both symbols presumably represent the response for 4.5cm/s targets.As was to be expected,this value lies between that for the 1.5cm/s and 7.5cm/s targets.The arrow (RT)indicates the mean reaction time during these trials.It is evident that the change in target velocity influenced the speed of this subject's hand even if it occurred when the hand was already moving.

If the acceleration of the hand is influenced during the movement,it is important to know how long it takes for visual information to influence this acceleration.The data in Fig.6suggest that the visuo-motor delay is about 200ms,because the mean time it took this subject to hit the targets was 564ms,whereas the last moment at which a change in target velocity had an influence on the final velocity of his hand was between 350and 400ms.

Constant acceleration

In order to examine how constant the acceleration is dur-ing the movements,we averaged the acceleration traces

of

Fig.5A,B Distribution of horizontal positions at which the screen was hit.The positions are determined either relative to the rod's ini-tial horizontal position on that trial (A )or relative to the spider's po-sition when the rod hit the screen (B ).The height of the bars indi-cates the number of occurrences.The axis at the top in A indicates about how much further the rod had to move when it hit the screen to the left or right of its initial position,rather than moving straight to the screen.The horizontal black bar in B shows which spiders were considered to have been hit.The spider and the tip of the rod are shown (approximately to scale)for

comparison

Fig.6Mean final velocity of subject J.S.'s hand for various inter-vals between the target appearing and its velocity changing.The ar-row marked RT indicates the mean reaction time during these trials.The bars show the standard errors.Data for the slower initial target velocity.Note that this subject's values for the final velocity of the hand are much smaller than those of subject J.B.(Fig.4)

472

the selected trials (about 40%of all trials)after synchroni-sing them with respect to the reaction time.Separate averages for the four conditions are shown in Fig.7.Of course,the acceleration is not really constant.However,after an initial increase,the mean acceleration did remain at approximately the same level until just before the hand reached the screen.

The differences between the conditions are also as ex-pected.Immediately after the reaction time,the accelera-tion traces are grouped by initial target https://www.wendangku.net/doc/3c12875017.html,ter in the movement,the traces for increases and decreases in velocity gradually diverge.The gradual shift is presum-ably caused by the change in target velocity influencing the acceleration of the hand at different times ±between 0and 150ms after the reaction time ±on different trials.Visuo-motor delay

In order to determine the visuo-motor delay,we averaged the acceleration of the hand after synchronising the accel-eration traces with respect to the moment at which the change in target velocity occurred.Figure 8shows the mean acceleration of the hand for each of the four condi-tions.In accordance with our initial estimate,the traces for targets that changed to a higher velocity (thick lines)and those for targets that changed to a lower velocity (thin lines)diverge about 200ms after the change in target ve-locity.

Because we selected trials on which the hand was not yet moving when the change occurred (see Materials and methods),the acceleration at the moment of the change is

almost zero.As the time from the change progresses,the hand is moving on ever more trials,so that averaging these trials gives rise to a more gradual increase in the mean ac-celeration of the hand than in Fig.7(and masks the depen-dence of acceleration on the initial target velocity).Comparison with the model

The mean equivalent target velocities are shown in Fig.9.The mean data show the same general pattern as the data shown in Fig.6.The analysis of variance revealed signif-icant influences of Initial target velocity (P `0.0001)and Change in velocity (P `0.0001),and a significant interac-tion between Change in velocity and Time of change in velocity (P `0.0001).There were also significant differ-ences between subjects (both main effects and interac-tions)and a significant interaction between Initial target velocity and Time of change in velocity (P =0.04).

In order to determine whether this pattern of results could be due to direct,continuous control of the acceler-ation of the hand (on the basis of the velocity of the tar-get),we compared the data to the prediction of our simple model.The predicted final velocity of the hand,as a func-tion of when the change in velocity took place,is shown by the thick lines.The model predictions have been trans-formed to equivalent target velocities in the same manner as the experimental data.Although the variability in the experimental data is large,the model captures the general pattern quite well:There was a difference between the conditions if the change in target velocity took place early during the trial,and this difference gradually disappeared as the moment that the change in target velocity took place shifted from 100to 300ms after the target appeared.The difference was absent if the change took place later during the

trial.

Fig.7Mean acceleration of the tip of the rod.The acceleration is shown for 100ms before and 150ms after the moment the hand reached a velocity of 0.2m/s.It is evident that the movement actu-ally started at least 50ms before the velocity exceeded this thresh-old.Mean of selected trials (ones in which the change in target ve-locity occurred between 200and 50ms before the hand reached the velocity threshold;546±560trials per condition)from all five sub-

jects

Fig.8Mean acceleration of the tip of the rod.The acceleration is shown for 250ms from the moment the target velocity changed.Mean of selected trials (see Fig.7)from all five subjects

473

That this model could account for much of the gener-al pattern is demonstrated by an analysis of the residual variability (the differences between the equivalent target velocities and the values predicted by the model).An analysis of variance on the residual variability no longer showed significant influences of Initial target velocity (P =0.19)or Change in velocity (P =0.23)and ±most im-portantly ±no longer showed a significant interaction between Change in velocity and Time of change in ve-locity (P =0.61).The significant differences between subjects and the significant interaction between Initial target velocity and Time of change in velocity were of course still present,because the model does not predict these effects.

Discussion

From these results we conclude that the acceleration of the hand is under continuous control.Our data are consis-tent with a direct dependence of the instantaneous accel-eration of the hand on the velocity of the target 200ms earlier.The visuo-motor delay of 200ms is considerably longer than the 110ms that has been found for adjust-ments to the direction of the hand (Brenner and Smeets 1997;Prablanc and Martin 1992).A possible reason for the longer visuo-motor delay is that determining the tar-get's velocity takes more time than detecting its position.The minimal time needed for detecting motion is between about 30and 70ms for the velocities at which our spiders moved (van Doorn and Koenderink 1982).However,the

time it takes to perceive a change in velocity is longer ±presumably because velocity signals are smoothed over time (McKee and Welch 1985;Snowden and Braddick 1991)±and increases with the initial speed (Dzhafarov et al.1993).The longer visuo-motor delay may therefore be inevitable.

In the Introduction we mentioned that separating the visual control of the direction and velocity of the move-ment of the hand could result in faster responses to unex-pected changes in a target's movement.The results of the present study support this suggestion,by showing that it takes almost twice as long for the velocity of the hand to react to a change in target velocity than it does for the direction of the hand to react to a change in target po-sition.Waiting for new velocity information before ad-justing to a change in position would therefore delay one's responses.However,if the longer delay is indeed caused by the time it takes to acquire new velocity infor-mation,then a combined control on the basis of the latest measures of position and velocity,despite them relating to different moments in time,need not increase the visuo-motor delay.

There are clear discrepancies between the model and the data in Fig.9.However,these discrepancies are not systematic.Despite averaging across five subjects,with a total of well over 5000trials,the experimental data do not show the smooth pattern we expected.The reason for this is that the influence of target velocity on the final velocity of the hand is modest in relation to the variability due to other causes.This is not surprising considering the large number of trials that were needed to obtain enough different values for the time at which the change occurred.Factors such as fatigue and responding to feedback (e.g.intentionally reducing the speed of the hit after repeatedly missing the target)certainly contributed to the variability in the final velocity of the hand.

Beside deviations from the model due to variability in the data,the model itself is clearly based on several over-simplifications.For instance,the acceleration of the

sub-

Fig.9A,B Mean normalised velocity of the hand (equivalent target velocity;see Fig.4)for various moments at which the target velocity changed.The dotted lines show the three target velocities involved.The points are means of all the data for all five subjects (and overall standard errors).The thick curves show the prediction of the simple model.A Initial target velocity of 4.5cm/s.B Initial target velocity of 7.5cm/s

474

jects'hands is not really constant from the reaction time until40ms before the hand stops on the screen(Fig.7). Moreover,if velocity signals are smoothed,as suggested above,the transition in the perceived target velocity must be gradual rather than abrupt.Furthermore,individual subjects'strategies appear to differ.Nevertheless this simple model catches the general trends in the data quite well,without any parameter being explicitly fit for this purpose.Thus we consider the agreement between the model and the data sufficiently good to act as support for the hypothesis that the control of the acceleration of the hand by the(perceived)velocity of the target could be very simple.

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