Perceptuo-motor compatibility in pursuit tracking of two-dimensional movements

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Reference

Perceptuo-motor compatibility in pursuit tracking of two-dimensional movements

VIVIANI, Paolo, MOUNOUD, Pierre

Abstract

In a previous article, we reported an investigation of visuomanual pursuit tracking of unpredictable two-dimensional targets. This article extends the study to the tracking of predictable stimuli. In both investigations, the target trajectory was elliptical.The experimental factors were the orientation of the major axis of the ellipses, the period of the movement and the law of motion (natural vs. transformed). The main results of the study are as follows: (a) Satisfactory performance is achieved only in the natural condition. Pursuit movements obey the same constraints observed in spontaneous movements. (b) Predictability affects significantly the average delay between target and pursuit.(c) Each component of the pursuit movements depends on both components of the target. (d) The simple model developed previously to describe performance with unpredictable targets can be generalized to cover the present case as well.

VIVIANI, Paolo, MOUNOUD, Pierre. Perceptuo-motor compatibility in pursuit tracking of two-dimensional movements. Journal of Motor Behavior , 1990, vol. 22, no. 3, p. 407-443

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1 / 1

Journal of Motor Behavior

1990, Vol. 22, NO.3, 407-443

Perceptuomotor Compatibility in Pursuit Tracking of Two-Dimensional Movements

Paolo Vivi ani Pierre Mounoud Department of Psychob iology

Un iversity of Geneva

ABSTRACT. I n a previous article, w e reported a n investigation o f visuomanual pursuit tracking of unpredictable two-dimensional targets. This article extends the study to the tracking of predictable stimuli. In both investigations, the target trajec­

tory was elliptical. The experimental factors we vari ed were the orientation of the major axis of the ellipses (horizontal or vertical), the period ot the movement (9.65

to 1.61 s), and the law of motion (natural vs. transformed). In the natural condition (L), the motion results from the combination of harmonic functions, as would be the case if the target were generated by a human. In the transformed (T) condition, the law of motion departs systematically trom this natural mode I. The main results of the study are as follows: (a) Satisfactory performance is achieved only in the natural condition. Pursuit movements obey the same constraints observed in spontaneous movements. (b) Predictability aftects significantly the average delay between target and pursuit. (c) Each component of the pursuit movements de­

pends on both components of the targets. Thus, two-dimensional tracking gener­

alizes significantly the classical one-dimensional condition. (d) The simple model developed previously to describe performance with unpredictable targets can be generalized to cover the present case as well.

A PR EVIOU S ARTICL E (Viv ian i, Campadel l i, & Mounoud, 1987) re­

ported a study of visuoman ual p ursuit trac king of un pred ictable two­

d i mensionai target s. As argued there, the two- d i mensional version of this task represents a signific ant general izatio n of the paradigm used in al most ali investigation s of the tracking ski l l . In deed, purs u it move­

ments are rather pec u liar among ali h uman motor actions, because they are at the same time under voluntary control and yet c onstrained both spatially an d tempo rally by an external d riv ing in put. I deal ly, per­

fect p u rsuit requ i res that the motor contro I system be u sed as a general-p u rpose p rocessor for rep l icating faithfully any vis ual i nput.

Whatever its intern ai structure, the processor shou ld be able to pre­

vent its inne r wo rkin g f rom man ifest ing itself in the o perating c harac­

ter istic s . In actual fact, of co u rse, suc h an independence of the oper­

atin g character i stics f rom the internai structu re i s never f ul ly ach i eved.

Eye movements rep resent perhaps the most typic al case in which the few poss i b le types of motor patterns ali carry a c lear i m pr int of the

407

P. Viviani & P. Mounoud

un derlyin g c ontro l processes . More generally, however, there is a growing body of evidence to suggest t hat voluntary movements pos­

sess certain unique features t hat sharply d i stingu ish t hem from mo­

tion s that can be produced by mechan ical contraption s . These fea­

tures are l i kely to ref lect both t he inner working of the motor control system and the p hysical propert ies of the effectors. In particular, in free han d movement s, the law of motion depends on t he geometry of the trajectory in a way that c annot be predict ed from bioc hemical con­

siderations alone ( Flash, 1987; Flash & Hogan, 1 985; Lacquan it i, Ter­

zuolo, & Vivian i , 1 983; Soecht ing & Terzuolo, 1 986; Viv i an i & Terzuolo, 1 982; Wann , Nimmo-Smith, & Win g, 1 988). Wit h respect to v i suoman­

ual pursuit, theref or e , t h e generai quest ion can be raised as to the extent to wh ic h the intr insic propert i es of t he control system interfere with the perceived kinemat ical an d geometr ical propert ies of the v isual st i mulus. C l assical one- d i mensional targets (sine waves, steps, ramps, etc . ) are intr ins ically inadequate to tac kle t h i s quest ion , be­

c ause the geometry of t hese movement s i s far too simple. Curved tra­

jector i es in t h e pian e represent, in stead, the simplest s ituation in whic h t h e n ot ion of geometr ical for m can be brought t o bear in t h e context of v i suomanual coor d in at i on .

Several results of the previou s study are summar ized here:

1 . Un predictable targets whose velocity sat i sf ies the con straints present in human movements can b e pursued wit h good accuracy and in a sur pri s ingly consistent manner.

2 . The strategy u sed to pursue t hese target s can b e represented adequately by a s imple veloc ity feedback scheme that inc ludes a cen­

trai p roc essin g delay.

3. Target s whose veloc ity is con st ant- a con d ition that i s nev er ver­

ified in n atural mov ements-can also be trac ked in a consistent man­

nero The feedbac k scheme c an be applied to t h i s c ase too, but t he resu lt s sug gest t h at coping wit h these "unnatural" target s requ ires a c h ange in t h e oper at ing c h aracteristics of t he motor contro l system .

Several quest ions were left unanswered by the exper i ments wit h un­

pred ictab le target s . The first question t hat remains open in the two-

This research was supported by Research Grant 1.073.0.85 of the Fonds Natianal Suisse de la Recherche. The experiments were can­

ducted at the Istituto di Fisiologia dei Centri Nervosi - CNR, Milan, with which the first author was former/y associated. We are grateful ta Dr.

Paola Campadelli for callaborating in the early stages af this praject.

We alsa wish to thank Anatal Feldman, Alan Wing, and an anonymous reviewer for suggesting a number of improvements ta the originaI man­

uscript. Correspandence concerning this article can be sent ta Dr. Pa­

% Viviani, FAPSE, University af Geneva, 24 rue du GeneraI Dufa ur, 1211, Geneva 4, Switzer/and.

408

Perceptuomotor Compatibility

d im ensionai case i s the inf luence of p red ictab i l ity on motor perfo rm­

ance. It h as often been argued (cf. Poulton, 1 974; Wickens, 1 986) that the pu rsu it respon se to perio d i c targets is partly contro l led by a centrai anticipation of their future cou rse . In the one-d imension al case, the law of motion and its derivat ives p rov ide the on ly b as i s fo r such an ant ici­

pato ry component of the respon se. In the case of planar trajecto ries, the d i rection of the movement also changes in an o rderly manner and, therefore, cou l d be antici pated . T h e extent to which this is actually the case is not known; nor do we know whether suc h an anticipation of the changes in d i rection can be carried out independently of a con cu rrent antic i pation about the target kinematics. F inally, we also i gnore the question of whethe r the co rrelation that m i g ht exi st b etween the law of motion and the g eometry of the trajecto ry can itself prov i de gu idance fo r p redicting the future cou rse of the target .

T he secon d i s sue that deserves further attention i s also rel ated to the f act, m entioned above, that vo luntary hand movements obey cer­

tain con straints in the relation between thei r form and kinem atics. Re­

m embe r that the constant velocity con dition explored in the previous study v io l ates these constraints but is sti l i com patib le with a reason ­ ably good trackin g perfo rm ance. The question then arises, whether a mo re d rastic departure of the t arget motion fram the regu larities pre­

sent in spontaneous movements can u ltimately resu lt in a measu rab l e d i s ru ption o f the perfo rmance.

A t hi rd issue concern s the valid ity of the feedb ack scheme u sed to represent the pu rsuit perfo rmance. The range of average v elocities explored in the experiments with un pred ictab l e targets was rather nar­

row ( 1 0 .42-1 6 . 67 cm/s). It is open to question, t herefo re, whether the conclusions reached by applyin g the scheme within suc h a narraw range rem ai n val id when one cons iders the fu l l spectrum of velocities observed in natu ral movements . In particular, the paramount im por­

tance g iv en to the velocity feedback deserves special attention be­

cause, with a few exception s (e . g ., Jagacin ski & Hah, 1 988; M i l ler, Jag acinski, N av alad e , & Johnson , 1 982), models of pu rsuit trackin g tend to emphasize in stead the feedb ack of position information (cf . Pou lton , 1 974 ) .

The fou rth an d f inal question arises a s a resu lt of recent f in d ings on the g eneration of h and trajecto ries. The re is, in fact, the following evi­

dence:

1 . The usefu l wo rkspace fo r mov ements invo lv ing wri st, elbow, an d shoulder jo ints is not homog eneous f rom the po int of v iew of the contro l strategy (F l as h, 1 987; Hollerbac h & Atkeson, 1 986).

2 . At l east one of the param ete rs that are relevant to the descri ption of the movement (t he mechan ical an d ref lex stiffness) depen d s on the position in the wo rkspace and the d i rection of the displacement ( Mussa- Ivaldi, Hogan , & Bizzi, 1 985).

3. When the availab le degrees of f reedom exceed the requi rements.

of the desired end- point trajectory, specific con straints are injected by

409

P. Viviani & P. Mounoud

the motor control system i nto the relation between elbow and shou lder angles (Soec hting, Lacquan iti, & Terzuolo, 1 986; Soec hting & Terzu olo, 1 986). T hese const raints resu lt i n equally specif ic d i stortions of the actual trajectory.

T aken together, th ese f i ndings suggest the possibility that, i n the pu rsuit-tracking task, the hand-arm system may respond differently to target motions i n different direct ions and different positions withi n the workspace. Comp lex targets, suc h as those u sed in the p revious ex­

periments, are not well su ited to expose this possible i nflu ence, be­

cause local efforts may cancel out. The pursu it-tracki ng experiment reported in this article addresses al i fou r issu es evoked above. A closed geometrical form-th e el lipse- provided in ali cases a pre­

dictable and simple target trajectory (Issu e 1 ). Two factors of the ex­

perimental design were related to the kinematics of the target. The first factor was the i nstantaneous modu lations of the veloc ity along the tra­

jectory. I t has been s hown-bot h i n adu lts (Lacquaniti et al . , 1 983) and c h i ld ren (Viviani & Sc h neider, 1 990 )-that a stab le relation exists be­

tween the tangential veloc ity and the radius of cu rvature of elli ptic movement s . A power law p rovides a satisfactory qu antitative desc ri p­

tion of such a relation . I n half of the trials, the veloc ity of the templates satisf ied this i nternai const raint of natu ral movements. In the other half , the constraint was g rossly violated ()ssue 2 ) . The second f actor was ave rage veloc ity of the target over one cyc le. The period of one motion cyc le was varied ove r a 1- 7 range ( I ssu e 3). T he t h i rd experimental factor was the orientation of the major axis of the e l l i pse, whic h def ines both the main d i rection of the movement and the range of covariation of the joint angles ( I ssu e 4 ) .

I n com parison with the one-d i mensional case, t h e analysis of two­

di mensionai pu rsu it tracking requ i res the development of spec ific methodological tool s . Here, these matters wi l l b e dealt with b rief ly and only insofar as t hey are necessary to make the p resentation self­

contained . The reader is referred to the p reviou s artic le (Viviani et al ., 1 987) for a d etai l ed d i scu ssion of these methodolog ical i ssue s .

Methods

Subjects. Ten subjects ( 7 males and 3 females, age rang e: 25-42 years) vol u n teered for the experi ment. They were ali rig ht-handed and had normal or c o rrected-to-normal vision .

Apparatus. The experimental apparatus and the data acq uisit ion were identical to those desc ribed in detai! i n the previou s artic le (Vivi­

ani et al . , 1 987). B riefly, the visu al target was p roduced by a computer­

controlled l aser b eam that was rear-p rojected on a transparent dig itiz ­ ing table. Subjects pu rsued the target with the standard h and-held cu rso r of the tab l e . The i nstantaneous position of both the target and the cu rsor were samp led at 60 Hz, with a nomi n ai accu racy of 0 .025 m m .

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Perceptuomotor Compatibility

Targets. I n ali cases, target trajectories we re e l lipses with the same form (Iabeled G for Geometrie; eccentricity, Ig ⁼ 0 . 9 ; perimeter, P ⁼ 58 .905 c m ) . I n half of the targets (type H) , the major axi s of the e l l ipse was horizontal ; in the other half (type V), it was vertical . Each type of target could fol low two diffe rent l aws of motion [the law of motion is the function I ⁼ l(t) which descri bes the i n c rease in ti me c:if the curvilinear coord inate I along the trajectory].

The f i rst law of motion (to be cal led L for Lissajous) is the one that obtains when the e l l i ptic trajectory is g enerated by composing vecto­

rially two s inusoidal functions as follows:

xg (t) ⁼ Axg . sin(nt) , Yg (t) ⁼ Byg . cos(nt) ,

where Q is the rhyth m of the movement. The eccentricity Lg of G is de­

fined as

Ig ⁼ y^{1 -} (Ayg/Axg)2, if Axg 2: Ayg (type H) , Ig = Y1 (Axg/Ayg)2, if Axg s; Ayg (type V).

The second law of motion (to be called T for transformed) was de­

fined by the fol l owing condition: At any instant, t, the tangential veloc ity along the trajectory must be equal to the tangential velocity that wou ld have , at the same i n stant, a point trac i n g a Lissajous ell ipse (called D for dynamic) whose major axi s was rotated by 90° with respect to that of the actual target. The ti me cou rse of the x and y components of the target in cond itions L and T is contrasted in the uppe r two panels of Figure 1 A. For each speed condition (see below), the cyc le period of L- and T-type targets is the same . Therefore , the averag e tangential speed and its time p rof i l e are identical in the two cases (see the t h i rd pane l i n F i g u re 1 A) . Whereas i n L-type targets, however, the maxi mum tangential velocity occu rs where the radi us of cu rvatu re i s maxim u m , a n d the m i n i m u m tangential veloc ity occu rs where t h e radi us o f cur­

vatu re i s m i n i m u m , the position of the same m inima and maxima along the trajectory is i nte rc hanged i n type-T targets. Thus, both the ti me cou rse and the polar distri bution of the ang u lar velocity (A ^{= V/R) are} different in the two cases (Fig u re 2). The g enerai mathematical pro­

ced u re for generat i n g e l l i ptic trajectories whose law of motion is that of any Lissajous ellipse is descri bed in Ap pendix A.

I n ali cases, p u rsuit movements were recorded for 1 9 . 3 s . The num­

ber of complete movement cycles with i n thi s period cou ld be 2 , 4 , 6, 8 , 1 0 , 1 2 , and 1 4 , which correspond to the followi n g average veloci­

ties: 6 . 1 0, 1 2 . 20, 1 8 . 3 1 , 24.42, 30.52, 36.62, and 42 . 73 cm/s , respec­

tively. One extra cycle was always inserted at the begi nni ng of the trial to allow su bjects to attai n a steady performance. This i n itial cycle was not recorded. Twenty eight d ifferent target types resu lted from the com bination of the two orientations , the two laws of movement, and the seven average velocities.

411

P. Viviani & P. Mounoud

Procedure. Eac h subject fol lowed eac h target type f ive times.

Thus, one complete experiment comprised 140 trial s . To avoid fatigue, we ad min i stered t h e t rials in two sessions. Half of the subjects were p resented with a random permutation of ali targets with 2, 6, and 1 0 cyc les i n t h e fi rst session and of ali targets with 4, 8, 12, and 1 4 cyc les in the second session . The session sequence was inverted for the other half . For b oth sessions, a different permutation was u sed for eac h session . Two of the subject s (the authors) were aware of the ex­

perimental design . Of the 8 other s ubjects to whom no information was pravided , some real ized in the course of the experi ment that the same target was p resented several t i mes. The average interval between rep­

etitions was too l arge for any motor learn ing to occ u r, however. The assignment was intrad uced verbal ly, and we stressed the fact that a good performance consisted of " keeping the target as c lose as pos­

sib le to the c rass- hai r of the c u rsor for as l ong as possib le." Before eac h session , subjects p racticed for a few trials at eac h of the veloci­

ties inc l uded in that session . No constraints were i m posed on either the generai body post u re adopted d u ring the task or the arm seg­

ments involved in t h e tracking movements. At the beginn ing of a trial, subjects placed the c u rsor c rass-hair on the laser spot that ind icated the start ing point of the target. This point was always in the 3-0'c loc k d i rection , but t h e d istance f ram t h e center of t h e table pravided a c l u e t o the target type ( H or V) about t o b e presente d . Two s after an acous­

tic warning signal, the target spot started moving, and the subject had to t rac k the spot for as many cyc les as were p resented in the tr ial . The pace of the p resentation was contral led by the experimenter, and, on the average, trial s were 10 s apart . Short periods of rest were inserted at the subject's request. Subjects could also ask to repeat a tri al if they felt that in the cou rse of the record ing they had lost the necessary concentration .

Resu/ts

Qua/itative ana/ysis. Thraughout this section , we wi l l cont rast the trac k ing performance in the two kinematic cond ition s , L (L issajous) and T (Transformed ) . Thus, we begin by emphasizing the natu re of the d ifference b etween these condition s. As detailed under M ethods, the ave rage speed and time course of the veloc ity p rof i le for a g iven rhyt h m are identical in b oth con d itions. L-type targets resu lt f ram the comb ination of pu re harmon ic f unctions, whereas the hor izontal and vertical components of T-type targets contain h i g h e r harmonics (Fig­

u re 1 A) . N evertheless, both components remain fai rly smooth and , eac h one ind e pendently, wel l within the range of commonly observed spontaneous movements of the han d .

A quantitative d ifference a l s o exists in t h e d istrib ution of t h e angular veloc ity ( Fi g u re 2). The main qual itative d ifferenc e , however, concerns the rel ation b etween the geometry and the kinematics of the targets . It c an be shown (Viviani & C enzato, 1 985) that in L-type targets, the tan-

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Figure 2. Angular velocity for the two types 01 targets. The lower plot compares the time course of the angular velocity for an L-type (thin line) and a T-type (thick line) targe!. The example shown (H-type at the slowest velocity) is the same as that in Figure 1. In the !wo upper plots, the angular velocity is displayed in the lorm 01 polar diagrams, to emphasize its relation to the geometry of the trajectory (AO = angular velocity at point P). Notice that, despite the surface difference be!ween the !wo plots, the average angular velocities are identical in both cases. A visual impression 01 the difference be!ween the profiles 01 tangential velocities is provided by plotting only a lew points 01 the trajectory equally spaced in time.

414

Perceptuomotor Compatibilrry

gential velocity, V(t), is re lated to the instantaneous rad i u s of c u rvature , R(t) , b y the form u l a

2 71" V(t) ⁼ n ^. P2/3F(�g) . R1/3(t) ,

where n ⁼ 271"/T i s the rhythm of the movement, P i s the perimete r of the elli pse, and F is a mild ly varying function of the eccentricity �g.

For any choice of the parameters n, P, and �g, V(t) is an i n c reasing li near function of R 1/3 For T-type targets, however, the V - R1/3 relation decreases monotonously and i s h i g h ly non l i near. Figure 1 B contrasts the two relations for ali tested values of the rhythm.

We di d not search i n d ividuai performances for statistical evidence of motor l earn i n g . Qualitative analysis of the performance ind ices (see be low) fai led to s how any systematic effect across subjects of the or­

der in which the various condition were ad m i n i stered, howeve r. In most cases, a stab le performance was reached after the very first fam i l i ar­

ization trial . Moreover, even the two authors, who had much more prac­

tice with the experi mental conditions then the other subjects , d i d not fare better in negotiating the d ifficult T condition (see later). T h e two experi mental factors that have the largest effect on the tracking per­

formance are the rhythm , n, and the kinematic condition (L vs. T).

Typical exam ples are used to i l lustrate the main q ual itative find i ng s in the s pace (Fig u re 3 ) , and time (Fig u re 4) domain s . Each trace i n Fig u re 3 represents the average trajectory for one cycle, computed from ali the data ava i lable in the indicated condition (for i nstance, in the 8- cycles condition , averages are computed on a 5 x 8 ⁼ 40 cyc les ) . As the rhythm of the target i n c reased , the enti re pursuit trace rotated in the direction of the target movement. A few control trials, recorded i n 3 su bjects , showed conc l u s ively that the angle of rotation was i n ­ verted w h e n the di rection o f the target movement was i nverted. To var­

ious degrees, al i su bjects p resented t h i s pec u l iar effect in cond ition T In condition L, i nstead, a systematic rotation was observed only with targets ori ented horizontal ly, and , on the average, its amplitude was smaller. It is remarkable that , althoug h the angle of rotation at the h i g h ­ est rhyt h m cou ld be a s l arge a s 1 r, no su bject attempted t o compen­

sate for the systematic erro rs that resulted from such rotation . I n deed , no one was ever reported to have noticed these errors. I n condition L, the average peri meter and the eccentricity of the pursuit track matched q u ite accu rately those of the target . In condition T, howeve r, both these parameters dec reased when the rhythm i n c reased . I n most su bjects, pursuit traces of Li ssajous targets were reasonable approx­

imations of an e l l i pse, at ali rhythms. Departures from the i ntended form i n c l uded occasionai di stortions and a tendency at the h i g hest rhythms to flatten out the l east curved portions of the trajectory. These types of d i stortions occu rred al so in the transformed cond itio n , but their ampl itude was larger. In addition , withi n the low-medi um range of rhythms, we al so observed anothe r type of d i stortion that was , in­

stead, q u ite systematic and man ifested itself as an antisymmetric

415

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P. Viviani & P. Mounoud

bending of the most curved portions of the trajectory (see upper right panel in Figure 3).

Gross qualitative difterences between performance in the two con­

ditions emerged in the time domain. Figure 4 shows the polar repre­

sentation of the instantaneous delay, 8, between the target and the pursuit for the same set of experimental parameters of Figure 3. ^The delay is defined and calculated as in the case of random targets (Vi­

viani et al., 1987). For each radiai direction, the indicated quantity, 8, is proportional to the time that elapses between the moments when the target and the pursuit pass through homologous points on the trajec-

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Figure _3. Pursuit trajectories. Data for one representative subject. For the three indicated

rhythms, left and right plots illustrate some of the typical features of the pursuit trajectories in the two experimental conditions. Also shown inset are the quantitative parameters used

to characterize the performances. As a generai rule, pursu it trajectories in the T condition

are more distorted, smaller, and less eccentric than those in condition L. A conspicuous

rotation of the axes of the pursuit ellipse is present in both conditions.

41 6

Perceptuomotor Compatibility

tory. When the polar plot l ies withi n the e l l i ptic reference, as in two of the examples on the right, the pursu it is i n advance with respect to the targ et . Delays were almost constant for ali d i rections and ali rhythms i n cond ition L. Instead , they osc i l l ated considerably and systematical ly i n condition T, the osc i l l ations being rel ated to the velocity profile but not to the orientation of the major axis of the targ et. Obviously, the pec u l iar modu l ation of the tangential velocity prevented the su bjects fram mai ntai n i n g a constant level of accuracy in the execution of the task.

Lissajous transformed

UFT RIGHT LEFT RIGHT

(�

Figure _4. Temporal delays. Data lor the same subject as in Figure _3. For the three indi­

cated rhythms, left and right plots illustrate the typical proliles 01 the instantaneous delay

in the two experimental conditions. The results shown are averages over ali the cycles within a trial and ali the trials lor the same condition . As i ndicated in the lower left panel, the delay at any one point of the trajectory is proportional to the radiai distance between

the target trajectory and the polar plot. Portions 01 the plot laying within the outline 01 the

target correspond to regions ₀₁ the trajectory in which the pursuit is ahead ₀₁ the target

(negative delay) . At ali rhythms and in ali subjects, the delay in condition L is lairly con­

stant across movement cycies. Large modulations ₀₁ the delay appear instead in condi­

tion _T.

417

P . Vlviani & P Mounoud

Finally, the effect of the relation between geometry and kinematics on tracking performance is dramatized by plotting the V ^- R1/3 relation for the pursuit movement. Because individuai plots are qualitatively similar, this relation has been calculated from the average pursuit tra­

jectory over ali subjects and ali cycles of movements within a trial. In condition L (Figure 5) the data points far one complete cycle are al­

most indistinguishable fram the superimposed straight lines that rep­

resent the behavior of the target (cf. Figure 1 B). The agreement is equally good whether the major axis is harizontal (panel A) or vertical (panel B). In condition T, the situation was quite different. Each panel in Figure 6 displays the V ^- R1/3 plot for the indicated rhythms and for the two orientations of the majar axis of the ellipse (A, harizontal; B, vertical) These plots demonstrate large, systematic departures of the pursuit movements fram the V ^- R 1/3 covariation that characterizes the targets The differences depend on both the rhythm of the move­

ment and the orientation of the axis. The results for the highest rhythms (10, 12, and 14 cycles) suggest that subjects modulated the velocity in accardance with the Lissajous model over a large partion of the 1 st and 3rd quadrant, but with a considerable hysteresis. Efforts to comply with the kinematics of the target were only partially successful over small segments of the 2nd and 4th quadrants.

To summarize, these qualitative results demonstrate that, when the relation between the kinematics and the geometry of the target differs fram that which is characteristic of the Lissajous model, the tracking task becomes unusually difficult. A serious deterioration of perform­

ance results, both in the space and temporal domain, but it is much more marked in the latter. In the next section we pravide a quantitative analysis of these findings.

Quantitative ana/ysis. The parameters of the pursuit trace to be considered are the perimeter, P, the angle of ratation, e, the eccentric­

ity, L, the distortion with respect to the elliptic form, O, and the average delay, ù. The perimeter was calculated by integrating numerically the data points of the average complete cycle. e and L were estimated by the corresponding parameters of the least squares best-fitting el­

lipse of the pursuit trace. The distortion was defined as the least mean quadratic error resulting from the fitting routine. Table 1 summarizes the results of a 2 x 2 x 7 (Orientation x Kinematic Condition x

Rhythm) analysis of vari ance (ANOVA) of these quantitative parame­

ters, with 10 replicates (subjects) per celI.

As mentioned in the previous paragraph, the performance parame­

ters and their variations as a function of the rhythm were mostly af­

fected by the kinematic modality. To some extent, however, they were also subject specifico This is particularly obvious for the average delay, ù (Figure 7 ) . The left panel in this figure compares the results of each subject in the two Lissajous conditions (O ⁼ H targets; • ⁼ V tar­

gets); the right panel shows the corresponding results far the trans­

farmed conditions. The botto m lines in each panel show the population

418

� (Cl

60 50 u Cl:) ti> '- 40 E 2 � .(3 30

o a; '" _{� E} Cl:) C>

20 c:: � 10,

J

&? ;;r

Jii'"

12

ti> Cl:) u >­^u E Cl:) E Cl:) 8 6 E

4

- o ^� Cl:) ..c E :::l c::

radius of curvature power one-third (cml

A 14 8 10 6 4 Figure 5. Covariation between lorm and kinematics in condition L. A. Covariation 01 the instantaneous tangenti al velocity with the radius 01 curvature 01 the pursuit trajectory lor the indicated rhythms (H-type ellipses). B. Analogous data lor V-type ellipses. The results shown are averages over ali cycles within a trial, ali trials lor the same condition, and ali subjects. Each cluser 01 points in this plots describes the lour alternations between minimum and maximum values 01 the radius 01 curvature in one cycle 01 movement. Thin straight lines that interpolate the data points represent the theoretical predictions in the case 01 a perfect pursuit (cf. Figure 1). The results demonstrate that, in condition L, the pursuit movement exhibits the same covaria­ tion between lorm and kinematics that is normally observed in spontaneous movements.

-o m

�

�

� I\) o

60 50 40 � 30

� cv � 20 E 8

10 >­

' 'E

0 1

..!2 ^{CV >} � t: CV CD C ca -

6 2 3

8 10 12 14 ..-:

�� 2)\

�

:u $; �. �. I/O :u s: g :J g Q.

Perceptuomotor Compatibilrry TABLE 1

Anatysis 01 the Effects 01 the Experimental Factors on Various Performance Parameters

dr a l p a D

A (1, 9) 45.09* 21.00* 33.52* 56.20* 24.57*

B (1, 9) 20.32* 3.62a 2.66a 31.87* 3,50a

C (6,54) 7 01 * 21,00* 8.85* 10.39* 14.37*

A x B (1,9) 1.55a 0.21a 0.01a 0.12a 2.90a

AxC (6,54) 9.17* 20.68* 21.81 * 7.93* 3.27**

B x _C (6,54) 0.69a 6.75* 1.21 a 6.06* 1.67a

Ax BxC (6,54) 1.69a 1.59a 2.09a 2.07a 0.59a

Note. ANOVA results (F values). Columns, parameters of the performance: _a, average delay; _l, eccentricity; _P, perimeter; _a, rotation; _D, distortion. Rows, main effects and interactions of the experimental factors: A, kinematic condition (Lissajous vs. trans­

formed); B, orientation of the major axis (horizontal vs. vertical); C, rhythm.

* p :::; ^{.001; ** p} :::; ^{.01; a =} nonsignificant.

ave rages. As al ready observed in the c ase of unp red ictable targets (Viv ian i et a l ., 1 987) , the average delay varied as m uc h as 1 00% be­

tween s u bjects. Moreover, the effect of the rhyt hm was som ewhat d if­

ferent among s u bjects. Those su bjects who had long delays also tended to inc rease them at the h i g hest rhythms; this was particu larly obv ious in con d ition T. Other su bjects maintained the delay fai rly con­

stantly throug h out the enti re range of rhythm s . The sign ificance of these d ifferences i s d ramatized by the striking s im i larity between the in d iv i d uai results for the horizontal an d vert ical orientation of the major axis,

Fig u re 8 summ arizes the quantitative results for ali the other descrip­

tive param eters of the performance . Eac h plot com pares the averages ac ross ali su bjects far the ind icated quantity. As in F i g u re 7, the results for the two orien tation s are com pared within the same p lot. Left and rig ht panels refer to cond ition s L and T, respectively. The most d ra­

matic difference between the two kinematic conditions is revealed by the perimeter d ata, whic h, in c ond ition T, show a large dec rease in the s ize of the pu rsuit movement. The g lobal measure of d istortion , O, also conf i rm s a q ual itative d ifference between L- an d T-type targ ets. It fai l s , however, t o resolve t h e d ifferent types of d istortion s doc umented in Fig u re 4 . With in both kinem atic cond ition s , a signif icant d ifference ex­

isted between the amount of rotat i on for the two orientations of the major axi s. As a g enerai rule, the effect of the rhythm on ali m easures of perform ance was g reater in c on d ition T than in condition L. The rhythm interacted with the orientation of the axes only for the purely geom etrie parameters (eccentric ity and rotation ) . Final ly, no signif icant interaction eme rged between or ientation an d kinematics .

421

P. Viviani & P. Mounoud

Modeling the Tracking Performance

Definition of the Model

I n the previous article on pu rsuit t racking of random targets (Viviani et al., 1987), we presented a simple formai scheme that accounts with reasonable accu racy for both the generai and the idiosyncratic fea­

tu res of the tracking performance. Briefly, the scheme proposed earl ier was based on the following ideas. Let T(t) ^and P(t) be two vectors, representing the instantaneous position of the target and .of the p'u rsu it, respectively. Moreover, let dS ⁼ T(t) - P(t) ^and dV ⁼ T(t) - P(t) ^be the displacement and vel ocity errar vectors. The basic intu ition, which we tested using random targets, is that ti me changes of P(t) are driven uniquely by the differences, dS and dV. By analogy with the behavior of a mass coupled to a displacement generator via a viscoelastic link, the acceleration of P(t) is supposed to be a l i near combination of dS and dV. We also hypothesized that the processing of the error signals

2 3 4

-CI) c..> 5 E' 6 CI) ::::l CI)

7 8 9 10

A

lissajous transformed

���---=::�..------'- -�- �� - + ---=-

�-==� ^___ -_�_�-�-=-_-:-_��=-=Z�::-:��

*'

�"� ^__ �'�--." '1\ ^,-O-.^, � ----�

��-=�" c��.:.:c+,:��� ���--�'�-./ *

t {+ ,,, ",�,,(L � � "'"

, t

I I I I I I I

2 4 6 8 10 12 14 2 4 6 8 10 12 14

number of movement cycles

o H

.v

]�

o N

Figure 7. Average delays. Data points (O for horizontal, • for vertical ellipses) represent

the average over ali movement cycles of the instantaneous delay between target and pursuit. Tics on the horizontal axis identify the baseline for the corresponding subject.

Averages and standard deviations across subjects are displayed at the bottom and refer to the indicated baseline. In condition L, no clear trend emerges in the relation between

the delay and the rhythm. In condition T, the delay increases significantly with the rhythm.

Large differences exist among individuai pertormances, however.

422

Perceptuomotor Compatibility

lissajous transformed

.90

*���t9+9'i

* � � � � � \

>- :<::::

.!::"!

.... - oH

c: .85

CV u

u ^CV

.v

.80

100 � � � ^O

,9

<:)

Q � ^{• O}

�

....

-? �

-

....

_{t9 �}

88

+?

O

� , q t � _�

Cl CV

"S

c: 5

-

-

3

::j

� � ^* r � t ^{# * J * U / �}

� 2

c: O

-....

-

.� "O

O 2 4 6 8 10 12 14 2 4 6 8 10 12 14

number of movement cyc/es

Figure 8. Descriptive parameters 01 the performance. Each panel displays the average

and standard deviations. across ali subjects. ₀₁ the indicated parameters lor ali rhythms

and both kinematic conditions. Empty and lilled symbols are relative to H- and V-type

targets. respectively. _Ali parameters are more inlluenced by the rhythm in condition T

than in condition L.

423

P. Viviani & P. Mou noud

is not i n stantaneo u s , as it is in a reai mechanical system , but takes i n stead a ti me, te' that may depend on the su bject.

To encom pass with i n the same scheme also the case of pe riod i c targ ets , w e now general ize t h e model i n two d i rections. Fi rst , g iven t h e pred ictab le natu re o f t h e sti m u l i , w e must take i nto account t h e possi ­ b i l ity that su bjects antici pate the futu re course o f t h e target. Specifi­

cal ly, we assume that the relevant errar vectors , �S and � V, are com­

puted between the actual position and ve loc ity of the p u rsuit (time t) and the position and velocity of the target at some point i n the futu re (time t + tf)' Second , the experi ments have s hown that the orientation of the major axi s of elongation of the pursuit t race undergoes a rotation i n the d i rection of the target movement, the amplitude of which de­

pends on the tempo (see Figu re 3 ) . Thus, we m ust al low for the pos­

s i b i l ity of cross-co u p l i ng between the Cartes ian components of the er­

rar vectors and those of the p u rsuit movement (Todosiev, 1 967): T h e errar s i g n a l s �S and � T are l i nearly combi ned by matrices with non­

ze ro off- d i agonal terms. Formal ly, the p u rsu it movement is then de­

scri bed by the d ifferential vector equation as follows:

. . o .

P(t + te) ⁼ a[T(t + tf) - P(t)] + �[T(t + tf) - P(t)], ( 1 ) where

P(t) == {xp(t),Yp(t)}, T(t) == {xr(t),Yr(t)},

� ==

I

I

With i n the s p i rit of the model , the transformation matrices a and � can be conceptu al ized as a v i rtual stiffness and a vi rtual vi scosity ten­

sor, respectively (al i the components of <X ^and � are normal ized to the u nspecified mass of the system) . If the link between the target and the p u rsuit behaves as a spring/dash pot syste m , it fol l ows that the two tensors m u st be skew-symmetric (<XXy ⁼ <Xyx; �xy ⁼ �yx) (cf Hogan , 1 985) . T h e components of the stiffness tensor i n c l ude the contri bution of centrai and peri pheral effects . Althou g h it is known ( M ussa- I vald i , Hog an , & B i zzi , 1 985) that the latter are d e pendent o n end-point po­

sition with i n the workspace (at least in i sometric cond itions) , we cannot evaluate the re lative weig ht of the two contri butions. T h u s , we w i l l make t h e s i m p l ify i n g ass u m ption that t h e workspace is isotro p i c , w h i c h i m pl i es the fu rther constraints <Xxx ⁼ <Xyy ^and �xx ⁼ �yy'

By definition, the "Iook-ahead" parameter, tf' shou ld be i n d ependent of the rhythm of the movement, whereas , of cou rse , the corres pond ing spatial look-ahead i s p roportional to the instantaneous velocity. By contrast, n ot only is there no reaso n , a priori , to assu me independence of the centrai p rocess i n g time te' buI, as we shall see later, the results

424

Perceptuomotor Compatibility

even sug gest a decrease of tc with target vel ocity. Likewise , several studies have shown that both stiffness and viscosity of actual muscles scale with speed. Nevertheless, we estimated that allowing for a mod­

u l ation of the m odel parameters by the rhythm of the target wou l d have inc reased the deg rees of freedom of the model too much . Con se­

quently, we assu med that a l i parameters depend on ly on the su bject and on the compat i b i l ity condition .

Because the model i s l i n ear, its behaviar is best analyzed i n the frequency domai n . Applying the Laplace transform and rearrang i ng one obtains

s2estc· P ^{= (a} + sp) . (est, . T ^- P),

[s2estc· I + (a + sp) ] . P ^{= est,} . (a + sp) . T, P ⁼ est, . [s2estc . I + (a + SP)]-1 . (a + ps) . T.

The compl ex matrix

r ⁼ est, . [s2estc _{. I +} (a + SP)]-1 ^{. (a + ps) =}

I

l

(fxx ⁼ fyy; fXY ⁼ fyx)'

represents g l obally the visuomanual transformation tensor. Both the diagonal and off-diagonal components of f can be descri bed by the corresponding ampl itude (A) and phase (<\» characteristics. Explicit formu l as far A(fxJ, <\>(fxx), A(fxy)' and <\>(fxy) are deri ved i n Appen­

dix B.

Va!idation or the Mode!

The model has six parameters; Uxx' uxy' I3xx' I3xy' tc' and tj' which are supposed to be i ndependent of the rhythm of the movement. Thus, for any choice of val ues far the parameters, the model yields a fu l l set of predictions about the effects of the rhythm on the motor performances.

Far the pu rpose of val idating the model , however, we concentrate only on fou r quantitative aspects of the pu rsu it track: the perimeter, P; the eccentricity, L; the rotation of the major axis, 0; and the average del ay, ù, between target and pursu it (see Fig u res 7 and 8). Spec ifically, we searched the parameter space for the val ues that si mu ltaneously m i n­

imize the mean square error between the predicted and observed val­

ues of P, L, <\>, ^and ù at al i tested rhythms. The validity of the model is then gauged joi ntly by the size of the resi dual errar and by the dis­

c repancy between mode I and data for other qual itative aspects of the performance that were not taken into account in the search pracess (see later) . It shou l d be st ressed that, althou g h the temporal parame­

ters, tc and tj' seem to have opposite effects on the pursuit accelera­

tion , in fact, no appreciable trade-off can exist between thei r l east squares estimates.

425