Principal components in three-ball cascade juggling

7/30/2019 Principal components in three-ball cascade juggling

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Abstract. To uncover the underlying control structureof three-ball cascade juggling, we studied its spatiotem-

poral properties in detail. Juggling patterns, performedat fast and preferred speeds, were recorded in the frontalplane and subsequently analyzed using principalcomponent analysis and serial correlation techniques.As was expected on theoretical grounds, the principalcomponent analysis revealed that maximally four in-stead of the original six dimensions (3 balls 2 planarcoordinates) are sucient for describing the jugglingdynamics. Juggling speed was shown to aect thenumber of dimensions (four for the fast condition, twofor the preferred condition) as well as the smoothness ofthe time evolution of the eigenvectors of the principalcomponent analysis, particularly around the catches.Contrary to the throws and the zeniths, and regardlessof juggling speed, consecutive catches of the same handshowed a markedly negative lag-one serial correlation,suggesting that the catches are timed so as to preservethe temporal integrity of the juggling act.

1 Introduction

Randomness is an inherent feature of biological systems.The combination of deterministic processes with randomuctuations results in a stochastic system that may be

measured by its degree of variability. Variability in thissense is a hallmark property of human motor perfor-mance. Depending on one's theoretical perspective, ithas been perceived as a curse or as a blessing. On the onehand, additional stochasticity can limit the ``predictionhorizon'' of an operational measure, i.e., randomnesscan reduce the controllability of a system. On the otherhand, random uctuations can also play a benecialrole. For instance, viewing motor control as an optimi-

zation problem, they may be useful to avoid spuriousstates, i.e., locally stable states in which the system might

get trapped. Random uctuations can provide escapesfrom such undesired states and, thus, support the searchfor the global optimum. Similarly, random uctuationsmay be required to induce switches from one state toanother (cf. critical uctuations, e.g., Haken 1983;Gardiner 1990; Kelso 1995) or may lead to resonancephenomena (cf. stochastic resonance, e.g., Benzi et al.1981; Jung and Ha nggi 1991; Hu Gang et al. 1996).Recent studies also considered the case of noise-inducedstabilization of unstable orbits (Wackerbauer 1998).Even in the absence of such dynamical aspects, stochas-ticity may generate correlations between observablesthat may provide a window into the underlying controlstructure. For example, statistically independent noisesources, dened by clock and motor variances, produceerror corrections in terms of negative serial correlations(Wing and Kristoerson 1973; Vorberg and Wing 1996;Thaut et al. 1998). In summary, variability should beviewed as an intrinsic and essential property of humanmovement systems and may be exploited to obtain vitalinformation about the control structure of these systems(Shannon and Weaver 1949; Haken 1988). The questionis how.

Usually, the description of a process by means of asmall number of variables is based on a priori assump-tions regarding the most relevant events in the dataobtained. In studying human movement, such events

might be the start of a limb movement, sign reversals inthe acceleration or velocity proles, or the moment atwhich the aperture of the thumb and the index nger ismaximal in a prehension task. The inherent danger ofthis approach is that, in the study of a certain task,particular events may be falsely assumed to be relevant,whereas more important features of the dynamics maybe ignored. For instance, Haggard and Wing (1997)found that the thumb is a more appropriate controlindex in prehension than the wrist, which has been usedin many studies as the main variable of interest. To avoidsuch unpleasant discoveries, methods for analysis areneeded that address the entire set of time series and

Correspondence to: A.A. Post (e-mail: [email protected],Tel.: +31-20-4448454, Fax: +31-20-4448509)

Biol. Cybern. 82, 143152 (2000)

Principal components in three-ball cascade juggling

A.A. Post, A. Daertshofer, P.J. Beek

Faculty of Human Movement Sciences, Vrije Universiteit, Van der Boechorststraat 9, 1081 BT Amsterdam, The NetherlandsInstitute for Fundamental and Clinical Human Movement Sciences, Amsterdam/Nijmegen

Received: 1 April 1999 / Accepted in revised form: 9 August 1999


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allow for the detection of the most signicant eventswithout presupposing them.

To the extent that such methods exist, they capitalizeon the intrinsic variability of natural phenomena. Vari-ous operational measures of variability (spectral ana-lyses, cross-correlations, dimensionality analyses,Gestalt methods, etc.) can be applied. Unfortunately,

the most commonly used estimates are standard devia-tions or other scalar values with a limited analyticalscope. Certainly, the standard deviation is a usefulmeasure of variability, assuming the underlying distri-bution is suciently symmetric. However, it does notprovide information about possible correlations betweensubsequent observations of the same variable or betweendierent variables. It may be the case that successivedata points are not simple random uctuations about amean value. Correlations within the dynamics of an in-dividual variable can be measured by its correspondingauto-correlation function. If the dynamics pertain to afew distinct points in time, it is expedient to consider the

auto-correlation function only at these points leading to(discrete) serial lag correlations. An estimate of thenumber of relevant variables, however, requires a studyof the cross-correlations between dierent time series.To account for such correlations in time, one canconsider the system's (cross-)covariance matrix. Theprincipal component analysis (also known as Karhunen-Loe ve expansion or singular value decomposition) is aparticularly powerful tool to decompose complex dis-tributed signals into lower-dimensional structures. Suchlower-dimensional structures may be more amenable toan analysis of the underlying dynamics than the originalpattern itself. In spite of its wide application in physics,engineering and biology (e.g., Fuchs et al. 1992; Oja

1992), this tool has seldom been applied to humanmovement. A notable exception is the analysis of pedaloriding movements (Haas 1995; Haken 1996), whichrevealed that the original 22-dimensional whole-bodymovement vector could be reduced to ve or fewerdimensions. This reduction allowed for the study ofchanges in dimensionality in the course of learning toride the pedalo, which turned out to occur predomi-nantly in the arm movements.

In the present paper, we apply the aforementionedmethods to gain insight into the control structure ofthree-ball cascade juggling. Juggling has proven to be avery useful experimental task for studying the dynamical

properties of human perceptual-motor organizations(Beek 1989; Beek and Turvey 1992; Beek and VanSantvoord 1992; Beek and Lewbel 1995). The signi-cance of juggling for the present context resides in thefact that a juggler, i.e., a highly dimensional controlsystem, has to accommodate rather severe task con-straints in order to juggle successfully. In view of thisrequirement, it is to be expected that a small number ofrelevant principal components are sucient to describethe entire spatiotemporal pattern of (cascade) juggling.

Juggling is a cyclic activity in which the hands movealong more or less elliptical trajectories while throwingand catching balls in a regular fashion (Beek 1989; Van

Santvoord 1995). In cascade juggling, one hand moves

clockwise and the other anti-clockwise (Fig. 1) with anaverage phase dierence of circa 180. The balls are, inthis particular pattern, released at the inside of theellipses and caught at the outside. Between throws andcatches, they travel through the air to the other handalong a parabolic trajectory. Figure 1 depicts the pathsof the moving hands and balls in the frontal plane. It is

worth noting that there is one point where the ball tra-jectories intersect, and where they can collide whenbadly timed.

Van Santvoord and Beek (1996) examined the vari-ability of both spatial and temporal juggling variables.With respect to the spatial variables, it was found thatthe dispersion of the points of release was smaller thanthat of the zeniths, which, in turn, was smaller than thatof the catches. With respect to the temporal variables, itwas found that the variability of ball ight intervals wassmaller than that of loaded hand intervals, which, inturn, was smaller than that of empty hand intervals.Furthermore, the temporal variability of full ball cycle

intervals (i.e., from catch to catch) was smaller than thatof full hand cycle intervals. These ndings were inter-preted by viewing juggling as a ``spatial clock'': Bythrowing the balls consistently to a specic height, jug-glers set up a stable time base for the hand movements.However, Van Santvoord and Beek (1996) only studiedvariability at discrete points of the movement. The an-alyses oered here build on their ndings by focusing onthe entire time-dependent trajectories. To gain insightinto the patterns of variability, linear analysis techniquessuch as cross-spectral and relative-phase analysis will beapplied. Without presupposing distinct sources of vari-ability identied as discrete events during the task(throw, catch, etc.), they can emerge depending on the

specic task constraints. Provided that this occurs, fur-ther correlation analyses will be used to explore speciccontrol features (e.g., error correction).

Movement frequency has been demonstrated to be acontrol parameter in many motor tasks, aecting thestability of performance. In doing so, it can inducequalitative changes in behavior. These changes may bereadily observable in the form of a qualitative change inthe performed pattern, but they may also be more subtle.In juggling, task constraints are so severe that a bifur-cation to a pattern with dierent phasing characteristicsis not to be expected. However, increase in frequency

Fig. 1. Paths of the three balls and the two hands during cascadejuggling. Between throw (lYr) and catch points (lYr the balls describea parabolic path which peaks at l, while the hands (subscripts lY r)

follow a more or less elliptic path

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induces an increase in variability of throwing locationand the corresponding throw velocity vector and resultsconsequently in a more variable parabolic ight path.This may have implications for the way in which taskcontrol is brought about. Thus, two juggling frequencylevels were used to investigate the inuence of frequencyon task performance.

2 Experiment

2.1 Methods

2.1.1 SubjectsFour jugglers of intermediate skill who could not jugglemore than three balls in a cascade pattern participated inthe experiment. All subjects were right-handed and hadnormal or corrected-to-normal vision. They were paid asmall fee for their participation.

2.1.2 Experimental setupA 16-mm high-speed motion picture camera (DBM 55,Teledyne Camera Systems, Arcadia, Calif.) was used fordata collection. The focus of the zoom lens was adjustedso that the juggler and the entire juggling pattern were inview of the camera. A ashing light with a xedfrequency was placed in view of the camera to checkthe actual frame rate against the nominal frame rate(125 Hz). The gravitational vertical was dened by aplumb line suspended from the ceiling. Proper lightingconditions for lming were created with the help of four2000-W stage lamps. The equipment of the subjectconsisted of three so-called stage balls (diameter 7.3 cm,mass 130 g).

2.1.3 ProcedureSubjects were placed in front of the camera at a distanceof about 5 m, which was sucient to avoid signicantdistortions of the image on the projection plane of thelm. They were asked to juggle the balls in cascadefashion, either at a preferred or at a fast frequency (i.e.,markedly faster than preferred). Before the start of eachtrial, the subject was allowed to juggle a few cycles toaccommodate to the task instructions in question. Whenthe subject reported that a satisfactory juggle wasachieved, the movie camera was started. During eachtrial, roughly 22 complete juggling cycles were recorded.

2.2 Pre-processing

After development, the lms (Kodak 7251, Ektachromehigh-speed daylight lm, 400 ASA) were projected ontothe opaque screen of a lm-motion analyzer (NAC typeMC OF) by means of a 16-mm projector (NAC type RH160F), connected to a computer. The position of theprojector was adjusted so as to align the plumb linevisible on the lm with the vertical axis of theprojection screen. Frame by frame, the horizontal andvertical coordinates of the centers of the balls were

digitized. Thus, each ball trajectory was represented by a

two-dimensional time series xjtYyjt with j 1Y 2Y 3.For each trial, these time series were rescaled to theinterval 1Y 1 by subtracting their mean and dividingby their maximal value.

3 Data analysis

During cascade juggling, a 90 rotated gure-8 move-ment pattern is generated (Fig. 2, left panel). In thecourse of a full revolution of a ball to its originalposition (e.g., a left-hand catch), the ball travels oncefrom left to right and vice versa while it travels twicealong a parabolic ight path. This can be viewed as 1:2frequency locking between the horizontal and verticalcomponents of each ball trajectory (Fig. 2, right panel).Furthermore, the balls are juggled equidistantlyin time, which can be viewed as phase locking betweenthe balls.

From the moment a ball leaves a hand, it follows a

simple ballistic path until it is caught by the other hand[y $ x x02 $ t t0Y

2; see Fig. 3]. To determine theindividual ight intervals, we therefore tted the ighttrajectories of the balls to a parabola.

Throws (r and l) and catches (r and l) wereidentied at the points at which the trajectories deviatedmore than 5% (relative to the maximal displacement ofthe ball) from the estimated parabola. The parabolas inquestion were determined with the help of the zeniths (land r) of the ball ight (see Figs. 1, 3). In addition, thecorresponding temporal variables were determined, thatis, the moments at which these spatial locations werereached (t, t, and t).

Fig. 2. Example of the recorded ball trajectories. The cascade jugglingpattern (left panel) consists of individual ball trajectories composed of

a horizontal evolution xkt and a vertical evolution ykt (right panel).xkt (solid line) oscillates with half the frequency as ykt (dash-dottedline)

Fig. 3. y-Component of the trajectory of one ball (dashed); piecewise

parabolic t (dash-dotted)

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3.1 Spatiotemporal analysis

In three-ball cascade juggling, the individual trajectoriesof the balls can be considered as (largely) identical butphase-shifted by 2pa3. Such symmetries reect theexistence of some redundancy in the acquired signalsthat can be eliminated by transforming the system onto

its principal axes. For this sake, we merged thetrajectories of the three balls and described each dataset by a six-dimensional signal qt X x1tYy1tY

x2tYy2tYx3tYy3t

. For a lower-dimensional appro-ximation of the data, we chose an optimal set of vectorsfvkg to obtain

qt %w 6k1

nktvk X 1

The choice of the basis vectors vk was realized byminimizing the least square error

Ew X qt wk1

nktvk

4 52

B C

3min X 2

In Eq. (2), hi denotes the time average given byhi 1a

0

dt, where is the length of the time seriesand t denotes time. In this approach, the vectors vk areassumed to be pairwise orthogonal so that the minimi-zation of Ew becomes equivalent to an iterativeconstruction of vectors along the direction of maximalvariance of the data. This is followed by a subtraction ofthe remaining subspace via projection (Karhunen 1946).The iteration is truncated after w steps and realized by

diagonalizing the covariance matrix R of the data setqt q1tY F F F Y q6t

. The components of the soobtained matrix are explicitly dened as

mn G hmtnti with t X qt hqti Y 3

and further rescaled by means of trR 1. Theeigenvectors of the covariance matrix dene the desiredvectors vk. The corresponding eigenvalues kk reect theamount of signal (variance) that is covered by the modevk. Because the vk are orthogonal the time-dependentcoecients nkt [cf. Eq. (1)] can be easily obtained byprojection.

Besides the equivalence of the balls, the symmetry of

three-ball cascade juggling patterns is manifested in thephase relation between the balls. The positions of twoballs fully determine the position of the remaining one.The ``extra information'' about this third ball in theform of its trajectory is therefore redundant. The origi-nally six dimensions eectively reduce to four. Thus, inthe principal component analysis, one expects theeigenvalues kk to drop drastically at k0 4, that is,kkbk0 ( kk0 (see also Appendix A). In other words,one expects that it would be sucient to describe thedata set as a maximally four-dimensional systemn1tY F F F Y n4t

. To illustrate this eect of symmetry

we discuss a -periodic set of piecewise parabolic tra-

jectories dened recursively as

~xt X

t a2 sha2 for 0 t` a2 sh

a2 sha2 for a2 sh t` a2

~xt a2 for t ! a2

VbbX

and ~yt X ~xt 2 X 4

species the horizontal periodicity and sh ` a2

denotes the time between two consecutive parabolas($ time that a hand is loaded with a ball). For thispiecewise parabolic juggling model,1 the entire signalreads q2j1 X ~xtja3 and q2j1 X ~ytja3 for

j 0Y 1Y 2.As expected, the corresponding principal components

vanish for k ! 5 (see Fig. 4). The projections nkt arepairwise equivalent by means of the amount of variancekk. They are associated with the horizontal and verticaltime series. The phase shift between n1 and n2 (andsubsequently between n3 and n4) is pa2 representing theorthogonality of the corresponding eigenvectors (cf.Appendix A).

In analogy with the analysis of the model describedabove, we investigated the principal components of therecorded juggling patterns. The calculated and normal-ized eigenvalues for each trial are depicted in Fig. 5(upper panel). In accordance with the preceding analy-sis, the last two eigenvalues almost vanished, that is,each pattern essentially reduced to four dimensions.Interestingly, the eigenvalue spectra diered qualita-tively between the two experimental conditions. Duringjuggling with a preferred tempo, the largest amount ofsignal was represented by just two modes (k3Y4 3 0),whereas for fast juggling the gap between k1Y2 and k3Y4was markedly smaller (Fig. 5, lower panel).

Thus, besides an increase of the overall variance re-

ecting the fact that performance was worse in the fastcondition, the cross-covariance between the vertical andhorizontal evolution of the balls increased. The questionarose as to the location of this increase in variability inspace and time. Since the eigenvalue spectra are basedon (cross-)covariances over time remember that, inEq. (2), hi denotes the time average each kk representsa dierent temporal variability. The correspondingeigenvectors or modes can reect the spatial propertiesof the system.

Since the results of the following analyses wereessentially the same in all trials and subjects, we discussonly two trials in detail. Trial p12 represents juggling in

the preferred condition, whereas trial f11 represents jug-gling in the fast condition. Figure 6 shows the rst foureigenvectors v1FFF4 for the two conditions. As is apparentfrom this gure, the largest amount of variability (therst two modes v1Y2) was associated with the y-direction.Evidently, the angle between the rst two eigenvectorswas larger in the fast condition, reecting an increase inspatial variability. The large angle between eigenvectorsv3 and v4 in the preferred condition was of minor im-portance since the corresponding eigenvalues were

1 This ``model'' is in fact a convenient formalization of the balltrajectories based on the physical laws for the free-ight portions

and ignoring the non-ight portions

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rather small (cf. Fig. 5). However, their contributionwas no longer negligible in the fast condition.

To study the temporal evolution of the balls, weprojected the signal onto the individual eigenvectors bymeans of nkt X v

k qt. The projections of the

eigenvectors on the experimental time series are shownin Fig. 7 (upper panels). These projections were used toconstruct the n2n1 and n4n3 planes depicted in Fig. 7

(lower panels). Both representations show steady graphsreecting the strong correlation between the dierentprojections. In good qualitative agreement with themodel (Eq. 4; Fig. 8), the projections of the eigenvectorson the data are ` jagged'', and the corresponding

nk1nk portraits triagonal or hexagonal.Similar projections of the eigenvectors on thepreferred time series are shown in Fig. 9 (upper panel).Remarkably, the plane representation of the rst twoprojections has a rather circular shape compared to thetriangular structure in Fig. 7 (lower left panel). Since n1Y2were associated with the vertical direction, the trajecto-ries ykt were smoother in the preferred condition. Thisphenomenon was not present for the x-direction (cf.Figs. 7, 9, lower right panels) where the hexagonalstructure ofn4n3 is preserved. It should again be notedthat the contribution of the x-direction was of minor

Fig. 6. Eigenvectors vjYj 1Y F F F Y 4 displayed on the subspace ofeach ball k 1Y 2Y 3. The vertical lines correspond to the rst twoeigenvectors (solidv1, dash-dottedv2) and the horizontal ones reectthe subsequent two eigenvectors (dashed v3, dash-dotted v4). Left

panelpreferred condition, right panel fast condition

Fig. 7. Data in the fast condition. Left panels depict n1Y2, right panelsdepict n3Y4. Upper panels projection of the eigenvectors on the data;lower panels nk1nk plane of projections in the upper panels

Fig. 8. Model in the fast condition (for legend see Fig. 7). ModelEq. (4) with 1X5 s, sh 0X09 s

Fig. 9. Data in the preferred condition (for legend see Fig. 7)

Fig. 5. Principal component analysis of all individuals trials. The sixeigenvalues kk for all subjects and all conditions are shown; on theordinate the trials fj and p

j are depicted in the upper panel(f fast,

p preferred, subject, j trial number). The averaged valuesKk X hkkitrials are displayed in the lower panel

Fig. 4. Principal component analysis of model Eq. (4) with 2;time resolution Dt 8 103 s. The dominant modes account forabout 80% of the data

2k1 kk % 0X8 and oscillate with the same

frequency f0 as the yj components. n3Y4t reect the xjt terms, i.e.,they oscillate with f0a2

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importance, given the fact that the corresponding ei-genvalues were small. This experimental observationcannot be accounted for by the model as formulated inEq. (4). Since the change in shape of the nk1nk planescan be interpreted in terms of smoothing of the under-

lying trajectories, the vertical components have to below-pass ltered (so that catch and throw points becomedierentiable). The projections thus obtained are inagreement with those of the data (cf. Fig. 9, lower leftpanel).

To quantify these changes in the shape and smooth-ness of the time evolution of the eigenvector projections,we analyzed the spectral properties of the dynamics.Therefore, we computed the cross-spectral density WkYlbetween the pairs of adjacent nkt

WkYlx X F nkt F nlt j j

2 Y 5

where F nt denotes the Fourier transform of nt. Inaddition, we computed the Hilbert phase Ht betweenthese pairs in order to assess the time-dependent relativephase between nk and nl (Rosenblum and Kurths 1998).

Hkt X1

p

II

nks

t sds A

/hkYl t X arctan

Hktnlt nktHlt

nktnlt HktHlt

& 'Y 6

note that /h is identical to the relative Fourier phase/f in the case where the signal readsnk G expfxt /kg XX. Given that the two signalsdeviate periodically from such a sinusoidal form, the

spectral power of/h shows a distinct peak at the basicfrequency. Cross-spectral density results for data and themodel in the fast condition are displayed in Figs. 11, 12(upper panels). The WkYls of the rst pair of projections(Figs. 11, 12, upper left panel) showed a strong peakwhich was identical to the juggling frequency in the y-direction, whereas the WkYls of the second pair ofprojections (Figs. 11, 12, upper right panel) showed asimilarly strong peak which was the same as thefrequency in the x-direction. These large peaks reectthe strong frequency locking between n1t and n2t, aswell as between n3t and n4t.

The time evolutions of the relative Hilbert phases for

data and model in the fast condition are shown in

Figs. 11, 12 (middle panels). As was already mentioned,

/h1Y2 equalled on average pa2 but displayed pronounced

oscillations. The spectral power of the relative Hilbertphase (cf. Figs. 11, 12, lower panels) showed the samedominant frequency as the main cross-spectral frequency(cf. Fig. 12, upper and lower left panels). These oscilla-tions of/h result from stable periodic deviations fromsimple sinusoids, suggesting that these deviations occurat xed points in the n2n1 representation (Figs. 7, 8, leftlower panels). Hence, the representations in questionremained steady (cf. Fig. 7). The power spectrum of theHilbert phase revealed a subharmonic of the dominantmovement frequency in the vertical direction (cf. Fig. 11,lower left panel). This frequency, however, was equiva-lent to the frequency of the horizontal components.

Therefore, this subharmonics accounts for the increasingvariability along the x-direction that now ``creeps into''the rst two principal axes.

In contrast with these eects for juggling in the fastcondition, Figs. 13, 14 show the spectral analyses fordata and model in the preferred condition. All peaks ofthe spectral densities were markedly shifted to the leftcompared to the fast condition, indicating a decrease infrequency. Again, the relative Hilbert phase /

h1Y2 oscil-

lated around pa2 but the amplitude of this oscillationwas markedly reduced in comparison with the fastcondition (cf. Figs. 11, 13, middle panels). This lowamplitude indicates the presence of rather synchronous

sinusoidal trajectories n1t and n2t (see Fig. 10).

Fig. 10. Model in the preferred condition (for legend see Fig. 7).Model Eq. (4) with 2X5 s, sh 0X15 s. The vertical components

ykt were low-pass ltered (second-order Butterworth at f 1 Hz)

Fig. 11. Data in the fast condition. Left panels refer to n1Y2, rightpanels to n3Y4. Upper panels logarithmic cross-spectral density plot ofthe projections in Fig. 7; middle panels relative Hilbert phase of theseprojections; lower panels logarithmic power spectral density plot ofrelative Hilbert phase

Fig. 12. Model in the fast condition (for legend see Fig. 11). ModelEq. (4) with 1X5 s, sh 0X09 s (cf. Fig. 8)

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In both conditions, the horizontal components werequalitatively equivalent. Analogous to the vertical parts,

the relative Hilbert phase showed small oscillationsrepresenting the hexagonal structure of the corre-sponding n4n3 planes. Especially for the preferredcondition, a broad-banded spectrum was observed at thelower frequencies which, however, could be neglectedgiven that the corresponding eigenvalues were verysmall.

3.2. Temporal correlations

Despite the marked dierences in smoothness of themovement trajectories in both conditions, the timing

properties were quite similar. That is, frequency andphase locking were equally present in fast and preferredjuggling. In the former case, however, the performancewas worse so that one would expect at least an increaseof the timing variability. Increasing variability impliesincreasing timing errors that, given the task constraints,can be expressed in terms of deviations of the 2pa3 phasedierence between the balls. The deviations predomi-nantly occurred at distinct locations as shown in Figs. 7and 8. These points dene the dierent geometricalproperties of the nk1nk representations. For example,n4n3 showed a hexagonal structure indicating siximportant events, i.e., two events per ball per cycle,

whereas n2n1 hinted at the presence of a single distinct

event per ball per cycle (triangular structure). All theseevents showed pronounced discontinuities in the deriv-atives of nkt with respect to time. Thus, they did notoccur along the parabolic part of the trajectories. Sincethese events are more or less xed in phase and position,it is most likely that they are associated with the throwsand the catches. A closer comparison between the

temporal evolutions nkt and the corresponding xjtand yjt conrmed this expectation. Explicitly, the mostimportant events for the dynamics ofn1 and n2 were thecatch points. The higher modes n3Y4 were additionallyaected by the throw points. Forthcoming studies oftemporal correlations can therefore focus exclusively onthese events. Notice that these discrete events emergedfrom the preceding analyses that involved no a prioriknowledge of the juggling pattern.

Besides the locations of the throws and the catches, acomplete determination of the parabolic part of thetrajectories requires either the initial force (velocity) ofthe throw or the location of the parabola's zenith.

Therefore, we computed time intervals between consec-utive zeniths, consecutive catches and consecutivethrows by, or referring to, the same hand (i.e., left,right). As shown in Fig. 15, the normalized variances ofthe time intervals between consecutive catches by thesame hand were consistently larger in the fast conditionthan in the preferred condition. Throw and zenith in-tervals did not show such a signicant dierencebetween the two conditions. This result can be readilyexplained: Every distinct point t F F F t contributes to thetemporal variability of the overall cycle. Consequently,the variance of the very last point, i.e., the catch, in-cludes the variability of all previous ones (throw, zenith),at least to some extent. All previous errors are combined

in the point of catch that, with its additional own vari-ability, shows the largest total variance.

The variance s20n X in2 in2 with i as ex-

pectation value of n reects the amount of errors but itdisregards any mechanism of error correction. To gaininsight into such mechanisms, we considered therst three lag covariances s2sn X int innts in. The lags s 0Y F F F Y 3 refer to the consec-utive balls (e.g., lag-zero: ball1, lag-one: ball2, lag-two:ball3, lag-three: ball1, etc.). Again, the most dominantlag-covariances were found at the catch point becausethe point of maximal variance requires larger errorcorrections. The increase of variance was accompanied

Fig. 13. Data in the preferred condition. Left panels refer to n1Y2, rightpanels to n3Y4. Upper panels logarithmic cross-spectral density plot ofthe projections in Fig. 9; middle panels relative Hilbert phase of theseprojections; lower panels logarithmic power spectral density plot ofrelative Hilbert phase

Fig. 14. Model in the preferred condition (for legend see Fig. 13).Model Eq. (4) with 2X5 s, sh 0X15 s (cf. Fig. 12)

Fig. 15. Normalized variances r20 X s20Dt

aDt of the intervalsbetween the catches for all individuals trials. Left panel catches with

left hand: right panel catches with right hand

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by an increase of the absolute covariance. For animmediate comparison between the dierent points ofinterest, we therefore used an appropriate normaliza-tion. Explicitly, we compared lag correlations of theform cs X r

2sar

20 s

2sas

20 denoting relative serial error

corrections. Analogous to the ndings of Wing andKristoerson (1973) regarding self-paced isochronous

rhythmic movements, we found markedly negative lag-one correlations in the range 0X5 c1 0 as shown inFig. 16. As this eect was symmetric for both hands, itsuggests that errors in the phase relation betweenconsecutive balls are immediately corrected by thecatches. The higher lags were centered around zero andcan thus be neglected as possible correction mechanisms.Interestingly, the lag-one correlations were only signi-cantly negative for the catches. For the throws and thezeniths, all serial lag correlations were negligible. That is,deviations in the timing of the throw do not aect timingof the subsequent throw whereas the catch is used tocorrect for the proper phase relation between the balls.

This correction of relative errors is independent of thecycle frequency.In a manner analogous with the temporal properties

of the system, we nally studied variances and correla-tions of the spatial locations of throw, zenith and catch.At the throw points, we further computed the balls'initial velocities. For all computed variables, the vari-ances for the preferred and the fast condition were notsignicantly dierent. In detail, no structure wasobserved in the variance of x- or y-directions of theposition of throws, zeniths, or catches, nor in the x- or

y-direction of the velocity at the throws. Similarly, thelag correlations did not hint at any spatial error cor-rection mechanism between consecutive balls, i.e. no

particular auto-correlation patterns could be discernedin these positions and velocities. However, the ndingsof Beek and Van Santvoord (cf. spatial clock; VanSantvoord and Beek 1996) indicate the presence of somemechanism that pronounces distinct points in space inaddition to the discontinuities of throwing and catching.Since the ball ight trajectories and, in particular, theirzeniths, are determined by the location of the throw andits force, one can expect signicant cross-correlations

between the corresponding variables, and possibly cor-relations at dierent lags. This analysis, however, isbeyond the scope of the present study.

4 Discussion

In the present article, we showed analytically that theinherent symmetries of three-ball cascade jugglingpatterns imply a reduction of the dimensionality ofthese patterns. This theoretical result was corroboratedexperimentally. Principal component analysis revealedthat the six-dimensional ball pattern can always betreated as a maximally four-dimensional system. In thefast condition, all these four dimensions were needed toreconstruct the signal. In the preferred condition,however, vertical and horizontal components werealmost perfectly correlated, i.e., xt determined ytand vice versa. As a consequence, the system's dimen-sionality reduced further to two in this case. In control

theoretical terms this indicates that, in the preferredcondition, the system has to accommodate fewerdimensions, implying a strong reduction of the controlproblem. It is important to note that the experimentalmanipulation of movement frequency from fast topreferred is completely unspecic to this reduction, i.e.,it does not entail it. Thus, a change in detail (frequency)causes a qualitative change in the movement as a whole(dimensionality of the control space), in a manneralready intuited by Bernstein (1984 p. 84). For bothconditions, the rst two eigenvalues k1Y2, that is, the twomost important modes, were always associated with the

y-direction. This implies that the main direction ofcontrol is along the (gravitational) vertical, which is

consistent with the previous ndings of Van Santvoordand Beek (1996). The corresponding cross-spectraldensities and Hilbert phases revealed that the timingproperties, i.e., the 1:2 frequency locking and the 2pa3phase locking were not aected by a change in tempo.For the fast condition, however, the power spectra of therelative Hilbert phase between n1 and n2 showed anincreased importance of the horizontal componentswhich coincided with increasing eigenvalues k3Y4, thatis, a higher dimensionality of the system (two 3 four).This nding might be associated with the decrease inangle of release in the fast condition.

Besides these global and time-dependent aspects, a

further important change in the movement pattern waslocated in space and time. Depending on the condition,the smoothness of the ball trajectories changed markedlyat the catch points. This reects the increasing dicultyat higher circulation speeds to precisely match the handvelocity vector to that of the ball. Thus, the degree ofsmoothing of the projections n1Y2t can be taken as anindex of the quality of task performance. An equivalentchange of smoothness in the horizontal direction wasnot observed. The projections n4n3 always showedhexagonal shapes with the angles related to the catchpoints (left and right hand). Concerning the timingcorrection, juggling frequency had also a large eect on

the variance of catch-catch cycles. Combined with the

Fig. 16. Serial correlations cs X r2sar

20 s

2sas

20 at the point of catch

t. On average, the lag-one correlation is negative "c1 0X2whereas the higher lags are not signicantly dierent from zero, i.e.,"c2Y3 % 0

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relatively large spatial variance, this fact underscoresthat the signicance of the catches is in correcting errorsin the timing of the act (i.e., the stabilization of the 2pa3phase dierence between the balls). The increase oftemporal variability is corrected by means of the ob-served negative lag-one serial auto-correlation. That is,errors in the phase relation between consecutive balls are

immediately compensated by the catches. Higher lags,i.e., forthcoming catches, are not relevant for this con-trol mechanism. It can be expected that the spatialvariability of the catch is exploited to reduce the afore-mentioned temporal variability in the catch-catch cycles.The exact form of this control mechanism, however,remains an issue for further study.

Negative lag-one correlations have been frequentlyobserved in the context of the unimanual, unpaced per-formance of rhythmic movements. To account for suchcorrelations, Wing and Kristoerson (1973) formulated amodel that presupposes two hierarchical levels involvingan internal timekeeper (clock) and some motor delay.

This model addresses statistical aspects (variability andcorrelations) of temporal structures. The juggling pat-tern, however, also depends on spatial constraints. Fur-ther modeling should therefore account for the explicitspatiotemporal forms ofxjt andyjt. Equation (4) doesnot elucidate all spatiotemporal phenomena, but de-scribes important features of three-ball cascade jugglingsuch as its inherent symmetry and the physical nature ofballistic throws. In a recent study by one of us, it has beenshown how to incorporate certain timing aspects intooscillatory dynamical systems (Daertshofer 1998). Inparticular, negative serial correlations can be generatedin a system of coupled oscillators by introducing addi-tional stochastic forces. Thus, an extension towards sto-

chastic dynamical models allows one to account for boththe stability properties and the variability in the timing.Various aspects of variability can now be addressedcoherently from the perspective of coordination dynam-ics. The results presented in this paper merit the conclu-sion that valuable insights into motor performance can begained by studying its spatiotemporal variability indetail. This, however, requires appropriate methods ofanalysis that do not involve any a priori assumptions withregard to the system under investigation. The methodsapplied in the present paper are examples of such unbi-ased tools since they take the entire spatiotemporalevolution of the movement pattern into account.

Appendix

A. Symmetries

In this appendix, we consider the consequences of the inherentsymmetries of juggling for the number of relevant principal com-ponents. Since juggling is an essentially rhythmic activity, we studya -periodic function ft ft . For the sake of generality wedo not further specify the explicit form of f ft. Without anyrestrictions, we assume 1 and subtract the mean of ft, i.e., weconsider the case hfti 0. Regarding the periodic nature of ft,we compute its corresponding Fourier transform. Explicitly, the

Fourier series reads

ft In1

n sin 2pnt n cos 2pnt f g X 7

The symmetries in juggling can be expressed in terms of phaselocking between the balls. With identical balls, we thus treat thecase of dierent sets of functions ft that are pairwise shifted inphase by 2pa3.

At rst, the signal is composed of three components, i.e., we

have

qt 2k0

f tk

3

ek Y 8

where ek are the unity vectors in R3. Given the symmetry of the

signal the corresponding covariance matrix becomes

R 1

3r2

r2 12 12

12 r2 12

12 12 r2

Hfd

Ige

A k1Y2 r2 12

3r2and k3

r2 212

3r2X 9

The two dierent covariances can be calculated as

r2

10

ft2dt 1

2

In1

2n 2n

12

10

ftft 1a3 dt r2

2X 10

Thus, the covariance matrix has only two non-vanishing but de-generated eigenvalues k1Y2 0X5. Orthogonalization of the corre-sponding eigenvectors yields

v1 2

1

1

Hd

IeY v2 01

1

Hd

IeY and v3 11

1

Hd

Ie X 11

This orthogonality v1cv2 leads to periodic projections n1Y2t that

are phase-shifted by pa2, i.e.,

nkt nkt 1 n2t n1 t1

4

X 12

Next, we extend the number of dimensions of the signal to six byintroducing a a2-periodic function gt gt a2. This prop-erty is actually present in the studied data set. Again, we use thesymmetry of the system and the aforementioned results in threedimensions and obtain

R GRh Rgh

Rgh Rg

X 13

Each sub-matrix Ra has equivalent symmetries as (9), that is, thediagonal elements are r2a and all o-diagonal components read

r2

aa2. Thus, this combined covariance matrix R has two van-ishing eigenvalues k5Y6 0 and two degenerated sets

k

1Y2

Y 3Y4

1

41

r2h r

2g

2 4r2gh

r2h r

2g

2vuuuut

PTTTTR

QUUUUS X 14

As a result, the originally six-dimensional system is reduced to fourdimensions. In the case of phase-locking between ft and gt, thecross-covariance r2gh tends to zero based on the orthogonality ofthe trigonometric functions. Thus, if the individual variancesr2h and r

2g are identical, both eigenvalues will become

kY 3 1a4. On the other hand, if one variance vanishes, e.g.,

r2h 3 0, thecorresponding eigenvalue will vanishas well (k 3 0).

In this case, the system essentially becomes two-dimensional.

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References

Beek PJ (1989) Juggling dynamics. PhD thesis, Vrije UniversiteitAmsterdam

Beek PJ, Lewbel A (1995) The science of juggling. Sci Am 273:9297

Beek PJ, Turvey MT (1992) Temporal patterning in cascade jug-gling. J Exp Psychol Hum Percept Perform 18:934947

Beek PJ, Van Santvoord AAM (1992) Learning the cascade juggle:a dynamical systems analysis. J Mot Behav 24:8594

Benzi R, Sutera A, Vulpiani A (1981) The mechanism of stochasticresonance. J Phys A 14:L453

Bernstein NA (1984) The problem of the interrelation of co-ordination and localization. In: Whiting HTA (ed) Humanmotor actions: Bernstein reassessed. North-Holland, Amster-dam, pp 77120

Daertshofer A (1998) Eects of noise on the phase dynamics ofnonlinear oscillators. Phys Rev E 58:327338

Fuchs A, Kelso JAS, Haken H (1992) Phase transitions in the hu-man brain:spatial mode dynamics. Int J Bifurc Chaos 2:917939

Gardiner CW (1990) Handbook of stochastic methods, 2nd edn.Springer, Berlin Heidelberg New York

Haas R (1995) Bewegungserkennung und Bewegungsanalyse mitdem synergetischen Computer. PhD thesis, Universita t Stutt-gart, Germany

Haggard P, Wing A (1997) On the hand transport component ofprehensile movements. J Mot Behav 29:282287

Haken H (1983) Synergetics. An introduction, 3rd edn. Springer,Berlin Heidelberg New York

Haken H (1988) Information and self-organization. Springer,Berlin Heidelberg New York

Haken H (1996) Principles of brain functioning. A synergeticapproach to brain activity, behavior, and cognition. Springer,Berlin Heidelberg New York

Hu Gang, Daertshofer A, Haken H (1996) Diusion of periodi-cally forced Brownian particles moving in space-periodicpotentials. Phys Rev Lett 76:48744877

Jung P, Ha nggi P (1991) Amplication of small signals viastochastic resonance. Phys Rev A 44:8032

Karhunen K (1946) Zur spektralen Theorie stochastischerProzesse. Ann Acad Sci Fenn A1, Math Phys 37

Kelso JAS (1995) Dynamic patterns. The self-organization of brain

and behavior. MIT Press, Cambridge, MassOja E (1992) Principal components, minor components, and linearneural networks. Neural Netw 5:927935

Rosenblum M, Kurths J (1998) Analysing synchronization phe-nomena from bivariate data by means of the Hilbert transform.In: Kantz H, Kurths J, Mayer-Kress G (eds) Nonlinear analysisof physiological data. Springer, Berlin Heidelberg New York,pp 9199

Shannon CE, Weaver W (1949) The mathematical theory ofcommunication. University of Illinois Press, Urbana Ill

Thaut MH, Miller RA, Schauer LM (1998) Multiple synchroni-zation strategies in rhythmic sensorimotor tasks: phase vsperiod correction. Biol Cybern 79:241250

Van Santvoord AAM (1995) Cascade juggling: learning, variabilityand information. PhD thesis, Vrije Universiteit Amsterdam,CopyPrint 2000

Van Santvoord AAM, Beek PJ (1996) Spatiotemporal variability incascade juggling. Acta Psychol 91:131151

Vorberg D, Wing AM (1996) Modeling variability and dependencein timing. In: Heuer H, Keele SW (eds) Handbook of percep-tion and action, vol 2. Academic Press, London, pp 181261

Wackerbauer R (1998) Noise-induced stabilization of one-dimen-sional discontinuous maps. Phys Rev E 58:30363044

Wing AM, Kristoerson AB (1973) Response delays and the tim-ing of discrete motor responses. Percept Psychophys 14:512

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