Journal of Experimental Psychology: Human Perception and Performance 1992, Vol. 18. No. 4,934-947

Copyright 1992 by the American Psychological Association, Inc. 0096-1523/92/S3.00

Temporal Patterning in Cascade Juggling M. T. Turvey

P. J. Beek

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Faculty of Human Movement Sciences, Free University, Amsterdam, The Netherlands, and Center for the Ecological Study of Perception and Action, University of Connecticut

Center for the Ecological Study of Perception and Action, University of Connecticut, and Haskins Laboratories, New Haven, Connecticut

A key variable in cascade juggling is the proportion of time that a juggler holds onto a juggled object during a hand cycle, that is, the time from catch to throw in relation to the time from catch to catch. Space-time constraints and principles of frequency locking suggest 3/4 as the primary ratio and 2/3 and 5/8 as the most accessible options. In 5 experiments, object number, mass, and type (ball or scarf) were manipulated together with the frequency at which the objects were juggled. With 5 or 7 balls, the ratio was 3/4, independent of frequency. With 3 balls, the ratio decreased with frequency, with 3/4, 2/3, and 5/8 tending to predominate independently of the force variations induced by variation in object mass. With 3 scarves, ratios varied inversely with frequency and often exceeded 3/4. Implications for a dynamical theory of juggling were discussed with the issue of relative timing in coordination and the manipulation of task constraints as an experimental strategy.

Juggling is a skilled act characterized by a coordination of cyclic motions of limbs and objects. In the cascade version (a figure-eight pattern involving an odd number of objects), a juggled object is alternately caught with one hand and thrown to the other. Through cyclic shoulder, elbow, and wrist motions, the two hands move more or less elliptically in the same time frame, mirroring one another at characteristic phase lags (in the order of 180°) and conforming tightly to the parabolic flights of the juggled objects (P. J. Beek, 1988; Raibert, 1986). The fingers are the end effectors that are responsible for the actual manipulation of the objects. In a stable, recurrent juggling pattern, such as the cascade, it is useful to distinguish three component times: the average time a juggled object is held in the hand between a catch and a throw, the average time a juggled object is in the air between the hands, and the average time a hand is in motion free of a juggled object between a throw and a catch. These three times can be referred to as time loaded (7~L), time of flight ( T f ) , and time unloaded (7"u), respectively. When these continuous time variables are coupled to the discrete variables number of hands (//; a person may juggle alone, using one or two hands, or with one or more other jugglers, using any number of hands) and number of juggled objects (N; one or more jugglers can juggle two or more objects provided that TV > //), a general constraint on the temporal organization of juggling can be derived, as was first demonstrated by Claude Shannon (Horgan, 1990; Raibert, 1986).

For convenience, assume that two objects are never in the same hand at the same time and that the pattern in which they are thrown is periodic in the sense that each TY-object//-hand juggling cycle, beginning and ending with the same event (e.g., catching object j in hand /), has a fixed duration (T). Define the cycle time TH of hand ; to be the average time between catching successive objects (TH = TL + Tv) by that hand during T. Because there are N objects, the total loop time for hand /' during T is N( TL + TV). Similarly, define the cycle time T0 of object j to be the average time between successive catches (T0 = TL + Tf) of that object during T. Because there are H hands, the total loop time for object j during T is H(TL + 7». Consequently, N(TL + TV) = T = H(TL + Tf), or, equivalently, T0/Ta = N/H. In other words, for juggling to be periodic with T, the juggling hands and juggled objects must satisfy, on average, a general timing requirement in which the ratio of an object cycle time to a hand cycle time equals the ratio of number of objects to number of hands. That the assumption of periodicity is not necessarily required for Shannon's equation to be true is shown in the Appendix, which provides a more formal derivation in which 71, Tv, and Tf are treated as truly random variables. In its generalized form, Shannon's equation applies beyond the cascade juggle to all patterns in which no hand holds more than one ball at any one time (e.g., reverse cascade, fountain, and shower) and may therefore be considered a universal field equation of juggling (Morgan, 1990). From the perspective of the theory of movement coordination, the significance of Shannon's equation of constraint for juggling lies in its universality and the precision with which it is formulated. Recently, Newell (1986) emphasized the need to consider three sources of constraints on patterns of coordinated movements—environmental, organismic, and task. It may be argued that optimal coordination patterns are determined by the interaction among these three constraint sources. What Shannon's equation makes clear is that task constraints are potentially definable independent of the particular environmental setting and the individual animal. Regardless of the sizes and shapes of the juggled objects, the

The present research was supported, in part, by National Science Foundation Grants BNS-8811510 and BNS-9109880. We thank Wiero Beek, Piet van Wieringen, Elliot Saltzman, and four anonymous reviewers for their helpful comments on earlier versions of this article, Tony van Santvoord for his assistance in data acquisition, and the fine collection of ball manipulators, who were willing to demonstrate their skills before the camera. Correspondence concerning this article should be addressed to P. J. Beek, Faculty of Human Movement Sciences, Van der Boehorststraat 7, Room A-622. Vrije Universiteit, 1081 BT Amsterdam, The Netherlands. 934

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CASCADE JUGGLING

postures and limb configurations of the juggler, and the species of the jugglers (humans, seals, monkeys, or robots), juggling is only attainable by satisfying Shannon's equation. The intimation that all tasks are, in general, like juggling in that they are typically invariant over environmental and organismic variations can be developed empirically only to the extent that their constraint structures can be expressed with mathematical precision. The precise mathematical formulation of the constraints that characterize skilled acts is in its infancy and awaiting the emergence of more appropriate analytic tools. In the meantime, however, the study of the precisely formulated juggling task can provide an understanding of the formative role of task constraints in the assembling of coordinated movement patterns. The goal of the present studies was to investigate the principles that determine the ratio of object-in-hand time to hand-cycle time under different environmental and task conditions. We denote this ratio by the letter k and define k as T1/(T1 + Tv). That k is a key variable in the temporal organization of juggling can be appreciated from the fact that the time-averaged number of juggled objects in the air over a complete N - //juggling cycle of duration T is equal to N* = N-Hk (P. J. Beek, 1988; Appendix). Thus k directly determines N*, which is an index of the task goal of keeping as many balls airborne at any instant as possible. What Shannon's equation shows is how any one of the three time quantities (71, 71, and TV) is constrained by the other two, given TV and H. Suppose that TV is fixed. Then the juggler's options with respect to the remaining temporal components are that 7"L can be lengthened (holding onto a juggled object for a longer period of time) or reduced (getting rid of a juggled object more quickly). Exaggerating the first option produces what is known as "delayed juggling," in which TL approaches TV,, k approaches unity, and N* approaches N H. Exaggerating the second option produces what is known as "hot potato juggling," in which 71 approaches zero, k approaches zero, and jV* approaches N. Neither extreme is physically possible, but together they define the juggler's freedom with respect to the proportion k of hand cycle time consumed by 71, given a fixed TV. The importance of the preceding is its relevance to ordinary juggling in which jugglers, during a bout of catches and throws, tend to throw the objects to a relatively fixed height relative to the throwing positions of the hands, thereby fixing the flight time of the objects at TV. This "spatial clock," however, leaves the, two remaining degrees of freedom (71 and 710 unaffected. A central question therefore is, if during ordinary juggling k lies between 0 and 1, then what value or values does it assume? By asking this question, we step inside Shannon's equation of constraint for juggling and open up inquiry into the basis for determining the durations of the component repetitive subtasks. The repetitive subtask (of duration 71) initiated by catching a juggled object and terminated by releasing that object is nested within the repetitive subtask (of duration TH) of the hand being moved to return repeatedly to the catch position. An intuitive impression, one that is reinforced by a preliminary analysis using the techniques of classical mechanics and nonlinear dynamics (P. J. Beek & W. J. Beek, 1988), is that the two subtasks are

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dynamically distinct. During the loaded part of the hand loop, the total mass in motion is the mass of the object plus the mass of the hand, whereas during the unloaded part of the hand loop, the total mass in motion is simply the mass of the hand. Considering the fact that the frequency of an oscillating system is predominantly determined by its stiffness and mass, it follows that the frequency of the hand and the object can only be equal to the frequency of the hand in the unloaded part of the hand loop if adjustments in stiffness are being made. Thus, it is possible to conceive of a shorter term dynamical regime (associated with the transport of the juggled object) superimposed on a longer term dynamical regime (associated with the rhythmic movement of the hand) with different overall parameters. One may hypothesize further that these repetitive subtasks can only be conducted together in a stable, consistent fashion according to the principles of frequency locking and phase entrainment (Berge, Pomeau, & Vidal, 1984). These principles would prescribe rational ratios of relatively prime integers (i.e., integers lacking a common divisor). If the component behaviors of juggling were governed by these principles, then the period of one repetitive subtask and the period of the other repetitive subtask must relate as such a ratio (P. J. Beek, 1988, 1989): T1/7V = (2, 3/2, 1, 1/2, 1/3) and, consequently, for // = 2 and N = 3 (because of Shannon's equation), 71/(T1 + Tl), or k = 1, 9/10, 3/4, 1/2, 3/8, and so on. Clearly, there are infinitely many such ratios. Is any one ratio or any set of ratios preferred? An affirmative answer follows from a consideration of experimental results and theoretical analyses. k Values for Highly Skilled and Novice Jugglers In an experimental investigation of cascade juggling with H = 2, N = 3, three juggling speeds, and four skilled jugglers, it was found that the duration of the subtask of carrying the juggled object between catch and throw expressed as a proportion of the hand-cycle time (k = TL/TH) ranged between .54 and .83, with a mean of .71 (P. J. Beek, 1989). A linear regression of k on TH (which is valid only up to the limit k — 1) reveals for P. J. Beck's (1989) data a significant effect of hand-cycle time, k = .21 TV, + .57, ^(11) = .42, p < .02; k was smaller at higher juggling frequencies and thus not strictly invariant. Although allowing for the possible existence of mode locks other than k — 3/4 in the work space of cascade juggling, P. J. Beek interpreted these results in terms of frequency modulation around an underlying mode-locked state of k being 3/4. Using Denjoy's (1932) method for the decomposition of frequency-modulated waves, P. J. Beek proposed a measure of the degree of quasiperiodicity evident in an instance of three-object cascade juggling that assumed that the primary value of k at which frequency locking is attained is .75. The significance of k = 3/4 was recently underscored in an investigation of the acquisition of three-ball cascade juggling (P. J. Beek & Van Santvoord, 1992) with 20 senior undergraduates. All participants were instructed by the same teacher, who was uninformed of the hypothesis under study. Following the initial three learning sessions, the participants were divided into two groups of equal ability, as defined by

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P. J. BEEK AND M. T. TURVEY

the mean number of cycles of cascade juggling achievable at this early stage of learning. Seven further sessions with the same instructor then followed. For one group, the learning was conducted with the aid of a computerized metronome that provided auditory signals for the timings of a throw (or catch) by one hand and a throw (or catch) by the other hand. At the outset of a session, the metronome beeps corresponded to k = 3/4 when the height reached by the balls was 1 m. During the course of a session, a participant could alter at will the interbeep interval to more closely match the timings that he or she was employing. For the other group, there was no metronome assistance. All participants were filmed at the beginning of the 4th session, during the 7th session, and at the end of the 10th session. Analysis revealed that for both groups of subjects, k decreased with practice in the same way: from an average of .77 at Session 4, to an average of .76 at Session 7, to an average of .74 at Session 10 (with betweensubjects standard deviations of .05, .05, and .04, respectively). Collectively, the results from all 10 sessions of this learning experiment suggest that the initial phase of learning to cascade juggle involves discovering the real-time requirements as expressed in the Shannon equation, with the subsequent phase directed at discovering the stability of A: = 3/4 (P. J. Beek & Van Santvoord, 1992). When the results of the learning experiment are coupled with the results from the study of highly skilled jugglers (P. J. Beek, 1989), a picture of threeobject cascade juggling emerges in which fluency is characterized by the ability to operate at k values other than 3/4. In three-ball cascade juggling, k = 3/4 may be the primary fixed point for stable juggling, but the task possesses other sufficiently stable regimes, marked by other k magnitudes, and these are discovered in the course of extended practice.

A "Tiling-of-Time" Perspective on k In the preceding discussion it was noted that specific values of k might be expected from general principles of frequency locking, but no hints were provided as to which ratios should occur and, relatedly, why 3/4 might be predominant. The desired hints can be found, however, in a consideration of the strategy that most effectively guarantees the absence of collisions among juggled objects. In actuality, several strategies are at the juggler's disposal. He or she may introduce a small systematic difference in the absolute values of the release angles at the two hands or an alternating change in the vertical and/or horizontal difference between the cycles of the two hands. Alternatively, the juggler can vary the vertical planes of the juggled objects by a small forward-backward sway. These strategies differ in effectiveness and ease of accomplishment. The most effective and easily accomplished strategy, however, is defined within the basic terms of the Shannon equation, namely, keeping 71 greater than 71. The result of this inequality is that in juggling three objects with two hands, for example, the mean number of objects in the hand will be more than one and the time-averaged number of objects in the air will be less than two (P. J. Beek, 1988). The smaller the average number of objects in the air, the less the chance of collision. Obviously, maintaining a smooth juggling performance will benefit from maintaining a near constant ine-

quality between 71 and 71. The time difference between 71 and Tv may define, therefore, a basic unit of time in the temporal patterning of juggling motions. Pursuant to the preceding, it may be hypothesized that the time difference of imbalance (71 - 71) is frequency locked to the hand-loop time (71 + 71) = TH, that is, W(TL - 71) = (71 + TV). By this hypothesis, the temporal structure of juggling contains a convenient principle addressing the question of how the juggler fills up or tiles the loop time of a hand and the loop time of an object with component times (e.g., those of the various repetitive subtasks) to ensure a smooth juggle. This "tiling of time" principle is as follows: Partition the loop times of objects and hands in multiples of (71 - 71). Given k = 71/( 71 + 71), the tiling principle can be written as W(lk- 1)= l , o r A : = 1/2+ \/2W(W> 1; l/2 2, frequency locking will occur at k values less than 3/4 (.75); for example, for W = 3, k = 2/3 (.67); for W = 4, k = 5/8 (.63); for W = 5, k = 3/5 (.60); and so on. Because N and // do not enter into its expression, the tiling principle should apply regardless of the number of objects being juggled and the number of hands doing the juggling. Likewise, it should apply regardless of the frequency of juggling. Importantly, however, the independence of the form of the tiling principle from TV and H does not mean that the limitations on its realization should be independent of TV and H. In two-handed juggling, as TV increases, the opportunity to operate at values of W other than 2 might be expected to decrease (see below). If the latter is the case, then observed k values for fluent jugglers should converge on 3/4 as TV increases and should do so regardless of frequency.

A "Stability Under Frequency Modulation" Perspective on k In the preceding section we were concerned with developing an understanding of a basis within the temporal structure of juggling for the ratios of 71 to TH. We can also approach the same issue from a different angle, namely, an analysis of the dynamics of the hand during a hand loop. An argument can be made that k follows from stability requirements associated with the oscillating hand as a vibratory system whose system parameters (e.g., inertia and stiffness) are not constant but change during an oscillation. The consequence of these parameter changes is that the rate of change in phase angle varies throughout the cycle. The system exhibits frequency modulation. An intuitive example of such a system is a pendulum bob of mass m attached to a string whose upper end (a) passes through a hole in a horizontally aligned surface and (b) is

937

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CASCADE JUGGLING

raised and lowered harmonically (den Hartog, 1934/1985). The restoring force constant (stiffness, elasticity) s of such a system is mg/l so that a periodic change in / (from the surface with the hole to the bob) means a corresponding change in the constant. It varies in time from a maximum of s + &s to a minimum of 5 — As. The equation of motion is thus mx + (s + As cosust)x = 0, where n/o)s, where con is the system's average natural frequency. With respect to the canonical pendulum case, for certain frequency locks between the pendulum's frequency of motion and the frequency of the parameter modulation, the motions of the pendulum are stable (roughly, disturbances dissipate in time rather than amplify). For other frequency locks the motions are unstable. The hand cycle during juggling is analogous to the pendulum with harmonically changing length in that its parameters are not constant but variable. In part, the variation follows simply from the variation in the mass of the oscillating system—from the mass of the hand alone to that of hand plus object. Inertia, however, is not the only changing parameter. P. J. Beek and W. J. Beek (1988) have shown that the linear and nonlinear stiffness coefficients together with the linear and nonlinear friction coefficients vary within the hand loop. Local stiffness changes would arise, for example, from the particular and different cocontractions of muscles (Feldman, 1986) acting about the various joints to effect the throws, transports, and catches. Given the multiplicity of sources of parameter change, a complex "ripple" on the phase progression during the hand loop is expected. There is, in short, considerable potential for instability in the hand loop. As with the pendulum example, whether stable motion results depends on the frequency lock between the average natural frequency of the juggling hand and the frequency of parameter modulation. An analysis is presented below showing that the frequency lock that governs the stability of the Mathieu-Hill dynamic of the hand loop in juggling is selective of k = 3/4. The analysis is prefaced by a consideration of the following question: What kind of stability is required, and how is it affected by N? In general terms, the spatiotemporal constraints of juggling require that the frequency and amplitude of the hand movements be relatively stable. With H = 2, these stability requirements will increase as N gets larger. Self-evidently, for a given N, the objects must be thrown to such a height that sufficient time is available to execute the juggling pattern. Shannon's equation implies that the set of ratios of 71/71 and 71/71 over which this can be accomplished decreases as a function of N/H. Figure 1 illustrates this fact. The intersections with the x-axis (71 = 0) represent "hot potatoes juggling," for which 71/71 = H/N, whereas the intersections with the yaxis (71 = 0) represent "delayed juggling," for which 71/71 = H/(N — H). The ratio of these two extremes expresses the juggler's freedom to vary his or her speed between the fastest and the slowest juggling times given a fixed 71 (Buhler & Graham, 1984). According to the ratio [H/N]/[H/(N - //)], which equals (N - H)/N, the range of possible juggling speeds

2_

LL 1_

VTF Figure 1. The implications of Shannon's equation of constraint depicted in the time loaded/time in flight (TL/TF) against time unloaded/time in flight (Tu/TF) space for three-, five-, and sevenobject juggling. (Note that for convenience the scale of the x-axis is twice as large as the scale of the y-axis.)

increases with H and decreases with N. Conversely, given a fixed hand-cycle time (71 + 7\j), the ratio of the shortest and the longest possible 71 (from 71 = 0 to 71 = 0) is expressed by [N/H]/[N/H - 1], which equals N/(N - H), showing that the range of possible Tf increases with H and decreases with N. Hence, as A' increases, the need for exact reproducibility of Tf becomes more severe. Not only do the objects need to be tossed away more forcefully to create the required time to execute the pattern, they also need to be thrown to an exact height for the juggler to be able to stay within a narrowly defined range of hand-cycle times. Small deflections in the desired velocity angle of the object at release will have a considerable effect on the space-time location at which it is to be caught. The juggler therefore has to control carefully both the amplitude and the frequency of the hand motion in relation to the velocity vector of the object. W. J. Beek and P. J. Beek (1991) have suggested the following Mathieu-Hill equation for cascade juggling based on the conceptualization of two hands as harmonic oscillators (e.g., Whittaker & Watson, 1962): d2 0) and vanishes at the positions in the hand loop at which catch and release occur. In other

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P. J. BEEK AND M. T. TURVEY

words, an accurate catch and a steady throw require that the modulated behavior of the hand with the ball is harmonic at ball interception and ball release. If n is the number of consecutive catches (or throws) after which the initial situation (an arbitrary point from which the measurement of the hand loop is begun, but defined most conveniently at the discrete temporal locations of ball interception and ball release) is regained in full, then the requirement can be met by obeying the following rule: m(A)u>T[ = ir/2 + mr, or T{/TU = I/4m + n/2m. where A —> 0. The preceding equations, it must be emphasized, are first-order approximations for the modulated hand loop. Nonetheless, they reveal the kinds of temporal structure that will arise from satisfying the accuracy demands of juggling. Specifically, they reveal that when modulation is to be restricted, discrete ratios between characteristic time scales have to be adhered to, resulting in particular ratios of TI./TH. For m = 1 and n = 1, T\JT\\ = 3/4—meaning that when the angular frequency of modulation is phase locked to the hand loop, stable, precise throwing and catching are characterized by holding onto a juggled object for 3/4 of the cvcle time of the hand.

and seven objects. We expected to see less variety in k and a convergence on 3/4 as N increased from three to seven. In addition, we evaluated the prediction, implicit in the tiling and frequency modulation perspectives, that observed timing ratios would be indifferent (within limits) to the actual force structures involved—that they would be kinematic constants achieved through variable kinetics. Finally, we investigated the upper limits on k in three-object cascade juggling at extremely slow frequencies.

Experiment 1 The first experiment focused on three-ball cascade juggling at three frequencies. The experiment was a replication of that reported by P. J. Beek (1989). In the previous experimental investigation of the condition N/H = 3/2, k was found to depend on the juggling frequency. We expected that a similar pattern of results would be found when the data were provided by another, different group of skilled jugglers.

Method Predictions From Considerations of Tiling and Frequency Modulation In the preceding sections, we raised the question of how a juggler knows how much of the cycle time of a hand should best be occupied by holding an object. Two complementary perspectives on this question have been developed, both shaped by Shannon's equation of constraint for juggling and by considerations of the physical requirements for fitting together repetitive subtasks to produce a smooth, stable dynamic. According to the tiling perspective, the conjunction of task constraint and general principles of frequency locking prescribes optimal performance (i.e., best catching and aiming) as holding a juggled object for 1/2 + l/2H 7 of the hand's cycle time, regardless of A', //, and frequency of juggling, where W vs, an integer greater than 1. The tiling perspective, therefore, identifies that although in principle, k can assume a variety of magnitudes, it should most commonly assume magnitudes associated with integer values of H7 because these are fundamental to a stable dynamic. In contrast, the frequency modulation perspective as developed thus far is mute on the potential forms of variation in k. Its primary focus is on the magnitude that k must take if task demands are such that any perturbation at release caused by parameter variation would impugn performance. The demands on aiming precision and reliability become more severe with increases in N, because the need to throw the objects higher (to provide time for the coordination) means that a slightly off-target throw will have larger consequences. Convergence of the two perspectives occurs with respect to what is expected at higher N. From the tiling perspective, the magnitude of W at which mode locking can be achieved is inversely related to N; when H7 assumes its least integer value, then k = 3/4, as predicted from the frequency modulation perspective. In the present experiments, we evaluated the preceding predictions with skilled jugglers performing with three, five,

Subjects. Four jugglers participated. They were selected for their skilled competency in juggling balls and clubs. All four were regular stage and street performers. Procedure. The participants were required to cascade juggle three tennis balls (weight = 120 g and diameter = 6.7 cm) in their preferred frequency mode. Subsequently, the jugglers were instructed to increase or to decrease the cycling rate from the preferred frequency mode to low- and high-frequency modes without allowing the smoothness of operation to be affected. Otherwise, the jugglers were free to choose the preferred, low, and high frequencies of juggling with which they felt comfortable. Data collection and reduction. The subjects were filmed during at least three cycles of juggling (one complete ball cycle consisted of Ar x H hand cycles) in each of the three frequency modes with a 16mm motion picture camera (Bolex type H 16 M, Paillard AG, SainteCroix, Switzerland) operating at a frame rate of 64 Hz. The optical axis of the camera was perpendicular to the plane of motion of the balls, directly in front of the subject at a distance of about 4 m. In order to later check the frame rate of the camera, a 2-Hz flashing light was placed near to the juggler. The gravitational vertical was defined by a plumb line suspended from a nearby support. The films (Kodak 4X Reversal 400 ASA) were projected by means of a 16-mm projector (NAC type DF-16b) onto an opaque screen. Mounted on the screen was an SAC 14-in. x-y graph, connected to an Apple lie microcomputer. Frame-by-frame analysis of the position of the balls and the finger landmarks allowed the coordinates of these points to be read into the computer and stored on floppy disk. For each of the three frequencies, three of the recorded cycles were digitized. Checks were made to determine whether there were any visible major adjustments during these cycles (such as an obviously bad catch, an off-target throw, or a collision of balls), which was never the case. Before digitizing the movie frames, we adjusted the orientation of the projection device to align the vertical orientation visible in the first frame of the selected trial with the y coordinates of the tablet. The data were transferred later to a much faster Cyber 175750 mainframe computer for further processing. The displacement data were filtered with a recursive, second-order Butterworth filter with a cutoff frequency of 8 Hz. This procedure was performed twice to eliminate the phase shift (Lees, 1980; Wood, 1982). The first

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CASCADE JUGGLING derivative of the smoothed displacement data was computed by means of a 5-point central-difference algorithm. A peak-finding algorithm was applied to the velocity data of the balls in the vertical ( y ) direction of motion in order to identify the temporal location of the moments of release (positive peak velocity) and the moments of catch (negative peak velocity). From these temporal locations, the individual values of the time components 71, 7\j, TV, and the timing ratio k were computed for each sequence of cycles, resulting in 18 values for each time component in each threeball condition (2 hands x 3 balls x 3 cycles). (Each subject, therefore, provided a total of 54 observations on each of the relevant time quantities, given that there were three frequency conditions.) In the computation of k, a hand cycle was said to begin with a catch. Subsequently, the mean values of 71 (= ZTl,/«), Tv (= £71,/n), TV (= £7Vi/n), and k (= £fc/rc) and their standard deviations were computed. The inaccuracy of measuring these time variables was one frame at most, which corresponded to maximal errors (i.e., for TVl) on the order of 5%; because of interdependency of errors, the maximal error in estimating the timing ratio k, was on the order of 1 %. Multiple checks on the film rate of the camera—with the help of the 2-Hz flashing light—secured the reliability of the frame-to-frame time interval used in the computation. From the mean flight time of the balls, the mean height of throw was calculated in meters and compared with the mean height of the throw in the arbitrary units of the x-y graph. All x- and y-coordinates were multiplied by the ratio of these two heights to obtain data in real space.

Results and Discussion The data of the 4 subjects are reported in Table 1. As can be seen, the subjects were able to comply with the instruction to perform at three distinct frequencies, one higher and one lower than the preferred. An analysis of variance (ANOVA) confirmed that the instructions to juggle at different frequencies did produce significant frequency differences: high = 2.35 Hz, preferred = 1.60 Hz, and low = 1.07 Hz, F(2, 9) = 6.62, p < .05. The corresponding frequency values for the subjects in P. J. Beek (1989) were as follows: high = 2.30 Hz, preferred

= 1.66 Hz, and low = 1.03 Hz. Inspection of Table 1 also reveals that the mean k values differed across juggling frequencies (range = .54 to .86). An ANOVA on the k values substantiated the main effect of frequency on k: high = .61, preferred = .69, and low = .78, F(2, 9) = 10.51, p < .01. The mean k was .69 (compared with the mean of .71 for the subjects in P. J. Beck's study). A linear regression of A: on TH yielded a k of .23TH + .54, r 2 ( l l ) = .57, p < .01. Linear regressions on the individual subject data revealed that all subjects showed similar relations of k to TV, the slopes varied between .13 and .61 and were all significant (p < .05). The slope and coefficient of the k-TH relation were closely similar to those of the linear regression on the 4 subjects studied by P. J. Beek (see Introduction). In sum, the results of Experiment 1 were in agreement with those obtained by P. J. Beek: For skilled juggling under N/H = 3/2, k assumed a number of values that appeared to be dependent on frequency.

Experiment 2 An essential aspect of the tiling perspective provided in the Introduction is that the principles governing k are invariant over variations in the magnitudes of the forces involved in the catching, carrying, and throwing of juggled objects. To investigate this prediction, in Experiment 2 we manipulated the weight of the juggled objects to induce variations in the forces jugglers required in handling the objects. As had been the case in Experiment 1, variation in k was expected. The particular values of k, however, were not expected to depend on object weight.

Method Subjects. Five skilled jugglers participated. Procedure. The participants were required to cascade juggle with three different sets of so-called stage balls at their preferred frequency

Table 1 Means and Standard Deviations for the Time Components (in Seconds) and the Timing Ratio (k; Dimension/ess) for Three Juggling Frequencies With Number of Objects/Number of Hands = 3/2 in Experiment 1 Time unloaded Subject/frequency HT High Preferred Low NN High Preferred Low

RD High Preferred Low BT High Preferred Low

Time loaded

Time of flight

k

M

SD

M

SD

M

SD

M

SD

.156 .206 .173

.033 .032 .016

.232 .482 .588

.030 .035 .021

.355 .550 .557

.027 .039 .017

.60 .70 .77

.055 .034 .020

.126 .169 .184

.022 .026 .024

.260

.018 .024 .028

.320 .495

.019 .020 .033

.67 .72 .70

.046 .030 .036

.212 .206 .136

.033 .035 .026

.244

.411 .812

.037 .034 .051

.334 .516 .609

.028 .040 .026

.54 .67 .86

.055 .042 .026

.170 .198 .292

.038 .044 .032

.299 .395 1.110

.040 .037 .045

.403 .494 .994

.025 .022 .026

.64 .67 .79

.059 .068 .020

.437 .438

.471

This document is copyrighted by the American Psychological Association or one of its allied publishers. This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.

940

P. J. BEEK AND M. T. TURVEY

mode, for at least five complete juggling cycles. The first set consisted of stage balls with the standard weight of 130 g, the balls in the second set weighed 300 g and contained a mixture of shell sand and lead shot, and the balls in the third set weighed 550 g and contained only lead shot. All balls had the same diameter (7.3 cm) and color (white). The order in which the different sets of balls were manipulated by the subjects was randomized. Data collection and reduction. The method of filming was as described in Experiment 1. The recording apparatus was a 16-mm high-speed motion picture camera (Teledyne DBM 55, Teledyne Camera Systems) operating at a frame rate of 100 Hz. We computed the values of 71, TU, TV, and k from the films in the same manner as in Experiment 1, using another motion analyzer (Supergrid Digitizer SPG-1212-RP, Summagraphics Corporation). Because of the higher sampling rate, the error of measurement was slightly smaller in this experiment than in Experiment 1.

Results and Discussion For each subject, 30 hand cycles and their corresponding ball flights were analyzed, leading to 30 values (2 hands x 3 balls x 5 cycles) for each time component and k in each condition. The data of the 5 subjects are reported in Table 2. As can be seen, there were individual differences in the k values attained by the subjects: 3 subjects attained values close to a k of .67 (W = 3), and 1 subject juggled close to a A: of .75 (W = 2); the remaining subject was the only 1 who showed a marked tendency to juggle at larger k values with the heavier balls. Analysis of variance revealed a significant subject effect on both frequency, F(4, 6) = 109.77, p < .0001, and k, F(4, 8) = 13.18, p < .001. The weight manipulation, however, proved to have no effect on either the preferred frequency elected, F(2, 8) < 1, or k, F(2, 8) = 1.67, p> .05.

It might be argued that the absence of a weight effect was due simply to the range of weights used. Given a larger range, weight effects might have appeared. Although this possibility cannot be ruled out, it is important to underscore that weights much heavier than the heaviest weight of 550 g used in Experiment 2 introduce special difficulties for jugglers. Juggling three objects weighing 550 g each means keeping 1.65 kg aloft continuously. If the objects each weighed 800 g, for example, then 2.4 kg have to be kept in the air. For jugglers of ordinary build, the requirements of juggling objects of approximately 2 kg or greater are fatiguing, stress the wrists, and lead to impaired coordination. The point is that the range of object weights used in Experiment 2 was close to the maximum range over which weight variations can be manipulated without introducing significant noncoordination factors. With the foregoing considerations in mind, the outcome of Experiment 2 may be interpreted as consonant with the hypothesis that, for the ordinary circumstances of cascade juggling, k is governed by principles that are invariant over variations in the forces associated with catching, throwing, and transporting the juggled objects. To summarize the results of Experiments 1 and 2 in the light of the theory proposed, the individual-trial k values for each subject in Experiments 1 and 2 were collected in a frequency polygon (Figure 2). Inspection of the k distribution in the two experiments suggests three overlapping subdistributions concentrated on k values of .61, .69, and .75. The trimodal distribution and the values of the peaks are consistent with the understanding that the tiling principle k= 1/2 + \J2W entails regions of frequency entrainment for integer values of W > 2. From the data presented in Figure 2, a

Table 2 Means and Standard Deviations for the Time Components (in Seconds) and the Timing Ratio (k; Dimensionless) for Juggling Three Different Weights (in Grams) With Number of Objects/Number of Hands = 3/2 in Experiment 2 Time unloaded Subject/weight LE 130 300 550 RB 130 300 550

Time loaded

k

Time in flight

M

SD

M

SD

M

SD

M

SD

.243 .232 .226

.027 .024 .019

All .471 .495

.034 .031 .026

.608 .580 .588

.039 .036 .028

.66 .67 .69

.033 .035 .033

.233 .220 .225

.026 .023 .017

.434 .455 .482

.018 .019 .022

.566 .558 .580

.014 .018 .021

.65 .67 .68

.033 .028 .021

.179 .172 .184

.030 .028 .033

.355 .343 .368

.033 .029 .030

.446 .430 .460

.032 .031 .035

.67 .67 .67

.049 .047 .045

.171 .145 .130

.025 .016 .022

.386 .419 .450

.021 .014 .025

.452 .427 .433

.024 .018 .026

.69 .74 .78

.038 .019 .020

.178 .167

.026 .034 .026

.596 .615 .565

.045 .042 .039

.568 .558 .545

.053 .050 .048

.77 .79 .76

.034 .049 .034

TN

130 300 550 BT 130 300 550 FR 130 300 550

.177

CASCADE JUGGLING 100-1

balls, 30 (instead of 18) values for each time component were obtained in each five-ball condition (2 hands x 5 balls x 3 cycles).

k = .69 -n

A' \

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Temporal patterning in cascade juggling.

A key variable in cascade juggling is the proportion of time that a juggler holds onto a juggled object during a hand cycle, that is, the time from ca...
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