Spatiotemporal Variability in Cascade Juggling

acta psychologica ELSEVIER Acta Psychologica91 (1996) 131-151 Spatiotemporal variability in cascade juggling a,b A. (Tony) A.M. van Santvoord a,*, Peter J. Beek a Department of Psychology, Faculty of Human Movement Sciences, Free University, Van der Boechorststraat 9, 1081 BT Amsterdam, The Netherlands b Center for the Ecological Study of Perception and Action, University of Connecticut, Storrs, CT, USA Received 14 March 1994; revised 3 June 1994; accepted 28 July 1994 Abstract Three expert jugglers and three intermediate jugglers performed a three-ball cascade pattern under spatially and temporally constrained conditions. In the spatially constrained conditions the balls were thrown to a specific height, whereas in the temporally constrained conditions the balls were thrown to the beeps of a metronome. The experiment was conducted to examine the hypothesis that juggling represents a spatial clock in that jugglers attempt to set up an invariant time base for the hand movements by throwing the balls consistently to a fixed height. Specifically, two expectations following from this hypothesis were examined: (1) that the spatiotemporal variability of the produced patterns would be less when juggling to an externally specified height than to an externally specified beat, because throwing to a height would be more in line with what jugglers actually do, and (2) that in both conditions the space-time trajectories of the balls would be less variable than the space-time trajectories of the hands. Examination of the observed patterns in terms of a set of theoretically motivated variables confirmed the second expectation. At the level of the individual variables the first expectation was not confirmed by the data: the spatiotemporal variability of the patterns was very similar under the two conditions. However, at the level of ensemble variables, the variability of the ball loop time (defined as the time that the ball was carded by the hand plus the subsequent flight time) was smaller when juggling to a height than when juggling to a beat, while the variability of the hand loop time (defined as the time that the hand carded a ball plus the time that it moved empty) was the same. These results were largely independent of skill level; only a few differences between expert and intermediate jugglers were found. The implications of the findings with regard to the development of a theory of perceptual-motor control in which spatial and temporal variables are linked in a task-specific manner are discussed. * Corresponding author. Fax: +31 20 4445867, Tel: +31 20 4448532. 0001-6918/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0001-691 8(94)00044- I 132 A.A.M. van Santvoord, P.I. Beek /Acta Psychologica 91 (1996) 131-151 PsyclNFO classification: 2330 Keywords: Motor control; Motor variability; Motor learning; Dynamical systems theory I. Introduction How the spatial and temporal properties of movements are controlled in the execution of actions is a fundamental problem in 'motor control'. Broadly speaking, two types of accounts may be distinguished: those referring to processes internal to the human motor system and those addressing the behavioral level of the movements themselves. To the former class belong inferential structures, such as cognitive constructs (i.e., motor programs, schemata) and neurophysiological mechanisms (i.e., CPGs, neural circuits), which are deemed responsible for the spatiotemporal regularities observed at the behavioral level. The latter class consists of a variety of non-inferential constructs varying from the lawful regularities formulated in the human performance literature (e.g., Fitts' law, Fitts, 1954), to the lawful regularities between perceptual variables and movement identified by exponents of the ecological approach to perception and action (e.g., Turvey, 1990), and the (conceptually related) mathematical models proposed by exponents of the dynamical approach to movement coordination (e.g., Haken et al., 1985). In the present article, which examines spatiotemporal patterns in cascade juggling, the concern is with the behavioral level per se. No inferential constructs will be proposed to account for the observed relations. The approach adopted is a descriptive one that seeks to identify general principles reflecting the organization of the human action system in accomplishing specific task goals. A theoretically important and challenging issue in understanding the space-time properties of movement is that of the spatiotemporal variability of performance. Every movement, simple or complex, is characterized by a certain variability in its spatial and temporal dimensions. With regard to this general theme, the study of discrete targeting movements has led to the identification of what is generally known as the speed-accuracy trade-off in movement: Movement is more accurate at low speeds and less accurate at high speeds. As noted by Newell et al. (1993), early investigations of the speed-accuracy trade-off focused almost exclusively on spatial error (e.g., Beggs and Howarth, 1972; Fitts, 1954; Keele, 1968; Woodworth, 1899), but by now the effects of movement speed on both spatial and temporal variability are well documented (e.g., Hancock and Newell, 1985, Newell, 1980; Newell et al., 1979, Newell et al., 1980, Newell et al., 1982; Schmidt et al., 1979). A general conclusion from this work is that the degree of spatial and temporal variability of movement in general and the exact form of the speed-accuracy trade-off in particular are a function of the task and the nature of the task instructions (e.g., Carlton, 1994; Newell et al., 1993; Zelaznik et al., 1988). Clearly, the prevailing impression that constant error shifts and speed-accuracy functions are largely task-dependent poses a problem for the formation of an encompassing theory of variability in motor control. As a prerequisite for the development of such a theory it seems important, if not necessary, to investigate the task-specificity of spatiotemporal variability in a variety of different task domains, including both discrete and continuous A.A.M. van Santooord, PJ. Beek /Acta Psychologica 91 (1996) 131-151 133 'T 1 h F q I D Fig. 1. Schematic representation of juggling three balls in a figure-eight pattern, momentary situation: ~, is the angle of release; Vo is the velocity of the ball at the moment of release; h is the height to which the balls are thrown relative to the point of release; F is the base width of the flight parabola, and D is the width of the elliptical hand movement. perceptual-motor actions. Experimental studies are called for that examine how the space-time patterns within each of these domains are adapted when relevant task constraints are manipulated. In the present study, space-time patterns in cascade juggling are examined and compared under both spatially and temporally constrained conditions. In the spatially constrained conditions jugglers throw the balls to a fixed height (indicated by target markers), while in the temporally constrained conditions jugglers throw the balls in synchrony with the beeps of a metronome. The basic question is: Are the spatiotemporal properties of a juggler's performance affected by these task manipulations, and, if so, how? In the temporally constrained condition, does the temporal variability decrease and the spatial variability increase, whereas the obverse is true in the spatially constrained condition? Our expectations with regard to the space-time relations that will be observed under these conditions are based on our previous analyses of cascade juggling. Cascade juggling is a complex, cyclic activity in which the hands move along more or less elliptical trajectories, one evolving clockwise and the other anticlockwise (see Fig. 1). The objects (hereafter referred to as 'balls') are released at the inside of the ellipses and caught at the outside. The balls fly to the other hand along a parabolic trajectory. In the simplest version, in which two hands juggle three balls, the two parabolic flight paths i intersect at a point close to the points of release. As a result, the balls travel in the figure-eight pattern that is characteristic for the cascade. Fig. 1 defines the essential spatial variables of juggling: the height (h) to which the balls are thrown relative to the points of release, the angle of release (9,), the base width (F) of the parabolic flight trajectories, i.e., the horizontal distance between throwing and We reserve the term 'path' for the curve in space traced out by the ball or the hand and the term 'trajectory' for the curve in space traced out by the ball or the hand and its time history. 134 A.A.M. van Santvoord, PJ. Beek /Acta Psychologica 91 (1996) 131-151 catching of the same ball, and the width (D) ('amplitude') of the elliptical hand trajectories, i.e., the horizontal distance between throwing and catching by the same hand. Essential temporal variables of the task are the flight time of a ball between the hands (TF), the time that a hand is filled with a ball between a catch and a throw (time loaded: TL), and the time a hand moves empty between a throw and a catch (time unloaded: T v). A relevant additional variable is the velocity of the ball at release (V0), because it determines, together with the angle of release and the acceleration due to gravity (g), the parabolic flight trajectory. Obviously, these variables are related. The basic spatial equation of juggling can be derived from the parabolic flight path of the balls, which is described by F = VxoTr, 2Vro = gT F, and 8h = gT2F, and reads F cot 9r = 4-h (1) assuming that catching and throwing occur at the same height and that air friction is negligible (Beek, 1989a). Both F and D are associated with the catch of a particular ball. However, D is related to the movements of one hand and connects two consecutive ball flights, whereas F is related to a ball flight and connects the movements of the two hands.

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