Temporal and spatial factors reflecting performance improvement during learning three-ball cascade juggling

Beek and van Santvoord [Beek, P. J., & van Santvoord, A. A. M. (1992). Journal of Motor Behavior, 24, 85-94] proposed a three-stage model of learning to juggle based on group analyses of temporal measures. Here, we examined in detail how the temporal and spatial features of juggling evolved in eight individual participants progressing from the second to the third stage of learning. During the second stage, the dwell ratio, defined as the ratio of the time that the juggler holds a ball between catch and toss and the hand cycle time (HCT), was stable when it was about 0.83. The subjects with a dwell ratio near this value and controlled throws exhibited stable juggling, whereas the subjects with a dwell ratio of 0.80 or smaller exhibited unstable juggling. Compared to the former group, the latter group had a longer time from the throw of a ball to the arrival at its zenith (TZ), and a shorter time between the arrival of an airborne ball at its zenith and the subsequent throw (IZR). The latter group also exhibited larger variability in the dwell ratio and IZR. With practice, the subjects appropriated, on average, the duration of TZ and IZR to the dwell ratio and improved the ability to accurately throw balls by changing the motions of the limb segments involved. Although these changes helped to stabilize the performance during the second stage, the variability problem was not sufficiently resolved. Only two out of eight subjects passed on to the third stage by the last (10th) Session. They achieved small variability in IZR, dwell ratio, and flight paths of the ball while juggling with short HCTs and small dwell ratios. These results suggest that the reduction of variability in these variables was essential to pass on to the third stage.     

Phasing and the Pickup of Optical Information in Cascade Juggling

Four experiments were conducted to examine the relationship between the phasing of hand movements and the pickup of optical information in cascade juggling. Three jugglers of intermediate skill juggled three balls while wearing liquid crystal (LC) glasses that opened and closed at preset intervals. The first experiment, in which the duration of the viewing window was gradually reduced to zero, revealed a preference for seeing the segment of the ball flight following the zenith in one subject; such a preference was hinted at in the other two subjects. The second experiment, in which the tachistoscopic rhythm of the glasses was perturbed, showed that, in the case of a stable phase lock, the phasing of the hand movements was adjusted to restore the visibility of the segment following the zenith when it was lost. The third experiment, however, revealed that, after practice, the jugglers did not become better attuned to the optical information contained in this segment. The fourth experiment, in which two jugglers per- formed a cascade together while viewing the ball flights intermittently, suggested that haptic information about the trajectories of the balls to be caught is not necessary for subsequent catching: Optical information picked up during brief intervals of viewing was sufficient to perform the task equally well as when they juggled alone (i.e., when haptic information about the throws was available). Although, admittedly, the results raised only a tip of the veil covering the perceptual basis of juggling, they testify to the potential power of the new technique that was used to let subjects themselves reveal what optical information is relevant for performance.     

Coordination changes in the early stages of learning to cascade juggle

The experiment was setup to examine the coordination changes in assembling the movement form of 3-ball cascade juggling. Eight adult participants learned to juggle over 4 weeks of practice. Juggling scores were recorded at each session and performance was videotaped at eight selected sessions for purposes of movement analysis. Once the basic spatial and temporal constraints on cascade juggling were satisfied, and the figure-8 juggling mode was established, temporal modulations of the relative motions of the hands were emphasized. All participants learned to juggle and the increase over practice in the number of consecutive balls caught was best fit with a power law. The non-proportional rate of performance increment was consistent with the qualitative changes in the form of the hand and ball movement kinematics that occurred over practice.     

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 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.     

Timing and Phase Locking in Cascade Juggling

A natural-physical approach is pursued in uncovering basic timing and phase relations in human rhythmic movement. The approach is based on the theory of nonlinear oscillatory motion, entrained by continuously and discretely distrib- uted forcing. In the context of juggling three balls in a figure-eight pattern, a preliminary modeling attempt of the cyclical hand motion suggested that the dynamics underwriting juggling are captured best by a discretely kicked, highly nonlinear, self-sustained oscillator. Discretely kicked, nonlinear oscillators may be characterized by regime diagrams that depict the periodic (phase-locked) and quasiperiodic (not phase-locked) regimes in which the system can operate depending on the magnitude of the kicks. This article provides evidence for 2-quasiperiodicity and near, but not perfect, phase locking between tl/tf and tu/tf (where tl is the mean time that the hands move loaded with a ball, tu is the mean time that the hands move empty, and tf is the mean flight time of the balls). Jugglers perform along the boundaries of Arnol'd tongues (representing complete phase locking) in a regime diagram without actually entering into them. With the help of Denjoy's decomposition of phase modulation into a fast and a slow mode, the deviation from the potential minimum defined by complete phase locking can be understood. The frequency ratios within the continuous relative phase between the two juggling hands reveal a Farey type of phase-locking structure, allowing a qualitative insight into which regimes jugglers position themselves when asked to speed up or slow down their act. Modulation of the hand movements increases when timing constraints become more severe (e.g., when the number of balls in the air increases). The modified standard map promises to he an adequate tool in analyzing the phase progression in juggling. All in all, the results favor an understanding of rhythmic movement in terms of discretely forced, nonlinear dynamics, rather than fully autonomous, self-sustaining oscillators.     

Goal-directed imitation: the means to an end

The effects of goal-directed imitation and observational learning were examined whilst learning a goal-directed motor skill (three-ball cascade juggling). An observational learning (OL) group observed a model and a control (CON) group received minimal verbal instructions regarding how to hold and release the juggling balls. The OL group performed more juggling cycles across practice and retention than the CON group. In addition, the OL group's upper limb coordination and ball flight trajectory pattern were more similar to the model's movements than the CON group. These data show that when the to-be-learnt movement pattern and end-goal are not specified by the task's mechanical constraints, or can be achieved by modifying a pre-existing motor skill, individuals have difficulty learning on the basis of discovery processes alone. Under these circumstances, observational learning is effective because it conveys to the individual the specific means by which the end-goal can be achieved. These findings lead us to suggest that when the end-goal and the means to achieve the end-goal are directly linked, the means are given sufficient weight in the goal hierarchy such that the model's movement is imitated.     

Transfer, control, and automatic processing in a complex motor task: an examination of bounce juggling

The authors evaluated the hypothesis that controlled and automatic processes are opposite ends of a continuum of learning (e.g., R. M. Shiffrin & W. Schneider, 1977) vs. an alternative, concurrent emergence hypothesis (e.g., J. M. Bebko et al., 2003; G. Logan, 1989). The authors also measured potential positive transfer effects of learning from one motor task to another. Four experienced cascade jugglers and 5 novices learned to bounce juggle, practicing regularly for 5 weeks. The experienced jugglers showed positive transfer of learning, maintaining a lead of approximately 6-10 days over the novices, even as both groups automatized the new skill. Measures of automatic and controlled processing were positively correlated, indicating that those processes emerge concurrently. The authors present a model in which controlled and automatic processes emerge orthogonally.     

Pattern Stability and Error Correction During In-Phase and Antiphase Four-Ball Juggling

The authors studied pattern stability and error correction during in-phase and antiphase 4-ball fountain juggling. To obtain ball trajectories, they made and digitized high-speed film recordings of 4 highly skilled participants juggling at 3 different heights (and thus different frequencies). From those ball trajectories, the authors determined and analyzed critical events (i.e., toss, zenith, catch, and toss onset) in terms of variability of point estimates of relative phase and temporal correlations. Contrary to common findings on basic instances of rhythmic interlimb coordination, in-phase and antiphase patterns were equally variable (i.e., stable). Consistent with previous findings, however, pattern stability decreased with increasing frequency. In contrast to previous results for 3-ball cascade juggling, negative lag-one correlations for catch-catch intervals were absent, but the authors obtained evidence for error corrections between catches and toss onsets. That finding may have reflected participants' high skill level, which yielded smaller errors that allowed for corrections later in the hand cycle.     

Pattern stability and error correction during in-phase and antiphase four-ball juggling

The authors studied pattern stability and error correction during in-phase and antiphase 4-ball fountain juggling. To obtain ball trajectories, they made and digitized high-speed film recordings of 4 highly skilled participants juggling at 3 different heights (and thus different frequencies). From those ball trajectories, the authors determined and analyzed critical events (i.e., toss, zenith, catch, and toss onset) in terms of variability of point estimates of relative phase and temporal correlations. Contrary to common findings on basic instances of rhythmic interlimb coordination, in-phase and antiphase patterns were equally variable (i.e., stable). Consistent with previous findings, however, pattern stability decreased with increasing frequency. In contrast to previous results for 3-ball cascade juggling, negative lag-one correlations for catch-catch intervals were absent, but the authors obtained evidence for error corrections between catches and toss onsets. That finding may have reflected participants' high skill level, which yielded smaller errors that allowed for corrections later in the hand cycle.     

One-Handed Juggling: A Dynamical Approach to a Rhythmic Movement Task

The skill of rhythmically juggling a ball on a racket was investigated from the viewpoint of nonlinear dynamics. The difference equations that model the dynamical system were analyzed by means of local and nonlocal stability analyses. These analyses showed that the task dynamics offer an economical juggling pattern that is stable even for open-loop actuator motion. For this pattern, two types of predictions were extracted: (a) Stable periodic bouncing is sufficiently characterized by a negative acceleration of the racket at the moment of impact with the ball, and (b) a nonlinear scaling relation maps different juggling trajectories onto one topologically equivalent dynamical system. The relevance of these results for the human control of action was evaluated in an experiment in which subjects (N = 6) performed a comparable task of juggling a ball on a paddle. Task manipulations involved different juggling heights and gravity conditions of the ball. The following predictions were confirmed: (a) For stable rhythmic performance, the paddle's acceleration at impact is negative and fluctuations of the impact acceleration follow predictions from global stability analysis; and (b) for each subject, the realizations of juggling for the different experimental conditions are related by the scaling relation. These results permit one to conclude that humans reliably exploit the stable solutions inherent to the dynamics of the given task and do not overrule these dynamics by other control mechanisms. The dynamical scaling serves as an efficient principle for generating different movement realizations from only a few parameter changes and is discussed as a dynamical formalization of the principle of motor equivalence.     

Multiple time scales and subsystem embedding in the learning of juggling

To gain insight into the multiform dynamics and integration of remote yet pertinent subsystems into the performance of complex perceptual-motor skills, we recently conducted a series of longitudinal and cross-sectional experiments on the acquisition of 3-ball cascade juggling in which we measured, next to the ball trajectories, postural sway, eye and head movements and respiration. The aim of the present paper is to review the main results and theoretical implications of these experimental studies for understanding skill acquisition. As regards the evolution of the quality of the juggling itself, we found that only certain aspects of throwing and catching were adjusted, while the goal behavior of sustained juggling (operationalized as the number of consecutive throws) and the degree of frequency and phase locking between the ball trajectories, indexing pattern stability, increased monotonically. The latter three aspects evolved at different rates, reflecting the existence of a temporal hierarchy in learning. Postural sway exhibited initial manifestations of task-specific, possibly mechanically induced, modes of 3:1 and 3:2 frequency locking with the ball trajectories and only few transitions between those modes. Functional stability appeared to be enhanced during practice by minimizing the sway amplitudes rather than by adjusting the sway dynamics itself. Eye and point-of-gaze movements also showed instances of 3:1 and 3:2 frequency locking with the ball trajectories; especially establishing a 3:1 locking (horizontal eye movements) appeared to be important. Expert behavior suggested that extended practice promotes reliance on multiple sources of information, allowing the proficient juggler to switch adaptively between functional organizations involving distinct perceptual systems. No consistent coordination between breathing and juggling was found. It was concluded that multiform dynamics, involving hierarchically ordered time scales, underlie the acquisition of complex skills and that the subsystems subserving realization of the task goal become assembled and embedded in a task- and subsystem-specific manner.     

Dynamical aspects of learning an interlimb rhythmic movement pattern

Learning a bimanual rhythmic task is explored from the perspective that motor skill acquisition involves the successive reparameterization of a dynamical control structure in the direction of increasing stability, where the intentional process of reparameterization is itself dynamical. Subjects learned to oscillate pendulums held in the right and left hands such that the right hand frequency was twice that of the left (2:1 frequency lock). Over 12 learning sessions of 20 trials each, we interpreted the decreasing fluctuations in the frequency locking to be an index of the increasing concavity of the underlying potential, a measure of stability; the time required to achieve the 2: 1 pattern was interpreted as indexing the relaxation time of an intentional dynamic. Power spectral analyses of the phase velocity ratio exhibited two strategies for acquiring the interlimb movement pattern: (a) adding spectral peaks at integer multiples of the left hand frequency or (b) distributing power across many frequencies in a l/f-like manner. Results are discussed in terms of the promise of a dynamical approach to learning coordinated movements.     

Does transfer of training help children learn juggling?

The effectiveness of four different learning sequences in teaching juggling to 5th grade students was investigated. Practice schedules using combinations of scarves, weighted scarves, beanbags, and balls allowed both identical task elements and learners' time-on-task to be varied during a 3-wk. practice period. The exclusive use of beanbags for practice, prior to testing with balls, resulted in significantly better juggling scores than did the other combinations. While transfer of training occurred for other groups, the amount of students' time-on-task during practice appeared to be a strong influence on final performance.     

Qualitative and quantitative change in the dynamics of motor learning

The experiments examined qualitative and quantitative changes in the dynamics of learning a novel motor skill (roller ball task) as a function of the manipulation of a control parameter (initial ball speed). The focus was on the relation between the rates of change in performance over practice time and the changing time scales of the evolving attractor dynamic. Results showed 3 different learning patterns to the changes in the dynamics as a function of practice that were mediated by the initial ball speed. Only participants who learned the task showed a bifurcation in coordination mode that was preceded by enhanced performance variability. The observed multiple time scales to motor learning are interpreted as the products of the dynamical stability and instability realized from (a) the continually evolving landscape dynamics due to bifurcations between attractor organization and (b) the transient phenomena associated with moving toward and away from fixed-point dynamics.     

Multi-segmental movement patterns reflect juggling complexity and skill level

The juggling action of six experts and six intermediates jugglers was recorded with a motion capture system and decomposed into its fundamental components through Principal Component Analysis. The aim was to quantify trends in movement dimensionality, multi-segmental patterns and rhythmicity as a function of proficiency level and task complexity. Dimensionality was quantified in terms of Residual Variance, while the Relative Amplitude was introduced to account for individual differences in movement components. We observed that: experience-related modifications in multi-segmental actions exist, such as the progressive reduction of error-correction movements, especially in complex task condition. The systematic identification of motor patterns sensitive to the acquisition of specific experience could accelerate the learning process.     

Constrained paths based on the Farey sequence in learning to juggle

In this article we report the results of a study conducted to investigate the learning dynamics of three-ball juggling from the perspective of frequency locking. Based on the Farey sequence, we predicted that four stable coordination patterns, corresponding to dwell ratios of 0.83, 0.75, 0.67, and 0.50, would appear in the learning process. We examined the learning process in terms of task performance, taking into account individual differences in the amount of learning. We observed that the participants acquired individual-specific coordination patterns in a relatively early stage of learning, and that those coordination patterns were preserved in subsequent learning, even though performance in terms of number of successful consecutive throws increased substantially. This increase appeared to be related to a reduction in spatial variability of the juggling movements. Finally, the observed coordination patterns were in agreement with the predicted patterns, with the proviso that the pattern corresponding to a dwell ratio of 0.50 was not realized and only a hint of evidence was found for the dwell ratio of 0.67. This implies that the dwell ratios of 0.83 and 0.75 in particular exhibited a stable coordination structure due to strong frequency locking between the temporal variables of juggling.     

Rapid motor adaptations to subliminal frequency shifts during syncopated rhythmic sensorimotor synchronization

The synchronization of rhythmic arm movements to a syncopated metronome cue was studied in a step-change design whereby small tempo shifts were inserted at fixed time points into the metronome frequency. The cueing sequence involved three stimulus types: (1) target contact in synchrony with the metronome beats, (2) syncopated target contact midway in time between audible beats, and (3) syncopated target contact following either a +2% or -2% change in stimulus frequency. Analysis of normalized and aggregated data revealed that (1) during the syncopation condition the response period showed a rapid adaptation to the frequency-incremented stimulus period, (2) response period was less variable during syncopated movement, (3) mean synchronization error and variability, calculated during syncopation relative to the mathematical midpoint of the stimulus cycle, were reduced during syncopated movements, and (4) synchronization error following the frequency increment showed trends to return linearly to pre-increment values which was fully achieved in the -2% change condition only. The results suggest that frequency entrainment to stimulus period was possible during syncopated movement with the response and stimulus onsets 180 degrees out of phase. Most remarkably, 70-80% of the adaptation of the response period to the new stimulus period was immediately attained during the second half cycle of the syncopated movement. Finally, a mathematical model, based on recursion, was introduced that accurately modeled actual data as a function of the previous stimulus and response intervals and a weighted response of period error and synchronization error, which showed dominance of frequency entrainment over phase entrainment during rhythmic synchronization.     

Understanding projectile acceleration

Throwing and catching balls or other objects is a generally highly practiced skill; however, conceptual as well as perceptual understanding of the mechanics that underlie this skill is surprisingly poor. In 5 experiments, we investigated conceptual and perceptual understanding of simple ballistic motion. Paper-and-pencil tests revealed that up to half of all participants mistakenly believed that a ball would continue to accelerate after it left the thrower's hand. Observers also showed a remarkable tolerance for anomalous trajectory shapes. Perceptual judgments based on graphics animations replicated these erroneous beliefs for shallow release angles. Observers' tolerance for anomalies tended to decrease with their distance from the actor. The findings are at odds with claims of the naive physics literature that liken intuitive understanding to Aristotelian or medieval physics theories. Instead, observers seem to project their intentions to the ball itself (externalization) or even feel that they have power over the ball when it is still close.     

Catching of balls unexpectedly thrown or fired by cannon

Although learned actions can be automatically elicited in response to expected stimuli for which they have been prepared, little is known about whether learned actions can be automatically initiated by unexpected stimuli. Responses of unwitting participants to balls unexpectedly thrown by an experimenter (n=10) or propelled by a hidden ball cannon (n=22) were recorded by motion capture. Experience of ball catching correlated negatively with hand movement distance, indicating most responses were defensive, but successful catches were made in response to both thrown and fired balls. Although reaction time was faster in response to fired balls, interception was more frequent in response to thrown balls, indicating that movement cues by the thrower facilitated unexpected ball catching. The latency to begin a catching action by the only successful catcher of an unexpectedly fired ball was 296 msec. Given current knowledge of reaction time tasks and latencies of neural substrates of conscious perception and deliberation, it is probable that there was insufficient time available for conscious preparation of catch attempts. Ball catching may represent an example of a learned response which can be rapidly and unconsciously initiated without contextual priming or expectation of the stimulus.     

Grasping tau.

In the present study a direct manipulation of the optical expansion pattern was carried out. What happens to the timing of the grasp movements involved in catching a ball when optical expansion information is not veridically provided? By using 2 luminescent balls of constant size and a luminescent ball that could change its diameter during flight, it was possible to manipulate the rate of optical expansion directly. The results of 2 experiments (binocular vision in Experiment 1 and monocular vision in Experiment 2) showed that the time of the maximal closing velocity of the hand--which conforms to the prediction if Ss use retinal expansion information--was later for the deflating ball than for the balls of constant size. Adjustments to the aperture of the hand in response to the different ball sizes, especially the adjustment of the hand to the deflating ball (even though Ss were not aware that the ball was deflating during its approach), point to a finely attuned perception-action coupling.