Bouncing a ball: tuning into dynamic stability

Rhythmically bouncing a ball with a racket was investigated and modeled with a nonlinear map. Model analyses provided a variable defining a dynamically stable solution that obviates computationally expensive corrections. Three experiments evaluated whether dynamic stability is optimized and what perceptual support is necessary for stable behavior. Two hypotheses were tested: (a) Performance is stable if racket acceleration is negative at impact, and (b) variability is lowest at an impact acceleration between -4 and -1 m/s2. In Experiment 1 participants performed the task, eyes open or closed, bouncing a ball confined to a 1-dimensional trajectory. Experiment 2 eliminated constraints on racket and ball trajectory. Experiment 3 excluded visual or haptic information. Movements were performed with negative racket accelerations in the range of highest stability. Performance with eyes closed was more variable, leaving acceleration unaffected. With haptic information, performance was more stable than with visual information alone.     

Bouncing between model and data: stability, passivity, and optimality in hybrid dynamics

Rhythmically bouncing a ball with a racket is a seemingly simple task, but it poses all the challenges critical for coordinative behavior: perceiving the ball's trajectory to adapt position and velocity of the racket for the next ball contact. To gain insight into the underlying control strategies, the authors conducted a series of studies that tested models with experimental data, with an emphasis on deriving model-based hypotheses and trying to falsify them. Starting with a simple dynamical model of the racket and ball interactions, stability analyses showed that open-loop dynamics affords dynamical stability, such that small perturbations do not require corrections. To obtain this passive stability, the ball has to be impacted with negative acceleration--a strategy that subjects adopted in a variety of conditions at steady state. However, experimental tests that applied perturbations revealed that after perturbations, subjects applied active perceptually guided corrections to reestablish steady state faster than by relying on the passive model's relaxation alone. Hence, the authors derived a model with active control based on optimality principles that considered each impact as a separate reaching-like movement. This model captured some additional features of the racket trajectory but failed to predict more fine-grained aspects of performance. The authors proceed to present a new model that accounts not only for fine-grained behavior but also reconciles passive and active control approaches with new predictions that will be put to test in the next set of experiments.     

Uncontrolled manifold analysis of joint angle variability during table tennis forehand

This study was conducted with the objective of evaluating the variance structure of the trunk and racket arm joint angles in table tennis topspin forehand using the uncontrolled manifold (UCM) approach, regarding racket orientation as the task variable. Nine advanced and eight intermediate male collegiate table tennis players performed the topspin strokes against backspin balls. The trunk, upper limb, and racket were modeled as six rigid-link segments with a total of 16 rotation degrees of freedom. The UCM analysis was conducted using 30 trial datasets per participant to quantify the degree of redundancy exploitation needed to stabilize the vertical and horizontal angles of the racket. Irrespective of the performance level, the variance of the joint angle vector increased towards ball impact. The degree of redundancy exploitation increased towards ball impact. As a result, the variability of the racket angles was minimal at impact. Both groups of players used the relative movement between the racket and the hand to stabilize the racket angles at ball impact. The variance of the joint angle vector that affected the vertical racket face angle at ball impact was significantly smaller for advanced players than for intermediate players, and the degree of redundancy exploitation to stabilize that angle at impact tended to be larger for the advanced players. The ability to use the redundancy of the joint configuration to stabilize the vertical racket face angle at impact may be a critical factor that affects performance level.     

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.     

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.     

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.     

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.     

The dynamics of perception and action

How might one account for the organization in behavior without attributing it to an internal control structure? The present article develops a theoretical framework called behavioral dynamics that integrates an information-based approach to perception with a dynamical systems approach to action. For a given task, the agent and its environment are treated as a pair of dynamical systems that are coupled mechanically and informationally. Their interactions give rise to the behavioral dynamics, a vector field with attractors that correspond to stable task solutions, repellers that correspond to avoided states, and bifurcations that correspond to behavioral transitions. The framework is used to develop theories of several tasks in which a human agent interacts with the physical environment, including bouncing a ball on a racquet, balancing an object, braking a vehicle, and guiding locomotion. Stable, adaptive behavior emerges from the dynamics of the interaction between a structured environment and an agent with simple control laws, under physical and informational constraints.     

Learning the cascade juggle: a dynamical systems analysis

How beginning jugglers discover the temporal constraints governing the juggling workspace while learning to juggle three balls in a cascade pattern was the subject of this investigation. On the basis of previous theoretical and experimental work on expert jugglers, we proposed a three-stage model of the learning process, for which objective evidence was sought. The first stage consists of learning to accommodate the real-time requirements of juggling, as expressed in Shannon's equation of juggling, which states that, averaged over time, the cycle time of the hands should be a fixed proportion of the cycle time of the balls. The second stage of learning consists of discovering the primary frequency lock of.75 between the shorter term dynamical regime underlying the repetitive subtask of transporting a ball and the longer term dynamical regime underlying the total hand loop cycle. The third and last stage of learning consists of discovering the principles of frequency modulation from.75 to lower (averaged) values of the proportion of time that a hand carries a ball during the total hand cycle time. Twenty subjects were taught to juggle three balls in a cascade pattern. Ten subjects were trained with the aid of an instructor and a metronome, and 10 with the instructor only. The metronome proved to be of no particular additional help, but the timing results obtained were in agreement with the proposed three stages of learning. The picture that emerged from this study was that learning a new motor skill involves the discovery of invariance's or fixed points in the perceptual-motor workspace associated with that skill, from which excursions can be made and the skill further refined. Because these fixed points afford stability of operation, discovering them logically and factually precedes the acquisition of the functional adaptability and flexibility of operation ("flair") inherent to frequency modulation.     

Stability of heartbeat interval distributions in chronic high altitude hypoxia.

Recent studies of nonlinear dynamics of the long-term variability of heart rate have identified nontrivial long-range correlations and scale-invariant power-law characteristics (l/f noise) that were remarkably consistent between individuals and were unrelated to external or environmental stimuli (Meyer et al., 1998a). The present analysis of complex nonstationary heartbeat patterns is based on the sequential application of the wavelet transform for elimination of local polynomial nonstationary behavior and an analytic signal approach by use of the Hilbert transform (Cumulative Variation Amplitude Analysis). The effects of chronic high altitude hypoxia on the distributions and scaling functions of cardiac intervals over 24 hr epochs and 4 hr day/nighttime subepochs were determined from serial heartbeat interval time series of digitized 24 hr ambulatory ECGs recorded in 9 healthy subjects (mean age 34 yrs) at sea level and during a sojourn at high altitude (5,050 m) for 34 days (Ev-K2-CNR Pyramid Laboratory, Sagarmatha National Park, Nepal). The results suggest that there exists a hidden, potentially universal, common structure in the heterogeneous time series. A common scaling function with a stable Gamma distribution defines the probability density of the amplitudes of the fluctuations in the heartbeat interval time series of individual subjects. The appropriately rescaled distributions of normal subjects at sea level demonstrated stable Gamma scaling consistent with a single scaled plot (data collapse). Longitudinal assessment of the rescaled distributions of the 24 hr recordings of individual subjects showed that the stability of the distributions was unaffected by the subject's exposure to a hypobaric (hypoxic) environment. The rescaled distributions of 4 hr subepochs showed similar scaling behavior with a stable Gamma distribution indicating that the common structure was unequivocally applicable to both day and night phases and, furthermore, did not undergo systematic changes in response to high altitude. In contrast, a single function stable over a wide range of time scales was not observed in patients with congestive heart failure or patients after cardiac transplantation. The functional form of the scaling in normal subjects would seem to be attributable to the underlying nonlinear dynamics of cardiac control. The results suggest that the observed Gamma scaling of the distributions in healthy subjects constitutes an intrinsic dynamical property of normal heart function that would not undergo early readjustment or late acclimatization to extrinsic environmental physiological stress, e.g., chronic hypoxia.     

Stability of heartbeat interval distributions in chronic high altitude hypoxia

Recent studies of nonlinear dynamics of the long-term variability of heart rate have identified nontrivial long-range correlations and scale-invariant power-law characteristics (1/f noise) that were remarkably consistent between individuals and were unrelated to external or environmental stimuli (Meyer et al., 1998a). The present analysis of complex nonstationary heartbeat patterns is based on the sequential application of the wavelet transform for elimination of local polynomial nonstationary behavior and an analytic signal approach by use of the Hilbert transform (Cumulative Variation Amplitude Analysis). The effects of chronic high altitude hypoxia on the distributions and scaling functions of cardiac intervals over 24 hr epochs and 4 hr day/nighttime subepochs were determined from serial heartbeat interval time series of digitized 24 hr ambulatory ECGs recorded in 9 healthy subjects (mean age 34 yrs) at sea level and during a sojourn at high altitude (5,050 m) for 34 days (Ev-K2-CNR Pyramid Laboratory, Sagarmatha National Park, Nepal). The results suggest that there exists a hidden, potentially universal, common structure in the heterogeneous time series. A common scaling function with a stable Gamma distribution defines the probability density of the amplitudes of the fluctuations in the heartbeat interval time series of individual subjects. The appropriately rescaled distributions of normal subjects at sea level demonstrated stable Gamma scaling consistent with a single scaled plot (data collapse). Longitudinal assessment of the rescaled distributions of the 24 hr recordings of individual subjects showed that the stability of the distributions was unaffected by the subject’s exposure to a hypobaric (hypoxic) environment. The rescaled distributions of 4 hr subepochs showed similar scaling behavior with a stable Gamma distribution indicating that the common structure was unequivocally applicable to both day and night phases and, furthermore, did not undergo systematic changes in response to high altitude. In contrast, a single function stable over a wide range of time scales was not observed in patients with congestive heart failure or patients after cardiac transplantation. The functional form of the scaling in normal subjects would seem to be attributable to the underlying nonlinear dynamics of cardiac control. The results suggest that the observed Gamma scaling of the distributions in healthy subjects constitutes an intrinsic dynamical property of normal heart function that would not undergo early readjustment or late acclimatization to extrinsic environmental physiological stress, e.g., chronic hypoxia.     

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.     

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.     

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.     

Anticipating action effects recruits audiovisual movement representations in the ventral premotor cortex

When table tennis players anticipate the course of the ball while preparing their motor responses, they not only observe their opponents striking the ball but also listen to events such as the sound of racket–ball contact. Because visual stimuli can be detected more easily when accompanied by a sound, we assumed that complementary sensory audiovisual information would influence the anticipation of biological motion, especially when the racket–ball contact is not presented visually, but has to be inferred from continuous movement kinematics and an abrupt sound. Twenty-six observers were examined with fMRI while watching point-light displays (PLDs) of an opposing table tennis player. Their task was to anticipate the resultant ball flight. The sound was presented complementary to the veracious event or at a deviant time point in its kinematics.Results showed that participants performed best in the complementary condition. Using a region-of-interest approach, fMRI data showed that complementary audiovisual stimulation elicited higher activation in the left temporo-occipital middle temporal gyrus (MTGto), the left primary motor cortex, and the right anterior intraparietal sulcus (aIPS). Both hemispheres also revealed higher activation in the ventral premotor cortex (vPMC) and the pars opercularis of the inferior frontal gyrus (BA 44). Ranking the behavioral effect of complementary versus conflicting audiovisual information over participants revealed an association between the complementary information and higher activation in the right vPMC. We conclude that the recruitment of movement representations in the auditory and visual modalities in the vPMC can be influenced by task-relevant cross-modal audiovisual interaction.     

Modeling the Dynamics of Risky Choice

Individuals make decisions under uncertainty every day. Decisions are based on incomplete information concerning the potential outcome or the predicted likelihood with which events occur. In addition, individuals' choices often deviate from the rational or mathematically objective solution. Accordingly, the dynamics of human decision making are difficult to capture using conventional, linear mathematical models. Here, we present data from a 2-choice task with variable risk between sure loss and risky loss to illustrate how a simple nonlinear dynamical system can be employed to capture the dynamics of human decision making under uncertainty (i.e., multistability, bifurcations). We test the feasibility of this model quantitatively and demonstrate how the model can account for up to 86% of the observed choice behavior. The implications of using dynamical models for explaining the nonlinear complexities of human decision making are discussed as well as the degree to which the theory of nonlinear dynamical systems might offer an alternative framework for understanding human decision making processes.