Elsevier

Brain Research

Volume 982, Issue 1, 22 August 2003, Pages 64-78
Brain Research

Research report
Actively tracking ‘passive’ stability in a ball bouncing task

https://doi.org/10.1016/S0006-8993(03)02976-7Get rights and content

Abstract

This study investigates the control involved in a task where subjects rhythmically bounce a ball with a hand-held racket as regularly as possible to a prescribed amplitude. Stability analyses of a kinematic model of the ball–racket system revealed that dynamically stable solutions exist if the racket hits the ball in its decelerating upward movement phase. Such solutions are resistant to small perturbations obviating explicit error corrections. Previous studies reported that subjects’ performance was consistent with this ‘passive’ stability. However, some ‘active’ control is needed to attune to this passive stability. The present study investigates this control by confronting subjects with perturbations where stable behavior cannot be maintained solely from passive stability. Six subjects performed rhythmic ball bouncing in a virtual reality set-up with and without perturbations. In the perturbation trials the coefficient of restitution of the ball–racket contact was changed at every fifth contact leading to unexpected ball amplitudes. The perturbations were compensated for within 2–3 bouncing cycles such that ball amplitudes decreased to initial values. Passive stability was reestablished as indicated by negative racket acceleration. Results revealed that an adjustment of the racket period ensured that the impacts occurred at a phase associated with passive stability. These findings were implemented in a model consisting of a neural oscillator that drives a mechanical actuator (forearm holding the racket) to bounce the ball. Following the perturbation, the oscillator’s period is adjusted based on the perceived ball velocity after impact. Simulation results reproduced the major aspects of the experimental results.

Introduction

The task of ‘juggling’ or bouncing a ball cyclically on a racket has received considerable attention in recent years in both the robotics and the motor control literature [1], [2], [3], [4], [6], [9], [16], [17], [18], [19], [20]. The reason for this interest is that this task constitutes an exemplary case where an actor, or more precisely an end-effector, i.e., the racket, interacts with an object in the environment, i.e., the ball. The movement of the racket and the racket–ball contact determines the flight trajectory of the ball to which, in turn, the racket has to synchronize again. The classical approach in control theory would suggest that the trajectories of the racket need to be planned and monitored based on feedback from the ball’s trajectory. Along this line, Koditschek and co-workers designed a robot actuator capable of bouncing ball in three dimensions [6], [16]. The racket movements were controlled by the ‘mirror algorithm’, which matched the actuator’s velocity to the velocity of the ball with opposite sign (with a gain), i.e., ball and actuator were tightly coupled at every moment in time.

Further analysis of the bouncing-ball system revealed that such continuous closed-loop control of the racket trajectory is not necessary as only the contact event has an effect on the ball’s trajectory. Further, the work of Wood and Byrne [22], Holmes [8] and Guckenheimer and Holmes [7] showed that a ball bouncing on a periodically driven planar surface exhibits dynamically stable solutions, i.e., stable performance is obtained in an open-loop fashion without continuous feedback from the ball trajectory (see also Ref. [21]). Schaal et al. [17] extended these analyses, which were initially valid for a table vibrating with a small amplitude, to the range of motion involved when human actors bounce a ball on a racket. Stability analyses revealed that stable bouncing can be achieved with an arbitrary periodic motion of the racket, provided that the successive impacts occur during upward movements of the racket that display negative acceleration. In the context of this study, we refer to this mode of performance as ‘passive stability’, since no explicit corrective control of the racket with respect to the ball’s trajectory is required.

Fig. 1 displays the periodic motion of a racket together with ball trajectories that were generated by using equations for ballistic flight and elastic impact. These trajectories, which were generated by the model described in the discussion below, illustrate how a decelerating upward movement of the racket at impact gives rise to passive stability. Three simulations are shown where for two of them a small perturbation was applied on the second impact shown. One of these perturbations (bold curve) results in a lower ball amplitude than that of the preceding unperturbed ball trajectory. As there is no modulation of the racket movement in the passively stable mode, the ball–racket contact of the succeeding contact occurs earlier than in the preceding contact. Since racket acceleration is negative in this segment of the trajectory, an earlier contact is associated with a higher racket velocity, which in turn gives rise to a higher ball amplitude in the next cycle. The opposite chain of events holds for perturbations with a higher than average ball amplitude. As a result, these small perturbations die out and converge back to the initial unperturbed amplitudes after few contacts. Formal stability analyses of the ball-bouncing map can be found in Ref. [19].

Bouncing the ball with ball–racket contacts in this decelerating racket movement phase, i.e., with passive stability, thus represents an alternative to the explicit specification of a desired trajectory with continuous coupling between the racket and ball movements. Sternad and co-workers showed in different task variations that subjects perform ball bouncing with negative acceleration of the racket at impact, indicative of a strategy that exploits passive stability to maintain stable bouncing [9], [17], [19], [20]. This can be interpreted that such open-loop control has less computational demands on the controller and is thus more efficient. However, these results do not exclude that other control strategies are involved over and above exploiting stability properties. In fact, Sternad et al. [20] showed that performance was more variable when visual information was excluded and when only haptic information about successive impacts was available. Furthermore, in order to establish this passive stability when starting periodic bouncing movement, the racket has to be controlled with respect to the ball. Active control is also expected to prevail when large perturbations are applied. To throw light on such control mechanisms the present study introduced sufficiently large perturbations that were outside the realm of passive stabilization. A virtual reality set-up allowed us to introduce such perturbations during a ball bouncing performance. Our aim was to examine how subjects recover from these perturbations. Subsequently, a model is developed which consists of a neural oscillator that drives a mechanical actuator (forearm holding the racket) that bounces a ball. This model exhibits dynamical stability but also includes a control algorithm that counteracts large perturbations. The model is shown to replicate the major experimental results.

Section snippets

Participants

Six volunteers participated in this experiment. Their age ranged between 25 and 43 years (four male, two female). All were right-handed and used their preferred right hand to bounce the ball with the racket. The participants were informed about the experimental procedure and signed the consent form in compliance with the Regulatory Committee of the Pennsylvania State University.

Virtual reality apparatus and data collection

In the virtual reality set-up the subject manipulated a real tennis racket in front of a large screen (1.4 m wide and

Results

To evaluate the overall performance in the different conditions a mixed-design 6×2 ANOVA was performed on the trial means AE. It revealed an interaction between perturbation and subject, F(5, 84)=5.07, P<0.01, a main effect of perturbation, F(1, 84)=62.06, P<0.01, and a main effect of subject, F(5, 84)=6.40, P<0.01. Fig. 4A shows that each subject performed better for UP, despite variations between subjects. The error bars indicate 2 standard errors of the mean. Overall, AE was significantly

Discussion

Bouncing a ball regularly on a racket is a rhythmic task that involves the intricate coordination between the movements of the racket and the movements of a ball. Two types of control have been suggested to perform this skill. The first strategy follows classical control theory where the racket trajectory is planned and continuously controlled on the basis of visual feedback from the movement of the ball. Koditschek and colleagues applied such type of control algorithm for the control of a

Acknowledgements

This research was supported by a grant from the National Science Foundation, Behavioral Neuroscience 00-96543 awarded to D.S.

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