The dialogue between data and model: Passive stability and relaxation behavior in a ball bouncing task

We investigate the skill of rhythmically bouncing a ball on a racket with a focus on the mathematical modeling of the stability of performance. As a first step we derive the deterministic ball bouncing map as a Poincaré section of a sinusoidally driven bouncing ball. Subsequently, we show the ball bouncing map to have a passively stable regime. More precisely, for negative racket acceleration at impact, no control of racket amplitude or frequency is necessary for stable performance. Support for the model comes from a motor learning study, where a decrease in variability covaries with a change of mean acceleration at impact towards more negative values. For a more fine-grained test of the model we develop a stochastic version of it, by adding Gaussian white noise to the dynamics. We then test the model predictions for the correlation functions. We find that the observed correlation functions match the theoretical ones quite well, lending new support for the model. Lastly, we compare the observed recovery from a sudden change with which the ball leaves the racket with model predictions. We find a mismatch between data and model in the sense that the model is too “slow”. We take this failure of the ball bouncing model as an impetus to further develop the model. In the perturbation study, we observe a significant modulation of the racket period but not of the racket amplitude. Thus, racket period seems a candidate state variable that should be included the ball bouncing map. 2000 Mathematics Subject Classification: Primary 37H20; Secondary: 93C55