Abstract
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 Poincar´e 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.
