Bouncing between Model and Data: Stability, Passivity, and Optimality in Hybrid Dynamics

ABSTRACT

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.

Keywords:

• dynamical system
• interactive task
• optimal content
• rhythmic cooperation
• stability

ACKNOWLEDGMENTS

Renaud Ronsse was funded by the European Union grant Evryon (FP7-ICT-2007.8.5 #231451). Dagmar Sternad was funded by the grants NIH R01-HD045639, NSF PAC-0450218, and NSF DMS-0928587.