How do we bounce a ball in the air with a hand-held racket in a controlled rhythmic fashion? Using this model task previ
Optimal Feedback Control of Rhythmic Movements: The Bouncing Ball Revisited Renaud Ronsse, Department of Biomedical Kinesiology, Katholieke Universiteit Leuven, Belgium Kunlin Wei, Rehabilitation Institute of Chicago, Northwestern University, USA Dagmar Sternad, Dpts of Biology, Electrical & Computer Engineering and Physics, Northeastern Univ., USA How do we bounce a ball in the air with a hand-held racket in a controlled rhythmic fashion? Using this model task previous theoretical and experimental work by Sternad and colleagues showed that experienced human subjects performed this skill in a dynamically stable manner [2]. However, the model predictions were based on the assumption that the hand movements are sinusoidal and control is open-loop. Racket trajectories, however, often significantly deviate from sinusoids and more recent results also identified active corrections, highlighting the presence of closed-loop control of the racket trajectory [4, 5]. The present study investigates how this active control may be governed by optimal control principles. To this end, the task is divided into two control layers with the objectives: (1) to achieve a given target height the racket-ball impact has to have a required velocity; (2) the racket trajectories should be smooth and efficient between two successive impacts. To test this control strategy we created new and relatively difficult conditions for the actor by changing gravity in the virtual environment. Eight subjects performed the task in a virtual-reality setup (see Fig. 1) where gravity acting on the virtual ball was manipulated to effectively slow the bounce-to-bounce periods. Seven values of g ranged from 0.61m/s2 (g1 ) to 9.81m/s2 (g7 ) leading to effective periods between 2.8s and 0.7s. Longer bounce-to-bounce periods are expected to induce more non-sinusoidal racket trajectories with increasing dwell intervals. The first layer of the controller determined the racket velocity required at impact k + 1 (i.e. the produced “output”) that compensates for the ball height error following impact k, i.e., the difference between the target height and the ball apex (i.e. the sensed “input”). The gain was tuned to achieve a correction of the ball height error in one impact, such that the correlation between the errors over successive impacts was minimized. This gain tuning produced correlation slopes between height error during and racket velocity were close to the observed subjects’ behavior (see Fig. 2). The second layer consisted of a feedback controller with gains designed to reach the desired impact velocity while minimizing the control cost. We applied an optimal control algorithm that computed time-varying gains to minimize a cost function penalizing discrepancies between actual racket velocity and desired racket velocity and large control inputs [3]. This controller predicted that with a decrease in gravity longer dwell intervals (i.e. resting periods characterized by very small velocity) should appear in the racket trajectory, as a consequence of the attempt to minimize the control effort. Comparison between model predictions and actual data (see Fig. 3) illustrated that subjects did not scale their racket trajectories as the bounce-to-bounce interval became longer, but modified the sinusoidal racket trajectory to a sequence of discrete movements, according to the control assumptions captured by the cost function. In conclusion, this study illustrates that intuitive optimality principles can predict human behavior in a simple rhythmic task. This work complements studies that have demonstrated optimal control principles for discrete movements [1] and postural control [6]. The controller consisted of two distinct layers in hierarchical order: the top layer computes the impact velocity required to cancel the ball height error in the following cycle; the subordinate layer executes a smooth and efficient racket movement to reach this impact velocity. Similar hybrid control for collision dynamics can be expected in other rhythmic movements including locomotion, for which the bouncing ball might serve as a benchmark.
Fig. 1: Experimental setup. The actor moves a real racket that is online displayed on the screen where it interacts with a virtual ball. Break pulses delivered to the real racket mimic the effect of ball-racket impacts.
Fig. 2: Regression slopes between height error and the next impact velocity, as predicted by the model (blue) and for the actual data (black). The error bars denote the between-subjects standard deviations.
Fig. 3: Average racket position (A) and normalized velocity (B) over normalized time. The racket kinematics were resampled and normalized between two consecutive impacts (two vertical gray lines). The left column illustrates the model predictions, the right column displays the actual data.
References [1] D. Liu and E. Todorov. Evidence for the flexible sensorimotor strategies predicted by optimal feedback control. J Neurosci, 27(35): 9354–9368, 2007. doi:10.1523/JNEUROSCI.1110-06.2007. [2] D. Sternad. Stability and variability in skilled rhythmic actions: a dynamical analysis of rhythmic ball bouncing. In M. L. Latash and F. Lestienne, editors, Motor Control and Learning, pages 55–63. Springer, New York, 2006. [3] E. Todorov. Stochastic optimal control and estimation methods adapted to the noise characteristics of the sensorimotor system. Neural Comput, 17(5): 1084–1108, 2005. doi:10.1162/0899766053491887. [4] K. Wei, T. M. H. Dijkstra and D. Sternad. Passive stability and active control in a rhythmic task. J Neurophysiol, 98: 2633–2646, 2007. doi:10.1152/jn.00742.2007. [5] K. Wei, T. M. H. Dijkstra and D. Sternad. Stability and variability: indicators for passive stability and active control in a rhythmic task. J Neurophysiol, 99(6): 3027–3041, 2008. doi:10.1152/jn.01367.2007. [6] T. D. J. Welch and L. H. Ting. A feedback model reproduces muscle activity during human postural responses to support-surface translations. J Neurophysiol, 99(2): 1032–1038, 2008. doi:10.1152/jn.01110.2007.
