State space analysis of timing: exploiting task redundancy to reduce sensitivity to timing

Abstract

Timing is central to many coordinated actions, and the temporal accuracy of central nervous system commands presents an important limit to skilled performance. Using target-oriented throwing in a virtual environment as an example task, this study presents a novel analysis that quantifies contributions of timing accuracy and shaping of hand trajectories to performance. Task analysis reveals that the result of a throw is fully determined by the projectile position and velocity at release; zero error can be achieved by a manifold of position and velocity combinations (solution manifold). Four predictions were tested. 1) Performers learn to release the projectile closer to the optimal moment for a given arm trajectory, achieving timing accuracy levels similar to those reported in other timing tasks (~10 ms). 2) Performers develop a hand trajectory that follows the solution manifold such that zero error can be achieved without perfect timing. 3) Skilled performers exploit both routes to improvement more than unskilled performers. 4) Long-term improvement in skilled performance relies on continued optimization of the arm trajectory as timing limits are reached. Average and skilled subjects practiced for 6 and 15 days, respectively. In 6 days, both timing and trajectory alignment improved for all subjects, and skilled subjects showed an advantage in timing. With extended practice, performance continued to improve due to continued shaping of the trajectory, whereas timing accuracy reached an asymptote at 9 ms. We conclude that skilled subjects first maximize timing accuracy and then optimize trajectory shaping to compensate for intrinsic limitations of timing accuracy.

Fig. 1.

Experimental setup. A: schematic of subject and virtual environment. B: 3 exemplary throws in workspace. The view of the ball, center post, and target is top-down, as it is presented to subjects on the large screen in front of them. (Subjects only see their current throw, but 3 throws are shown here for illustrative purposes.) PC, personal computer.

Fig. 2.

Three example trajectories and calculation of dependent measures. For all subplots, the optimal release moment is represented with an asterisk, and the actual release is indicated with a filled dot. A: execution space spanned by angle and velocity at release. Background shading denotes the error (distance to target) calculated for each pair of release variables. The broad white band denotes the set of release angles and velocities that lead to a target hit (error < 1.2 cm), and the thin black strip denotes the solution manifold, the set of combinations of angles and velocities that lead to zero error. The 3 trajectories shown were executed by 1 subject on different days. B: calculation of Timing Error. The example trajectory from day 1 was transformed into error over time so that the y-axis shows the error that each throw would have yielded had the subject released the ball at that moment in the trajectory. The difference between the actual release and the ideal release quantifies the Timing Error. Dt, Δt (| actual release time − ideal release time | = timing error). C: calculation of Integrated Error. The example trajectory from day 6 was transformed into error over time. The gray shaded zone indicates the integrated error over the time window of 25 ms centered on the ideal release moment. D: calculation of Time in Hit Zone. The example trajectory from day 15 was transformed into error over time. The horizontal line indicates the threshold below which a target hit occurred (the target hit zone). The total amount of time that the trajectory spends below the horizontal line is the Time in Hit Zone (TIZ).

Fig. 3.

Exemplary trajectories in execution space of 3 subjects. For 3 subjects in the Expert group, 20 trajectories from 3 selected days (days 1, 6, and 15) are shown. The 20 trajectories are from the middle block. The blue asterisks denote the release moments. Trajectories for throws that resulted in a target hit are shown in red; those that did not are shown in black.

Fig. 4.

Result measures. A: average Error over practice days. Each point is an average across 180 throws per day. B: Target Hits (error < 1.2 cm) per block, averaged across 3 blocks per day. Error bars indicate standard error across subjects. Note that for the expert group, the number of hits per block continues to climb after the average error per day has leveled off.

Fig. 5.

Timing accuracy. Absolute difference between the actual and optimal release times for each trial averaged across 180 trials per day. Error bars indicate standard error across subjects.

Fig. 6.

Trajectory shaping. A: Integrated Error: average difference in error between actual trajectory and solution manifold during a 25-ms window centered on the release moment. B: Time in Hit Zone: average time per trial that subjects spent in regions of the execution space where releasing the ball would have led to a successful hit (Error < 1.2 cm). Error bars indicate standard error across subjects.