Exploiting the geometry of the solution space to reduce sensitivity to neuromotor noise

Abstract

Throwing is a uniquely human skill that requires a high degree of coordination to successfully hit a target. Timing of ball release appears crucial as previous studies report required timing accuracies as short as 1-2ms, which however appear physiologically challenging. This study mathematically and experimentally demonstrates that humans can overcome these seemingly stringent timing requirements by shaping their hand trajectories to create extended timing windows, where ball releases achieve target hits despite temporal imprecision. Subjects practiced four task variations in a virtual environment, each with a distinct geometry of the solution space and different demands for timing. Model-based analyses of arm trajectories revealed that subjects first decreased timing error, followed by lengthening timing windows in their hand trajectories. This pattern was invariant across solution spaces, except for a control case. Hence, the exquisite skill that humans evolved for throwing is achieved by developing strategies that are less sensitive to temporal variability arising from neuromotor noise. This analysis also provides an explanation why coaches emphasize the "follow-through" in many ball sports.

Fig 1. Model throwing task, experimental set-up and analyses of timing measures.

(A) Real skittles task. (B) Experimental setup of the virtual throwing task; manipulandum and screen display. (C) Top down view of skittles workspace. Red circle is the post, green circle is target and pink bar represents the lever arm. The ball trajectory shown by the solid line incurred a non-zero error; the two dashed ball trajectories were successful throws hitting the target. The two insets illustrate the definition of the error around the target (top) and the release angle (below). (D) Execution space of the skittles task is spanned by two execution variables, release angle and velocity. Assuming each point in the space is a ball release, the error is calculated and shown in color code: green denotes the solution manifold, where each point leads to a hit below the error threshold of 1.1cm. The adjacent yellow band is the error threshold, denoting throws that hit the target but with an error larger than 1.1cm and smaller than 2.5cm (no intersection of ball trajectory with target). Two representative arm trajectories of the same subject are plotted in execution space, red from Day 1, blue from Day 6; white dots represent ball releases. The release on the red trajectory resulted in a large error; the release on the blue trajectory in a successful hit. (E) Each arm trajectory can be represented as an “error trajectory”, assuming each point on the trajectory is a ball release associated with an error. The timing error was calculated as the difference between the time of actual and ideal release. (F) Using the same “error trajectory” the timing window was quantified as the time that the trajectory would yield errors below the success threshold of 1.1cm, i.e. within the solution manifold.

Fig 2. Solution spaces for the four different tasks and results of performance and timing measures for the four tasks.

(A) Solution spaces with 60 exemplary arm trajectories and release points on Day 1 (dark purple) and Day 6 (light purple) in each of the four tasks. White dots represent the points of release. (B) Task success (percent of hits per day with 240 throws) across the 6 practice days of all subjects in the four tasks. (C) Average performance error across the six practice days of all subjects for each of the four tasks. (D) Average timing error across the six practice days of all subjects in four tasks. (E) Average timing window across the six practice days of all subjects in the four tasks. For all four figures the shaded regions represent ±1 standard error of the mean.

Fig 3. Results of performance and timing measures separated by task.

Performance measures, success rate and performance error, are combined into one figure; timing error and timing window are combined into a separate figure. The units are scaled differently for each task to highlight the relative changes. Shaded regions represent ±1 standard error of the mean.

Fig 4

Results from multiple regressions of performance error on timing error and timing window for the J-Shape (A-D) and the I-Shape (E-H) task across practice days. Each dot represents performance of a single subject. The x-, y- and z-axis represent the normalized timing error, normalized timing window, and normalized error respectively. All variables were transformed into z-scores prior to each regression to make units comparable.

Fig 5. Results from multiple regressions of performance error on timing error and timing window.

(A) Regression coefficients of the U-Shape task. Blue bars represent the values of the standardized regression coefficients of timing error (β₁); the red bars represent the values of the standardized regression coefficients of timing window (β₂). The gray error bars represent the 95% confidence intervals for β₁ and β₂. The light blue shaded bars indicate that the coefficients were not significantly different from 0 (p≥0.05). (B) Regression coefficients of the J-Shape task. (C) Regression coefficients of the Box-Shape task. (D) Regression coefficients of the I-Shape task.

Fig 6. Skittles tasks represented in Cartesian coordinates.

(A) Work space of the skittles task with the two execution variables defined in Cartesian coordinates. The variables are now velocity in orthogonal x- and y-directions and also fully define the trajectory of the ball. (B) Solution spaces of the four tasks spanned by the two Cartesian velocity variables together with the same exemplary arm trajectories as in Fig 2. The nonlinear transformation of the variables also transforms the solution manifolds and the arm trajectories. However, as can be seen from the release points, their errors remain unaffected.

Fig 7. Workspaces and solution spaces of the four tasks.

(A) Workspaces of the four tasks. The red circle in the middle is the post, the yellow circle is the target, and the pink bar is the lever arm; the black arrow indicates the direction of the arm movement. Each task has a different target location. The Box-shape task also located the post away from the center (0, 0 cm). (B) Solution spaces of the four tasks. Each of the four panels corresponds to the four workspaces in panel A. The error of all combinations of release angles and release velocities were calculated and coded by color; green denotes the set of points that hit the target with errors smaller than 1.1cm; yellow denotes errors larger than 1.1cm and smaller than 2.5cm; grey shades represent errors larger than 2.5cm; black denotes throws when the error is larger than 40cm or the ball trajectory hit the post.