Fig 1.
Unconstrained throwing task and experimental performance.
(A) Schematic representation of the unconstrained overarm throwing task considered in this study as an example of a complex motor skill. The release parameters characterizing a throwing action are six-dimensional (three position components and three velocity components). (B) Performance (mean squared error ± SE) across participants (n = 20) and four targets.
Fig 2.
Definition of variables (action, outcome, and score) and schematic overview of the proposed approach to decompose the mean score (performance).
(A) An action, represented by a vector a in a high-dimensional action space (illustrated by a red circular marker in a three-dimensional action space), leads to an outcome, x = f(a), i.e. a vector typically represented in outcome or goal space with less dimensions. The outcome is then associated with a corresponding score, π = sx(x, xT), based on its relation with respect to the aimed target. The set of points in action space associated with an outcome achieving the goal constitute the solution manifold (illustrated by the black mesh). (B) Illustration of the Hessian-based decomposition of the mean score in the case of two-dimensional actions. The action-to-score mapping is represented by the gray-shaded areas. A set of actions (small red markers, first panel) has a distribution characterized by the mean (large blue marker, second panel) and covariance (Σ, blue ellipse in the third and fifth panels). The tolerance of the score to variations of the actions around their means is characterized by the Hessian (H, green ellipse in the fourth panel). The mean score (E(π)) is decomposed as the sum of the score of the mean action (α) and a tolerance-variability index (β) expressed as the product of three terms: the reciprocal of tolerance (τ), noise (η), and alignment (θ), i.e. a scalar characterizing whether the most sensitive directions of the Hessian (green arrows uH, fifth panel) are aligned to the directions of highest action variability (blue arrows uΣ).
Fig 3.
Examples of different throwing strategies in a simulated 2D throwing task.
(A) Toy model of a throwing task in which the action (a) is characterized by only two parameters, i.e. the horizontal (vy0) and the vertical (vz0) release velocity components, the outcome (x) is the arrival position of the projectile on a vertical plane at a distance d from the release position, and the score (π) is the squared distance of the arrival position from a target (xT). (B) Action score (gray-shaded background), solution manifold (yellow line), and Hessian (red ellipses) in a region of release parameters. The score sensitivity for different throwing strategies is captured by the Hessian, which is represented with red ellipses whose major axis indicates, locally, the direction of maximum sensitivity (smaller tolerance to noise). (C) Five (simulated) individual strategies. For each strategy (S1-S5, different panels and colors) the individual throwing actions are shown together with mean (large black circle), their covariance (black ellipse, 95% C.L.) and the Hessian at the mean action (red ellipse). (D) Hessian-based decomposition of the mean score of the five strategies. For each strategy, the mean release parameters (vy0 and vz0), the mean score (), and the indicators from the decomposition (score of the mean action α, tolerance-variability index β, tolerance (τ), noise η, and alignment θ) are presented in the table. The four scatter plots illustrate how these indicators allow to differentiate strategies with different performance as well as strategies with same performance. See text for more details.
Fig 4.
Comparison of the Hessian-based decomposition with existing computational methods for the analysis of throwing strategies in a 2D throwing task.
(A) Comparison with TNC-Cost analysis. Three strategies (S1 blue, S2 green, S3 magenta) are simulated by generating a distribution of 50 release parameters (vy0, vz0) by randomly sampling from a bi-dimensional Gaussian distribution with a given mean and covariance (see parameters in the table on the left). The strategies have the same performance because they trade-off bias (α) with tolerance-variability (β), as can be clearly observed in the α-β scatter plot. Moreover, the strategies have equal noise (η). However, the strategies have equal T-cost but different N-Cost, which then represent a recombination of the indicators extracted from the Hessian-based decomposition. Moreover, as the TNC costs do not sum up to the performance, the α-β tradeoff cannot be observed. (B) Comparison with UCM analysis. Three simulated strategies with no bias (i.e. with a mean action on the solution manifold) differ in performance because of differences in β, due to differences in the alignment of the covariance (black ellipses) with the Hessian (red degenerate ellipses), since the noise is equal. The decomposition of the action variability onto an uncontrolled manifold (UCM) subspace and an orthogonal (ORT) subspace does not discriminate the three strategies. (C) Comparison with GEM analysis. Three simulated strategies with equal noise but different performance due to differences in α are not discriminated by the goal-relevant sensitivity ΣGR and the goal-relevant variability fraction ΦGR.
Fig 5.
Examples of distribution of release velocity components for five representative participants (target T1).
Each row illustrates a pairs of release velocity components (,
,
). Colored circles represent release parameters of individual throws. Black circles and ellipses represent mean and covariance (95% c.l.) of each parameter distribution. The gray-level shading of the contours indicates the local score, as a function of the release parameters. Note that in each row, the range of horizontal and vertical axes are the same for all participants, and the different shapes of the score across participant, reflect individual differences in the average action (mean position and velocity vectors at release). The wider the white area around the mean action, the more tolerant the score is to noise. Participants have been sorted from left to right according to their average release speed.
Fig 6.
Validity of the quadratic approximation.
Mean score vs. quadratic approximation (8) across targets (T1-T4, different panels) and participants (P1-P20) as in Fig 1. Note that P9 is the only participant deviating substantially from the identity line (dashed line).
Fig 7.
Decomposition of the mean score across participants for target T1.
Circular markers are filled with areas that are gray-shaded according to the score (lighter gray for lower mean score). The edge of the markers corresponding to the five participants of Fig 5 are colored with matching colors. To increase the readability of the figure, P9 is not shown. (A) The α − β plane and the iso-performance lines. The dashed line represent the direction of maximal change in performance (orthogonal to the iso-performance lines). (B) The β − η plane. The lines have slopes corresponding to different values of . Participants along the same line, such as P1, P2, and P12, have the similar alignment-to-tolerance ratio and the difference in their β are only due to the noise η. (C) The θ − τ (tolerance-alignment) plane. The lines have slopes corresponding to different values of
. The negative correlation between θ and τ indicates that participants with mean release action in a region with higher tolerance also tend to have lower alignment with the most noise sensitive directions. (D) The log(η)-log(θ/τ) plane and iso-β lines. As β is the product of θ/τ and η and log(β) = log(θ/τ) + log(η), in the log(η)-log(θ/τ) plane participants with the same β lie on a line with a −1 slope and maximal change of the tolerance-variability index occurs along the orthogonal direction.
Fig 8.
Decomposition of the mean score across participants for all targets.
In each panel, the quadrilateral connects the points corresponding to the values of a pair of indicators for the four targets of each participant (excluding P9) and they are colored according to the mean score of that participant (color scale on the top right). The targets are indicated by different colors of the circular markers and the participant number is indicated close to the marker for T1 (except for P7 and P14). (A) The α − β plane and the iso-performance lines. (B) The β − η plane. (C) The θ − τ (tolerance-alignment) plane. (D) The log(θ/τ)-log(η) plane.
Table 1.
Robustness of decomposition parameters across targets.
Fig 9.
Error distribution between experimental score and score predicted with the no-drag model in Eq (29).
Black and red vertical dashed lines indicate mean and ±1SD of the distribution, respectively.