Data Supplement
Trajectory of the overdamped stick balancing by a human subject. The trajectory corresponds to the six-second fast stick balancing by Subject 1. The video is slowed down two-fold.
Article Figures & Data
Figures
- Figure 1.
One-degree-of-freedom overdamped inverted pendulum.
- Figure 2.
Cart velocity dynamics of three representative subjects. Each trajectory represents the randomly selected 10 s period of fast stick balancing without stick falls. The values of %drift (calculated based on the presented 10 s time series) are shown for reference. (Online version in colour.)
- Figure 3.
Phase trajectories of the overdamped stick balancing. Coloured trajectories correspond to the fast stick condition and represent the randomly selected 15 s time fragments. Same subjects' trajectories obtained in the slow stick condition are shown in black. (Online version in colour.)
- Figure 4.
Distribution of the action points (the values of stick angle triggering the operator's response) in the fast stick condition. The angle value was counted as an action point if it corresponded to the instant the mouse velocity switched from zero to a non-zero value. In (a) and (b), coloured lines represent the distributions for each subject, and black lines correspond to the average distributions across two groups. The high skill group consists of subjects 1, 2, 3, 7, 8; the low skill group includes subjects 4, 5, 6, 9, 10. Panel (c) illustrates the average distributions of the absolute value of action point for each group in the logarithmic scale; the standard normal distribution truncated at zero is represented for reference. (Online version in colour.)
- Figure 5.
Experimentally obtained distributions of the stick angle and cart velocity. Panels (a) and (c) depict distributions for the fast stick condition in logarithmic and linear scales, correspondingly. Panels (b) and (d) illustrate the slow stick condition distributions in the same way. Coloured lines represent the distributions for each subject. Solid black lines represent the average distributions calculated based on the aggregated data for all the subjects. Dashed lines reproduce the Laplace distribution (zero mean, unit variance) for reference. (Online version in colour.)
- Figure 6.
Typical dynamics of the model (4.7). Panels (a)–(c) show the phase trajectories (solid black lines) generated by the system (4.7). Faint arrowed lines represent the force field of the linear system (4.8). The values of parameter γ used for simulations were (a) γ = 12, (b) γ = 22, (c) γ = 48. The values of other parameters remained unchanged, σ = 7.5, ɛ = 0.2. Frame (d) presents the phase diagram of the model (4.8) in the σγ plane. The solid line represents the stability boundary, the dashed line marks the border between the node-type and focus-type dynamics, and the dotted line corresponds to the feedback optimality condition. Dynamics of the model (4.7) in regions (a–c) of the diagram are illustrated in the corresponding panels. (Online version in colour.)
- Figure 7.
(a) Action point, (b) peak velocity, and (c) phase duration distributions exhibited by the model (4.7) for γ = σ2/2, ɛ = 0.2. Solid lines represent distributions for different σ. Dashed lines represent the experimentally obtained distributions averaged across all subjects. In (b) all positive local maxima and negative local minima of the cart velocity ν are treated as peak values. (Online version in colour.)
- Figure 8.
(a) Stick angle and (b) cart velocity distributions exhibited by the system (4.7) for different values of parameter σ. The parameter γ was set to σ2/2 to match the optimality condition (4.10); the noise intensity was fixed, ɛ = 0.2. (Online version in colour.)
- Figure 9.
(a–d) Phase trajectories of the model (4.7) for σ = 7.5 and (a) ɛ = 0.001, (b) ɛ = 0.13, (c) ɛ = 0.8, (d) ɛ = 2.0. Panels (e) and (f) depict the stick angle and cart velocity distributions for σ = 7.5 and different ɛ. Panel (g) presents the dependence of velocity kurtosis excess κ on the parameter ɛ. Black dashed line represents the double power law with exponents α1 = −0.2 and α2 = −1.5. The kurtosis excess is defined as κ = μ4/μ22 − 3, where μi is the i-th central moment of the cart velocity. In all panels γ = σ2/2. (Online version in colour.)
Tables
- Table 1.
Fast and slow stick conditions.
condition parameter τ parameter l slow 0.7 1.0 fast 0.3 0.4 - Table 2.
Balancing characteristics of the subjects. nfall is the average number of stick falls per minute, std(θ) is the standard deviation of the stick angle and %drift is the proportion of total balancing time the mouse velocity υ was equal to zero. In the slow stick condition, no stick falls were registered in all subjects.
subject sex age fast stick slow stick std(θ) nfall %drift (%) std(θ) %drift (%) 1 M 22 0.07 0.00 42 0.03 62 2 M 21 0.21 1.87 22 0.04 48 3 M 25 0.19 0.93 25 0.07 45 4 F 61 0.36 6.40 31 0.04 59 5 M 20 0.32 3.67 10 0.12 16 6 M 58 0.38 5.73 31 0.08 46 7 F 27 0.25 2.73 35 0.03 59 8 M 29 0.18 0.93 36 0.03 56 9 F 58 0.32 4.93 31 0.04 43 10 F 21 0.28 4.27 25 0.06 37
Volume 11, issue 99
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