Fig 1.
The controlled system is a pendulum with mass m hanging from a weightless rod of length L.
The angle between the pendulum and its vertical axis is denoted by θ, where θ = 0 rad indicates the pendulum’s downward position. The goal of the task assigned to participants is to swing the pendulum in order to balance it in its upright position for as long as possible (i.e., θ ≃ π rad and ).
Table 1.
Action and state entropies’ Spearman coefficients ( and
, respectively) for all pendulum lengths L.
Fig 2.
(a) Utility Ui of trials i during the balancing task plotted as a function of H(Ai|Si) for all participants and pendulum lengths L. Larger values of utility Ui correspond to better performance in balancing the pendulum. Spearman correlation ρ = 0.944; Pearson correlation r = 0.967. In (b) the trials are separated by pendulum length in different subplots.
Fig 3.
Action entropy H(A|S) averaged over trials with the same pendulum length L.
The abscissa represents the action entropy and the ordinate represents the pendulum length to allow direct comparison with Fig 2.
Fig 4.
(a) Utility Ui of trials i during the balancing task plotted as a function of H(Si) for all participants and pendulum lengths L. Spearman correlation ρ = −0.982. In (b) the trials are separated by pendulum length in different subplots.
Fig 5.
Utility Ui of trials i during the balancing task plotted as a function of H(S′i|Ai) in (a) and of I(Ai;S′i) in (b), for all participants and pendulum lengths L.
(a) Spearman correlation ρ = −0.979. (b) ρ = −0.907.
Fig 6.
State entropy H(S) averaged across trials with the same pendulum length L.
The abscissa represents the state entropy and the ordinate represents the pendulum length to allow direct comparison with Fig 4.
Fig 7.
Support of Si within the phase space of four representative trials: Two participants swinging the longest pendulum ((a) and (b)) and other two participants swinging the shortest pendulum ((c) and (d)).
Trials start at the center of the phase space.
Fig 8.
Proportion of the time spent by participants in the balancing region (± 0.27 rad from the vertical axis) for each pendulum length L.
Fig 9.
Key press activity within and outside balance region (± 0.27 rad from the vertical axis) for each pendulum length L.
Fig 10.
Distribution of key presses in the balancing region for three trials and distance traveled by the pendulum during the key presses.
Blue segments represent actions that push the pendulum in the clockwise direction, red segments indicate actions that steer the pendulum towards the anti-clockwise direction. Key presses (“balancing episodes”) are reported subsequently from the top to the bottom of the plot. In (a) and (b) the key presses of two skilled participants are reported, where in (c) the key presses of a less skilled participant is shown.
Fig 11.
Utility Uj of simulations j plotted as a function of H(Aj|Sj) ((a)) and H(Sj) ((b)) for all artificial controllers and pendulum lengths L.
These can be compared to Figs 2A and 4A respectively. No observation and transition noise is employed here (i.e., ). (a) Spearman correlation ρ = 0.67 (p = 0.334, not significant, due to the marked non-monotonicity of the data). (b) ρ = −0.96 (p < 0.01).
Fig 12.
Utility Uj of simulations j plotted as a function of H(Aj|Sj) ((a,c,e)) and H(Sj) ((b,d,f)) for all pendulum lengths L and artificial controllers with different observation disturbances η (here ).
(a) Observation standard deviations ,
; Spearman correlation ρ = 0.474 (p < 0.01). (b)
,
; ρ = −0.951 (p < 0.01). (c)
,
; ρ = 0.908 (p < 0.01), Pearson correlation r = 0.980 (p < 0.01). (d)
,
; ρ = −0.908 (p < 0.01). (e)
,
; ρ = 0.954 (p < 0.01).(f)
,
; ρ = −0.954 (p < 0.01).
Fig 13.
Utility Uj of simulations j plotted as a function of H(Aj|Sj) ((a,c,e)) and H(Sj) ((b,d,f)) for all pendulum lengths L and artificial controllers with different observation disturbances η and transition disturbances δ.
(a) Observation standard deviations ,
; transition standard deviations
; Spearman correlation ρ = 0.694 (p < 0.01). (b)
,
;
; ρ = −0.957 (p < 0.01). (c)
,
;
,
; ρ = 0.67 (p < 0.01), Pearson correlation r = 0.961 (p < 0.01). (d)
,
;
,
; ρ = −0.885 (p < 0.01). (e)
,
;
,
; ρ = 0.582 (p < 0.01).(f)
,
;
,
; ρ = −0.574 (p < 0.01).