Fig 1.
Experimental task and protocols.
A) The task of carrying a ‘cup of coffee’ was simplified to a ball sliding inside a semicircular cup and modeled as a cart-pendulum system. B) The cup-ball task was displayed on a screen. Participants controlled the cup shown on a screen by moving the handle of a robotic manipulandum. The cup was constrained to move horizontally in 1-D delimited by two target boxes. C) Exemplary trial. Participants ‘jiggled’ the cup to explore and prepare the cup-ball system in the ‘Preparation Stage’ before executing rhythmic oscillations of the cup between the two boxes at their choice of frequency in the ‘Rhythmic Stage’. The initial ball angle represented preparation of the cup-ball system and the cup frequency in the rhythmic stage represented continuous interaction. D) Three different pendulum lengths were used to manipulate the uncertainty of the ball dynamics: short (0.3 m), medium (0.6 m), and long (1.2 m). Two experimental protocols were tested: i) Random protocol: the pendulum length was varied from trial to trial within a block without providing any explicit cues, and ii) Blocked protocol: the pendulum length remained constant within a block of trials. Each participant completed 40 trials per pendulum in both random and blocked protocols. The order of random and blocked protocol was counterbalanced as indicated by the bidirectional arrows. The grip force exerted by the participant on the handle was measured using a grip force sensor as shown in B. The grip force was used as a proxy for mechanical impedance of the arm.
Fig 2.
Exemplary trial visualizing the computation of relative phase between the cup and ball. The phases of the cup and ball were computed using the Hilbert transform. The phases were then unwrapped and subtracted to obtain the relative phase or phase difference. The cup and ball oscillate in phase, as indicated by the relative phase hovering around 0 deg, a relative phase of 180 deg represents anti-phase oscillations. The circular variance of relative phase that represents relative phase variability is 0.002.
Fig 3.
A) Relative phase plotted against trials for the three pendulum conditions in the two protocols. Different lines represent different participants. The distributions of relative phase across all participants and trials are plotted at the right margin. Participants mostly exhibited in-phase (0 deg) or an anti-phase (180 deg) relations between the cup and ball. B) Relative phase variability, computed as the circular variance of relative phase, represented the stability of the dynamics between the cup and ball. Mean relative phase variability across subjects is plotted against trials for the three pendulum conditions in the two protocols. The error ribbons represent standard error. C) Summary results of relative phase variability. Each data point in the box plot represents a participant average across 40 trials. Relative phase variability was similar between the two protocols, despite the increased uncertainty in the random protocol.
Fig 4.
Behavior in preparation stage.
A) Heatmaps of relative phase plotted against trial time from all participants and trials. Preparation stage is shown as negative time and rhythmic stage as positive time. The color scheme for the heatmap is mapped to the normalized probability and is different for the preparation and rhythmic stages. Relative phase converges within the preparation stage and remains fairly constant in the rhythmic stage in both random and blocked protocols. B) Summaries of duration of preparation and time taken to stabilize relative phase. Each data point in the box plot represents a participant average across trials. Participants utilized more preparation duration, and took more time to stabilize the relative phase in the random protocol in comparison to the blocked protocol.
Fig 5.
Preparation and interaction strategies.
A) Cup frequency plotted over trials for the three pendulum conditions in the two protocols. Different colored lines represent different participants. Individual participants adopted different cup frequencies that did not converge to any particular common value over trials. The cup frequency distributions across participants and trials are shown in grey at the right margin. B) Initial ball angle plotted over trials for the three pendulum conditions in the two protocols. There were no observable trends in the initial ball angles or cup frequencies.
Fig 6.
Forward simulations to explain preparation and interaction choices.
A) A human-inspired controller with mechanical impedance was used to run forward simulations of the cart-pendulum model. The controller computed a force proportional to deviation from the desired states, related through the impedance gains K and B. The impedance gains were held constant at K = 40 and B = 70. B) One forward simulation for initial ball angle of and cup frequency
. The relative phase hovered close to
with a circular variability of 0.012. C) Heatmaps of relative phase variability for combinations of initial ball angles and cup frequencies plotted in grey shades in the background. Participant choices of initial ball angle and cup frequency from all trials are plotted as colored points in the foreground. Colors represent different participants. Participants’ choices fell into the dark regions of the heatmap, showing that they nonlinearly covaried preparation and interaction strategies to maximize relative phase stability. D) Heatmaps of absolute force, smoothness of force, and risk of ball escape plotted for the medium pendulum condition. Participant choices of initial ball angle and cup frequency from the random protocol plotted in the foreground. Kullback-Leibler divergence between the distributions of participant data and the underlying heatmaps were lowest for relative phase variability, i.e., stability, (random-medium plot in C) compared to the three alternative costs. Relative phase variability captures covariation of preparation and interaction strategies better than absolute force or smoothness of force.
Fig 7.
A) Grip forces and applied forces from all trials of one representative participant are plotted as thin faint lines in the background. The mean grip force and applied force are plotted as thick solid lines. The participant increased the grip force during the preparation stage and reached a higher level in the rhythmic stage in the random protocol compared to the blocked protocol. In contrast, the applied forces were similar between the random and blocked protocols. B) Grip force and applied force from all trials for a participant plotted as individual points. Grey lines connect data points within the same participant. Grip forces were higher in the random protocol compared to the blocked protocol, both in the preparation and rhythmic stage. In contrast, applied forces were similar between the two protocols.
Fig 8.
Predictions of open-loop optimal control on mechanical impedance.
A) State space variables from one simulation run with the medium pendulum, and low noise simulating the blocked protocol, setting the desired initial ball angle to , and cup frequency to
. The desired cup trajectories and initial ball angle are highlighted in red. The stiffness K rises monotonously in the preparation stage and plateaus in the rhythmic stage. B) Stiffness and applied force plotted against time for the blocked (low noise) and random (high noise) conditions, for the same desired initial ball angle and cup frequency as in A. Predicted stiffness values are higher in the random protocol in comparison to the blocked protocol, similar to the experimentally reported grip forces. In contrast, applied forces are similar between the protocols. C) Average stiffness and applied force from all runs with combinations of desired initial ball angles and cup frequencies. Stiffness values are higher in the random protocol compared to the blocked protocol. In contrast, the applied forces are of similar magnitude between the two protocols. D) Model predicted stiffness (K) for various combinations of initial ball angle and cup frequency plotted in the background. Participant choices from different trials from either the blocked protocol or random protocol are plotted in the foreground. Colors represent different participants. Participants covaried their choice of preparation and interaction to minimize mechanical impedance in both the random and the blocked protocol. E) Model predicted average absolute feedforward force (Fff) for various combinations of initial ball angle and cup frequency plotted in the background. There are some regions of overlap between low average absolute forces and low stiffness.