Figure 1.
Model of the task and experimental setup.
A: The actual task of manipulating a cup of coffee. B: Conceptual model of a cup of coffee as a ball in a cup. C: Mechanical model of the cart-and-pendulum used as a simplified two-dimensional model for the ball-in-the-cup system. D: Model dynamics. E: Task performed in the virtual environment: the haptic manipulandum provides the real-time mechanical interaction with the object; the behavior of the system is displayed on a projection screen.
Figure 2.
Examples of force profiles and task kinematics in simulation and experiment, with corresponding definitions of the main variables.
The left column of panels shows the simulations: the cup kinematics is defined as a pure sine wave with fixed amplitude
and frequency
; ball angle
and angular velocity
are fully defined by their initial values
and
. The force F is found by solving the inverse dynamics for these kinematic profiles (
= 1.0 Hz,
= 0.2 m,
= 1.0rad,
= 0 rad). The right column shows the experimental kinematics: cup position
, ball angle
and ball angular velocity
are used to estimate the frequency
, amplitude
, initial ball angle
, and initial ball angular velocity
at each cycle of the trajectories. Red dots and vertical red lines indicate the location and time of the peak cup positions, at which the initial ball angle and angular velocity values for each cycle were determined,
and
, marked by blue dots. The experimental force profile is the net force applied by the subject to the manipulandum.
Figure 3.
Two profiles of force and cup and ball displacement exemplify unpredictable dynamics (left) and predictable dynamics (right).
The strobing procedure is illustrated by the dots and the vertical lines: at every peak of the cup displacement, the value of force is picked; the strobed force values are then projected onto the vertical axis to show the distribution for each simulation. The left example shows a scattered distribution, while the right periodic profile only shows one value. The bifurcation diagram summarizes all force distributions as a function of the parameter , the initial ball angle. The vertical axis displays the stroboscopic samples of force values
, simulated from a 1.0 Hz sinusoidal cup displacement
with 10 cm (full peak-to-peak) amplitude and
rad/s. The horizontal axis shows the initial ball angle
. The plot shows that when the simulation starts from the initial angle
= 1.0rad, the strobed force values do not change in successive cycles (blue), corresponding to the time profile on the right. At
rad, there is variance in the strobed values of force
, corresponding to the plot on the left (purple). The variance of
was used to define a measure for the predictability of object's dynamics (see Predictability Index in the supporting Text S1).
Figure 4.
Map of Mutual Information in the 2D result space spanned by amplitude and initial ball angle
(left) and examples of force-kinematics relationship (right).
The color map shows that realization of a 1.0 Hz sinusoidal cup trajectory with different amplitudes and initial angles differ in the resulting predictability of the object dynamics. Mutual Information, coded by color shades, describes the predictability of the object's behavior for a 1.0 Hz sinusoidal cup trajectory with different and
. The light blue circles indicate the strategies with the highest predictability of object's dynamics. The force-cup displacement plots on the right correspond to 2 representative points (strategies) in the result space. The profile on the top right shows high Mutual Information with consistent force-kinematic relationship over successive cycles (arrows indicate the traversal on the same constant curve over consecutive cycles). The profile on the bottom right shows a strategy with low Mutual Information, where the force-kinematic relationship changes in every cycle.
Figure 5.
Predicted values in the 2D result space for alternative criteria.
A: Global Lyapunov Exponent. B: Mean Squared Force. C: Normalized Ball Jerk. Note that the maps of the Global Lyapunov Exponent and Mutual Information are remarkably similar, while the Mean Squared Force and the Ball Jerk maps predict very different preferred regions in the result space.
Figure 6.
Experimental setup, the robotic arm (HapticMaster), rear-projection screen, and a subject sitting on a chair, while holding the manipulandum of the robotic arm.
The task instruction was to move the ball-and-cup object rhythmically between the two green targets at a frequency of 1 Hz. As long as the excursion maxima alternated between the wide target regions, the task was satisfied. Hence, the subjects could choose their movement amplitude.
Figure 7.
Representative profiles of raw experimental data from early and late practice.
is the interaction force between the subject and robot,
is the displacement of the cup (end-effector), and
is the angle of the ball. The time of peak cup positions (shown by red dots) served as time reference to strobe the ball angular displacement
and define
. Notice that the relatively irregular patterns of cup displacement and especially force in early practice became more regular and also larger in amplitude later in practice. An example of a highly unpredictable or chaotic behavior is shown on the right. Notice the negative values of
, which indicate the chaotic regime, shown in Figure 4. The Mutual Information measure for these continuous experimental data are 0.1256nat (early practice), 0.1693nat (late practice), and 0.1295nat (unpredictable behavior).
Figure 8.
Execution variables across 50 practice trials: frequency of cup displacement shows that the task was mostly performed at the instructed frequency of 1.0 Hz, amplitude of cup displacement
showed a significant increase across practice, ball angle at peak cup position
was approximately 1rad, and ball angular velocity
was close to zero throughout practice.
The bold lines show the averages across subjects (n = 8); the shaded bands represent one standard error across subject means; the thin lines show one representative subject.
Figure 9.
Strategies for all subjects and all trials displayed in the 2D result space.
The horizontal axis corresponds to the simulation variable or its average experimental estimates
in each trial. The vertical axis corresponds to the simulation variable
or its average experimental estimates
. Each point represents the average strategy in one 45 s trial. Darker red indicates early practice and lighter red indicates late practice. The left panel shows that subjects explored regions with lower Mutual Information and lower Mean Squared Force; however, the majority of trials converged to areas with higher Mutual Information and lower Sensitivity. The right panel shows the same data separated by subject: the red arrows mark how each subject's average strategy changed from early practice (first 5 trials) to late practice (last 5 trials). All subjects increased their movement amplitude, associated with an increase in overall exerted force. The majority of subjects switched from low- to high-predictability regions in the result space. None of the subjects moved toward the minimum force strategy.
Figure 10.
Strategy measures Mutual information and its Sensitivity, Mean Squared Force, and Smoothness plotted across all 50 trials.
Thick lines are the across-subject average, the shaded bands show one standard error around the subjects' mean; the thin lines show one representative subject. Left column: model-based strategy measures, simulated at experimental execution variables that at each point show the behavior of the object in the upcoming cycle(s) for perfect realization of a sinusoidal cup trajectory. Right column: strategy measures determined from continuous experimental force and kinematics that indicate the combined behavior of the object dynamics and the continuous online control by subjects. Note that Sensitivity can only be determined based on the simulated maps.
Figure 11.
Schematic of the control loop between the central nervous system and the object dynamics.
Mutual Information between the force and kinematics quantifies the complexity of the object dynamics. At the same time, Mutual Information between the sensory information and motor commands quantifies the level of the required sensorimotor information processing. This can be interpreted as a reflection of the complexity of the sensorimotor information processing required by central nervous system for performing the movement task.