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The Journal of Neuroscience, February 1, 2000, 20(3):1066-1072
Human Arm Movements Described by a Low-Dimensional Superposition
of Principal Components
Terence David
Sanger
Department of Brain and Cognitive Sciences, Massachusetts Institute
of Technology, Cambridge, Massachusetts 02139
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ABSTRACT |
A new method for analyzing kinematic patterns during smooth
movements is proposed. Subjects are asked to move the end of a two-joint manipulandum to copy a smooth initial target path. On subsequent trials the target path is the subject's actual movement from the preceding trial. Using Principal Components Analysis, it is
shown that the trajectories have very low dimension and that they
converge toward a linear superposition of the first few principal
components. We show similar results for handwriting on an electronic
pen tablet. We hypothesize that the low dimensionality and convergence
are attributable to combined properties of the internal controller and
the musculoskeletal system. The low dimensionality may allow for
efficient descriptions of a large class of arm movements.
Key words:
reaching; human; movement; motor control; convergence; principal components; linearity
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INTRODUCTION |
For unconstrained human hand
movements, there is a very large set of possible trajectories
determined both by the path as well as by the time at which each point
on the path is reached. Accessing or computing such trajectories would
require significant motor memory storage and computational power. This
consideration leads to the question of whether the human motor system
might use a simplified strategy that restricts the set of possible
movement trajectories (Bernstein, 1967 ). Several properties of
human movement patterns are well known, including smoothness (Hogan,
1984; Flash and Hogan, 1985; Uno et al., 1989), interjoint torque
constraints (Gottlieb et al., 1996), and the "2/3-power law"
relating speed to curvature (Viviani and Terzuolo, 1982; Lacquaniti et
al., 1983). It is not evident how these observed properties of movement
might lead to computational simplifications.
Using Principal Components Analysis (PCA), recent studies of human
kinematics have suggested that in several different cases movements can
be described as a linear combination of a small number of components.
Eighty percent of the variance of grasping hand movements is captured
using only two principal components (Santello and Soechting, 1997;
Santello et al., 1998), and similar results have been found for typing
(Soechting and Flanders, 1997). PCA has also been used to analyze lip
motion during speech (Ramsay et al., 1996), trunk bending (Alexandrov
et al., 1998), and gait (Loslever et al., 1994; Borghese et al., 1996;
Olney et al., 1998), and in all cases the number of components needed
to describe the movements was significantly fewer than the total number
of possible degrees of freedom. Recent results suggest that 94% of the
variance in the pattern of forces generated in the frog hindlimb in
response to vestibular stimulation can be explained using only five
components (d'Avella and Bizzi, 1998), and 75-90% of the variance of
rabbit jaw muscle EMG during voluntary chewing can be explained using only three components (Weijs et al., 1999). This data, along with earlier results on superposition of fields of forces with spinal cord
microstimulation (Bizzi et al., 1991; Mussa-Ivaldi et al., 1994; Sergio
and Ostry, 1995), provides a physiological correlate to the
psychophysical results.
We hypothesize that the trajectory followed by the hand during planar
curved movements can be approximated as a linear combination of a small
number of principal components. Because human movement has great
flexibility and may change depending on the specified task, we
developed a new practice methodology to investigate the unconstrained
behavior of the human arm. As a subject attempts to copy a curved
figure, each movement that he or she makes is used as the target figure
for the next trial. We refer to this technique as "iterated
practice." Although an initial target is provided, subsequent
movements can modify this target toward a class of preferred movements.
The trajectory followed on successive movements becomes progressively
smoother with an increasing fraction of the variance accounted for by a
decreasing number of principal components. We show that this behavior
is a property of smooth linear low-dimensional systems, and we suggest
that it arises as a combined effect of an internal human movement
controller and mechanical properties of the arm.
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MATERIALS AND METHODS |
Experimental techniques were approved in accordance with MIT's
policy statement on the use of humans in experiments, and informed consent was obtained from eight right-handed subjects. Subjects grasped
the end of a two-joint horizontal planar manipulandum (Fayé,
1986) with their dominant hand while seated in a chair aligned with the
center of the workspace. A point on a 14 inch video screen placed
slightly above eye level tracks the movement of the end of the
manipulandum. To start a trial, the subject must move the manipulandum
so that the point on the screen lies within a small box corresponding
to a 2 cm square in the center of the manipulandum workspace. As soon
as the point comes to rest within this box, a smooth target curve is
displayed in the lower left-hand corner of the screen. Subjects are
asked to copy the approximate shape of the figure smoothly and rapidly
within 0.5 sec. There are no other timing constraints, and in
particular, subjects are free to choose movement velocities at each
point on the curve. The subjects perform their copy in the center
of the screen at a distance of several inches from the displayed target
curve so that tracing is not possible. The trial starts when the
manipulandum leaves the start box and ends when the manipulandum is
again brought to rest, at which time the full path of the subject's movement is displayed. Trials lasting >0.5 sec sound a warning buzzer
and are discarded. After the first few minutes of practice, subjects
had no difficulty in accomplishing movements in <0.5 sec. The
horizontal (tangential) and vertical (radial) positions of the
manipulandum endpoint were sampled at 100 Hz by a 66 MHz Intel Pentium
computer. In some cases, the x, y, and z
components of elbow position were also sampled using a Polhemus
position sensor. The elbow movement was not constrained. In a related
experiment, four subjects moved a pen on the surface of a smooth tablet
(Calcomp), and pen tip positions were recorded at 30 Hz. The experiment
was otherwise similar.
For each block of trials, the first 2 target paths are identical, and
they are generated automatically using:
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(1)
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(2)
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where  (t) and  (t) are the
path velocities in centimeters per second, the amplitudes  are
chosen randomly between 0 and 1, and the phases  are chosen randomly
between   and . The dynamic trajectory information is arbitrary,
and the display does not include timing or velocity information. Note
that although the static shape of the target is described by only 12 variables, the set of possible trajectories that accomplish this path
has very high dimensionality because the subjects are free to choose from an infinite number of possible sets of hand velocities.
On trials beyond the second, the target hand path is the average of the
recorded hand trajectories on the previous two movements, where the
average x and y coordinates are computed
point-by-point at each time during the movements. Averaging of two
previous paths reduces the effect of small hand tremors or
unintentional movements. Points are aligned in time by scaling pairs of
movements with linear interpolation to have the same total time from
movement onset to return to a stationary position. The size of the
displayed target path is normalized so that the larger of the
x or y deviation is constant.
Data was collected in blocks of 20 trials, with each block initiated by
a new random target path. An example set of hand paths from a single
experiment is shown in Figure
1A. Experimental
sessions lasted 1 hr, during which time 60-80 movement blocks
(1200-1600 movements) were performed. Subjects were seen once or twice
weekly over a period of 3 months.
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Figure 1.
Example of iterated practice and the calculated
principal components. A, Each row of the figure
shows the recorded hand movements for a block of 20 trials. Each block
was started with a new random target shape presented during the first
two trials. Errors on the first trial are often the result of
accommodation to a new shape. B, The shape of the first five
eigenvectors (principal components) computed over 2400 movement trials.
The first row shows the path in x-y coordinates,
and the second and third rows show the
x and y components as a function of time.
C, Log-log plots of the instantaneous velocity versus
curvature of the components, with numeric values of the average slope
of the plot.
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PCA was used to analyze the variability in trajectory shape. For each
movement, 50 points spaced equally in time from the start to the end of
the trajectory were extracted. Thus, all movements are effectively
scaled to the same duration. A 100 element column vector
 was constructed for each movement by joining the
x and y coordinates mx(t)
and my(t) of the 50 points so that
= < mx(1), ... , mx
(50), my (1), ... , my(50) >. (If
elbow motion is recorded, then 50 x, y, and z
components of elbow position are appended to the hand movement vector
.) To compute the principal components, the mean
vector is subtracted from each of a set of K movement
vectors j, and the covariance
matrix is formed:
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(3)
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The eigenvectors  i (principal
components) are then calculated using Matlab (version 5.0, MathWorks).
To allow comparison across movement trials, all 1200-1600 movements
made by a subject on a given day are used to calculate a single set of
eigenvectors. The x and y components of the
eigenvectors are then separated to yield cix(t)
and ciy(t), and example components are shown in Figure 1B. Note that the components themselves
represent smoothly curved trajectories. They also satisfy an
approximate power law relating velocity to curvature (Viviani and
Terzuolo, 1982; Lacquaniti et al., 1983), as shown in Figure
1C. There is a high baseline correlation between principal
components from different subjects because most movements are
smooth, and the early components are thus dominated by low-frequency
smooth movement terms.
Any movement vector can be exactly reconstructed as a superposition of
the full set of principal components i
according to:
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(4)
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where the number of dimensions is 100 (250 if elbow motion is
included). If the movements are approximated with a smaller number of
components n < 100, then we can write:
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(5)
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so that:
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(6)
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(7)
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where n is the number of necessary components and is
much less than the maximum possible N (100). For any given
number of components n, the approximation error is:
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(8)
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and the error en decreases as the number
of components n increases.
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RESULTS |
Figure 2A shows
histograms over all subjects and all days of the approximation error
en2 for one to five components. The
principal components are calculated from the entire set of trajectories
on each day. The results are not significantly different if components
are calculated from only the final movement of each block, but this
leads to increased variability caused by the decreased number of
samples. The first five components account for an average of 90% of
the hand movement variance. Figure 3
shows the decrease in component variance (eigenvalue) with increasing
component number for one subject on a typical day. Note the rapid
decrease after the first few components, with two orders of magnitude
decrease after the first 10 components and 10 orders of magnitude after
the first 20.
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Figure 2.
Histograms of movement approximation variance
using from one to five principal components for all subjects. Each
point in the histograms represents 1 d for one subject.
The horizontal axis shows the fraction of the total movement
variance accounted for. A, Approximation for two dimensions
of hand movement. B, Approximation for all subjects using
the components from subject 1. The similarity of components between
different subjects makes the approximation possible, but slightly worse
than in A. C, Approximation for two dimensions of
hand movement combined with three dimensions of unconstrained elbow
movements for all subjects. The number of components remains low
despite 250 dimensions in the input data. D, Approximation
for pen tip position on an electronic drawing tablet. The same
phenomenon holds, although in general each subject uses different
components than in the manipulandum task.
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Figure 3.
Magnitude of the eigenvalues versus eigenvalue
number for a single subject on 1 d (1200 movements). The same data
is plotted for the first 10 components on a linear scale and for all
100 components on a log scale.
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Because of the high dimensionality of the space of possible movements,
there is no single measure that describes the similarity between
components from different subjects. In fact, we are most interested in
the similarity between the space of components from each subject. To
investigate this, we can examine the ability for the components from
one subject to describe movements from a different subject. Low error
for a small number of components indicates similar approximation
abilities and a similar space of components. Figure
2B approximates the movement error for all subjects
using the components from subject 1, thereby demonstrating the
similarity between component spaces from different subjects and at
different times. The actual components may be very different, because
there may be shuffling or varying linear combinations of the first few components.
Figure 2C shows that inclusion of three additional
dimensions of elbow position at each time point does not significantly increase the required dimensionality. This implies that the same small
set of components can be used to describe both hand and elbow movement,
and that the hand and elbow trajectories are correlated for this task.
Figure 2D shows similar results for the approximation error using a pen tablet instead of the manipulandum to record movements.
Figure 4 shows a typical trial block from
subject 5, as well as approximations of each trial movement using from
one to five components. As the iterated practice progresses, the actual
subject movements change to shapes that are better approximated by a
smaller number of components. Note that the first component does not
contribute significantly to this particular block of trial
movements.
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Figure 4.
Approximation of trajectories within a single
trial block using from one to five components. The top row
shows a single iterated block of 20 trials. Components are calculated
from the movements of the entire day, and the second through
sixth rows show approximation of the movement trajectories
using progressively fewer components. Note that the similarity between
the actual movement and the approximations increases as the iteration
continues.
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On average, the approximation error en2 for
the first few components decreases with iterated practice, indicating that these components represent progressively more of the data variance
and that the movement converges toward a superposition of fewer and
fewer components. The effect is shown for all subjects in Figure
5A. The figure shows the
percentage of variance accounted for, averaged over blocks of 20 trials
(400 movements) on multiple days for each subject. Here, principal
components are computed separately within each block of 400 movements
to allow comparison of the average errors. In all cases, the
combination of the first few components accounts for progressively more
of the variance. In some cases, although the percentage accounted for
by the first five components does not change significantly, the
movements converge so that the first 3-4 components contain a larger
proportion of the variance than initially. Also note that the first
component may actually lose power in some series, but that this power
is recovered in the second component as the set of movements
"rotates" in the high-dimensional space toward the low-dimensional
projection.
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Figure 5.
Change in approximation error with iterated
practice for each subject. A, From top to
bottom in each figure, the lines indicate the
decrease in error with from one to five components, showing that the
first few components account for progressively increasing percentages
of the total movement variance. The top line shows the
decrease in error with approximation using one component, the
second line shows the decrease for two components, and so
on. The error is shown normalized to the total movement variance. Each
plot is averaged over 20 movement blocks. Note that, for subject 1, the
first two components combined account for an increasing percentage of
the total variance, although the first component percentage decreases.
B, Lack of convergence when the same target is presented at
each trial (Subject 2). This shows that convergence is not
simply an effect of repeated practice.
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Convergence is not simply an effect of practice, because Figure
5B shows that repeated presentation of the same target does not lead to this phenomenon. Figure 6
shows the average convergence behavior for all subjects for 2 and 3 components, with error bars indicating 1 SD from the
between-subject mean. Although the slope of the average curve is not
significant, the average downward trend is significant (p < 0.01). Figure 7 shows the distance
from each movement in an iterated sequence to the final movement in the
block. Although the progression toward this final movement is steady,
it does not stabilize at any time. If movements progressed toward a
particular set of movements, we would expect that at least in some
cases this curve would approach zero at a low iteration number and then
remain near zero. This pattern is not seen. Figures 6 and 7 show that
although iterated practice converges toward a linear subspace, it does
not converge to a fixed set of preferred trajectories such as those in
a memory-based controller (Atkeson, 1989, 1990).
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Figure 6.
Average across subjects of the decrease in
approximation error with increasing iteration, using two and three
components. Error is averaged over blocks of 400 trials for eight
subjects on multiple days. Error bars indicate 1 SD of the
between-subject variation for a single iteration number.
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Figure 7.
Average sum squared distance from the final
movement in a block, by iteration number. Distance is averaged over
blocks of 400 trials for 8 subjects on multiple days. Error bars
indicate 1 SD of the between-subject variation for a single iteration
number. Note that the curve shows steady slow progression toward the
final movement, but it does not show evidence of stabilizing near the
final curve.
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DISCUSSION |
We have shown that the trajectories followed by the hand while
performing smooth planar movements can be described by a small number
of principal components. These components satisfy an approximate power
law relating speed and curvature. The approximation by a small number
of components is accentuated with iterated practice, for which an
increasing fraction of the variance of the movements can be described
by a decreasing number of components. These results therefore show that
arm movement trajectories provide a further example of low
dimensionality in human motor behavior.
The methods used here operate within the Cartesian coordinate system.
Although there is some evidence to suggest that visual-motor tasks are
in fact planned in Cartesian coordinates (Morasso, 1981; Wolpert et
al., 1995), our results do not address this question, and the same type
of analysis could have been performed in joint or body-centered polar coordinates.
Trajectories with a small number of components are a property of
certain classes of very smooth movements. For example, trajectories that are generated by splines, Fourier series, or from linearly filtered noise will all have low dimensionality. In particular, a
second-order mechanical system with mass, stiffness, and damping may
generate low-dimensional trajectories when driven by white noise. We
now provide a mathematical analysis of iterated practice for linear
systems, followed by a simulation of iterated practice for a two-joint
planar arm with a smooth equilibrium point controller (Feldman, 1974;
Bizzi et al., 1984).
Consider a system in which the mapping from one movement to the next
can be described by a linear system A: k  k+1. A represents the entire process of seeing
the target path generated from the previous movement, planning the next
trajectory, sending motor commands to the muscles to execute the
trajectory, and then performing the next movement. If the initial
target movement is 0, the
kth movement in the iterated practice series is
k = Ak0. (There is an additional
effect from rescaling of the movements to be at approximately the same
size. We will ignore this because it does not affect the conclusions.)
This process is known to converge toward the first eigenvector of the
matrix A (Golub and Van Loan, 1983). If we perform a
principal components decomposition k =  iai(k)i at each
step, then the change in each component for step k is given
by:
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(9)
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where  i is the ith eigenvalue
of A, and ci is its ith
eigenvector. This means that if we order the components from largest to
smallest with 1 > 2 > ... , then during iterated practice the higher-numbered components
will decrease fastest so that the lower components
a1, a2, and so on
account for an increasing percentage of the data variance. This effect
is greatest for low-dimensional A with rapidly decreasing
eigenvalues i. Therefore, the property of
convergence toward a low-dimensional representation can be explained if
the combination of movement controller and musculoskeletal mechanics
behaves as a low-dimensional linear system.
This raises the question of whether the musculoskeletal mechanics alone
can account for the data. The mechanics are not able to perform the
required translation from the visual representation of a path (which
has no timing or velocity information) into the temporal description of
a trajectory, but the mechanics can act as a temporal smoothing
operator that reduces the apparent dimensionality. However, we now show
that simulation of a two-joint mechanical arm with equilibrium point
control is not sufficient by itself to explain the convergence results.
For the simulation we used standard equations for the dynamics of a
two-joint planar robot arm (for example, see McCarthy, 1986) with Euler
integration. Mechanical parameters were taken from Amis et al. (1979),
and an equilibrium-point controller was used to map from a desired trajectory  at each joint (in joint coordinates) into torques according to  = K(  ) B (Flash, 1987). Stiffness K and damping
B coefficients were set as constant at 2 Nm/rad and 0.4 Nm · rad 1 · sec1,
respectively, based approximately on human measurements from Katayama
and Kawato (1992) and Bennett et al. (1992). Initial target
trajectories were generated from Equations 1 and 2, and 20 blocks of
iterated trials were simulated. At each iteration, the target for the
next movement was the average of the two previous trajectories
(including timing information), because the mechanical system could
otherwise not make use of the visual representation of a path.
Figure 8 shows the first 10 iteration
trials, and Figure 9 shows the decrease
in approximation error for 1-5 components as a function of trial
number. Although the general features of increasing smoothness and
decreasing number of components are present, the simulated trajectories
in Figure 8 do not always converge and frequently appear to jump
between different alternatives. In particular, note the increase in
error on the last two trials in Figure 9. This occurs despite the
apparently low variability in the trajectories shown in Figure 8
because of the fact that the final trajectories in each block of trials
are poorly represented by the set of principal components computed for
the entire set of movements. This effect was seen on multiple
simulations and suggests that although the movements are converging
toward a low-dimensional space, it is not the same low-dimensional
space as that computed for the initial movements. This behavior differs
from the human data and the analysis of linear systems discussed above,
for which convergence is expected to be approximately monotonic, as in
Equation 9. The simulation results therefore suggest that the
convergence behavior cannot be explained solely on the basis of the
mechanical system with an equilibrium point controller. Therefore, the
smoothness and low dimensionality may be attributable to the
combination of an internal movement controller and the musculoskeletal
mechanical system.
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Figure 8.
Example of iterated practice for simulation of an
equilibrium-point controlled two-joint planar mechanical arm, shown as
in Figure 1. Each row of the figure shows the simulated hand
movements for a block of 20 trials.
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Figure 9.
Change in approximation error with iterated
practice for simulation of equilibrium-point controlled two-joint
planar mechanical arm, plotted as in Figure 5. From top to
bottom in each figure, the lines indicate the
decrease in error with from one to five components, averaged over 20 movement blocks.
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In conclusion, the human data shows that the final resulting movements
are a set of trajectories of low dimensionality relative to the
100-dimensional complete set of possibilities. Similar results hold for
handwriting as for arm movements, and when elbow movements are measured
they are found to correlate with the hand movements in such a way that
the dimensionality does not significantly increase. Simulation results
suggest that the low dimensionality and convergence of the trajectories
are attributable to a combination of the movement controller and the
musculoskeletal system, but further research will be needed to
determine the relative contribution of each.
We emphasize that our results provide only a description of the
movements, and therefore they do not indicate how these trajectories are generated. Furthermore, our results do not indicate whether this
phenomenon represents an intended behavior of the system or merely a
necessary but unintended consequence of the mechanics. Nevertheless, it
is possible that the low dimensionality may allow the controller to use
a simplified strategy to reduce the storage and computational
requirements needed to express variability within the set of curved
hand trajectories.
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FOOTNOTES |
Received Sept. 20, 1999; accepted Nov. 15, 1999.
This research and Dr. Bizzi's laboratory were supported by National
Institutes of Health Grant NS09343 and ONR Grant
N00014-95-1-0445 to Dr. Emilio Bizzi. I was supported during part of
this research by a McDonnell-Pew postdoctoral fellowship. I thank
Emilio Bizzi, Simon Giszter, Francesca Gandolfo, Brian Benda, and Margo
Cantor for discussions and assistance with these experiments. I also acknowledge the diligence and insight of the anonymous reviewers who
contributed significantly to revision of this manuscript.
Correspondence should be addressed to Terence D. Sanger, Department of
Movement Disorders, Toronto Western Hospital, MP-11, 399 Bathurst
Street, Toronto, Ontario M5T 2S8, Canada. E-mail: tds{at}ai.mit.edu.
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