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The Journal of Neuroscience, September 15, 2002, 22(18):8201-8211
Comparing Smooth Arm Movements with the Two-Thirds Power Law and
the Related Segmented-Control Hypothesis
Magnus J. E.
Richardson1, 2 and
Tamar
Flash1
1 Department of Applied Mathematics, Weizmann Institute
of Science, Rehovot 76200, Israel, and 2 Laboratoire de
Physique Statistique, Ecole Normale Supérieure, 75231, Paris
Cedex 05, France
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ABSTRACT |
The movements of the human arm have been extensively studied for a
variety of goal-directed experimental tasks. Analyses of the trajectory
and velocity of the arm have led to many hypotheses for the
planning strategies that the CNS might use. One family of control
hypotheses, including minimum jerk, snap and their generalizations to
higher orders, comprises those that favor smooth movements through the
optimization of an integral cost function. The predictions of each
order of this family are examined for two standard experimental tasks:
point-to-point movements and the periodic tracing of figural forms, and
compared both with experiment and the two-thirds power law. The aim of
the analyses is to generalize previous numerical observations as well
as to examine movement segmentation. It is first shown that contrary to
recent statements in the literature, the only members of this family of
control theories that match reaching movement experiments well are
minimum jerk and snap. Then, for the case of periodic drawing, both the
ellipse and cloverleaf are examined and the experimentally observed
power law is derived from a first-principles approach. The results for
the ellipse are particularly general, representing a unification of the
two-thirds power law and smoothness hypotheses for ellipses of all
reasonable eccentricities. For complex shapes it is shown that velocity
profiles derived from the cost-function approach exhibit the same
experimental features that were interpreted as segmented control by the
CNS. Because the cost function contains no explicit segmented control,
this result casts doubt on such an interpretation of the experimental data.
Key words:
motor control; two-thirds power law; arm; planning
strategies; cost function; minimum jerk; segmentation
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INTRODUCTION |
Movements of the human body are
restricted both by mechanics and computational resources. However, the
repertoire of movements that is actually observed comprises a small
part of those that are possible, because the CNS uses planning
strategies that impose further restrictions. The determination of these
strategies is nontrivial because they are not measurable directly. The
approach has been therefore to observe the behavior under various
conditions and to construct mathematical theories that describe the
output of the motor system. The past few decades of experimentation
have lead to a number of (apparently) different hypotheses for planning strategies; for example, it has been proposed that the CNS plans in the
coordinate system of the joint angles (Uno et al., 1989 ; Nakano et al., 1999) or in the coordinates of the
hand's position (Morasso, 1981; Flash and Hogan,
1985) or that the CNS plans movements that are robust against
the inherent noise in the motor system (Harris, 1998;
Harris and Wolpert, 1998).
A successful description of an experiment by a theory is usually taken
as evidence that the CNS does indeed operate in such a manner. But what
if two different theories have the same predictions? Obviously in this
case no conclusions can be firmly drawn until further experiments are
performed, outside the region of agreement. However, hypotheses can
also be compared mathematically to reach an understanding of their
interrelation. In this paper such a comparative study is undertaken
with respect to two simple, but widely applicable theories: the
optimization of smoothness-based cost functions and the two-thirds
power law between curvature and velocity. Predictions of these control
hypotheses will be examined for two standard experimental paradigms:
point-to-point reaching movements and the periodic tracing of simple
figures. For the case of point-to-point movements it is shown that only a few members of the family of smoothness-based cost functions can
match experiments well. For the case of the periodic drawing of figural
forms, a number of novel results will be derived. A mathematical form
for the exponent relating hand velocity to curvature of the two-thirds
power law will be obtained from the cost functions for an ellipse and
shown to be indistinguishable from the experimental value of ~0.33
(the name "two-thirds" power law comes from the original
formulation in terms of curvature and angular velocity). This result
holds for all practical eccentricities, generalizing and formalizing
previous numerical evidence obtained for specific examples (Wann
et al. 1988; Harris and Wolpert, 1998). As has been pointed out previously, the ellipse is a special case with respect
to the two-thirds power law because it implies coupled harmonic motion.
For this reason, the same analysis was performed for the cloverleaf,
and again, an exponent was found that was in good agreement with
experiment. Finally, the case of movement segmentation in complex
shapes is examined. The velocity-curvature relationship is derived from
the smoothness-based cost function for a figure-of-eight. It is shown
that the same features of segmentation seen in experiment are seen in
the mathematically derived velocity profile. Because the cost function
contains no segmented control, this suggests that this apparent
segmentation is an epiphenomenum of smooth movements, in agreement with
the experimental work of Sternad and Schaal (1999).
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MATERIALS AND METHODS |
In this section, the two motor-control hypotheses to be studied
will be defined mathematically, and the notion of segmentation with
respect to the two-thirds power law will be reviewed.
The mean squared derivative cost functions. Each member of
this family of motion-planning hypotheses is a generalized version of
the minimum-jerk approach used initially to model velocity profiles
generated by elbow movements (Hogan, 1984) and later extended to trajectory prediction for reaching movements between visual
targets in the horizontal plane and to curved and obstacle-avoidance movements (Flash and Hogan, 1985). Since then, the
minimum-jerk hypothesis has been applied to a great many other motor
tasks, including the drawing of complicated figural patterns
(Viviani and Flash, 1995; Todorov and Jordan,
1998) for which a numerical approach was used to obtain
velocity profiles in excellent agreement with experiment. The
predictions of these mean squared derivative (MSD) hypotheses are
derived through the minimization of the time integral of the squared
nth-derivative of the coordinates of the hand (x(t),
y(t)):
|
(1)
|
where T is the duration, or period of the movement.
Each order n, which can take any integer value between unity
and infinity, corresponds to a different member of the family and is an
independent hypothesis for how the CNS might plan movement. For
n = 2, n = 3, and n = 4 the
corresponding hypotheses have been named minimum acceleration, jerk,
and snap, respectively. The case n = 1 of minimum
velocity is not compatible with the observed velocity and acceleration
data and will not be considered here.
For completeness it should be stated that the trajectories derived miss
some of the fine detail seen in point-to-point movements. The paths
measured are actually slightly, but systematically, curved depending on
their position with respect to the body. This observation has led to
further hypotheses (Flash, 1987; Uno et al.,
1989; Harris and Wolpert, 1998) that deal with
the problem of motor execution and include explicit details of the
mechanical properties of the arm. Nevertheless, the simpler cost
function of minimum jerk gives a satisfactory prediction of the
observed movements and is more than sufficiently accurate to provide
the basis for the analyses below.
Given that minimum jerk is just one member of the MSD cost function
family (with n = 3), it could be asked if the
theoretical velocity profiles derived for other orders give an equally
good prediction of experiment. Recently it was stated in the literature (Harris, 1998) that this was indeed the case. If this
were true it would of course imply a fundamental problem with
MSD-generated velocity profiles: if they all fit the data equally well
there are no criteria for choosing one hypothesis over another. In the first part of Results it will be shown that in fact this is not the
case. By calculating the predicted velocity profiles for each order, it
will be shown that only minimum jerk (and to a lesser extent minimum
snap) is compatible with the experimental data.
The two-thirds power law. In the extensive analysis of
handwriting and drawing motion, where the trajectories of the hand are
curved, it was seen that the velocity is not constant but varies
strongly with the instantaneous curvature. It was found (Viviani
and Terzuolo, 1982; Lacquaniti et al., 1983)
that this phenomena could be expressed in terms of a power law between
velocity  and curvature  as follows:
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(2)
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where the accepted value of the exponent   0.33. The
factor g is known as the gain factor and is set by the tempo
of the movement. The name "two-thirds" power law comes from the
original formulation of the law in terms of the angular velocity. The
tracing of ellipses of various eccentricities is the most extensively analyzed task used in the examination of the power law. However, many
results exist for more complex shapes, and although the power is not
indistinguishable from 0.33 in every case (Wann et al. 1988; Viviani and Schneider, 1991;
Viviani and Flash, 1995) the power-law form still holds
with an exponent close to this value.
An intriguing aspect of the power law is seen in the tracing of
extended shapes and also in free scribbling. In such cases a single
value of the gain factor g is not sufficient to fit the data. However, a power law between velocity and curvature can still be
shown to exist if a piecewise constant gain factor is used. A second
and separate issue is that the regions of zero curvature in such shapes
are incompatible with the power law in its usual form, and so an
altered form involving an effective radius of curvature R*
(the standard radius of curvature is R = 1/) was
also introduced to avoid divergences (Viviani and Stucchi, 1992). Thus the velocity , in this case, takes the form:
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(3)
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with  = 0.05. The discontinuous changes throughout the
movement of the gain factor are compelling evidence for segmentation of
movements in the CNS (Viviani and Cenzato, 1985) and are
seen clearly in log-log plots of velocity versus curvature. Recently, this view of segmentation was challenged in the paper of Sternad and Schall (1999). Subjects were asked to trace increasingly
large ellipses with the point of their finger. It was found that the larger the ellipse traced, the clearer the evidence of segmentation in
the velocity of the finger. However, at the same time it was shown that
the motion at the joints was purely oscillatory. Based on these
observations, it was concluded that the apparent segmentation of the
hand trajectories was a result of the nonlinearities in the forward
kinematics of the human arm (which become significant for large
movements). However, it is also true that segmentation is seen in the
tracing of small, but complex shapes, which would be in the linear
regime of the kinematic transformations. The aim here will be to
examine if a smoothness maximization of the end-point trajectory around
such a complex shape also yields apparent segmentation [it should be
stressed at this point that unlike in Viviani and Flash
(1995), no experimental data is used in obtaining the
prediction: the approach is from first principles].
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RESULTS |
In this section, explicit mathematical forms for the
order-n predicted velocity profiles will be given for the
two contrasting experimental tasks of point-to-point and periodic
drawing movements.
Reaching movements in the plane
The aim of this section is to analyze the forms of the predicted
velocity profiles for each order n of the derivative of the coordinates of the hand. The mathematical form for the
nth-order profile will be obtained, and the limit
n   will be examined. It will be shown that the
profiles diverge in this limit, and therefore not all MSD profiles can
fit the data equally well.
The position vector of the hand is written as r = (x,
y). Using the Euler-Lagrange formalism for the functional
minimization of the cost function given in Equation 1 yields the
following differential equation for the trajectory r as a
function of time t:
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(4)
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This implies that r is an order (2n  1)th polynomial in time and therefore that the velocity is an
order (2n 2)th polynomial in time. The constants in
this polynomial can be found by applying the boundary conditions. The
first set of boundary conditions are those that enforce the maximal
allowable stationarity at the start and at the end of the movement. For
an nth order MSD velocity profile this requires setting the
first (n 1)th time derivatives of r to
zero at t = 0 and t = T. As an example,
minimum jerk with n = 3 has boundary conditions of zero
velocity and acceleration at the start and end of the reach. It should
be noted that these boundary conditions are the only choice that does
not require input of experimental data (e.g. the velocity) at the
beginning and end of the movement. The differential Equation 4 for the
trajectory can be integrated using the boundary conditions, allowing
all but one of the 2n 1 constants in the velocity
profile n to be found:
where the only unknown is the peak velocity
pn (occurring at  = 1/2) which is
different for each order n of the cost function. The
notation = t/T (where T is the movement
duration) has also been introduced to simplify the appearance of the
equation. The final quantity needed, pn,
can be found by integrating the velocity over time and setting the
result equal to the total amplitude of the movement L. The
resulting integral can be found in a standard book of tables and gives
the following form for the peak velocity, and its limit when
n is large:
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(5)
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The n limit was obtained using Stirling's
formula for factorials of large numbers. This result should be
interpreted as the average velocity L/T multiplied by an
order-n-dependent factor. The ratio between the peak and
average velocities implied by this result is a simple (but sufficient
for the purposes of discounting the higher-order MSD cost functions)
measure of the shape of the order-n velocity profile.
Experimentally, the mean value of the peak to average velocity ratio
has been measured over 30 movements and found to be close to 1.8 (Flash and Hogan, 1985) with an SD of ~10%.
This value should be compared with the predicted results from Equation 5:
Clearly the predictions of the minimum-jerk cost function fall
between the bounds mentioned above, but as can be seen in Figure
1, minimum snap also gives a reasonable
fit of the data. From this figure and also Equation 5 it can be seen
that the ratio of peak to average velocity increases as a function of
n. This is a result of the large n form given in
Equation 5 above. It is therefore apparent that high-order MSD velocity
profiles are incompatible with experiment: because as n
tends to infinity the predicted peak velocity goes to infinity,
diverging with the square-root of n. The functional form of
the velocity profile in this limit can be found using the following
simple argument: the distance traveled by the hand is constrained to be
L and is equal to the area under the velocity curve.
Therefore to compensate for the diverging peak velocity, the width of
the bell-shaped velocity profile must become increasingly small for
larger n (although still remaining centered at = 1/2). In the extreme limit of n this forces
the velocity profile to have a vanishingly small width to
counterbalance the increasingly high peak velocity. Therefore the
velocity curve takes the form of a Dirac delta function centered on
time T/2, which is clearly unacceptable experimentally and physically. This result provides the proof that only the low-order MSD
velocity profiles could be compatible with experiment.
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Figure 1.
The MSD profiles for n = 2, 3, 4,
and 10 with the only experimental input being the time scale
T (given by the experimental data). The profiles become
narrower and taller as n increases. The experimental curve
is best fitted by minimum jerk n = 3 and snap
n = 4.
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However, the conclusions reached above need to be reconciled with the
results previously reported that stated that the MSD velocity profiles
converge to an experimentally more realistic Gaussian function in the
large n limit (Harris, 1998). There the velocity profiles found at each order n were rescaled by
different (n-dependent) amounts so that their peaks all
coincided at the same point and their widths passed through a value of
25% of the normalized peak velocity at the same times. Such a
rescaling of the profiles in Figure 1 has been performed, with the
results plotted in Figure 2. In this way,
it appears that the curves become similar as n takes higher
values, with the n = limit converging to a
Gaussian. Nevertheless, because the velocity tails become increasingly
large, a cut-off must also be introduced to fix the time of the
movement to be T (which is given by experiment). Essentially what that approach amounted to was to treat each order-n MSD
velocity profile as a curve that can be fitted to experiment by
stretching the time axis and compressing the velocity axis.
Curve-fitting is a valid and useful approach for examining the shape of
the experimental curves (Plamondon et al., 1993).
However, it is in spirit a very different approach to the derivation of
a curve from a hypothesis as was done here and elsewhere (Flash
and Hogan, 1985), with a minimal number of free parameters.
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Figure 2.
The same profiles as in Figure 1 but rescaled as
by Harris (1998). Each profile is rescaled such that
they all pass through the points marked by arrows. A cut-off
for the start and end of the movements must also be supplied.
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Periodic tracing of figural forms
In this section the case of continuous drawing of closed shapes is
analyzed. The approach will be to substitute into the MSD cost
function, written in Equation 1, a specific path (the template of
the shape) and then use a minimization procedure to derive the
predicted velocity profile. It is important to note that this approach
differs from the one taken in an earlier study (Viviani and
Flash, 1995) where the experimental values of the velocity of the hand and acceleration at several via-points were used in the minimum-jerk prediction. By contrast, in the present study the
velocity profiles are derived from a first-principles approach for
general classes of shapes, necessitating minimal experimental input
(only the path itself and the tempo). In fact the mathematical approach
used here is more closely related to the detailed numerical study used
in Todorov and Jordan (1998) in which the jerk was minimized along a prescribed path.
In contrast to the previous experimental task, the motion is periodic,
and because of this it is natural to describe the motion in the
language of Fourier series. The time dependence of the x and
y coordinates of the hand can each be expanded in Fourier series, where for convenience the normalized time = t/T is again used (where T is now the period of the
motion):
![x(&tgr;)=<FR><NU>a<SUB>0</SUB></NU><DE>2</DE></FR>+<LIM><OP>∑</OP><LL>k=1</LL><UL>∞</UL></LIM>[a<SUB>k</SUB> <UP>cos</UP>(2&pgr;k&tgr;)+b<SUB>k</SUB> <UP>sin</UP>(2&pgr;k&tgr;)]](/content/vol22/issue18/fulltext/8201/img009.gif)
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(6)
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These forms are quite general and can be used to describe the
continuous and repeated tracing of any closed shape. The infinite set
of variables {a, b, p, q} are fixed both by the path and
the velocity profile along the path taken by the hand. When the
x and y coordinates are written in this way, the
integral (Eq. 1) can be performed to give:
|
(7)
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The Fourier coefficients are of course subject to many auxiliary
conditions that ensure that the path taken has the given shape, which
still makes minimization very difficult. Nevertheless, this form for
the cost function allows for a straightforward and revealing analysis
in terms of the frequency components present in the movement. If the
first few terms of the series (Eq. 7) are written out
explicitly, it becomes clear that the higher-order terms contribute a
cost that is amplified by a factor of
k2n. This means that, if the cost
function is to be minimized, the higher-order Fourier coefficients must
be small (as would be expected for smooth movements). In fact, just
restricting the cost to be finite means that the square of these
coefficients must decay with the order k of the expansion
faster than 1/k2n. For low
order-n MSD derivative cost functions like acceleration and
jerk there is some balance between this amplifying factor and the
magnitude of the Fourier coefficients. However, for the higher-order
MSD profiles the amplifying factor becomes increasingly significant.
This means that in the limit n the corresponding order n MSD trajectory converges to the one that has the
smallest allowable values of Fourier coefficients in the range
k > 2. This is the central result of this section: in
contrast to the case of point-to-point movements, for periodic
movements the limit n predicts a convergence to the
maximally smooth velocity profile that follows the given template. This
result often allows the minimizing trajectory for the limit n to be found with little mathematical effort. Moreover, as the
MSD velocity profiles converge in this case it also provides a good
first guess for the lower-order n profiles. This first guess
can be used for numerical minimization because it would reduce the time
taken to search through the space of possible profiles. The
implications of this mathematical analysis are examined for two simple
figural forms: the ellipse and cloverleaf.
The ellipse
The general parametric equations for an ellipse are:
|
(8)
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where for various values of the constants A, and
B and the function  () the Fourier expansion
coefficients in Equations 6 take different values. In particular, the
simplest choice of = 2 would imply
a1 = A, q1 = B, and all
other Fourier coefficients would be zero. The Fourier expansion
corresponding to this choice has the lowest frequency components in the
range k > 2 (they are all zero), and therefore
following the reasoning above, this choice for defines a velocity
profile that minimizes the MSD cost function in the limit n . As is well known (Wann et al., 1988) this harmonic form:
|
(9)
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automatically satisfies the power law between velocity and
curvature given in Equation 2 with =  . Despite the simplicity of the harmonic forms of Equations 9 this is an
important result because it unifies mathematically the predictions of
the two-thirds power law and the MSD cost function in the limit n
for ellipses. This is despite the fact that the two laws were originally formulated independently and have different
"philosophical" underpinnings: the two-thirds power law is an
empirical law relating instantaneous velocity to the local curvature,
whereas the MSD cost function implies the full trajectory is planned as
a whole before implementation, by virtue of the integral formulation
and was originally designed to model point-to-point movements.
This equivalence between the two laws only holds for the somewhat
abstract limit of n . It would be more interesting to know what the predictions for the exponent are from MSD hypotheses with orders that are more closely related to quantities with physical meaning (as was shown for point-to-point movements) like jerk or snap.
In the remainder of this section, the results of such a calculation are
presented with the mathematical details given in Appendix A. The method
used is a perturbative approach whereby the calculation is performed
for ellipses that are very close to circles, with a small parameter  measuring deviation of the ellipse from a circle. After performing this
calculation, the power-law exponent predicted from the
order-n MSD cost function can be found to have the following
form:
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(10)
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It is clear from the above form that as n the
exponent tends to the idealized value of  as was argued above. However, this does not mean that the n is the
most appropriate order to choose. The numerical values of the exponents predicted by the lower-order n MSD cost functions are given
from Equation 10 as:
The latter two, jerk and snap, are experimentally
indistinguishable from 0.33. In fact all orders n > 2
predict an exponent that is compatible with this accepted value. It is
interesting to note that it was previously thought that only minimum
jerk would give predictions similar to the two-thirds power law by virtue of the form of the jerk normal to the velocity tangent vector.
As has been shown (Todorov and Jordan, 1998), setting the normal jerk to zero automatically enforces the power-relation between curvature and velocity. However, the result (Eq. 10) shows that the close relation to the power law is present for
almost all order-n MSD velocity profiles and is not just a consequence of the special form of the normal jerk for the order n = 3 MSD cost function. It should be further noted
that the theoretical result exhibits other properties seen in
experiment. The exponent derived is independent of the size of the
ellipse and also the time period of the drawing of the ellipse. More
significantly, the predicted exponent is, for practical purposes,
independent of the eccentricity of the ellipse by virtue of the weak
dependence of n on (see Appendix A).
Taken together, these features represent a robust reproduction of the
experimental results by a first-principles approach.
The cloverleaf
In the previous section, the case of an ellipse was analyzed.
Although the choice of this shape is ubiquitous in the experimental literature it is, from the theoretical point of view, a special case.
This is because choosing the exponent to be exactly implies
drawing with coupled harmonic motion. It is therefore worth examining
another shape that has also been measured in experiments: the
cloverleaf. This is a more interesting shape than an ellipse (no
harmonic solution exists), but still shares the same property of a
single segment on a log() versus log() graph.
Unfortunately, the form of the cloverleaf used in experiment is far
removed from a circle. Because of this, a similar perturbative approach
to that used above yields bad results. For example, a general form for
a cloverleaf that extrapolates between a circle ( = 0) and the
figure used in experiment ( = 1) is:
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(11)
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Examples of this generalized cloverleaf for different values of
are given in Figure 3. The
perturbative expansion, used in the previous section and detailed
in Appendix A, gives the following exponent for the cloverleaf:
|
(12)
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which does not agree well with experiment: cloverleaves that were
traced with periods of 2.5 and 3.0 sec (Viviani and Flash, 1995) were found to have average values of of 0.35 ± 0.03 and 0.36 ± 0.03, respectively. (For completeness it should
be noted that for faster tracing, with a period of 2 sec, the value was lower = 0.33 ± 0.04, but this was attributable to a
somewhat-anomalous data point.) However, the inaccuracy of the result
(Eq. 12) is a breakdown of the perturbative approach only. A numerical
method can be used to obtain the velocity profile from the minimum-jerk cost function to an arbitrary degree of accuracy. Once the velocity profile has been obtained, the predicted exponent can be found from a
least-squares fit of Equation 2: the same approach used for
experimental velocity curves. Such a velocity curve was obtained from
Equation 1, and the fit can be seen in Figure
4. The exponent obtained was:
|
(13)
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which compares well with the two-thirds power law result and the
experimental values given above. This result cannot be a trivial
consequence of coupled harmonic motion, because for the cloverleaf
there are no coupled harmonic solutions. The result therefore
represents a success for the first-principles approach for calculating
the exponent, particularly because the only input into the theory is
the curvature profile given by Equations 11.
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Figure 3.
Examples of the generalized cloverleaf for various
values of the perturbation parameter . See Results (Eq. 11) for the
mathematical definition.
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Figure 4.
The log(curvature) versus log(velocity) curve as
derived numerically from the minimum-jerk cost function for a
cloverleaf with = 1 (in this case the units are arbitrary as
the fit is to theory, not experiment). The value of the exponent
measured from this curve is = 0.36, reproducing the results
found in the experiment.
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General one-segment shapes
Two specific shapes were tested above. However, it is possible to
generalize to an arbitrary one-segment shape. The approach is to write
the curvature as a Taylor expansion in a small parameter and solve
the optimization problem in terms of the derivatives of the curvature
with respect to this variable. Details of the method of obtaining this
generalized form for the exponent are given in Appendix B.
The drawing of complex figures
In the previous section the cases of the ellipse and the
cloverleaf were analyzed. Both of these shapes have a common feature: their curvature-velocity profiles are well matched by the two-thirds power law with a single gain factor. This means that in a log-log plot
of the velocity versus curvature only a single segment is seen. In this
section the more complex case of a multiple-segment shape is analyzed,
specifically the asymmetric lemniscate, or figure-of-eight with two
different sized lobes. Again comparison is made with experimental data,
but it should be stressed that here quantitative accuracy is not
required, just a qualitative reproduction of the features of
segmentation. Because of the complexity of the analysis of
multiple-segment shapes, the approach in this case is numerical. An
algorithm similar to that described in Todorov and Jordan
(1998) was used, in which the time the movement passes through
10 via-points is varied until the trajectory with the lowest
minimum-jerk cost is found. The optimal times were obtained using a
stochastic descent algorithm, to an accuracy of 0.25% of the total
movement duration.
Before examining the results, it is worth underlining again the
importance of using a first-principles approach in the derivation. In a
related paper (Viviani and Flash, 1995), similar shapes
were analyzed, but experimental information in the form of velocity and
acceleration was also supplied at the via points. The argument put
forward here is that the minimization of an integral cost function
(e.g. minimum jerk) gives a movement that mimics the segmentation seen
in experiment. This argument would be undermined if experimentally
derived values of the velocity and curvature for the via-points were
used as they could contain information about possible segmented
planning of the movements by the CNS.
The figure-of-eight
The shape to be analyzed has the following mathematical form:
with a = 3.26 cm and b = 2.80
giving a large and a small lobe (Fig. 5).
The drawing of shapes of this form has been extensively analyzed
(Viviani and Flash, 1995), and it has been shown that when the lobes in the figure-of-eight have different sizes, the log(curvature) versus log(velocity) plots show two apparent segments (corresponding to the large and small lobes) with distinct gain factors. Using the form of the power law given by Equation 3 in Materials and Methods, measurements have shown (Viviani and
Flash, 1995) that across a sample of three subjects the gain
factors and power law fits were found to be:
for the large and small lobes, respectively (the gain factors are
written in unitless form here), and the average period of tracing was
T = 1.43s ± 0.05s. The errors in the gain factors mainly represent between-subject variance. Individually, the three subjects consistently traced the figure-of-eights with
gL > gS with ratios
gL/gS = {1.35 ± 0.04, 1.23 ± 0.03, 1.12 ± 0.03}. Thus, the experimental
data can be interpreted as having two segments.
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Figure 5.
A plot of the figure-of-eight template. The
numerical derivation of the minimum-jerk trajectory was obtained using
10 via-points. The labels M1, M2, and M3 mark the
curvature maxima, whereas m1 is the curvature minimum. It is
these extrema and their relation by symmetries that determine the
number of apparent segments (Fig. 6).
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The numerically derived minimum-jerk trajectory is plotted in Figure
6, in velocity versus
(R*)1/3 form. In this format, different segments
lie on radial lines passing through the origin, with gradients
corresponding to the different gain factors gL
and gS. The duration of the movement has been
scaled to agree with the average experimental period of T = 1.43s. As can be seen, the theoretically derived curve shows
qualitatively the same features of segmentation as the experimental data. The exponents of the power law are not too well defined for
the theoretical curve because of the curvature of the two segments
(unlike the experimental data, the theoretical curve has not been
smoothed, a process which tends to increase the likelihood of measuring
an exponent close to 0.33), and, as stressed above the aim here is to
show that features of segmentation are seen in theoretical predictions,
not to show close quantitative agreement. However, to provide a level
of comparison, the tangential dotted lines in Figure 6 (meant as guides
for the eye) represent gain factors corresponding to
gL = 20.5 and gS = 18.4, which implies a ratio
gL/gS = 1.1 (the choice
here of using tangents, rather than a fit, tends to underestimate the
value of this ratio). Though this value is about 10% smaller than the
average found experimentally (~1.2, the corresponding values of
gL = 21 and gS = 17 are represented as dashed lines in Fig. 6), it is not
inconsistent with the range of experimental values seen across the
three subjects.
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Figure 6.
The predicted minimum-jerk velocity versus
effective radius of curvature plot, for the tracing of the
figure-of-eight. The positions of the curvature extrema (M1, M2,
M3, and m1) are marked and can be compared with Figure
5. The dotted tangential lines give an indication of the
gain factors for the two segments (see Results), whereas the
dashed lines represent the average experimental values. The
theory reproduces the features of segmentation seen in the experimental
curves, despite no explicit segmental planning in the structure of the
theory.
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The theoretically generated trajectory captures all the qualitative
features of segmentation, despite the absence of any piecewise algorithm in the minimum-jerk hypothesis. In fact even the hook-like feature (corresponding to the region of the trajectory between the
curvature maxima M2 and M3 via the curvature
minimum m1) can be seen in a detailed analysis of the
experimental data. In examining this feature, it becomes clear what
criteria determine the apparent segmentation of the velocity versus
curvature plots: it is the extrema of the curvature and the symmetries
of the shape. Thus, the segmentation features seen in such log-log
plots can be directly accounted for by the geometric aspects (curvature
profile) of the drawn path: segmented planning in the CNS need not be assumed.
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DISCUSSION |
A family of movement-planning strategies for the human arm was
analyzed in mathematical detail and compared with experiment, with a
number of results derived. The planning strategies are defined through
the minimization of cost functions that favor smoothness, to differing
degrees, and are known collectively as the MSD cost functions. Two
standard and contrasting motor tasks were analyzed: point-to-point
reaching movements and the periodic tracing of closed shapes. For the
first task of point-to-point movements the predicted velocity profile
from each order n of the MSD family was derived explicitly
and compared with experiment. By comparing the ratio of the peak to
average velocities it was concluded that the best fit (from the MSD
family) was by the minimum-jerk cost function, corresponding to order
n = 3. It was further shown that as the order
n was increased the corresponding velocity profile diverged,
reaching the highly unphysical limit of a Dirac delta function when
n . This result corrects conclusions previously reached in the literature (Harris, 1998). However, it is
acknowledged here that this was not the main point of that paper, and
therefore the current work should not be mistaken as a criticism of the minimum-variance principle, for which the paper of Harris
(1998) laid the foundation. Rather, the aim here was to clarify
the mathematical understanding of the smoothness-based cost functions.
The theory of minimum variance was applied to both arm and eye
movements and may seem, at first view, to be at odds with the smoothness-based cost function approach. However, there are many overlaps between the two theories. An obvious example is the
independence of the velocity profiles for fast movements from physical
parameters of the arm. This suggests an interpretation of the
minimum-jerk theoretical framework as a simplified limit of a more
complete theory. As has already been noted, minimum-jerk predictions
for point-to-point movements miss some of the fine structure of the trajectories. Experimentally measured trajectories show a small but
systematic curvature, a feature captured by other models that take into
account both the problems of trajectory planning and motor execution
(Flash, 1987; Uno et al., 1989;
Harris and Wolpert, 1998). In this context, it would be
very interesting to examine theoretically the similarities between the
MSD cost functions and other more detailed hypothesis such as the
minimum-variance hypothesis (Harris and Wolpert, 1998)
or the optimal feedback control hypothesis of Todorov
(2001). In the latter work, it is demonstrated that greater
control of a noisy goal-dependent variable can be obtained at the
expense of allowing variance to increase in redundant degrees of
freedom. In their formulation, features such as simplifying rules,
control parameters, and synergies emerge as epiphenomena from the
optimal feedback control.
Several neural implementations have been proposed for the minimum-jerk
cost function (Hoff and Arbib, 1992; Jordan et
al., 1994) and neurophysiological studies have shown that the
fine kinematic details of the movements are represented within cortical neuronal populations (Schwartz and Moran, 1999).
Nevertheless, given the present level of knowledge of the neural
correlates of movement and also the success that different modeling
approaches have in describing the same psychophysical experiments (as
was shown here in the context of minimum-jerk and the two-thirds power law), it is clear that the data are not yet sufficiently fine to
restrict the space of the possible neurally based algorithms that might
generate the observed behavior. As an example, the optimal feedback
hypothesis produces movements that are smooth, and in some limits may
be reproduced by the minimum-jerk predictions, but the underlying
structure of the controlling system is very different.
The second experimental task analyzed was the periodic drawing of
closed shapes. In contrast to point-to-point movements the order-n velocity profile was shown to converge to a smooth
function as n was increased. For the limit n it was shown that the predicted velocity profile is the one
for which the Fourier series for the trajectory of the hand has the
lowest values in the high-frequency range. For the specific examples of
an ellipse and cloverleaf it was shown that the power-law exponent
derived from a MSD cost function, with any order greater than minimum
acceleration, gives values that are indistinguishable from the accepted
value of 0.33. Further comparison was then made between
the two theories for the case of a complex figural form (a
figure-of-eight with different sized lobes). The velocity profile
derived from the MSD smoothness-based cost functions was seen to have
features that implied segmented control. However, the cost function
contains no explicit segmented control. Therefore, the conclusion must be that these types of experimental features can no longer be taken as
evidence for segmented control in the CNS, in the absence of further
supporting evidence.
Despite the success of the smoothness-based cost function in fitting
different aspects of the two-thirds power law, there are still many
questions unanswered. In earlier studies several different reasons and
explanations for the observed two-thirds power law were offered. These
reasons included either the mechanical properties of muscles
(Gribble and Ostry, 1996), acceleration, or noise
constraints (Harris and Wolpert, 1998) or the coupling between oscillatory joint rotations (Sternad and Schaal,
1999). The analysis performed by Schwartz (1994)
and more recently by Schwartz and Moran (1999), has
shown that the two-thirds power law fits neural trajectories derived by
using a population vector analysis of the neural activity data recorded
from motor cortical cells in monkeys during drawing movements. The fact
that the two-thirds power law has been found in motor cortical
representations of drawing movements suggests that this relation is
manifested in the planning stages of hand trajectories (Schwartz
and Moran, 2000), consistent with the idea that central neural
representations have evolved or are acquired with age (Viviani
and Schneider, 1991) or through learning to achieve the
smoothest and/or the least variable movement. Nevertheless,
explanations for the two-thirds power law based on central mechanisms
or on peripheral factors, such as the mechanical properties of muscles
or neuromotor noise, may not be mutually exclusive, and both peripheral
and central factors might have evolved to work in tandem to guarantee
the smoothest possible or most accurate movement. Other questions of
interest that relate to the work presented here concern the nature of
the relation between the two-thirds power law and visual perception of
motion. There is compelling evidence from a number of studies that the
two-thirds power law is related to both the motor production and visual
perception of movement (Pollick and Sapiro, 1996;
de'Sperati and Viviani, 1997). Such connections were
not examined here and represent an interesting case for further study.
Another significant topic for future study that was addressed here is
that of movement segmentation. Many earlier hypotheses concerning the
nature of the underlying movement segments have been made (Krebs
et al., 1999; Doeringer and Hogan, 1998). Among them, one of the most compelling hypotheses, which attempted to suggest
a criterion for motion segmentation, is that based on the existence of
a piecewise constant gain factor in the relation between velocity and
curvature (Lacqaniti et al., 1983). Using a similar
approach, the boundaries of movement segments during drawing movements
were defined kinematically as places in the trajectory where speed is
maximal (Schwartz and Moran, 1999). These are points at
which the velocity gain factor in the power law changes
instantaneously, where curvature is minimal, and where curvature
inflections also occur. The conclusion, based on the mathematical
analysis presented here is that the presence of sharp changes in the
velocity gain factor does not necessarily suggest a noncontinuous mode
of motion planning and therefore cannot provide sufficient evidence for
segmented control [supporting the findings reached in the experimental
study of Sternad and Schall (1999)]. This conclusion
does not undermine the idea that complex movements are constructed from
simpler units of action or movement segments that are then concatenated
together or are temporally overlapped to generate longer movement
sequences. It does, however, raise the need for new ideas concerning
the nature of the underlying movement segments or strokes.
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FOOTNOTES |
Received Nov. 27, 2001; revised April 23, 2002; accepted May 17, 2002.
This work was supported in part by the Minerva Foundation
(Germany), by a grant from the Israeli Ministry of Science, Culture, and Sport, and by the Moross Laboratory. M.J.E.R. acknowledges the
support from a Feinberg research fellowship.
Correspondence should be addressed to Magnus J. E. Richardson,
Laboratoire de Physique Statistique, Ecole Normale Superieure, 24 Rue
Lhomond 75231, Paris Cedex 05, France. E-mail:
Magnus.Richardson{at}lps.ens.fr.
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APPENDIX A |
This appendix contains the details of the calculation leading to
the predicted exponent for the power law of the ellipse, Equation 10,
as derived from the general order-n MSD cost function. One
route to obtaining the exponent n of the
power law is to substitute the velocity form given in Equation 2 into
the integral (Eq. 1) and change the integration variable from time to
arc-length. This can be done directly for very small values of
n by using the Frenet-Serret formulation for curves (see
Appendix B). However, to get general results for arbitrary order, a
less direct approach is necessary. The route taken to find
n is to obtain the Fourier coefficients for
the cost function (Eq. 7) in terms of the unknown exponent . Once
this has been done it is straightforward to minimize
Cn by differentiation with respect to the exponent.
The first step is to write the coordinates of the hand (quite
generally) as parametric functions of a time dependent angle ():
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(14)
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with boundary conditions (0) = 0, (1) = 2 and = t/T, where T is the
period. Next, the time dependence of the angle must be found by using
the power law (Eq. 2). It is the time dependence of that encodes
the velocity profiles. Once this is done the Fourier coefficients
required for the cost function can be easily derived. Writing the
velocity as the rate-of-change of arc-length s with respect
to time allows Equation 2 to be rearranged to give:
The integral can now be performed, and the gain factor
g is fixed by the condition that = 2 when
normalized time = 1:
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(15)
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In the second equation, the following geometric properties of the
ellipse have been used:
The quantity is related to the eccentricity of the ellipse and
is a measure of how close the ellipse in question is to a circle, for
which = 0. To obtain the time as a function of angle, the
integrals in Equation 15 must be performed. Unfortunately, this is not
possible exactly because of the nature of the integrals and to proceed
further it is necessary to make an approximation. It will be assumed
now that the quantity is very small (more will be said about the
validity of this approximation later). This allows the arguments of the
integrals to be expanded as a series in the small parameter using
standard formulas, and then the integration performed. The result is
also a series expansion in which, in all the following steps, will
be written up to order 2:
This expansion can now be inverted to give the angle as a function
of time, the quantity needed for Equations 14. The inversion can be
performed by substituting the following into the previous equation:
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(16)
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The trigonometric functions can then be expanded again, keeping
terms only up to order 2. Then by matching terms with
the same powers of the following functions of time are found:
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(17)
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As expected, the first contribution f() is the
result for the angular dependence of time when a circle is drawn. The
higher order terms g() and h() represent
the corrections caused by the elliptical perturbation.
What remains now is to obtain the Fourier coefficients in Equation 6 of
the x and y coordinates. This is achieved by
substituting Equation 16 with the results (Eq. 17) into Equation 14,
for example:
The argument of the cosine can be expanded in allowing the
integral to be performed and the Fourier coefficients obtained, accurate to order 2. The calculation is similar for
bk, pk, and
qk, although for the latter two it must
be remembered that B = A(1 )1/2
and that this should also be expanded to the appropriate order. The final results for the squares of the Fourier coefficients are:
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TOP
ABSTRACT
INTRODUCTION
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