Optimal integration of gravity in trajectory planning of vertical pointing movements.

Abstract

The planning and control of motor actions requires knowledge of the dynamics of the controlled limb to generate the appropriate muscular commands and achieve the desired goal. Such planning and control imply that the CNS must be able to deal with forces and constraints acting on the limb, such as the omnipresent force of gravity. The present study investigates the effect of hypergravity induced by parabolic flights on the trajectory of vertical pointing movements to test the hypothesis that motor commands are optimized with respect to the effect of gravity on the limb. Subjects performed vertical pointing movements in normal gravity and hypergravity. We use a model based on optimal control to identify the role played by gravity in the optimal arm trajectory with minimal motor costs. First, the simulations in normal gravity reproduce the asymmetry in the velocity profiles (the velocity reaches its maximum before half of the movement duration), which typically characterizes the vertical pointing movements performed on Earth, whereas the horizontal movements present symmetrical velocity profiles. Second, according to the simulations, the optimal trajectory in hypergravity should present an increase in the peak acceleration and peak velocity despite the increase in the arm weight. In agreement with these predictions, the subjects performed faster movements in hypergravity with significant increases in the peak acceleration and peak velocity, which were accompanied by a significant decrease in the movement duration. This suggests that movement kinematics change in response to an increase in gravity, which is consistent with the hypothesis that motor commands are optimized and the action of gravity on the limb is taken into account. The results provide evidence for an internal representation of gravity in the central planning process and further suggest that an adaptation to altered dynamics can be understood as a reoptimization process.

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102:786-796, 2009. First published May 20, 2009; doi:10.1152/jn.00113.2009
J Neurophysiol
Frédéric Crevecoeur, Jean-Louis Thonnard and Philippe Lefèvre

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Optimal Integration of Gravity in Trajectory Planning of Vertical
Pointing Movements
Fre ´de ´ric Crevecoeur,1,2Jean-Louis Thonnard,2and Philippe Lefe `vre1,3
1Center for Systems Engineering and Applied Mechanics,2Rehabilitation and Physical Medicine Unit, and3Laboratory
of Neurophysiology, Universite ´ catholique de Louvain, Louvain-la-Neuve, Belgium
Submitted 6 February 2009; accepted in final form 15 May 2009
Crevecoeur F, Thonnard J-L, Lefe `vre P. Optimal integration of
gravity in trajectory planning of vertical pointing movements. J
Neurophysiol 102: 786–796, 2009. First published May 20, 2009;
doi:10.1152/jn.00113.2009. The planning and control of motor ac-
tions requires knowledge of the dynamics of the controlled limb to
generate the appropriate muscular commands and achieve the desired
goal. Such planning and control imply that the CNS must be able to
deal with forces and constraints acting on the limb, such as the
omnipresent force of gravity. The present study investigates the effect
of hypergravity induced by parabolic flights on the trajectory of
vertical pointing movements to test the hypothesis that motor com-
mands are optimized with respect to the effect of gravity on the limb.
Subjects performed vertical pointing movements in normal gravity
and hypergravity. We use a model based on optimal control to identify
the role played by gravity in the optimal arm trajectory with minimal
motor costs. First, the simulations in normal gravity reproduce the
asymmetry in the velocity profiles (the velocity reaches its maximum
before half of the movement duration), which typically characterizes
the vertical pointing movements performed on Earth, whereas the
horizontal movements present symmetrical velocity profiles. Second,
according to the simulations, the optimal trajectory in hypergravity
should present an increase in the peak acceleration and peak velocity
despite the increase in the arm weight. In agreement with these
predictions, the subjects performed faster movements in hypergravity
with significant increases in the peak acceleration and peak velocity,
which were accompanied by a significant decrease in the movement
duration. This suggests that movement kinematics change in response
to an increase in gravity, which is consistent with the hypothesis that
motor commands are optimized and the action of gravity on the limb
is taken into account. The results provide evidence for an internal
representation of gravity in the central planning process and further
suggest that an adaptation to altered dynamics can be understood as a
reoptimization process.
I N T R O D U C T I O N
The investigation of the skillful and smooth movements that
we achieve so many times a day intends to clarify how
movements are controlled by the CNS. There are infinite
numbers of feasible trajectories and kinematics that allow
individuals to perform a basic movement. However, in general,
little variability is observed between repeated movements.
Why is this? Is there any reason why one particular movement
is realized among an infinite number and why this movement
is stable across repeated executions? To answer these ques-
tions, one should identify the criteria taken into account when
the movements are planned and controlled and how the system
responds to a change in the environment.
For this purpose, the concept of optimality has long been
associated with mechanical parameter adjustments (Burdet
et al. 2001) and with movement planning and control, assum-
ing that motor actions were optimal with respect to a cost
defined along the movement. Kinematics and dynamics costs
have successfully modeled real movements (Flash and Hogan
1985; Nakano et al. 1999; Uno et al. 1989) as well as the costs
related to the movement variability and sensorimotor noise
(Harris and Wolpert 1998). Feedback control has also been
proposed to rely on optimal strategy, maximizing locally the
expected reward related to the next state at minimum cost by
taking into account an estimate of the current state (Todorov
2004; Todorov and Jordan 2002). The optimal feedback con-
trol framework, supported by the neural basis, finely models
the movement strategies, flexibility, and learning properties of
motor control (Izawa et al. 2008; Liu and Todorov 2007; Scott
2004; Shadmehr and Krakauer 2008).
These models assume that the CNS has access to the internal
representations of the limb dynamics and state (position, ve-
locity, etc.), as a result of internal models (Kawato 1999;
Wolpert and Ghahramani 2000), that are necessary for the
computation of motor commands in the both feedforward and
feedback control schemes (Wagner and Smith 2008). Indeed,
the generation of an appropriate motor command must rely on
the knowledge of limb dynamics and its interaction with the
environment through external constraints, including that om-
nipresent force field in which we have learned to move from
our earliest childhood: the gravity field.
Gravity has multiple implications for motor control. It in-
fluences reference frames for body orientation in the environ-
ment and for interaction with moving objects (McIntyre et al.
1998, 2001; Pozzo et al. 1998). Changes in motor responses
following changes in gravity were observed from various
contexts such as the synchronization of grasping forces and
isometric force production (Augurelle et al. 2003; Crevecoeur
et al. 2009; Girgenrath et al. 2005; Mierau et al. 2008; White
et al. 2005). In the context of vertical pointing movements, the
characteristics of arm kinematics in Earth’s gravity led to the
hypothesis that the gravitational constraint is internally repre-
sented in the planning process of motor actions (Atkeson and
Hollerbach 1985; Papaxanthis et al. 1998). Vertical pointing
movements typically exhibit skewness in the velocity profile,
with the peak velocity occurring before the middle of the
movement. This is in contrast to the horizontal pointing move-
ments, which demonstrate symmetric velocity profiles (Gentili
et al. 2007). In addition, simulations of arm trajectories with
minimal absolute mechanical work reproduce the skewness in
the velocity profiles (Berret et al. 2008), suggesting that the
Address for reprint requests and other correspondence: P. Lefe `vre,
CESAME, Universite ´ catholique de Louvain, 4 Ave Georges Lemaı ˆtre, 1348
Louvain-la-Neuve, Belgium (E-mail: philippe.lefevre@uclouvain.be).
J Neurophysiol 102: 786–796, 2009.
First published May 20, 2009; doi:10.1152/jn.00113.2009.
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CNS accounts for the action of the gravitational torque on the
limb to optimize the motor commands.
In the present study, we test the hypothesis that motor
commands are optimized in the gravity field by investigating
how subjects adapt vertical pointing movements in a perturbed
environment where the terrestrial gravity was nearly doubled.
The analyses combine experimental and modeling approaches
with simulations of arm trajectories with minimal control input
to theoretically test the effect of an increase in gravity on the
optimal trajectory. In particular, the simulations predicted
changes in the kinematics profiles in hypergravity compared
with the optimal trajectories computed in normal gravity. The
data strongly indicate that the subjects’ behavior in hypergrav-
ity was adjusted accordingly, providing further evidence to the
theory that the arm motor commands are optimized with
respect to the action of gravity on the limb and that adaptation
can be seen as a reoptimization process (Izawa et al. 2008).
M E T H O D S
Subjects
Ten right-handed volunteers (four males, six females) between the
ages of 25 and 52 yr with no neurological disorders gave their
informed consent to take part in this experiment. They complied with
the medical requirements to participate in the parabolic flights (Bel-
gian Center for Aerospace Medicine, class II medical examination).
The experimental protocol was approved by the ethical and biomed-
ical requirements for experimentation on human subjects from the
European Space Agency (ESA) Medical Board Committee and the
French Comite ´ pour la Protection des Personnes, which reviews life
science protocols in accordance with French law.
Parabolic flight
The experiment was performed during the 47th and 48th ESA
Parabolic Flight Campaigns. Parabolic maneuvers generate sequences
of 20 s of hypergravity (1.8 g), followed by about 22 s of weightless-
ness (0 g) before another period of 20 s of hypergravity. The aircraft
ran a sequence of 30 parabolas per flight, which permitted us to
evaluate each subject during 15 consecutive parabolas. In the follow-
ing sections, one block refers to the set of trials performed during one
parabola.
Experimental procedures
The subjects sat in front of three visual targets that were aligned
vertically with respect to the aircraft floor and that were separated by
18 cm. The center target was in front of the subject’s shoulder and
defined the horizontal arm position. They were asked to grasp a
manipulandum (mass 250 g, grip aperture 4.5 cm) with the right hand
and to perform visually guided pointing movements toward the
current target with arm-straight rotations around the shoulder. Upward
movements (from the center to the top) and downward movements
(from the center to the bottom) were randomly interleaved to avoid
anticipatory movements. All of the subjects performed control exper-
iments in normal gravity conditions (1 g) prior to the in-flight
experiment. The subjects performed the task during the 0 g and
subsequent 1.8 g phases of each parabola (they did not perform the
task under normal gravity conditions during the flight). Each subject
performed from 60 to 80 trials in each direction and in each gravita-
tional condition. The analysis reported in the present study focuses on
the data acquired during the hypergravity phases. The data collected
during the 0 g phases of the flights will be addressed in another study.
Indeed, previous studies suggest that the adaptation to other gravity
levels does not generalize to zero gravity (Mireau et al. 2008; White
et al. 2008), which means that this condition should be considered as
a singularity where the adaptation relies on distinct mechanisms.
Data acquisition and postprocessing
Gravity was sampled at 800 Hz with a three-dimensional (3D)
accelerometer (ADXL330, Analog Devices). This signal was used to
check that the gravity was stable during the movement executions.
The trials performed at the transition phases between 0 g and hyper-
gravity were removed because the gravity was unstable. The 3D
positions of the shoulder and the hand held load were acquired at 200
Hz (Codamotion, Charnwoods Dynamics, Leicestershire, UK).
The gravity and position signals were digitally low-pass filtered
with a zero phase-lag Butterworth filter of order four with a cutoff
frequency set to 20 Hz. The position of the held load and the shoulder
permitted us to compute the elevation angle of the arm with respect to
the horizontal axis, as illustrated in Fig. 1A. The black dots represent
the infrared markers placed on the shoulder and the manipulandum
for the acquisition of the position signals. Figure 1B represents a
two-dimensional reconstruction of a single upward movement, with
each line joining the held load and the shoulder marker every 25 ms.
The angular velocity and angular acceleration signals were computed
from numerical differentiation of the position signals. Figure 1, C–H
shows the elevation angle, angular velocity, and angular acceleration
of two trials presenting typical overshoot (Fig. 1, C–E) and under-
shoot (Fig. 1, F–H) profiles.
The movement onset and end were computed with the following
procedure illustrated in Fig. 1D. The velocity profile was linearized
around the intersection with the threshold equal to 10% of the peak
velocity (VT) computed for each trial (linear regressions computed on
five time steps). The onset (t0) and offset (tf) were then determined by
the intersection of the linear regressions with the horizontal axis (red
dots). The movement duration was defined as the elapsed time
between t0and tf. The procedure to detect the overshoots and under-
shoots was based on the acceleration profile. For an overshoot, as
indicated in Fig. 1E, the acceleration presented a local maximum (red
dot) occurring after the movement end (Fig. 1E, tOSH). We let ?(t) be
the elevation angle of the arm as a function of time and defined the
overshoot as the percentage of the displacement from tfto tOSHwith
respect to the total displacement from t0to tf
OSH ???tf? ? ??tOSH?
??tf? ? ??t0?
? 100(1)
The elapsed time between tfand tOSHwas limited to 150 ms to exclude
corrective movements.
Similarly, an undershoot strategy typically presented a local max-
imum in the acceleration profile before the end of the movement
(tUSH, Fig. 1H). When such a maximum was detected, the undershoots
were defined exactly like the overshoot in Eq. 1, replacing tOSHby tf
and tfby tUSH. With these definitions, the overshoots and undershoots
are positive and negative, respectively. Trials presenting an under-
shoot detected before 150 ms prior to movement end were considered
missed trials, given that the amplitude of the correction was not
negligible compared with the total movement amplitude. Such trials
must therefore be considered to be composed of at least two distinct
movements realized prior to reaching the target. This criterion re-
vealed that one subject actually did not meet the task requirements.
The success rate for this subject was 57% for the upward movements
and ?15% for the downward movements. Accordingly, this subject
was removed from the subsequent analysis and in the RESULTS the data
are reported concerning the nine remaining subjects. A direct appli-
cation of these procedures may provoke the detection of an under-
shoot and overshoot for the same movement. To avoid this situation,
the undershoot strategy was considered to take precedence over the
overshoot, which means that the detection of an undershoot implied
that no overshoot was recorded for the corresponding trial.
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Finally, this study analyzes the skewness of the velocity profiles
defined as the ratio of the acceleration duration to the total movement
duration (AD/MD). Equivalently, this ratio is equal to the relative
time from the movement onset to the peak velocity. For a symmetric
velocity profile with a peak velocity arising at the middle of the
movement, the AD/MD ? 0.5. For analysis of the AD/MD, the
trajectories were interpolated with cubic splines on discretized time
intervals of identical size (200 samples) to compare the AD/MD ratios
measured with identical resolution.
Mechanical and physiological model
The model of the arm was composed of three rigid bodies: the arm,
the forearm, and the hand. The length and mass of the three bodies
were computed for each subject as a fraction of the total body height
and mass. The ratio was obtained from classical average anthropo-
metric tables (Winter 1979). The hand length was divided in half to
mimic the grasp posture. The held load was added to this model as a
point mass body located at the same coordinate as the hand center of
mass. We assumed that three torques were acting on the limb: the
muscular torque T(t), the gravitational torque, and a viscous friction
torque. In addition, the model assumes that the muscular torque is a
first-order, low-pass filtered response of the motor command u(t)
being the control signal. This control can take positive and negative
values to model a pair of agonist–antagonist muscles. Let us first
consider the unperturbed system in continuous time; the equations for
the deterministic case are
I?¨? T ? mgl cos ??? ? kv?˙
(2)
T˙?1
??kuu ? T?
(3)
The explicit dependence on time was omitted for clarity. The viscous
constant kvwas set to 0.63 ? 0.095TM, where TMis the average joint
torque across the movement (Nakano et al. 1999); the time constant ?
was set to 40 ms; and g was equal to 9.81 m/s2for the 1 g simulations
or 17.658 m/s2(1.8 g) for the hypergravity simulations. The parameter
m is the mass of the whole system, I is its inertia, and l is the distance
between the center of mass and the shoulder rotation axis. The control
gain kupermits us to have consistent units from the physical perspec-
tive. Assuming that the control input u represents the motor neuron
discharge, then kuis the steady-state gain relating the torque output to
the control input, which can be expressed in spike density (s). In
practice, the control was normalized between ?1 and 1 and kuwas
equal to 50 Nm/s (after the simulations, we verified that the bounds
were never active).
Optimal control problem and reoptimization hypothesis
The control problem for the unperturbed system is to find an
admissible control function u(t), which drives the system described in
Eqs. 2 and 3 from its initial to final position in a given amount of time.
The minimum control input solution considered in this study is the
solution of the control problem that minimizes the following cost
J?u? ??
t0
tf
?u?t??2dt
(4)
The solutions were computed with sequential quadratic program-
ming (SQP) methods applied to direct multiple shooting (Bock and
Plitt 1985). The multiple shooting algorithm works as follows. The
time span of the simulation is divided into n intervals tk–tk?1, where
0 ? t0? t1? . . . ? tn? tf. In each interval, the control policy is
ACF
DG
EH
B
FIG. 1.
the center of the held load is drawn every 25 ms. C, D, and E: the elevation angle, angular velocity, and angular acceleration as a function of time of a movement
presenting a typical overshoot profile. VTis the threshold of 10% of the peak velocity computed to linearize the speed profile and estimate the movement onset
and movement end. F, G, and H: the elevation angle, angular velocity, and angular acceleration as a function of the time of a movement presenting a typical
undershoot profile. The local maxima of the angular acceleration used to compute the overshoot and undershoot are shown in E and H, respectively.
A: illustration of a subject performing the task. B: 2-dimensional reconstruction of an upward movement. One line joining the shoulder marker and
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approximated by a constant value (say u ¯k). In the discretized formu-
lation, the cost function defined in Eq. 4 becomes
J?u? ??
k
u ¯k
2
(5)
and the optimal control problem becomes a discrete constraint opti-
mization problem. We initialize a guess of the state at each point of
the time grid x ?k? x ?(tk), where x ? represents the state variables
(position, velocity, and shoulder torque). Knowing the initial and final
positions, the problem to solve is minimization of the cost defined in
Eq. 5 such that the trajectory ends at x ?kwhen the system is integrated
from x ?k?1with control u ¯k?1. These are nonlinear constraints (given
the nonlinear dynamics) ensuring the continuity of the state at each
point tk. The size of the time subintervals was equal to 7.5 ms and the
system was integrated in each interval with an adaptive step-size
Runge–Kutta integrator of order four. The optimal control sequence
u ¯kconstitutes the open-loop control policy, which drives the unper-
turbed system along the optimal trajectory in the absence of noise or
unexpected perturbation. However, the CNS must be able to correct
for sensorimotor noise as a result of feedback control. To model this,
we now consider a discrete time formulation enabling us to introduce
additive and signal-dependent noises characterizing the properties of
the noise in the sensorimotor system. To this end, the system was
linearized around the optimal open-loop solutions, which permits us to
derive a feedback law as a linear function of the state deviations. The
optimal feedback gains were computed at each time step from the
local linear approximation of the system dynamics. The stochastic
dynamics of the deviations from the optimal trajectories are
?x ? x ??k?1? Ak?x ? x ??k? Bk?1 ? ?k??u ? u ¯?k? ?k
(6)
where
Ak? 1 ? ?t?
?f
?x ?k?
Bk? ?t?
?f
?u ¯k?
(7)
and
f?xk, uk? ??
xk
?2?
I?1?xk
?3?? mgl cos ?xk
??1?kuuk? xk
?1?? ? kvxk
?3??
?2??
?
(8)
defines the system dynamics. The noise terms ?kand ?kare Gaussian
random variables with zero mean and variances equal to 0.02 and 0.5,
respectively. The cost per step associated to the motor cost and state
deviation was
Jk? r?u ? u ¯?k
2? wT?x ? x ??k
2
(9)
andtheoptimalfeedbackgainswerecomputedusingthederivationgiven
by Todorov (2005). The weighting factors for the control and state
deviation in the cost function were: r ? 10?6and wT? [10, 1, 0.2]. In
the second term, the square is computed component-wise. In summary,
the stochastic system is driven by the following control policy
uk? u ¯k? Lk?x ? x ??k
(10)
The first term is the open-loop control sequence and the second term
is the feedback correction to the local perturbations. This method is a
simplified version of optimal control for nonlinear stochastic systems
(Li and Todorov 2007). For simplicity, we considered that there was
no uncertainty about the state of the system, i.e., the state is fully
observable and the feedback correction is computed on the true state.
This model still makes the hypothesis that there is a feedforward
optimization of the motor plan represented by u ¯k. Indeed, the nonlin-
ear dynamics do not allow computing an optimal control that is a
linear function of the state only and the feedforward component of the
control must be added if one desires that the average trajectory of
the stochastic system tends to the unperturbed optimal solution. Under
the hypothesis of reoptimization, the feedforward component of the
control law cost must be recomputed in hypergravity. Thus in prac-
tice, the same criterion (Eq. 4) is applied to the system (Eqs. 2 and 3)
with a distinct value of g.
Nonoptimizing strategy
As an alternative, we simulated the following nonoptimizing adap-
tation scheme. The solution of the control problem in normal gravity
is primarily added to an offset to compensate for the static postural
adjustment and then corrected to attain the desired final position. That is
u ¯?1.8g?? u ¯?1.0g?? ?
(11)
This formulation considers that the open-loop component of the
control policy (u ¯kin Eq. 10) is not reoptimized in hypergravity, but
adjusted only in response to the increase in arm weight.
Comparison between models and data
To compute the optimal trajectory in normal gravity, the simulation
duration was adjusted in such a way that, for each individual subject,
the durations measured in the data and in the simulation were
identical. This adjustment corresponded to about the average move-
ment duration plus 50 ms. To test the effect of hypergravity on the
optimal trajectory, the simulations were computed with all parameters
kept constant, except the gravity. Thus the same simulation duration
was used to compute the optimal trajectory in hypergravity. The
simulations were computed for each subject with the inertia, mass,
length, movement amplitude, and duration adjusted to their individual
values. The same procedure was used to measure the ratio AD/MD
and the movement duration in the simulations and in the data.
R E S U L T S
Testing the model
The vertical trajectories performed in the normal gravity
condition presented asymmetric velocity profiles. The AD/MD
ratio was significantly ?0.5 (t-test, P ? 0.01). Our experiment
also confirmed the difference between upward and downward
movements: AD/MD was equal to 0.451 ? 0.053 for the
upward movements and 0.463 ? 0.060 for the downward
movements, with a significant effect of the movement direction
(all subjects pooled, ANOVA, P ? 0.01). Despite its simplic-
ity, the model captures this property. The asymmetry was
computed on simulated trajectories of identical duration for
upward and downward movements in normal gravity. The
AD/MD ratio was equal to 0.475 ? 0.028 for the upward
movements and 0.482 ? 0.027 for the downward movements
(mean ? SD computed across the simulations of the model,
with the parameters fitted to each individual subject). Figure 2
presents the velocity profile for one subject computed from the
model simulation (solid black, left) and the velocity profile
collected in normal gravity condition averaged across the
individual trials (solid gray, right). Both traces present a
maximum velocity (vertical line) occurring before the middle
of the movement. For comparison, Fig. 2 shows the symmetric
velocity trace corresponding to the minimum jerk trajectory
(dashed black), with an AD/MD ratio equal to 0.5. According
to the model, the asymmetry can be directly attributed to the
gravitational torque since the simulations for the horizontal
movements (with g ? 0) lead to an AD/MD ratio equal to 0.5
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without the effect of the movement direction, as observed in
the horizontal movements performed in normal gravity.
The effect of hypergravity on the optimal trajectories is
illustrated in Fig. 3 showing simulations of upward and down-
ward movements in both gravitational conditions for the same
subject. The red and black traces correspond to the open-loop
trajectories in the two gravitational conditions. The shaded
areas represent the SDs from the average trajectory when the
system was simulated with noise (Eq. 6, deviations computed
on 100 simulations). The difference between the optimal solu-
tions is due to the reoptimization of the feedforward compo-
nent of the control policy. This difference in optimal trajecto-
ries can be directly attributed to the change in gravity since the
only difference between the simulations was the value of g.
Qualitatively, the main effects predicted by the model on the
kinematics properties is an increase in the peak angular accel-
eration and in the peak angular velocity. As a consequence, the
velocity profile in hypergravity changes, which leads to a
difference in the estimation of the movement end, even for
identical simulation durations. Therefore the measurement of
the movement duration applied to the simulations records a
decrease in hypergravity. This is emphasized in Fig. 3B, where
?tfillustrates the difference between the estimations of move-
ment’s end in normal and hypergravity computed on the
simulations.
Learning in hypergravity
To investigate the effect of hypergravity on unperturbed and
uncorrected trajectories, particular attention should be paid to
the undershoot strategies, which typically reflect a distinct kind
of movement planning and control. The detection of missed
trials based on the presence of undershoots occurring 150 ms
prior to movement end led to the removal of 5% of the trials for
upward movements and 10% for downward movements. This
failure rate was comparable to the behavior observed in normal
gravity conditions where 8 and 4% of the trials had to be
removed for the up and down trials, respectively. The evolution
of the percentage of success trials across the blocks and a
comparison with the average success rate in normal gravity are
shown in Fig. 4 (averaged across the subjects ? SE). The
performance was stable from the beginning for the upward
movement, whereas a stable performance for downward move-
ments was observed from the third block. The difference for
the downward movements between normal and hypergravity in
the first two blocks pooled together was significant (paired
t-test, P ? 0.01, gray rectangle in Fig. 4).
A
B
C
FIG. 3.
hypergravity (black) conditions for upward (left) and downward (right) move-
ments. The dashed traces in A represent the optimal control policy and solid
traces are the muscular response to the control input. B: the angular velocity
of the simulated trajectories as a function of time. C: the optimal trajectories
as a function of time. The shaded areas in B and C represent the SD of the
stochastic system around the optimal trajectories (solid black and red traces).
The SD was computed across 100 simulations.
The plots of the optimal trajectories in normal gravity (red) and
FIG. 4.
subjects (n ? 9) in both directions as a function of the block number. The
horizontal dashed line is the average success rate in normal gravity for both
movement directions. The gray rectangle illustrates the significant difference in
the success rate between hyper and normal gravity for downward movements.
Success rate for hypergravity movements averaged across the
FIG. 2.
model considered in this study (solid black) and observed in the data for one
subject (solid gray). The velocity traces were normalized in time and ampli-
tude. The minimum jerk trajectories with a symmetric velocity profile are
shown for comparison (dashed black). The vertical dashed lines are aligned on
peak velocity to illustrate the skewness of the corresponding curve.
The skewness of the velocity profiles of the minimum control input
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Effect of hypergravity
The elevation angle and angular velocity as a function of
time are shown in Fig. 5 in the two gravitation conditions.
These data are shown for the same subject whose simulations
were plotted in Figs. 2 and 3. All of the trajectories were
aligned with respect to the movement onset to compute the
average trajectory and SE at every time step. For the upward
movements (Fig. 5, A and B), any single trajectory (light gray)
rapidly deviates from the average control trajectory (repre-
sented in red) and goes faster toward the target. This tendency
is further confirmed by the average trajectory in hypergravity
(black trace). This logically provokes the change in the speed
profile illustrated in Fig. 5B. Similar comments may describe
the effect on the downward trajectories (Fig. 5, C and D),
although the effect of hypergravity seems to have less of an
impact on the downward movements than on the upward
movements. Clearly, the effect is qualitatively similar to the
model predictions shown in Fig. 3. The quantitative compari-
son will be addressed later in this analysis.
The statistical analysis revealed that the tendencies observed
qualitatively in Fig. 5 were significant. The Wilcoxon rank-
sum test was chosen for testing whether the variations were
significant, given that the data were not normally distributed.
For the upward movements, there was a significant increase in
the peak angular acceleration followed by an increase in peak
angular velocity and a significant decrease in the movement
duration (P ? 0.01). The same significant effects on the
kinematics parameters were observed for the downward move-
ments. In addition, the effect of the movement direction was
significant. This was evaluated by comparing the difference
between the hypergravity data and the average in normal
gravity across the two movement directions. In all of the cases
(peak angular acceleration, peak angular velocity, and move-
ment duration), the variations in absolute value were larger for
upward movements than those for downward movements (P ?
0.01). The magnitude of the effect is illustrated in Fig. 6A for
the peak angular acceleration, in Fig. 6B for the peak angular
velocity, and in Fig. 6C for the movement duration. All
differences across the gravitational conditions were significant
(the data of the peak angular acceleration and velocity for the
downward movements are presented as absolute values).
The effect of the movement direction revealed that adapta-
tion was not symmetrical for the two movement directions.
Analysis of the overshoots and undershoots shown in Fig. 7
provides interesting insight on the strategies and enables us to
better understand the difference observed between the up and
down movements. For the upward movements, the subjects
reinforced their tendency to overshoot with 77% of the trials
against 73% in normal gravity. In addition, the amplitude of the
overshoots was significantly increased in the 1.8 g condition
(P ? 0.01), as confirmed by the shift in the overshoots
histogram of Fig. 7A. In this direction, the undershoots were
detected in 6% of the trials in hypergravity compared with 11%
in normal gravity. For the downward movements, there was no
difference concerning undershoot trials (6% in hypergravity
compared with 7% in normal gravity without variation in
amplitude), whereas the overshoots were observed in only 43%
of the trials compared with 72% in normal gravity (Fig. 7B).
AC
BD
FIG. 5.
subject in the 2 directions. The red traces are the average 1-g
trajectory plus and minus the SE computed at each time step.
The gray traces show all of the hypergravity trials for this
subject, aligned on individual movement onsets. The black
trace is the average trajectory in 1.8 g. The top panels indicate
the angle as a function of time; the bottom panels indicate the
angular velocity as a function of time.
The elevation angle and angular velocities for one
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In summary, the subjects’ strategy was to globally reach the
target faster in hypergravity and with a significant difference
between the up and down variations and a preference for
overshooting when going upward.
Comparison with model simulations
For the upward movements, the simulations in hypergravity
predict a relative increase of 25% in the peak acceleration and
8% in the peak velocity. The prediction matched the data for
peak acceleration: we observed a relative increase in the peak
acceleration of 24% (averaged across subjects). The relative
increase in the peak velocity was equal to 19%, which is larger
than the prediction. Predictions for the downward movements
were similar to the upward movements: the model predicts 28
and 11% of relative increase in the peak acceleration and
velocity, respectively. Subjects’ behavior varied less: we ob-
served 9 and 3% of increase in peak acceleration and velocity,
respectively.
To compare the changes in the kinematics profiles across the
two gravitational conditions, we computed the areas between
the trajectories performed in the hypergravity condition and the
average 1 g trajectory (used as a reference trajectory). As
illustrated in Fig. 8A, the areas A1and A2were defined with
respect to the sign of the difference between the reference 1 g
trajectory (red trace) and the trajectories performed in hyper-
gravity (black trace; example of one single trajectory taken
from data).
Similarly, the areas between the optimal simulations were
computed to compare the magnitude of the effect with the
experimental data. Figure 8, B and C indicates that the areas A1
and A2in the two directions averaged across bins of three
movements and across subjects. The dotted lines in each plot
indicate the areas computed on the simulations of an identical
duration as the 1 g reference trajectory and were averaged
across the subjects.
As shown in Fig. 3, for the upward movements, the optimal
trajectory in hypergravity is “above” the optimal trajectory in
the normal gravity condition. Therefore for simulations A2
equals 0 (blue dotted line, Fig. 8B) and A1is a positive constant
(black dotted line). The opposite is observed for the downward
movements: Fig. 8C indicates zero for A1and a positive value
for A2for the simulations. The tendency observed in the data
is clearly consistent with the simulations. The trajectories
performed in hypergravity were neither strictly above nor
under the average trajectory in normal gravity (see example in
Fig. 8A; A1and A2are both nonzero), although the general
tendency confirms the prediction of the model. According to
the model simulations, for upward movements, the difference
between the predicted and the observed values for A1demon-
strates that the subjects tended to overcompensate (A1was
larger in data compared with the simulations). Similarly, the
effect of movement direction is illustrated in Fig. 8, B and C.
The downward movements present more variability, although
the global tendency is in accordance with the simulations.
A
B
C
FIG. 6.
tional conditions and in the 2 movement directions. The means of the peak
angular acceleration (A), peak angular velocity (B), and movement duration
(C) were averaged across the subjects. The error bars represent the intersubject
SE. The data of the peak acceleration and velocity for the downward move-
ments are presented as the absolute value. All of the differences between 1 and
1.8 g were statistically significant.
The variation of the kinematics parameters across the 2 gravita-
AB
FIG. 7.
normal and hypergravity conditions for the upward (A) and the
downward (B) movements. The histograms show the frequency
of the observation of trials with the corresponding percentage of
displacement. Recall from METHODS that the overshoots are
positive and the undershoots are negative.
The overshoots (OSH) and undershoots (USH) in the
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To further argue for a reoptimization of the motor com-
mands, we computed the trajectories obtained when the feed-
forward component of the control law corresponding to normal
gravity is adjusted to compensate for the increase in the arm
weight (Eq. 11), but not reoptimized. The simulation of this
alternative model predicted that the kinematics of movements
in hypergravity is the same as that in normal gravity condition,
yielding for A1and A2values very close to zero in Fig. 8. This
demonstrates that the model based on the nonoptimizing strat-
egy can be rejected.
D I S C U S S I O N
Our results describe the effect of an increase in gravity on
vertical pointing movements and compare the subjects’ behav-
ior with the simulation of arm trajectories using a model based
on minimum integrated control input. The proposed model
considers a first-order low-pass filter between the motor neuron
discharge and the muscular torque as a physiological model
and Newton’s second law for the mechanical equation. The
analysis reveals that after an adaptation to hypergravity, the
subjects’ behavior was consistent with the model predictions.
This was in accordance with the hypothesis that adaptation to
altered dynamics is achieved after a reoptimization of the
motor plan in the new environment (Izawa et al. 2008).
In addition, data reveal a significant effect of movement
direction, which was not predicted by the model simulations. It
takes more time to learn the downward movements and, on
learning them, the variation in the kinematics parameters was
larger for the upward movements. This was accompanied by a
clear preference for the overshoot strategy reinforced in hyper-
gravity, despite the increase in the arm weight. The fact that the
subject’s behavior is different depending on the direction of the
movement may reflect the strategy with respect to the goal of
the task, i.e., to stabilize next to the target for which gravity
either enables or hinders the braking acceleration. This result
suggests that reoptimization is not the only process that ensures
an adaptation to altered dynamics. Rather the goal of the task
and the strategies that rely on cognitive factors modulate the
adaptation of the motor plan.
The model was kept as simple as possible to focus on the
effect of a change in gravity and to avoid other factors coming
into play in the simulations. Several limitations can be easily
removed to obtain a better match between the predictions and
the data. First, one can introduce a bias to account for the effect
of the movement direction. Then, more sophisticated mechan-
ical and physiological models can improve the quality of the
prediction, by considering more carefully the properties of the
musculoskeletal system. Finally, keeping the same simulation
duration across the two gravitational conditions was a strong
constraint, which can be relaxed to improve the correspon-
dence between the predictions and the data. In particular,
matching the simulation duration with the movement duration
in hypergravity yields better predictions for upward move-
ments.
Our modeling approach follows the models proposed by
Gentili and colleagues (2007), which consisted of minimal
torque changes and minimal commanded torque changes
(Nakano 1999). These models assume that the brain has an
access to the joint torque, given that it is the variable being
controlled. We added the physiological model (Eq. 3) to for-
mulate a control variable that is closer to the neural motor
command, which is presumably accessible within the CNS as
AB
C
FIG. 8.
ity. A: the illustration of the computation procedure; the am-
plitudes were normalized to the movement amplitude from
movement onset to movement end. The time was unchanged.
A1and A2are the areas between the hypergravity traces and the
reference 1 g trace when the integrated differences were posi-
tive and negative, respectively. B (upward) and C (downward)
show the areas A1and A2averaged across bins of 3 movements
and across the subjects vs. the number of the bin.
The quantitative analysis of the effect of hypergrav-
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a corollary discharge. Our results obtained in the normal grav-
ity condition are fully compatible with the results from Gentili
and colleagues (2007), despite some differences between the
protocols. The main differences were that their subjects per-
formed reaches of various amplitudes and from various starting
positions; moreover, the movements were performed in dark-
ness without visual feedback. Our protocol was restricted to
one movement amplitude from a single starting position, to
collect a sufficient number of trials per subject and per flight.
Then, our subjects were provided with continuous visual feed-
back to prevent motion sickness. This was presumably the
origin of the high success rate observed early, in particular for
the upward movements. However, in normal gravity, we ob-
tained very similar values for the AD/MD ratio. This suggests
that asymmetry in the velocity profiles is a robust feature of the
vertical movements across various testing conditions. More
generally, our model accounts not only for the difference
between vertical and horizontal movements in normal gravity,
but also for the difference between the vertical movements
performed in normal and hypergravity.
Our main finding is that an increase in gravity is properly
compensated in a feedforward manner in accordance with the
hypothesis of minimum control input. In particular, the effect
of hypergravity on the peak acceleration (precisely matched by
the simulations for upward movements) is in direct support of
a change in the central planning process, which must account
for the effect of gravity on the limb. In addition, our results
suggest that the CNS uses the internal representation of the
gravitational torque to optimize the motor commands. Our
sample does not permit the evaluation of whether the optimal
feedback gains (Lkin Eq. 10) are also reoptimized in hyper-
gravity. However, the simulations predict distinct sensitivities
to the position and speed errors across the two gravitational
conditions. It follows from our result that the CNS takes into
account the effect of external forces on the limb and takes
advantage of the dynamic interaction between the body and the
environment. White and colleagues (2008) reported a similar
result in the context of oscillatory movements performed under
different gravity conditions. They found that the spontaneous
movement frequency was driven by gravity as the resonance
phenomenon of a pendulum, which is the signature of a system
maximizing energy transfer with the environment. The inter-
action with the environment is emphasized by the changes in
the movement frequency in response to the changes in gravity.
Altogether, these results provide evidence for an optimal in-
teraction with the environment in both rhythmic and discrete
movements depending on the action of gravity on the limb.
Previous studies argued for an optimization of the dynamic
forces only (Nishikawa et al. 1999; Soechting et al. 1995),
which appears contradictory with the hypothesis of integration
of the gravitational torque in the computation of a motor plan.
Nishikawa and colleagues (1999) showed that the final arm
posture after pointing movements depended on the initial
position and these postures were invariant, even for slow
movements, where the antigravity component of the joint
torque should be dominant compared with the peak joint
torque. In our study, the initial and final arm postures were
prescribed and the movements were constrained to one degree
of freedom in the vertical plane. Therefore for such move-
ments, the minimization of the dynamic variations only (as
supported by Nishikawa and colleagues) implies that the grav-
itational torque is taken into account, as supported by the
present study. Other kinds of movement with horizontal com-
ponents and more degrees of freedom may rely on distinct
strategies.
The results of the present study have two main implications.
First, the data allow us to reject the hypothesis of an invariant
desired trajectory, which assumes that internal models build a
new map between a desired trajectory and the motor command,
taking the unperturbed trajectory as the desired trajectory. In
our experiment, the optimal trajectory in normal gravity is of
course a feasible solution in hypergravity, although the simu-
lations reveal that, although possible, this trajectory is no
longer optimal. Accordingly, the subjects change the kinemat-
ics profile without any tendency to recover the characteristics
of the movements performed in normal terrestrial gravity.
However, many studies in various contexts do provide evi-
dence for a tendency to return to the performance of the
unperturbed condition (Burdet et al. 2001; Lackner and Dizio
1994; Papaxanthis et al. 2005; Shadmehr and Mussa-Ivaldi
1994), whereas other studies, including the present one, sug-
gest that reoptimization produces definitive changes in the
movement kinematics (Izawa et al. 2008). It seems the question
of which adaptation process (recalibration of the internal mod-
els or reoptimization) will be observed depending on the
experimental conditions remains open to debate.
Second, under the hypothesis that the motor actions are
reoptimized in response to a change in the environment, this
experiment suggests that the estimation of movement costs can
be based on the motor command corollary discharge and
proprioceptive feedback. Indeed, gravity acting vertically with
respect to the aircraft floor was not a divergent force field for
vertical movements. Therefore it is very unlikely that the
reoptimization observed in hypergravity is based solely on
visuomotor adjustments. The movements were visually guided
and the movement control relied on visual feedback (e.g.,
corrective adjustments as observed in trials presenting large
undershoots). However, visual feedback by itself fails to ex-
plain why the subjects changed the kinematics motor plan and
performed faster movements. Therefore the sources of infor-
mation available for an estimation of the movements’ costs
(necessary for optimization) are the motor command corollary
discharge, which is represented in the model by u(t), and the
proprioceptive feedback. Such an estimation of movement
costs based on these variables compared with an expected cost
from prior knowledge can represent a signal that generates the
update of the kinematic and dynamic properties of the move-
ment.
A potential region for the generation of this signal is the
posterior parietal cortex (PPC), which plays a fundamental role
in visuomotor transformations for reaching and in the plan-
ning and control of visually guided movements (Buneo and
Andersen 2006; Buneo et al. 2002). Downstream to the visuo-
motor transformation, PPC activity correlates with kinematics
parameters such as target location, movement direction, and
velocity. The PPC activity further has little effect for various
load conditions for a given invariant trajectory (Ashe and
Georgopoulos 1994; Hamel-Paquet et al. 2006; Kalaska et al.
1990). In addition, the neural correlates with kinematics errors
in reaching were found in the parietal area 5 (Diedrichsen et al.
2005). It was further found that transcranial magnetic stimu-
lations of this region alter the learning of new dynamics (Della-
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Maggiore et al. 2004). However, the fact that continuous visual
feedback was provided suggests that the alteration of learning
could be related to visuomotor transformation instead of a
kinematics adaptation.
Of course, the neural correlates with kinematics parameters
are present in other brain regions such as the premotor areas
(Kakei et al. 2003) and the primary motor cortex whose
activity is not solely related to force output (Graziano 2006;
Scott 2003). However, in addition to the evidence that the PPC
codes kinematics features of the movement, the anatomical
situation of the PPC is consistent with our hypothesis that both
internal and proprioceptive feedback are necessary to adapt the
kinematics motor plan. Indeed, the PPC receives projection
from the somatosensory cortex for proprioceptive feedback and
from the cerebellum via the thalamus (Amino et al. 2001),
whose activity relates to state estimation and prediction (Miall
et al. 2007; Wolpert et al. 1998), which could contribute to the
prior estimate of the movement cost. Altogether, the charac-
teristics of the PPC render it a good candidate as a sensorimo-
tor interface for updating the kinematics motor plans.
A C K N O W L E D G M E N T S
We thank the subjects for kindly participating in this study.
G R A N T S
This work was supported by grants from the PRODEX project, Fonds
National de la Recherche Scientifique, Action de Recherche Concerte ´e
(Belgium), and the European Space Agency of the European Union. This paper
presents research results of the Belgian Network DYSCO (Dynamical Sys-
tems, Control and Optimization), funded by the Interuniversity Attraction
Poles Programmes, initiated by the Belgian State, Science Policy Office. The
scientific responsibility rests with its authors.
R E F E R E N C E S
Amino Y, Kyuhou S, Matsuzaki R, Gemba H. Cerebello-thalamo-cortical
projections to the posterior parietal cortex in the macaque monkey. Neurosci
Lett 309: 29–32, 2001.
Ashe J, Georgopoulos AP. Movement parameters and neural activity in motor
cortex and area-5. Cereb Cortex 4: 590–600, 1994.
Atkeson CG, Hollerbach JM. Kinematic features of unrestrained vertical arm
movements. J Neurosci 5: 2318–2330, 1985.
Augurelle AS, Penta M, White O, Thonnard J-L. The effects of a change
in gravity on the dynamics of prehension. Exp Brain Res 148: 533–540,
2003.
Berret B, Darlot C, Jean F, Pozzo T, Papaxanthis C, Gauthier J-PP. The
inactivation principle: mathematical solutions minimizing the absolute work
and biological implications for the planning of arm movements. PLoS
Comput Biol 4: e1000194, 2008.
Bock HG, Plitt KJ. A multiple shooting algorithm for direct solution of
optimal control problems. In: Proceedings of the 9th International
Federation of Automatic Control World Congress, Budapest. Oxford,
UK: Pergamon Press, 1984, p. 243–247.
Buneo CA, Andersen RA. The posterior parietal cortex: sensorimotor inter-
face for the planning and online control of visually guided movements.
Neuropsychologia 44: 2594–2606, 2006.
Buneo CA, Jarvis MR, Batista AP, Andersen RA. Direct visuomotor
transformations for reaching. Nature 416: 632–636, 2002.
Burdet E, Osu R, Franklin DW, Milner TE, Kawato M. The central nervous
system stabilizes unstable dynamics by learning optimal impedance. Nature
414: 446–449, 2001.
Crevecoeur F, Thonnard J-L, Lefe `vre P. Forward models of inertial
loads in weightlessness. Neuroscience 161: 589–598, 2009.
Della-Maggiore V, Malfait N, Ostry DJ, Paus T. Stimulation of the posterior
parietal cortex interferes with arm trajectory adjustments during the learning
of new dynamics. J Neurosci 24: 9971–9976, 2004.
Diedrichsen J, Hashambhoy Y, Rane T, Shadmehr R. Neural correlates of
reach errors. J Neurosci 25: 9919–9931, 2005.
Flash T, Hogan N. The coordination of arm movements: an experimentally
confirmed mathematical model. J Neurosci 5: 1688–1703, 1985.
Gentili R, Cahouet V, Papaxanthis C. Motor planning of arm movements is
direction-dependent in the gravity field. Neuroscience 145: 20–32, 2007.
Girgenrath M, Gobel S, Bock O, Pongratz H. Isometric force production in
high Gz: mechanical effects, proprioception, and central motor commands.
Aviat Space Environ Med 76: 339–343, 2005.
Graziano M. The organization of behavioral repertoire in motor cortex. Annu
Rev Neurosci 29: 105–134, 2006.
Hamel-Paquet C, Sergio LE, Kalaska JF. Parietal area 5 activity does not
reflect the differential time course of motor output kinetics during arm-
reaching and isometric-force tasks. J Neurophysiol 95: 3353–3370, 2006.
Harris CM, Wolpert DM. Signal-dependent noise determines motor plan-
ning. Nature 394: 780–784, 1998.
Izawa J, Rane T, Donchin O, Shadmehr R. Motor adaptation as a process of
reoptimization. J Neurosci 28: 2883–2891, 2008.
Kakei S, Hoffman DS, Strick PL. Sensorimotor transformations in cortical
motor areas. Neurosci Res 46: 1–10, 2003.
Kalaska JF, Cohen DAD, Prudhomme M, Hyde ML. Parietal area-5
neuronal activity encodes movement kinematics, not movement dynamics.
Exp Brain Res 80: 351–364, 1990.
Kawato M. Internal models for motor control and trajectory planning. Curr
Opin Neurobiol 9: 718–727, 1999.
Lackner JR, Dizio P. Rapid adaptation to Coriolis force perturbations of arm
trajectory. J Neurophysiol 72: 299–313, 1994.
Li W, Todorov E. Iterative linearization methods for approximately optimal
control and estimation of non-linear stochastic system. Int J Control 80:
1439–1453, 2007.
Liu D, Todorov E. Evidence for the flexible sensorimotor strategies predicted
by optimal feedback control. J Neurosci 27: 9354–9368, 2007.
McIntyre J, Berthoz A, Lacquaniti F. Reference frames and internal models
for visuo-manual coordination: what can we learn from microgravity exper-
iments? Brain Res Rev 28: 143–154, 1998.
McIntyre J, Zago M, Berthoz A, Lacquaniti F. Does the brain model
Newton’s laws? Nat Neurosci 4: 693–694, 2001.
Miall RC, Christensen LOD, Cain O, Stanley J. Disruption of state estima-
tion in the human lateral cerebellum. PLoS Biol 5: 2733–2744, 2007.
Mierau A, Girgenrath M, Bock O. Isometric force production during
changed-Gz episodes of parabolic flight. Eur J Appl Physiol 102: 313–318,
2008.
Nakano E, Imamizu H, Osu R, Uno Y, Gomi H, Yoshioka T, Kawato M.
Quantitative examinations of internal representations for arm trajectory
planning: minimum commanded torque change model. J Neurophysiol 81:
2140–2155, 1999.
Nishikawa KC, Murray ST, Flanders M. Do arm postures vary with the
speed of reaching? J Neurophysiol 81: 2582–2586, 1999.
Papaxanthis C, Pozzo T, McIntyre J. Kinematic and dynamic processes for
the control of pointing movements in humans revealed by short-term
exposure to microgravity. Neuroscience 135: 371–383, 2005.
Papaxanthis C, Pozzo T, Stapley P. Effects of movement direction upon
kinematic characteristics of vertical arm pointing movements in man.
Neurosci Lett 253: 103–106, 1998.
Pozzo T, Papaxanthis C, Stapley P, Berthoz A. The sensorimotor and
cognitive integration of gravity. Brain Res Rev 28: 92–101, 1998.
Scott SH. The role of primary motor cortex in goal-directed movements:
insights from neurophysiological studies on non-human primates. Curr Opin
Neurobiol 13: 671–677, 2003.
Scott SH. Optimal feedback control and the neural basis of volitional motor
control. Nat Rev Neurosci 5: 534–546, 2004.
Shadmehr R, Krakauer JW. A computational neuroanatomy for motor
control. Exp Brain Res 185: 359–381, 2008.
Shadmehr R, Mussa-Ivaldi FA. Adaptive representation of dynamics during
learning of a motor task. J Neurosci 14: 3208–3224, 1994.
Soechting JF, Buneo CA, Herrmann U, Flanders M. Moving effortlessly in
3-dimensions: does Donder’s law apply to arm movement? J Neurosci 15:
6271–6280, 1995.
Todorov E. Optimality principles in sensorimotor control. Nat Neurosci 7:
907–915, 2004.
Todorov E. Stochastic optimal control and estimation methods adapted to the
noise characteristics of the sensorimotor system. Neural Comput 17: 1084–
1108, 2005.
Todorov E, Jordan MI. Optimal feedback control as a theory of motor
coordination. Nat Neurosci 5: 1226–1235, 2002.
795OPTIMAL INTEGRATION OF GRAVITY IN TRAJECTORY PLANNING
J Neurophysiol • VOL 102 • AUGUST 2009 • www.jn.org
on August 24, 2009
jn.physiology.org
Downloaded from

Page 12

Uno Y, Kawato M, Suzuki R. Formation and control of optimal trajectory in
human multijoint arm movement: minimum torque change model. Biol
Cybern 61: 89–101, 1989.
Wagner MJ, Smith MA. Shared internal models for feedforward and feed-
back control. J Neurosci 28: 10663–10673, 2008.
White O, Bleyenheuft Y, Ronsse R, Smith AM, Thonnard J-L, Lefe `vre P.
Altered gravity highlights central pattern generator mechanisms. J Neuro-
physiol 100: 2819–2824, 2008.
White O, McIntyre J, Augurelle AS, Thonnard J-L. Do novel gravitational
environments alter the grip-force/load-force coupling at the fingertips? Exp
Brain Res 163: 324–334, 2005.
Winter DA. Biomechanics of Human Movement. New York: Wiley, 1979.
Wolpert DM, Ghahramani Z. Computational principles of movement neuro-
science. Nat Neurosci 3: 1212–1217, 2000.
Wolpert DM, Miall RC, Kawato M. Internal models in the cerebellum.
Trends Cogn Sci 2: 338–347, 1998.
796F. CREVECOEUR, J.-L. THONNARD, AND P. LEFE`VRE
J Neurophysiol • VOL 102 • AUGUST 2009 • www.jn.org
on August 24, 2009
jn.physiology.org
Downloaded from