Springer-Verlag 1998
Exp Brain Res (1998) 120:233-242
RESEARCH ARTICLE
Charalambos Papaxanthis ´ Thierry Pozzo
Annie Vinter ´ Alexander Grishin
The representation of gravitational force during drawing movements
of the arm
Received: 19 May 1997 / Accepted: 3 November 1997
Abstract The purpose of the present experiment was to
study the way in which the central nervous system
(CNS) represents gravitational force (GF) during vertical
drawing movements of the arm. Movements in four different directions: (a) upward vertical (0), (b) upward oblique (45), (c) downward vertical (180) and (d) downward oblique (135), and at two different speeds, normal
and fast, were executed by nine subjects. Data analysis focused upon arm movement kinematics in the frontal plane
and gravitational torques (GTs) exerted around the shoulder joint. Regardless of movement direction, subjects
showed straight-line paths for both speed conditions. In
addition, movement time and peak velocity were not affected by movement direction and consequently changes
in GT, for both speeds tested. Movement timing (evaluated through the ratio of acceleration time to total time)
changed significantly, however, as a function of movement direction and speed. Upward movements showed
shorter acceleration times when compared with downward movements. Concerning the four directions, movements made at 0 and 45 differed significantly from
those made at 135 and 180. Drawing movements executed at rapid speed presented similar acceleration and
deceleration times compared with movements executed
at normal speed, which showed greater acceleration than
deceleration times. In addition, the form of velocity profiles (assessed through the ratio of maximum to mean velocities), was significantly modified only with movement
speed. Results from the present study suggest that GF is
efficiently incorporated into internal dynamic models that
the brain builds up for the execution of arm movements.
)
C. Papaxanthis ( ) ´ T. Pozzo
Groupe dAnalyse du Mouvement (G.A.M), U.F.R. S.T.A.P.S.,
Campus Universitaire, UniversitØ de Bourgogne, B.P. 138,
F-21004 Dijon, France
e-mail: [email protected], Fax: +33-3 80396702
A. Vinter
LEAD, UniversitØ de Bourgogne, B.P. 138, F-21004 Dijon, France
A. Grishin
Institute for Problems in Information Transmission (I.P.I.T.),
Academy of Sciences, Ermolova St.19, GSP-4, Moscow, Russia
Furthermore, it seems that GF not only is a mechanical
parameter to be overcome by the motor system but also
constitutes a reference (vertical direction), both of which
are represented by the CNS during inverse kinematic and
dynamic processes.
Key words Gravitational force and torque ´ Drawing ´
Arm kinematics ´ Planning ´ Human
Introduction
Drawing movements are performed by transforming cognitive representations into appropriate arm motions.
These transformations concern not only the position of
the hand (or the pen) in space but also the spatiotemporal
coordination of the joints and the muscles involved in the
motion. Apart from considering the coordinate systems
(Cartesian, joint or muscular) in which a movement could
be planned, an interesting question arises concerning the
relationship between gravitational force (GF) and arm
movement control. It is well known that GF plays an important role in spatial orientation (Young 1984), proprioception (Worringham et al. 1987; Fisk et al. 1993) and locomotion (Pozzo et al. 1990). Furthermore, gravity can either accelerate or decelerate three-dimensional (3D) arm
movements and consequently should be represented in
the motor command.
How is gravity represented by the brain for arm movement production? To approach such a question, the motor
act can be considered as a consequence of sensorimotor
transformations (Saltzman 1979; Atkeson 1989; Soechting and Flanders 1991). If the trajectory of the hand is
primarily planned (the preliminary stage of sensorimotor
transformations), inverse kinematics (the second stage)
is required to compute the time sequence of the joints involved in the motion. Plans of joint angle changes are
transformed into joint torques and muscle forces (the third
stage), by inverse dynamic computations. An alternative
hypothesis proposes, however, that, instead of inverse dynamic computations, the CNS may take advantage of the
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mechanical features of the musculoskeletal system (the
equilibrium-point hypothesis) in order to produce movements (Feldman 1986; Bizzi et al. 1992). In such a case,
only the endpoints of the movement are specified and the
intermediate trajectory is dynamically determined. Finally, the fourth stage translates the motor programs into patterns of neuromuscular activity. The first three stages
(termed ªplanningº) occur before motion and concern
central mechanisms that operate in the association and
sensorimotor cortices, the supplementary motor area, the
cerebellum and the putamen loop of the basal ganglia.
The final stage, which can be called ªexecutingº, integrates neural and biomechanical mechanisms (Brooks
1986). The neural mechanisms concern central inputs
(originating from motor, premotor cortical as well as subcortical areas), and afferent inputs (originating from muscle spindles, Golgi tendon organs and Renshaw cells), addressed to a- and/or g-motoneurones. Biomechanical
mechanisms are referred to visco-elastic properties of
muscle and tendon tissue, which contribute to the applied
torques (Lestienne 1979).
To treat gravity only at the executing level would suggest that the brain plans hand trajectories without taking
into account gravitational torques (GTs) during inverse
dynamic transformations. Nevertheless, even if it seems
difficult to accept such a hypothesis, owing to the strong
presence of gravitational force throughout life and the
transmission delays of feedback mechanisms, it cannot
be ruled out. For example, Gordon et al. (1994b, 1995)
have suggested that directional-dependent changes in inertial load in the horizontal plane are not taken into account during trajectory planning. According to the same
authors, compensatory changes in movement parameters
may arise either from sensory inputs from muscle receptors and/or spring-like properties of the muscle. Thus, following the same reasoning, gravity-dependent torques
could be compensated for during the motion by an increase in arm stiffness and/or feedback mechanisms that
involve spinal and supraspinal neural circuits. In such a
way, considering gravity as a mechanical ªdisturbanceº,
would avoid complicated calculations of GTs before the
movement but would also have a significant perturbing
effect upon movement kinematics.
Alternatively, the representation of GF at the planning
level suggests that the CNS would take into account GTs
for a specific movement before its execution. Indeed, the
CNS could establish relationships between torques and
movement directions and thus integrate GF into internal
dynamic models. Such an hypothesis is in agreement with
previous studies that have shown that the brain constructs
internal dynamic models of the mechanical properties of
the arm and the environment (Atkeson and Hollerbach
1985; Atkeson 1989; Lacquaniti 1993; Shadmehr and
Mussa-Ivaldi 1994; Virji-Babul et al. 1994). To propose
only that gravity is represented at the planning level
leaves unanswered questions surrounding the nature of
any such representation.
Mechanically speaking, the constant direction of gravity generates torques whose values vary as a function of
movement amplitude and direction in space. If one assumes that gravity is centrally treated then two forms of
representation could be hypothesised. According to the
first, gravity could be considered as a ªmechanical loadº
that must be overcome. This implies that, whatever the direction or amplitude of the movement, GTs will be anticipated during inverse dynamic processes in order to produce desired arm motions. Such a representation of gravity implies that trajectory parameters, planned during inverse kinematic processes, should remain invariant with
respect to variations in GTs. For instance, previous studies made in the horizontal and vertical planes have shown
that hand paths (Morasso 1981; Soechting and Lacquaniti
1981; Laquaniti et al. 1982) and velocity profiles (Atkeson and Hollerbach 1985; Flash and Hogan 1985; VirjiBabul et al. 1994) remained invariant regardless of arm
movement direction, speed or load. According to the second hypothesis, gravity could be centrally represented (1)
as an ªorientation referenceº (vertical direction), remaining invariant throughout movements made with differing
orientations or speeds, and (2) as a mechanical load
(GTs). In this way, gravity could be represented at two
levels: (1) during inverse kinematic computations as a kinematic parameter (vertical direction), and (2) during inverse dynamic computations as a dynamical parameter
(GTs). This double representation of GF may suggest that
the CNS will plan differently arm movements with varying directions by taking advantage of these two properties
of GF. Recent results obtained during vertical pointing
movements in normal and weightless environments (Papaxanthis et al. 1996) have shown different planning processes for upward and downward directions and long adaptation processes to weightlessness.
The purpose of the present experiment was to study the
level of the integration of GF within the neural motor
command and to more fully understand the nature of its
representation by the brain. Geometrical drawings in the
frontal plane were performed and kinematic characteristics analysed in order to infer planning and control processes underlying vertical drawing movements. Different
movement directions (which would imply changes in
GTs), and different speeds (changes in inertial but not
in GTs) were tested in order to understand the manner
in which movements are executed under the influence
of GF. Our results seemed to provide evidence that GF
is not a simple mechanical parameter, programmed for
during inverse dynamics processes, but also constitutes
a ªreferenceº, which is represented in the earlier stages
of sensorimotor transformation.
Materials and methods
Subjects
Nine healthy, right-handed adults (six men and three women), ranging in age from 19 to 39 years, having had no previous neuromuscular disorders, participated in this experiment. All subjects were naive as to the purpose of the experiment and gave their consent. All
235
Fig. 1 Apparatus and the motor
task. Tracing directions and locations of recorded markers
during experimental measurements. Marker (M) 1, shoulder
(acromion); M 2, elbow (lateral
epicondyle); M 3, wrist (cubitus
styloid process); M 4, wrist (radius styloid process); M 5, hand
(metacarpophalangeal joint);
M 6, pen
experiments were conducted in accordance with legal requirements
and international norms.
Procedure and apparatus
Subjects stood erect. They were asked to draw, on a vertical board
(1 m in height and 70 cm in width), predefined geometric patterns
of an amplitude of 30 cm in the frontal plane. Four different directions (see Fig. 1) were chosen: (1) upward vertical (0), (2) upward
oblique (45), (3) downward vertical (180), and (4) downward oblique (135). In addition, subjects were asked to conduct movements
at two different speeds (normal and as fast as possible). The drawing
board had no marks that may have provided subjects with a reference as to the distance to be drawn. Subjects could, however,
throughout the whole experimental period, see a diagram, located
at 2 m in front and to the right of them, indicating the required drawing (real length 30 cm in all directions). They were instructed to reproduce the figure without making corrective movements and to
move when they were ready, after a ªgoº signal. At the beginning
of each trial, subjects positioned a pen, held in the right hand, on
the board and waited for the starting signal. After the go signal, subjects performed a single drawing movement. Accuracy was not the
primary constraint imposed upon subjects during experimentation.
Three movements were performed in each experimental condition.
In order to minimise variations in movement execution, subjects
were asked to start upward and downward movements from the
same positions on the board. These positions corresponded to
15 cm below shoulder level (projection of the shoulder on the board)
for upward (0 and 45), and 15 cm above shoulder level for downward directions (0 and 45). In this way, variations of subjects initial posture, due to differences in their heights, were minimised. The
experimental protocol was presented in four blocks. Subjects executed drawing movements in the following order: the vertical downward direction (180), the oblique downward direction (135), the
vertical upward direction (0), and, finally, the oblique upward direction (45). In each direction, movement speed was randomly ordered. A pause of approximately 10 s was permitted between trials
as well as a 2-min one between blocks. In order to familiarise themselves with the experimental protocol, subjects performed 16 practice trials before movement acquisition, 2 in each experimental condition. Subjects had no feedback as to their performances during the
experiment.
Arm movements were recorded and analysed using an optoelectronic system (ELITE), set at a sampling frequency of 100 Hz. Two
TV cameras were placed at a 45 angle, 4 m away from the apparatus. Accuracy was in the order of 1/2500 of the working field and the
spatial resolution for measurements of movements in the present experiment was less than 0.5 mm. Six markers (plastic spheres of
0.4 cm in diameter), covered with reflecting material, were placed
on the joints of the arm (shoulder, elbow, wrist and hand; see
Fig. 1). Their positions during the movement were recorded and
their centroides underwent 3D reconstruction.
Data analysis
After completion of the experiment, kinematic parameters (position,
velocity and acceleration) in three dimensions were computed for
pen movements, anatomical angles of the elbow and wrist and spatial angles (vertical elevation and horizontal yaw of the arm, upper
arm and wrist). For both linear and angular values, movement onset
was defined as the moment at which linear or angular velocity exceeded 10% of its peak and the end of movement the point at which
linear or angular velocity dropped below the 10% threshold.
Linear parameters chosen to describe subjects motor performance in the present experiment were: (1) movement time (MT),
(2) acceleration time (AT) and deceleration time (DT), and (3) peak
velocity (Vpeak).
The symmetry of velocity profiles for each experimental condition was evaluated through the comparison of AT and DT. It was hypothesised that, if the comparison between AT and DT showed significant differences, a conclusion could be made that velocity profiles were asymmetric. However, if the comparison between AT
and DT gave no significant differences, velocity profiles were symmetric.
In order to test the invariance of the velocity profiles as a function of movement speed and direction, we used the following two
parameters: (1) the movement timing (B), and (2) the form of the velocity profile (C). The movement timing was defined as the ratio of
acceleration time to total movement time (B = AT/MT). A ratio
greater than 0.5 indicated an acceleration duration longer than the
deceleration duration. The form of the velocity profile was defined
as the ratio of peak velocity to mean velocity (C = Vpeak/Vmean). If
subjects produced equivalent velocity profiles, indicating an independence of required speed and movement direction, values of C
and B would have remained constant (Soechting 1984; Ostry 1987).
GTs produced around wrist, elbow and shoulder joints were calculated according to methods outlined in the Appendix. In order to
simplify the interpretation of GTs in the present study, only GTs of
the shoulder joint will be shown and discussed. Two components of
shoulder gravitational torque are represented here: (1) maximal
shoulder gravitational torque (MSGT), the greatest value during
the motion, and (2) the amplitude of shoulder gravitational torque
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Fig. 2 General features of arm
tracing movements for the four
directions, from one typical
subject. Stick diagrams, drawn
every 40 ms, representing the
upper arm, forearm and hand are
shown in both sagittal and horizontal planes. Data are taken
from movements executed at
normal speeds
(ASGT), the difference between maximum and minimum values
during the motion. In the present study, GTs will be considered to
have positive values (g = 9.81 m/s2).
Statistical analysis
Means and standard errors for each subject in all experimental conditions and for all parameters described above were calculated. Measures were subjected to multivariate analysis of variance (ANOVA).
The factors examined included four directions and two speeds. Newman-Keuls post hoc and paired t-tests were also performed on the
parameters within experimental conditions and movement directions.
Results
Before presenting kinematic results of pen movements,
hand, forearm and upper arm motions, as well as joint
GT values, will be presented in order to define the global
context in which arm drawing movements were performed.
General characteristics of drawing movements
Figure 2 shows stick diagrams of the upper arm, forearm
and hand in the sagittal and horizontal planes from one
typical subject. Prior to the onset of an upward movement
(0 and 45), the subject stood with his upper arm (elevation 155 and azimuth 65) and forearm (elevation
68 and azimuth 36) flexed and with the pen positioned on the board. Before a downward movement
(180 and 135), elevation and azimuth of the upper and
forearm were 125, 35, 40 and 60, respectively.
Averaged shoulder GTs (SGT) along x- and y-axes
from one typical subject are represented in Fig. 3. Two
important characteristics of SGT during drawing movements are illustrated in this figure. Firstly, MSGT was
greater in the y- than in the x-axis, regardless of movement direction. MSGT in the y-axis had approximately
similar values for the four directions. This was somewhat
in contrast to MSGT along the x-axis, which was greater
for the oblique (45 and 135) than vertical directions (0
and 180). Secondly, for both axes, ASGT depended upon
movement direction. ASGT in the x-axis demonstrated
higher variability than ASGT did in the y-axis, across
the four directions. Mean values of MSGT and ASGT
of all subjects are shown in Table 1. For each direction,
Fig. 3 Shoulder gravitational torques (GT, in newton-metres) around y- and x-axes during movement, for all directions from one typical subject. Traces are the mean (thick lines) and SEs (thin lines)
of movements executed at normal and rapid speeds
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Table 1 Shoulder gravitational torques of all subjects. Means (SE)
of: maximal shoulder gravitational torques around the X- and Y-axes
and total maximal shoulder gravitational torques; amplitude of
shoulder gravitational torques around the X- and Y-axes and total
amplitude of shoulder gravitational torques are shown
Movement direction
0
135
180
1.75 (0.46)
6.59 (0.94)
6.62 (0.94)
0.71 (0.47)
6.62 (0.98)
6.67 (0.99)
Amplitude of gravitational torque (Nm)
x
0.59 (0.39) 2.50 (0.56) 1.65 (0.53)
y
1.83 (0.46) 1.07 (0.42) 1.97 (0.50)
Total 1.88 (0.48) 1.67 (0.46) 1.67 (0.48)
0.48 (0.36)
2.27 (0.56)
2.29 (0.61)
Maximal
x
y
Total
45
gravitational torque (Nm)
0.71 (0.71) 2.75 (0.73)
6.77 (1.04) 5.95 (0.94)
6.84 (1.06) 6.60 (0.90)
values of MSGT and ASGT are mean ones for both normal and rapid speeds.
Kinematic features of pen movements
Fig. 4 Typical drawing paths. Individual trials (n = 6) from one subject in the frontal plane have been superimposed for both speed conditions. Arrows indicate movement directions
As has been already mentioned in the Introduction, kinematic parameters of the pen motion were analysed in order to understand the way in which arm drawing movements are planned and executed under the influence of
gravitational force. The results showed that GF differentially affected kinematic parameters.
Movement direction and speed do not affect paths,
movement time and peak velocity
Paths of the pen for the two speed conditions and the four
movement directions from one typical subject are shown
in Fig. 4. For each direction, paths with normal and fast
speeds have been superimposed. This one particular subject produced straight-line paths with little variability between trials, independently of movement speed and direction. Straight-line paths are, however, representative of all
the nine subjects tested. In order to present the straightness of pen movements across subjects, paths were scaled
and then rotated so as to align the starting points of the
movements. Mean paths and their SEs were calculated
for normal and fast speeds, in each direction. Figure 5
shows mean pen paths for all subjects in the four movement directions. It can be seen that, for all directions,
pen paths were straight. Mean pen path amplitudes for
all subjects were 31.58, 33.08, 34.05 and 31.50 cm for
the 0, 45, 135 and 180 directions, respectively.
Mean MTs of the pen for all the experimental conditions are shown in Fig. 6. Subjects presented approximately the same MT for the 0, 45 and 180 directions
in both speed conditions. For the 135 direction, however,
there was a tendency (although insignificant) for MT to
increase compared with the other directions for both nor-
Fig. 5 Mean (thick lines) and SEs (thin lines) of pen paths calculated for all subjects. Each direction shows means of both speed conditions
mal and rapid speeds. MT decreased highly significantly,
however, with speed constraints (F1,8 = 138.04,
P = 0.0001). Movement direction and the interaction between movement direction and speed gave no significant
effects (P > 0.05).
Movements executed in different directions showed
approximately the same peak velocities (see Fig. 7).
The magnitude of the peak pen velocity was not affected
by the direction of the movement (P > 0.05). In contrast,
peak pen velocity was dependent upon movement speed
(F1,8 = 183.04, P = 0.0001).
To summarise, present results showed that pen movement paths shape, time and peak velocities for both speed
238
Fig. 6 Means and SEs of movement time of all subjects and experimental conditions
Fig. 8 Typical tangential velocity profiles of the pen for the two
speeds and the four directions tested. Data represent trials from
one typical subject
Table 2 Means (SE) of acceleration and deceleration times (ms) of
all subjects and experimental conditions
Direction
Fig. 7 Means and SEs of peak velocity of all subjects and experimental conditions
conditions were not related to movement direction and
consequently to modifications in GTs.
Effects of movement direction and speed upon
velocity profiles
Representative tangential velocity profiles (normalised in
time) of the pen in the frontal plane for both direction and
speed conditions can be seen in Fig. 8. Velocities showed
unimodal profiles, with one acceleration and one deceleration phase for all experimental conditions. The symmetry of velocity profiles was tested statistically using paired
t-tests, in which AT was compared with DT, for each experimental condition. Using this analysis, symmetric velocity profiles were found for the 180 and 135 directions at rapid speeds (see Table 2).
In order to test the effects of movement direction and
speed upon velocity profiles (see Materials and methods),
movement timing (B = AT/MT) and velocity profile
shape (C = Vpeak/Vmean) were submitted to statistical analysis. A two-way ANOVA (four directions ” two speeds)
for B, revealed highly significant main effects for move-
Normal speed (ms)
Rapid speed (ms)
AT
DT
AT
DT
0
289.62*
(52.80)
334.23
(80.56)
151.92**
(25.61)
173.08
(24.29)
45
347.60*
(92.12)
300.80
(60.20)
159.23 *
(31.87)
173.85
(24.99)
135
375.19 **
(76.68)
306.30
(81.44)
190
(39.67)
187.08
(40.05)
180
329.62 **
(76.71)
266.54
(79.60)
170
(27.13)
166.15
(41)
* P < 0.01, ** P < 0.001
ment direction (F3,24 = 9.53, P = 0.0002), and speed
(F1,8 = 37.87, P = 0.0002). Their interaction did not, however, reach significance (P > 0.05). Post hoc analysis
showed a significant difference between with- and
against-gravity directions. Across the four directions, both
0 and 45 differed significantly from 135 and 180. Upward direction movements showed significantly smaller
relative times to peak velocity than downward direction
movements (on average, 0.47, 0.49, 0.53 and 0.53 for
0, 45, 135 and 180 directions, respectively). Movements executed at normal speeds showed greater relative
times to peak velocity than those at rapid speeds (on average, 0.53 and 0.49 for normal and rapid speeds, respectively).
A two-way ANOVA (four directions ” two speeds) for
C, gave a significant main effect only for movement
speed [F1,8 = 17.61, P = 0.0030). While C-values increased with movement speed (on average, 1.56 and
1.63 for normal and rapid speeds, respectively), they remained approximately constant with movement direction
239
Fig. 9A±D Averaged and normalised velocity profiles of the
pen from all subjects. The velocity traces were normalised
using procedures described by
Soechting (1984). Pen velocity
profiles from all movement
directions are plotted for normal
and rapid speeds in A. Pen velocity profiles from both movement speeds are plotted for the
four movement directions in B.
Velocity profiles of normal and
rapid speeds (A) are replotted in
C by scaling the time of rapid
speed by a factor of 1.07. Velocities of the pen for the four
movement directions are replotted in D by scaling the time of
upward directions by a factor of
1.08
(on average, 1.62, 1.59, 1.59 and 1.59 for 0, 45, 135
and 180 directions, respectively). The interaction effects
of direction and speed did not reach significance
(P > 0.05).
Figure 9 (upper row) qualitatively illustrates the effects
of speed and movement direction upon the pen velocity
profiles, averaged and normalised for all subjects. It can
be noted that, for rapid compared with normal speeds
(Fig. 9A), peak velocity was attained earlier, and that
the two velocity profiles were different. This result has
also been confirmed by the significant effects of movement speed upon both B- and C-values. Concerning the
effects of movement direction (Fig. 9B) and consequently
the effects of GT upon velocity profiles, it may be observed that upward directions achieved peak velocity earlier than the downward directions did. Nevertheless, velocity profiles seemed to be equivalent, if upward directions are expanded in time. This can be seen in Fig. 9D
in which velocity profiles in downward and upward directions are replotted together, the latter being expanded in
time by a factor of 1.07, which corresponds to the ratio
of downward to upward acceleration time. The same scaling in time for movement speed (rapid speed is scaled by
a factor of 1.08) gave no similar velocity shapes (Fig. 9C).
The fact that velocity profiles of different directions
were similar after rescaling in time, indicates that subjects
made equivalent movements (C = constant), but they
changed their relative times to peak velocity (B). The
same cannot therefore be said of movement speed.
Changes in both B and C values indicate that subjects
made different movements under different movement
speeds.
Discussion
In the present experiment, arm drawing movements under
the influence of GF were considered in order to characterise the level and the nature of the representation of GF
within the neural motor command. The principal results
of our study enable the formulation of three conclusions.
Firstly, they suggest that GTs, which are important components of any inverse dynamic procedure for vertical
movements, are effectively incorporated into internal dynamic models. Secondly, they support the hypothesis that
gravity has not only the status of a load, acting on the
limbs centre of mass, but constitutes a reference, the basis upon which arm orientation in space, proprioception
and motor commands are referred to. Thirdly, they emphasise the idea that any models of arm trajectory formation
in the vertical plane must include the difference between
upward and downward directions, by considering gravity
as both an orientation reference and a mechanical load.
Evidence for the representation of gravitational force
in internal dynamic models
Results showed that movement speed and GTs did not affect pen paths in the four directions tested. Subjects produced straight-line paths with little variability between
240
trials. This finding suggests that mechanical effects during arm motion are taken into account before movement
onset, during inverse dynamic calculations. If subjects
were not able to correctly predict changes in GTs and interactional torques produced during arm motion, curved
paths should have been observed, at least in the early part
of pen paths. Additionally, this curvature would have
been greater for the 45 and 135 directions, a situation
where gravitational and movement vectors did not correspond.
Trajectory parameter findings can also be used to
emphasise the idea that GTs are incorporated at the inverse
dynamic stage. Subjects showed approximately the same
MTs and peak velocities for all tracing directions and
equally for both speeds tested. We feel it important to note
that the possibility of the brain containing an accurate internal model of limb dynamics cannot be considered
alone. Our findings are in contrast with results obtained
by Gordon et al. (1994b) concerning planning for limb inertia in the horizontal plane. These authors found that subjects executed movements with higher peak velocities and
with smaller MTs and ATs in certain directions compared
with others. They were able to conclude that the same
force level is planned for both directions without considering changes in arm inertia within the workspace. In our experiment, however, if the CNS had planned equivalent
hand trajectories, without correctly taking into account
changes in GTs during the motion, movements executed
with gravity should have shown smaller MTs and greater
peak velocities than movements executed against gravity.
In addition, greater ATs should also have been observed
for upward compared with downward movements.
From a mechanical perspective, both limb inertia and
GTs can be modulated by changing the configuration of
the upper limb in space (Hogan 1985). Why then would
the CNS take into account GTs and not limb inertia during planning of arm movements? Unfortunately, we cannot make direct comparisons between our results and
those of Gordons and colleagues, owing to the different
motor tasks and workspaces used between experiments.
However, we postulate that some ideas can be proposed
and discussed with the precaution that more experiments
are needed in order to better understand how inertia and
GF are represented by the brain.
One explanation could be the possibility that gravity is
more directly estimated than inertia and consequently is
more efficiently represented and updated into internal dynamic models of the upper limb. GTs can be directly measured before and after arm motion from Golgi tendons organs. In contrast, arm inertia must be indirectly calculated
from relations between force and acceleration, comparing
proprioceptive signals from muscles spindles and Golgi
tendons organs to intended motion. Another, more complementary explanation, could be the fact that gravity provides the vertical direction whatever the position of the
limb in space. This important property of GF can be used
be the CNS in order to organise different frames of reference as well as different internal dynamic models concerning movements in visual or body space. These differ-
ent properties of gravity and inertia may clarify perhaps
why the CNS represents better the former than the latter.
To summarise so far, results showed that kinematic parameters (path form, MT and peak velocity) were independent of differing movement directions, indicating that
GF and its mechanical effects on arm joints are included
in dynamic models for arm movement execution.
Differing planning with respect to gravitational force
and movement speed
An interesting point of discussion does, however, concern
the relative time to peak velocity (movement timing) of
the pen that did not remain invariant with respect to
movement direction and speed. This raises the question
as to why the CNS chooses to modify rather than using
similar movement timing for different directions and
speeds? In other words, if the CNS plans pen trajectories
before inverse dynamic computations, there would be no
obvious reason to observe variations in movement timing.
For instance, several studies made in the horizontal (Flash
and Hogan 1985; Ostry et al. 1987; Gordon et al. 1994a,
1995) and vertical (Atkeson and Hollerbach 1985) planes
have shown that velocity profiles remained invariant
throughout differing directions and speeds, suggesting
that the same planning processes can be used and adapted
for similar kinds of movements.
By considering only movement timing results, it could
have been hypothesised that GTs are misrepresented or
disregarded during inverse dynamic computations, providing evidence, firstly, against a serial sensorimotor
transformation (from kinematics to dynamics) and, secondly, against a central representation of GF. Such a hypothesis becomes unacceptable if all kinematic parameters are taken into account. Thus, a misrepresentation of
GTs would suggest significant differences in MT, as well
as in peak velocities and path profiles, related to movement direction. Such differences were not found in the
present experiment. Results also indicated greater relative
times to peak velocity for downward compared with upward movements. An opposite pattern should have been
observed if GTs were inadequately taken into account
during joint torque computations. Furthermore, our results
showed no effects of movement direction upon velocity
shape (C parameter), indicating an equivalent movement
production.
A different representation of GF during inverse kinematic and dynamic processes may explain changes in
movement timing between upward and downward directions, as well as changes in velocity profiles (both movement timing and form), between normal and rapid speeds.
Representing the constant direction of gravity during inverse kinematic transformations for movements in variable directions may explain why the CNS plans hand trajectories with different ATs and DTs. In addition, by including GTs in inverse dynamic processes, which can decelerate and accelerate upward and downward movements, respectively, we may explain why relative AT
241
was greater for downward compared with upward movements. Moreover, while a movement is executed at normal speed, GTs, which dominate inertial torque, are sufficient to initiate or brake joint motions. In contrast, however, at rapid speeds, GTs are insufficient and subjects
must increase muscular force in order to initiate and brake
upward and downward movements. This may explain
why subjects did not use a scaling strategy for movement
speed by simply scaling joint torques, as has previously
been suggested (Hollerbach and Flash 1982; Soechting
1984; Atkeson and Hollerbach 1985). Other experimental
data have also shown that velocity profile shapes change
with speed (Nagasaki 1989), duration (Gielen et al. 1985),
stimulus size (Corradini et al. 1992) and movement goal
(Marteniuk and MacKenzie 1987), suggesting that the
movement context can affect planning and control processes.
The idea of a central representation of gravity, as an
orientation reference (the vertical direction) and as a mechanical effect, is also consistent with neurophysiological
studies. Neuronal populations that encode the direction of
the movement have been found in the motor, premotor
and parietal cortex (Georgopoulos 1990; Caminiti et al.
1991), suggesting an explicit representation of the movement in terms of its direction. Besides a kinematic representation, other alternative hypotheses that cells in the
motor cortex encode both dynamics and kinematics
(Kalaska 1991) have been proposed. Using a task in
which trained monkeys performed arm movements in
eight different directions, whilst applying external loads,
these authors found that the discharge of many cells in
the motor cortex (area 4) was affected by the direction
of the applied load. Loads that pulled the arm in the opposite, rather than the preferred, direction of movement
of the cell produced a large increase in cell discharges,
while loads that pulled the arm in preferred directions reduced cell activity. GF is such a directional ªloadº, which
can accelerate (downwards) or decelerate (upwards) arm
movements and could be represented during arm motion
planning.
Furthermore, the integration of GF in the early stages
of representation and transformation of intended arm
movements has also been suggested by several authors.
For instance, Lacquaniti and Maioli (1989a, b), using a
task consisting of catching a ball released from different
heights, have found that the CNS is capable of estimating
the time course of an object undergoing constant acceleration due to gravity. In addition, psychophysical experiments (Soechting and Ross 1984; Flanders et al. 1992)
have also supported the idea that the position of the arm
in space is more highly represented in an absolute frame
of reference (angular elevation and yaw).
Conclusions
Several studies (Atkeson 1989; Lacquaniti 1993; Shadmehr and Mussa-Ivaldi 1994; Ghez et al. 1995) have already suggested the existence and significance of internal
models that include the dynamic properties of the arm and
environment. Subjects, growing up and learning to perform movements in a gravito-inertial environment, build
up motor plans by integrating GF into these models. A
fundamental difference, however, between mechanical
parameters (stiffness, viscosity, inertia, etc.) and GF, despite the fact that all are represented in dynamic models,
is that the latter (GF) is an invariant reference (vertical),
which can be represented by the CNS at different levels
of the motor process. Such a representation of GF by
the CNS facilitates sensorimotor transformations for vertical arm movements and permits the calibration of different spaces of movement representation (head-centred,
arm-centred and retinotopic). Thus, while studying the
motor control of movements with respect to gravity, it
seems important to consider GF as an important component of the motor plan and not as a mechanical element
to be simply overcome.
Appendix
Gravitational torques exerted around wrist, elbow and shoulder
joints (Tw, Te and Ts, respectively) were calculated using the following formulae:
T w ˆ ‰Rww mw gŠ
…1a†
T e ˆ ‰Rew mw gŠ ‡ ‰Ref mf gŠ
…1b†
T s ˆ ‰Rsw mw gŠ ‡ ‰Rsf mf gŠ ‡ ‰Rsu mu gŠ
…1c†
Here, Rww is a radius-vector drawn from the centre of the wrist joint
to the mass centre of the hand. Rew and Ref are radius-vectors drawn
from the centre of elbow joint to the mass centre of the hand and
forearm, respectively. Rsw, Rsf and Rsu are radius-vectors drawn
from the centre of the shoulder joint to the mass centre of the hand,
forearm and upper arm, respectively. Masses of hand, forearm and
upper arm are, respectively, mw, mf and mu and g is the vector of
free-fall acceleration. Square brackets indicate vector multiplication.
Supposing that markers are placed near the centres of respective
joints and applying the ELITE system of coordinates (axis 0-z directed upward), it is possible to replace Eq. 1 with following formulae:
Twx ˆ …CMwy ÿ MKwy †mw g
…2a†
Twy ˆ ÿ…CMwx ÿ MKwx †mwg
…2b†
Tex ˆ …CMwy ÿ MKey †mw g ‡ …CMfy ÿ MKey †mf g
…2c†
Tey ˆ ÿ…CMwx ÿ MKex †mw g ÿ …CMfx ÿ MKex †mf g
…2d†
Tsx ˆ …CMwy ÿ MKsy †mw g ‡ …CMfy ÿ MKsy †mf g
‡ …CMuy ÿ MKsy †mu g
…2e†
Tsy ˆ ÿ…CMwx ÿ MKsx †mw g ÿ …CMfx ÿ MKsx †mf g
ÿ …CMux ÿ MKsx †mu g
…2f†
Here, Twx, Twy, Tex, Tey, Tsx and Tsy are x and y components of wrist
(Tw), elbow (Te) and shoulder (Ts) gravitational torques, respectively. z components of these torques are equal to zero. MKwx, MKwy,
MKex, MKey, MKsx and MKsy are recorded positions of x and y components of wrist, elbow and shoulder markers, respectively, and g is
the absolute value of free-fall acceleration. CMwx, CMwy, CMfx,
CMfy, CMux and CMuy are x and y coordinates of mass centres of
hand, forearm and upper arm, respectively. These values were calculated as:
242
CMwx ˆ MKwx ‡ lw …MKfx ÿ MKwx †; CMwy ˆ MKwy ‡ lw …MKfy ÿ MKwy †
CMfx ˆ MKex ‡ lf …MKwx ÿ MKex †;
CMfy ˆ MKey ‡ lf …MKwy ÿ MKey †
CMux ˆ MKsx ‡ lu …MKex ÿ MKsx †;
CMuy ˆ MKsy ‡ lu …MKey ÿ MKsy †
Here MKfx and MKfy are coordinates of finger marker; lw, lf and lu
are the ratios of the distance between the proximal end of the segment and its mass centre position to the length of the segment for
hand, forearm and upper arm, respectively. The values for mw, mf,
mu, lw, lf and lu were calculated using the mean anthropometric parameters given by Winter (1990).
The total gravitational torque (TGT) applied to the centre of
mass of the upper arm was calculated by the formula
q
2 ‡ T2 :
TGT ˆ Tsx
sy
Acknowledgements This work was supported by Centre National
dEtudes Spatiales (CNES). We thank Paul Stapley for his valuable
comments.
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