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The Journal of Neuroscience, December 15, 1998, 18(24):10724-10734
Control of Grip Force When Tilting Objects: Effect of Curvature
of Grasped Surfaces and Applied Tangential Torque
Antony W.
Goodwin2,
Per
Jenmalm1, and
Roland S.
Johansson1
1 Department of Physiology, Umeå University, S-901 87 Umeå, Sweden, and 2 Department of Anatomy and Cell
Biology, University of Melbourne, Parkville, Victoria 3052, Australia
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ABSTRACT |
When we manipulate objects in everyday tasks, there are variations
in the shape of the grasped surfaces, and the loads that potentially
destabilize the grasp include time-varying linear forces and torques
tangential to the grasped surfaces. Previous studies of the control of
fingertip forces for grasp stability have dealt principally with flat
grip surfaces and linear force loads. Here, we studied the regulation
of grip force with changes in curvature of grasped surfaces and changes
in tangential torque applied by the index finger and thumb when humans
lifted an object and rotated it about the horizontal grip axis through
an angle of 65°. The curvatures of the matched pair of spherical
surfaces varied from  50 m 1 (concave with
radius 20 mm) to 200 m1 (convex with radius 5 mm).
The applied tangential torque at the orientation of 65° was varied
sixfold. Regardless of the values of curvature and end torque, grip
force and tangential torque were coordinated, increasing in parallel
throughout the tilt with an approximately linear relationship; the
slope of the line increased progressively with increasing surface
curvature. This parametric scaling of grip force was directly related
to the minimum grip force required to prevent rotational slip,
resulting in an adequate safety margin against slip in all cases. We
conclude that surface curvature parametrically influences grip force
regulation when the digits are exposed to torsional loads. Furthermore,
the sensorimotor programs that control the grip force apparently
predict the effect of the total load comprising linear forces and
tangential torques.
Key words:
object shape; grasp stability; fingertip forces; human
hand; precision grip; fingertip torque; grip force; friction; rotational slip; tangential torsional loads
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INTRODUCTION |
We use our digits to manipulate
objects with widely diverse properties in a variety of dexterous
maneuvers and readily maintain stable grasps by controlling against
accidental slips. There have been many studies of the control of grasp
stability during manipulation in which the object had flat grasp
surfaces, and the destabilizing loads primarily involved linear load
forces tangential to the grasp surfaces. Approximately 15 years
ago, it was demonstrated that when objects are lifted using a precision
grip between the tips of the index finger and thumb, subjects maintain
grasp stability by automatically changing the normal grip forces
applied to the grasp surfaces in parallel with changes in the
applied tangential load forces (Johansson and Westling, 1984 ; Westling
and Johansson, 1984). Thus, the grip force in each instance is
constrained by the active sensorimotor program to vary in proportion to
the load force. This linkage of forces, which is expressed in a variety of grasp configurations and tasks involving tangential loads of various
complexities, ensures that appropriate grip forces are applied to
prevent frictional slips, irrespective of the tangential load
(Johansson and Westling, 1988a,b; Flanagan and Wing, 1993, 1995, 1997;
Flanagan and Tresilian, 1994; Kinoshita et al., 1996; Burstedt et al.,
1997). Moreover, based on tactile information, humans adjust the ratio
between the grip and load forces to the prevailing friction to maintain
an adequate safety margin against frictional slips (Johansson and
Westling, 1984, 1987; Cadoret and Smith, 1996). In common tasks, the
loads that potentially destabilize grasp typically include torques
tangential to the grasp surfaces. For example, in a precision grip
task, tangential torques occur whenever we tilt an object around a grip
axis (line joining the fingertips) that does not pass through the
center of gravity of the object. Tangential torques can also
arise because the normal force is distributed across the skin-object
contact area rather than being focused at a point (Buss et al., 1996; Howe and Cutkosky, 1996). In a "rotational slip and hold" task in
which a single digit generated tangential torsional friction that
stabilized a disk subject to torque loading, Kinoshita et al. (1997)
showed that such a load dramatically affected the grip forces required
to prevent slip. Furthermore, to perform this particular task,
primarily designed to measure torsional friction, subjects regulated
the grip forces to the tangential torque. However, it is not known
whether subjects regulate the grip forces to torques tangential to the
grasp surfaces that develop when hand-held objects are maneuvered in
common natural tasks.
The above findings are all based on studies in which the manipulated
objects had flat grasp surfaces, but most objects that we handle in
everyday activities have grasp surfaces that are curved. We have
demonstrated recently that when the grasp is subjected to linear load
forces, surface curvature substantially influences neither used
fingertip forces nor the minimum grip force required for grasp
stability (Jenmalm et al., 1998). It is not known whether this also
applies to conditions in which the fingertip loads involve torques
tangential to the grasp surfaces. It may be that changes in curvature
influence the rotational friction more than the linear friction and
that torsional and linear fingertip loads may require different
regimens in terms of grip force control. Although the viscoelastic
fingertip will, to some extent, mold to the curvature of the surface of
an object, changes in surface curvature are likely to influence the
effective contact area that is relevant for torsional friction. With a
marked convex surface, the torque-related tangential frictional forces
are likely to be distributed over a smaller contact area than with a
flat or concave surface, and thus the required grip forces may be
substantially increased.
In this study, we first analyze how subjects control the grip forces to
match changes in tangential torque as they develop in a highly natural
task, i.e., when subjects grasp an object between the thumb and index
finger, lift it, and then rotate it about the grip axis. We then
explore possible effects of the curvature of the surface of an object
on the coordination between grip forces and torque and the extent to
which these effects reflect changes in the coefficient of rotational
friction with changes in surface curvature. To this end, we examined a
range of curvatures, extending from spherically concave grasp surfaces
(radii, 20 mm) for large digit-object contact areas to markedly convex
surfaces (radii, 5 mm) for small contact areas.
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MATERIALS AND METHODS |
Subjects
Eight right-handed healthy human volunteers (three females and
five males) ranging in age from 24 to 53 years participated in these
experiments. Informed consent was obtained from all subjects, and the
local ethics committee approved the experimental protocol. Approximately 5 min before the experiments commenced, the subjects washed their hands with soap and water. During the experiments, they
sat in an office chair with the right upper arm parallel to the trunk
and the forearm extended. A test object, resting in a holder on the
floor, had its grasp surfaces located 35 cm above the ground and ~10
cm to the right of and 20 cm anterior to the subject's right hip.
Subjects grasped the object in a precision grip using the right index
finger and thumb, lifted the object, and then tilted it through
~65° by principally using a combination of elbow flexion and radial
flexion of the wrist. The subjects could see the object and their hand
throughout the experiments.
Experimental setup
The test object had two symmetrical grasp surfaces and a 31 cm
long aluminum rod that protruded orthogonal to the axis between the
centers of the grasp surfaces (Fig.
1A). Pairs of
exchangeable matching grasp surfaces were attached to the test object.
All surfaces were spherically curved (Fig. 1B); two
pairs were concave with radii of 20 and 40 mm (curvatures, 50 and
25 m1), one was flat (curvature, 0 m1), and three were convex with radii of 20, 10, and 5 mm (curvatures, 50, 100, and 200 m1).
Surfaces were coated with silicon carbide grains (50-100 µm) covered
with a thin layer of cyanoacrylate to give a finish similar to that of
a fine grain sandpaper. When attached, the lateral edges of the
surfaces were separated by 59 mm.
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Figure 1.
Description of the task. A, Front
view of the object held in the vertical position and side view of the
tilted object. The load force is the vector sum of the forces
Fx and Fy; c.g, center of
gravity. B, Side view of the six spherically curved
grasp surfaces. C, Time traces of the major parameters
measured during the various phases of a trial in the first experiment.
Grip force is averaged for the two digits; tangential torque is shown
independently for the thumb (broken line) and index
finger (solid line); load force is averaged. The
measurements representing the static hold phase before tilting are
shown by a (1 sec period), the tilting phase is shown by
b, the static tilt phase is shown by c (1 sec period), and the slip test is shown by d. Curvature,
50 m1; target torque, 66 mNm. D, In
the second experiment, there was no slip test; instead, the object was
returned to its holder. Curvature, 100 m1; target
torque, 74 mNm.
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Each surface was attached to the object via a six-axis force-torque
sensor (Nano F/T transducer; ATI Industrial Automation, Garner, NC)
that measured forces and torques in three dimensions. Grip force was
measured along the grip axis defined by the line through the centers of
the two grasp surfaces, and two force components were measured
orthogonal to the grip axis, oriented along the long axis of the object
(Fy) or at right angles to it
(Fx). The vector sum of these components was defined
as the load force, the major component of which supported the weight of
the object. Grip forces were measured with a resolution of 0.05 N, and
Fy and Fx were measured with a resolution of
0.025 N. Torques were measured about the three orthogonal force axes,
intersecting at the center of the grasp surface, with a resolution of
0.125 milli-Newton-meters (mNm); torques about the grip axis
corresponded to the tangential torques applied to the grasp surfaces by
the contacting digits. An electromagnetic position sensor (FASTRAK;
Polhemus, Colchester, VT) was attached to the object to measure its
vertical position, its horizontal position in the anteroposterior
direction, and its angle of tilt in relation to the vertical. In the
environment of operation of the sensor, the linearity of the position
measurement was better than 0.5 mm (resolution, 0.12 mm) and the
linearity of the angle measurement better than 1° (resolution,
0.025°).
The rod extending from the object allowed exchangeable masses to be
placed 32 cm below the grip axis. The resulting center of gravity of
the object varied with the magnitude of the added mass and was between
4 and 14 cm below the grip axis when the rod was oriented vertically.
Rotating the object about the grip axis thus produced torques
tangential to the grasp surfaces. Nine different masses were used to
arrive at the following target torques when the tilt angle was 65°:
31, 44, 55, 66, 74, 88, 114, 156, and 189 mNm. The corresponding masses
of the object were 88, 93, 96, 104, 108, 111, 121, 136, and 147 gm,
respectively, and the load forces applied to support the object were
thus 0.86, 0.91, 0.94, 1.02, 1.06, 1.10, 1.19, 1.33, and 1.44 N,
respectively. The aim of the long aluminum rod was to produce large
target torques with small linear load forces and to minimize the errors
resulting from variation in the contact points between the digits and
the curved surfaces as detailed below. An increase in target torque was
accompanied by only a small change in the vertical load force on the
digits (target torque increased by a factor of 6.1, whereas the mass of
the object increased by a factor of 1.7). From the data of Kinoshita et
al. (1997), the linear force loads observed in our experiments
(generally <1 N) would have had negligible influences on grip force
requirements compared with those imposed by the torques used.
Experimental procedure
Each subject performed two experiments. In the first experiment,
a trial consisted of the following temporal sequence of events. (1) The
subject lifted the object vertically from its support through a few
centimeters and held it in a stationary position with the rod oriented
vertically. (2) After 5 sec, a 1 kHz tone signaled the subject to tilt
the object through 65° by rotating it around the grip axis; the tone
ceased when the target angle was reached. For additional guidance, the
target angle was indicated by a line drawn on a board located in a
sagittal plane ~10 cm to the right of the subject's right arm. (3)
The object was held in the tilted position for 4 sec; this phase was
termed the static tilt phase. (4) Then, an auditory cue (1 kHz tone for
0.1 sec) signaled commencement of a rotary "slip test" in which the
subject decreased the grip force slowly and allowed the object to
rotate smoothly back to a vertical orientation, under the influence of gravity. (5) After this slip test, the subject replaced the object in
its support. During phase 3, the subject was required to maintain the
tilt angle between 60 and 70° for at least 4 sec. If these bounds
were exceeded, which happened occasionally, an auditory cue indicated
that a correction was required, and the 4 sec static tilt interval was
reset. Each trial lasted for ~20 sec, and a break of ~10 sec was
left between trials. A test series comprised 24 trials and consisted of
six blocks of four trials each. In each block, one of the six surface
curvatures was used, and in the four successive trials, four selected
exchangeable masses were used, resulting in four different nominal
target torques. The six surfaces and the four torques for each surface
were presented in a random order. Three such series were run on each
subject. Table 1 shows the four target
torques used for each of the six curvatures. As seen in Table 1, the
tangential torques during phase 3 varied slightly from trial to trial
because of variation in the tilt angle and in the location of the
contact points; in the remainder of the paper, target torques are given
as the nominal values (that would have resulted from the nominal tilt
of 65°). The selection of target torques for each pair of surface
curvatures was such that the range of grip forces used was comparable
for all surfaces, as determined in pilot experiments.
The second experiment comprised a single series of 39 trials consisting
of 13 blocks of three trials each. A single mass was used corresponding
to a target torque of 74 mNm. In the odd blocks, the grasped surfaces
were flat (curvature, 0 m1) for all three trials,
and in the even blocks, the curvature was 100 m1.
In this series, the subject initially lifted and tilted the object in
an identical manner to phases 1-3 in the first experiment. Then, in
phase 4, after maintaining the tilt for 4 sec, an auditory cue
signaled the subject to rotate the object back to the vertical orientation (instead of performing the rotational slip test) by a
combined elbow extension and ulnar flexion of wrist. In phase 5, the
object was held in this position for another 5 sec after which, after
an auditory cue, the object was replaced in its support.
Before the experiments, each subject was specifically instructed to
grasp the surfaces at their centers, and the experimenter pointed out
the center of one convex surface and one concave surface. After the
experimenter had demonstrated the task, the subject performed five
practice trials using the flat surfaces with a mass resulting in a
target torque of 66 mNm. The subjects performed the tilting movement at
their preferred speed. During all experiments, the subject was observed
carefully to ensure compliance with instructions.
Data collection and analysis
All data were digitized using a flexible data acquisition and
analysis system (SC/ZOOM; Department of Physiology, Umeå University, Umeå, Sweden) and stored on a personal computer. Signals were recorded
with a resolution of 12-bits (400 samples/sec) from the force-torque
sensors and 14-bits (120 samples/sec) from the position-angle sensor.
If the only tilt of the object were that produced by rotation about the
grip axis (as was requested of the subjects), then the grip force would
have been the same for the thumb as for the index finger. Observation
of our subjects revealed no obvious violation of this instruction; over
all trials and all subjects, the ratio of the two grip forces during
phase 3 above was 1.01 ± 0.01 (mean ± SD; n = 888). Thus, in Results, we present data for the average grip
force of the two digits. Tangential torques are either specified for
each digit independently or are given as the total torque (sum for the
two digits) as appropriate.
Time traces of the measured parameters are illustrated for the
various phases of the task in Figure 1C. For each trial, the grip forces, tangential torques, linear load forces, and tilt angle
were measured as their mean values during a 1 sec interval when the
object was held steady before the tilting movement (Fig. 1C,
a). This interval commenced 4 sec after the object was first contacted (defined by the time at which the sum of the grip forces for
the two digits first exceeded 0.3 N). Mean values of the same signals
were also calculated during a 1 sec interval, located in the middle of
the 4 sec static tilt phase (Fig. 1C, c).
During the tilt phase (Fig. 1C, b), grip force
measurements were taken at times when the total tangential torque at
the two digits had increased by 10, 50, and 90% of the increase in
total tangential torque that occurred from the period before tilt (Fig. 1C, a) to the period when the object was
maintained tilted (Fig. 1C, c). In addition, the
maximum grip force was measured as the peak value within ±1 sec after
the end of the tilt phase. Angular velocity and angular acceleration
were assessed from the first and second time derivatives of the tilt
angle, respectively, using a ±6 point numerical differentiation (±50
msec window). The rise time of the angle during the tilting movement
(Fig. 1C, b) was taken as the interval from 10 to
90% of the increase in tilt angle that occurred from phase
a to phase c in Figure 1C.
For each trial in the first experiment, the relationship between
grip force and tangential torque over the window of rotational slip
(Fig. 1C, d) was established independently for
each digit by linear regression. This window was determined off-line by
visual inspection of the time traces of tilt angle, grip force, and
tangential torque and by inspection of the corresponding force-torque
plots. The regression constants and the measured tangential torques
were used to estimate, for each digit, the minimum grip forces required to prevent slip (slip forces) during the static tilt phase. The safety
margin was calculated as the difference between the static grip force
and the larger of the slip forces for the thumb and index finger.
Grasp points and errors in torque measurements. Subjects
were instructed to grasp the surfaces of the test object at their centers. Although they performed remarkably well in this respect, there
were small deviations that introduced errors in the target torque, as
well as in our estimates of the true torques, applied by the digits.
First, subjects tended to displace the grip axis toward the center of
mass of the object, decreasing the effective length of the lever arm by
~2 mm and thereby decreasing the target torque (at a tilt angle of
65°) in relation to its nominal value (cf. Table 1). Second, if the
grasp points deviated from the axis intersecting the centers of the
grasp surfaces, the torque measured about this axis could have differed
from the true torques at the fingertips. That is, linear load forces
(orthogonal to this axis) could have contributed to the measured
torque. These linear forces resulted from the vertical forces applied
to counterbalance the weight of the object and from "internal"
forces applied to prevent the object from spinning if the grasp points
were located asymmetrically on the grasp surfaces. The linear forces
were measured and were only of the order of 1 N. Moreover, when the
object was held tilted, the deviation of the equivalent point of
contact between the digit and the surface, from the center of the
surface in the direction along the lever arm of the object, was
3.2 ± 2.3 mm for the thumb and 2.0 ± 2.1 mm for the index
finger (mean ± SD; data pooled across all trials; location of
contact points calculated as in Kinoshita et al., 1997). In the
direction orthogonal to the lever arm, the deviation was 0.0 ± 1.9 mm for the thumb and 1.7 ± 2.1 mm for the index finger. From
these deviations and the magnitude of linear forces, we can safely
conclude that the errors in torque measurements were small in relation
to the magnitudes of the target torques. In fact, they were of the same
order of magnitude as variations in torque resulting from variations in the effective length of the lever arm.
Statistical analysis. Pearson product-moment correlations
and repeated-measures ANOVA were used to evaluate the influence of
tangential torque, surface curvature, and digit; details are provided
in Results. The level of probability selected as statistically significant was p < 0.05. All reported correlation
coefficients were significant. Unless otherwise stated, where data were
pooled across subjects the following procedure was used. For each
subject, the trials in which the experimental parameters (target torque and surface curvature) were identical were combined, providing a
subject mean for each measurement. Values for these "average trials" were used in the ANOVA analyses, and in many figures subject means and SEM (n = 8) are presented.
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RESULTS |
The results are presented in three sections. In the first, we
characterize the task based on data for which the grasp surfaces were
flat and the target torque was held constant. In the second, we explore
the effects of changing target torque for flat surfaces. Finally, we
analyze the effects of changing both the curvature of the grasped
surfaces and the target torque. We stress that the grasp behavior
observed and described here emerged automatically in the sense that
subjects were not given any instructions pertaining to grip force.
Force-torque coordination with flat parallel grasp surfaces
When the subject tilted the test object by flexing the elbow and
wrist, the resulting increase in tangential torque at the digits was
accompanied by an increase in the grip force with no apparent time lag.
In Figure 2A, the
subject tilted the object at a moderate speed. There was a continuous
increase in the torque at each digit accompanied by a coordinated
parallel increase in the grip force. Figure 2E, which
shows the grip force plotted against the tangential torque at each
digit, emphasizes their coordination throughout the tilting movement.
In Figure 2, B and F, the subject tilted the
object rapidly (see angular velocity in B)
and the torques generating the tilt showed a more complex profile.
During the initial phase of the tilt, the grip force increased in
parallel with the applied torques until they reached their initial
peaks associated with maximum angular acceleration of the object (Fig.
2B,F, arrowheads). After
the peak angular velocity of the object, its angular deceleration was
caused by a marked decrease in the torque drive, with minimum values
close to zero. Despite this decrease in torque, the grip force
continued to increase, and when the "ringing" in torque had ceased,
the grip force had reached a value close to that maintained during the
subsequent static tilt phase. That is, the grip force was coordinated
to the overall increase in torque but did not change with the rapid
torque fluctuations during the tilting movement. Accordingly, the ratio
of grip force to tangential torque decreased markedly during the
initial torque peak and then increased during the subsequent dip in
torque. Interestingly, if rotational slips occurred at the minimum
force-to-torque ratio (Fig. 2F,
arrowhead), they would have occurred when the object had a
high positive angular velocity so that during such a slip, the tilting
movement would have continued.
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Figure 2.
Coordination of force and torque illustrated for
selected single trials in experiment 2. A-D, Time
traces showing (from top to bottom) the
average grip force, the total torque on the digits (solid
line), the torque predicted from regression (thin
line), the torque on each digit, the angle of tilt, and the
first and second time derivative of the tilt angle.
E-H, Plots of grip force versus torque for each digit.
In A and E, the object was tilted slowly
(rise time, 0.90 sec). In B and F, the
tilt was more rapid (rise time, 0.41 sec). In C and
G, the object was returned to the vertical slowly (fall
time, 1.11 sec). In D and H, the return
was more rapid (fall time, 0.38 sec). The regression predicting torque
had correlation coefficients of 0.997, 0.996, 0.998, and 0.996, in
A, B, C, and
D, respectively. Target torque was 74 mNm, and grasp
surfaces were flat.
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When the object was returned at a moderate speed to its original
vertical orientation (second experiment), the grip force and torques
decreased smoothly and in parallel throughout the replacement phase
(Fig. 2C,G). In trials with a rapid rotation of
the object back to the vertical position (Fig.
2D,H), the angular movement
was initiated by a sharp decrease in tangential torques at the digits,
lasting until the angular deceleration of the object was maximal. This
decrease was coupled with a parallel decrease in grip force. Then, the
subject increased the torques to brake the angular deceleration,
whereas the grip force continued to decrease. After the trough in
torque terminated by a local peak (Fig.
2D,H, arrowheads), the
torques and grip force decreased together until the object was oriented
vertically. Thus, during the replacement of the object, there could
also be a temporary decoupling of grip force and torques, but the grip
force was coordinated with the overall change in torque. Interestingly,
as judged from the ratio of grip force to tangential torque, potential
rotational slips would have occurred close to the minimum angular
velocity and not in the final phase of the replacement (Fig.
2D,H, arrowheads).
For object rotation at any speed, the trajectory of the tilt angle was
close to critically damped. A simple mechanical model that predicted
the time-varying torque from the motion of the object verified that the
trajectory of the tilt angle resulted from the torque dynamics. We used
the following linear regression model: T = a + b sin  + c ", where T is the
total torque (sum of the torque at each digit); a,
b, and c are regression coefficients; is the
tilt angle; and " is the angular acceleration. In mechanical terms, the expressions sin and " account for the static torque and the moment of inertia of the object, respectively. Torques resulting from linear acceleration of off-axis mass in the vertical and
anteroposterior directions were negligible because of the relatively
small accelerations. Likewise, torques resulting from off-axis load
forces (grasp points not at the centers of the surfaces) were
negligible and did not improve the regression significantly. In Figure
2, A-D, the time traces representing the total torque on
the digits show the recorded torques (solid lines) and
the torques predicted from the regression (thin lines)
for representative examples of single trials. Because the model is a
direct consequence of Newton's Laws of Motion, the fit was very good
with coefficients of correlation close to unity (i.e., >0.99 for all trials).
The speed of the tilting movement varied between the first and the
second experiments; in the first experiment, each trial was terminated
by the slip test, whereas in the second experiment, the subject rotated
the object back to the vertical orientation in an ordinary manner. The
rise times of the tilt angle in the first experiment were 1.05 ± 0.41 (mean ± SD; n = 96) and in the second
0.61 ± 0.22 (n = 168; p  0.001;
Mann-Whitney U test). The example in Figure 2,
A and E, illustrates a trial in which the rise
time of the tilt was representative of the trials used by our subjects
in the first experiment, and Figure 2, B and F, illustrates one of the fastest trials that was observed in the second
experiment. The fall times in the second experiment were 0.76 ± 0.37 (n = 312).
Effect of target torque
In the first experiment, four different exchangeable masses were
used, with the flat grasp surfaces resulting in four different target
torques. The effect of different torques on the coordination between
grip force and tangential torque is illustrated for single trials in a
single subject in Figure 3, A
and B. An increase in target torque resulted in an increase
in grip force and, for all four torques, the instantaneous grip force
was coordinated to the instantaneous torque throughout the tilting
movement. The trajectories for the higher torques in Figure
3B overlie the trajectories for the lower torques,
indicating a consistent force-torque relationship, regardless of final
torque. This pattern was consistent for all subjects and is illustrated
for an additional three subjects in Figure 3, C-E. A second
feature seen in Figure 3A is that the rise time of the
tilting movement did not vary with changes in the target torque (see
the consistency of the angle traces). This was true for each subject,
but the mean rise time varied between individuals (range of mean
values, 0.7-1.8 sec). Accordingly, there was no statistically
significant effect of target torque on rise time
(p = 0.246).
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Figure 3.
Effect of target torque on force-torque
coordination with flat grasp surfaces. A, Time traces
for average grip force and total tangential torque on the digits, the
angle of tilt, and the first time derivative of the tilt angle during
four trials for subject 1. The four trials were aligned at the time
when tilt angle was 10°. B, Force-torque plots of the
same data as in A. C-E, For three other
subjects, the force-torque coordination is shown for four single
trials with the four target torques (66, 114, 156, and 189 mNm)
indicated by the arrows.
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Safety margin preventing rotational slips
When the object was tilted toward the target orientation of 65°
and held tilted in this position, there were no obvious accidental rotational slips between the object and the fingertips. Apparently, the
grip force used was greater, by some safety margin, than the minimum
force required to prevent slip, termed the slip force. The slip force
is determined by the tangential torque, the coefficient of rotational
friction, the linear load force, the coefficient of linear friction,
and to some extent by interactions between the effects of torque and
load force (Kinoshita et al., 1997). In the present experiments, the
linear load forces were relatively small and would have contributed
little to the slip force when the object was held tilted (cf. Kinoshita
et al., 1997).
Rotational slip coefficient
A rotational slip coefficient for each digit was determined as
follows. At the end of the static tilt phase, the subject reduced the
grip force slowly, allowing the object to undergo slow rotational slip
back to the vertical orientation. During the slip phase, indicated in
Figure 4A,
vertical lines, the force and torques decreased in
parallel and, as seen from the angle trace, the object slipped smoothly
back to the vertical orientation. For both digits, the relationship
between grip force and tangential torque was linear during the slip
phase (Fig. 4B,C). The slope of the
regression line is termed the rotational slip coefficient and
corresponds to the inverse of the coefficient of rotational friction
used by Kinoshita et al. (1997). The fact that force and torque follow the same straight line throughout the slip phase shows that the rotational slip coefficient is independent of tangential torque. All
subjects showed linear relationships comparable to those in Figure 4
for all four target torques with the flat grasp surfaces. The 192 (4 target torques × 3 repetitions × 2 digits × 8 subjects) correlation coefficients ranged from 0.969 to 0.999. Although ANOVA showed that there was no significant difference between the
coefficients for the two digits (p = 0.08) and
no significant effect of target torque (p = 0.16), we cannot exclude a modest difference in friction for the digits
(Kinoshita et al., 1997).
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Figure 4.
Rotational slip illustrated for a single trial in
a single subject with flat surfaces and a target torque of 66 mNm.
Thin vertical lines demarcate the slip phase.
A, Time trace of average grip force, tangential torque
on the thumb (broken line) and index finger
(solid line), and tilt angle. B,
Force-torque plot (thick line) and linear regressions
(thin line) during slip for the index finger.
Correlation coefficient is 0.999, slope is 0.185, and intercept is
0.134. C, Force-torque plot (broken
line) and linear regressions (thin line) during
slip for the thumb. Correlation coefficient is 0.999, slope is 0.171, and intercept is 0.333.
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The rotational slip coefficients were measured during a phase of
continuous slip in which the operating frictional coefficient would be
a dynamic coefficient of rotational friction. Two factors indicate that
in our experiments there was little difference between the static and
dynamic coefficients of rotational friction. First, as grip force
decreased to the slip point, the curve in the force-torque plot (Fig.
4B,C) at the onset of slip never
dipped below the linear relationship, as would have occurred if the
static rotational friction had exceeded the dynamic friction. Second,
on some occasions, some of our subjects showed distinct slip-stick
behavior during the slip test, but the relationship between the grip
force and torque did not shift appreciably between the slip and stick phases.
Safety margins
The safety margin in grip force during the static tilt phase, when
the object was held at an orientation of 65°, was estimated in each
trial as follows. The slip force at each digit was estimated from the
regression constants of the force-torque relationships obtained as
described above and from the value of the static tangential torque. The
digit that showed the greater of the two slip forces was termed the
critical digit; rotational slip would have occurred if the grip force
had been less than this slip force. The safety margin was defined as
the difference between the grip force used by the subject and the slip
force of the critical digit. In Figure 5,
the grip and critical forces are shown for the four target torques.
Safety margins, shown in Figure 5, shaded area, were approximately invariant with target torque (p = 0.066). It is also evident that, on average, the slip force on the
noncritical digit was smaller than that of the critical digit.
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Figure 5.
Safety margin in grip force during the static tilt
phase for flat grasp surfaces. Data averaged over the eight subjects.
Solid lines show grip force (mean + SEM) and critical
force (slip force on the critical digit; mean SEM).
Broken line shows the slip force on the noncritical
digit. Shaded area shows the safety margin (difference
between grip force and critical force). Target torques were 66, 114, 156, and 189 mNm.
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In 53% of trials, the thumb was the critical digit. Partitioning of
torque between the two digits was the major factor determining which
digit was critical, because the rotational slip coefficients were
similar, as indicated above. In the trials exemplified in Figure 2, the
torques on the two digits were approximately the same, but the
partitioning of torque varied widely. The ratio of torque on the thumb
to torque on the index finger in the static tilt phase had a mean of
1.17 and a SD of 0.64. In 80 of 96 cases, the digit with the
higher static torque was the critical digit. In the remaining 16 cases,
the ratio of torques on the two digits was close to 1, in which case
the slight difference between the rotational slip coefficient for the
two digits determined the critical digit.
Force-torque coordination with curved surfaces
The grip force used by the subjects changed markedly with changes
in the curvature of the grasp surfaces. Figure
6, A and B,
illustrates four single trials for one subject tilting the object with
four different curvatures, all to the same target torque. For all
curvatures, the grip force and the tangential torque increased in
parallel throughout the tilt (Fig. 6B), but a change
in surface curvature changed the balance between grip force and
tangential torque throughout the trial. This parametric effect on the
force-torque coordination was already evident at the onset of the
tilting movement. The force-torque coordination is shown for three
other subjects in Figure 6, C-E. Changes in surface
curvature had weak effects on the rise time of the tilt angle, which
increased with increasing surface curvature. For a target torque of 66 mNm, the rise times at 50 m1 and 200 m1 were 0.94 ± 0.39 and 1.20 ± 0.46 sec, respectively (mean ± SD; n = 24;
p = 0.001).
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Figure 6.
Coordination of mean grip force and total
tangential torque with changes in surface curvature. A,
Four single trials for one subject with a target torque of 66 mNm;
superimposed trials were aligned at the time when tilt angle was 10°.
The data illustrate the most concave surface (curvature, 50
m1), the flat surface (0 m1),
and the two most convex surfaces (100 and 200 m1).
B, Force-torque plots of the same data as in
A. C-E, Comparable data for three
additional subjects.
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The coordination of force and torque during the tilt phase was examined
more closely by plotting the grip force at five successive time points
during the tilt (Fig. 7A). The
first three points were the times at which the total torque at the two
digits had increased by 10, 50, and 90% of the increase in total
torque that occurred during the tilting movement. The remaining two
torques were the torque at peak grip force and the torque during the
static tilt phase. For all six curvatures, the grip force increased in parallel with the tangential torque up to the peak torque, and the two
variables had an approximately linear relationship. Throughout the
tilt, an increase in curvature scaled the ratio of grip force to
tangential torque upward; that is, the higher the surface curvature, the higher the grip force at any given tangential torque. As early in
the trial as the point at which torque was 10% of the static value,
there was a significant effect of surface curvature
(p < 0.001).
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Figure 7.
Effect of surface curvature and
tangential torque on grip force. A, Coordination shown
for time points during the tilt in which total tangential torque had
increased to 10, 50, or 90% of the static tilt phase value, the torque
at peak force, and the static torque. Average for all subjects
(n = 8), with a target torque of 66 mNm. Parameters
show surface curvature in units m1.
B, Mean + SEM grip force during the static tilt phase
for the eight subjects. For each of the six surface curvatures
(indicated by the parameters), there were four target torques (see
Table 1 for details). C, For each of the eight target
torques (values in milli-Newton-meters shown by the
parameters), the number of curvatures tested varied from one (at
31 and 55 mNm) to six (at 66 mNm). Same data as in B
with the abscissa and parameter interchanged.
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Grip forces during the static tilt phase pooled for all eight subjects
are shown in Figure 7, B and C, for all six
surfaces; for each surface, four different levels of target torque were used (Table 1). Grip force was scaled by both target torque and surface
curvature. Curvature significantly affected the grip force at the
target torque of 66 mNm, which was common for all curvatures (p < 0.001). The effect on grip force was
significant for both curvature and target torque in the block indicated
by bold entries in Table 1 (curvature, 50, 25, 0, and 50 m1; torque, 66, 114, and 156 mNm;
p < 0.001 in both instances). Because the curvature
influenced the slope of the relationship between grip force and
tangential torque, the interaction term of these two factors was also
significant (p < 0.001).
Effect of surface curvature on safety margin
Rotational slip coefficients for curved surfaces. For
each pair of curved surfaces, the relationship between grip force and tangential torque during the slip phase of each trial was examined independently for the two digits. In all cases, the relationship was
highly linear as in the illustration for the flat surfaces in Figure 4.
The 1152 correlation coefficients ranged from 0.906 to 1.00 (mean ± SD; 0.993 ± 0.009). The rotational slip coefficient, given by
the slope of the regression line, increased with increasing convexity
of the surfaces (Fig.
8A). There was some
variation between subjects in the magnitude of the coefficients,
particularly for the highly curved convex surfaces. Rotational slip
coefficients for all six curvatures and for all four target torques
used at each curvature are shown for the index finger and thumb in
Figure 8, B and C, respectively (data averaged
across subjects). Coefficients were smaller for the thumb than for the
index finger, particularly for the most concave and convex surfaces.
Torque did not affect the slip coefficient (Fig.
8B,C). Two-way ANOVA showed a
significant effect on rotational slip coefficient of curvature
(p < 0.001) and digit (p = 0.01).
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Figure 8.
Rotational slip coefficients for the six curved
surfaces. A, For the target torque of 66 mNm,
coefficients for the index finger are shown for each of the eight
subjects. B, At each curvature, coefficients (mean over
eight subjects) are shown for the index finger for each of the four
target torques used. Note the overlap in data points for the four
torques. Lines join points at common torques; for
example, there are only two torques common to the surfaces at 100 and
200 m1. C, As for B,
except for the thumb. Details of the forces and torques are given in
Table 1.
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Safety margin. This was defined as the difference between
the grip force used by the subject during the static tilt phase and the
corresponding slip force for the critical digit, as defined above. The
safety margin is shown as a function of curvature in Figure
9A, hatched
area, for a target torque of 66 mNm. Safety margin increased
as the curvature of the surfaces increased (p < 0.001). In agreement with the data for flat grasp surfaces, at all
curvatures the slip force on the noncritical digit was, on average,
less than the slip force on the critical digit, primarily as a result
of the unequal partitioning of the torque. Absolute grip force safety
margins and relative safety margins (safety margins as fraction of grip
force) are shown for all curvatures and torques in Figure 9,
B and C. In broad terms, absolute safety margin
is more dependent on surface curvature than on target torque, whereas
relative safety margin is more dependent on torque than on curvature.
ANOVA indicated a reliable effect of curvature on absolute safety
margin at a target torque of 66 mNm (p < 0.001) and of target torque on relative safety margin for the flat surface (p < 0.001). For the block (curvature, 50,
25, 0, and 50 m1; torque, 66, 114, and 156 mNm;
see Table 1, bold entries), the only significant effect was for target
torque on relative safety margin.
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Figure 9.
Safety margins used to prevent
rotational slip. Data averaged over the eight subjects.
A, Solid lines show grip force (mean + SEM; n = 8) and critical force (slip force on the
critical digit; mean SEM) for a target torque of 66 mNm.
Shaded area shows safety margin (difference between grip
force and critical force), and broken line shows slip
force on the noncritical digit. B, Absolute safety
margin (mean + SEM; n = 8) for all six curvatures
(values in units m1 indicated by the
parameters). For each pair of surfaces, four target torques were used.
C, Relative safety margins (absolute safety margin
divided by grip force) for the six curvatures, each at four
torques. Note the overlap in data points for the various surface
curvatures. For B and C, the combinations
of curvatures and torques are given in Table 1.
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Effect of previous surface curvature. A major reason for
including experiment 2 in which the blocks (of three trials) alternated between surface curvatures of 0 and 100 m1, all at
a target torque of 74 mNm, was to examine whether coordination of grip
force was influenced by the surface curvature used in the previous
trial. For both surfaces, the test series included twelve trials for
which the preceding tilt occurred with the same curvature (condition 1)
and six trials for which the preceding tilt occurred with a different
surface curvature (condition 2). To study the effect of the previous
surface on force coordination, we compared five measures of grip force
for the two conditions for both surfaces. The measures were the grip
force at points at which the total torque at the two digits had
increased by 10, 50, and 90% of the increase in total torque that
occurred during the tilting movement, the peak grip force, and the
static grip force (compare with Fig. 7A). There was no
statistically significant effect of previous surface (i.e., no effect
of "condition") on any of these measures for either of the two surfaces.
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DISCUSSION |
In the experiments reported here, we used a tilting task that is
similar to natural maneuvers we use commonly. For such maneuvers, there
may be large variations in tangential torques at the fingertips. We
have shown that the sensorimotor system controlling grasp stability effectively adjusts the grip forces to match the demands imposed by
varying torsional loads. Furthermore, we found that the required grip
force was highly dependent on the curvature of the surface of the
object and that this too was efficiently compensated for.
Parallel coordination of grip force and tangential torque
Throughout the tilt, tangential torque and grip force changed in
parallel, with an approximately linear relationship. Thus, accidental
rotational slips were prevented by the grip force increasing in
parallel with the tangential torque. This coordination between grip
force and torque resembles the coordination between grip force and
linear load force that constitutes the basic strategy for the
prevention of accidental linear slips (see introductory remarks). In
particular, it resembles the coordination observed by Flanagan and Wing
(1993) in which subjects transported a hand-held object so that
variations in load force arose from inertial forces related to the
movements. Indeed, this coordination appears to be an integral part of
the process of planning arm movements and is also expressed under other
forms of load, e.g., viscous, elastic, and composite loads (Flanagan
and Wing, 1997).
The present study reveals that the sensorimotor programs controlling
manipulative actions account not only for linear loads as described
previously but also for torsional loads. Apparently, these programs
model the effect of the total load comprising linear forces and
tangential torques. However, there were brief moments during which this
coupling was violated. These occurred when the subject generated fast
fluctuations in torque to produce smooth angular movements during rapid
rotations of the object. Such decoupling was expected, because it is
known that rapid self-paced changes in linear load forces are not
paralleled by correspondingly rapid changes in grip force (Flanagan and
Tresilian 1994).
Parametric adjustment of the relationship between torque and grip
force with changes in surface curvature
The balance between the grip force and the tangential torque was
influenced by the curvature of the grasped surfaces throughout the
trial; the grip force at any given torque increased (parametrically) with increasing curvature, resulting in an adequate safety margin against rotational slips, regardless of curvature. The magnitude of
this safety margin (generally, ~30-40% of the grip force) is comparable to the safety margin against linear slips found in vertical
lifting tasks. The safety margin is also comparable to that found
against rotational slips by Kinoshita et al. (1997) during the hold
phase of their "rotational slip and hold" single digit task with
flat grasp surfaces. Thus, curvature influenced the ratio of grip force
to torque in a manner similar to the influence of the friction at the
digit-object interface (Kinoshita et al., 1997). Furthermore, the
changes in coordination of grip force and tangential load with changes
in the properties of the object also resemble those that take place
when objects are lifted vertically, subjecting the digits principally
to linear force loads; the adopted grip-to-load force ratio is adjusted
to the frictional condition (Johansson and Westling, 1984) and the
shape of the object (Jenmalm and Johansson, 1997), such that an
adequate grip force safety margin against slips is achieved.
Interestingly, in a previous study, the curvature of the grasped
surfaces had only a small effect on the grip force in tasks in which
the object was lifted vertically, inducing primarily linear force loads
(Jenmalm et al., 1998). Despite the small changes in grip force,
subjects still maintained an adequate safety margin against slips. This is explained by the fact that the curvature has only a small effect on
grip forces required to prevent slips under linear loads, in contrast
to the large effect during torsional loads demonstrated in the current study.
Rotational friction
The coefficients of rotational friction measured by us for the
flat surfaces are consistent with those measured by Kinoshita et al.
(1997). (Note that the rotational slip coefficient reported in Results
corresponds to the inverse of the coefficient of rotational friction.)
In general terms, the contact mechanics that account for friction with
loads comprising tangential torques, linear forces, and combinations
thereof, is poorly understood. This applies to soft artificial fingers
(Buss et al., 1996; Howe and Cutkosky, 1996) and even more so to human
fingertips (El-Shimi, 1977; Han et al., 1996). In the latter
case, the distribution of normal and shear stress within the contact
area is unknown, even when a digit contacts a flat surface. With curved
surfaces, the frictional limits reflect additional factors related to
the complex mechanical contact between the surfaces and the digits.
Indeed, the coefficient of rotational friction was influenced by the
surface curvature; it decreased with an increase in curvature. However,
there was a pronounced variability across subjects with regard to the
effect of curvature. This variability is probably attributable
to differences in digit mechanics, including anatomical and
biomechanical factors such as nonlinear anisotropic elastic compliance,
sudomotor activity, and degree of greasiness and hydration of the skin
(Jenmalm et al., 1998). Unfortunately, current models of human
fingertip mechanics are not yet sophisticated enough to allow a
synthesis of such data for comparison with ours (cf. Srinivasan and
Dandekar, 1996).
The coefficients measured by us were similar for the index finger and
the thumb when the flatter surfaces were used but were slightly greater
for the index finger when the more curved surfaces were used. This is
probably attributable to differences in the nature of the contact for
the two digits, but how factors like contact area would affect the
coefficient of friction is not clear. A more pronounced consequence of
differing contact was the partitioning of torque between the two
digits. There was a wide range of torque ratios for the two digits, and
the digit with the greater torque was usually the critical digit from
the point of view of motor control and grasp stability.
Control mechanisms adapting force and torque output to the mass
distribution and surface curvature of the object
For individual subjects, the rise time of the tilting movement was
nearly constant across all trials, although the mass distribution of
the object, and thus the target torque, varied between trials. The
temporal development of torque and the coupled development of grip
force was thus scaled parametrically by the mass distribution of the
object to generate an angular trajectory that was approximately constant. The intended angular position was reached in a smooth and
apparently critically damped manner, with an angular velocity profile
that was essentially unimodal (Figs. 2A-D,
3A, 6A). This behavior indicates that the
motor output during the tilt reflected predictive control in which both
the mass distribution and the target angle of the object were taken
into account. The angular velocity profile resembled the
"continuous" (Brooks, 1984) or "bell-shaped" (Bizzi and Abend,
1983) velocity profiles frequently reported for programmed intended arm
and hand movements toward a target. Furthermore, the generation of
complex torque waveforms during rapid rotations to obtain a smooth
tilting movement in our task indeed indicates that the CNS used
a refined internal model of the entire task dynamics. In general terms,
there is abundant evidence that the CNS utilizes internal models of
relevant limb mechanics, environmental objects, and task properties to structure the motor commands appropriately before their execution (Johansson and Westling, 1988b; Ghez et al., 1991; Johansson and Cole,
1992; Lacquaniti, 1992; Johansson, 1996; Miall and Wolpert, 1996;
Flanagan and Wing, 1997).
When the mass distribution of the object changed, the appropriate
modification of the torque and grip force was present from the
beginning of the tilt (Fig. 3A). This implies that subjects used information related to the mass distribution of the object from
the onset of the tilting movement. This information must have been
obtained at the beginning of each trial, because the mass distribution,
and therefore the target torque, was always changed between trials. One
obvious cue in this respect was object weight, which covaried with the
mass distribution. However, we cannot exclude the use of small
"probing tilts" in the hold phase before the tilting movement. It
is also possible that visual cues obtained by observing the
interchangeable mass of the object could have provided cues on the
current mass distribution. It has been shown that in lifting tasks,
visual cues related to the mass of the object are used for forward
parametric control of the force output (Gordon et al., 1991, 1993).
Similarly, when the surface curvature was changed, the appropriate
modification of the grip force was present from the beginning of the
tilt (Fig. 6A-E). There was no effect on the grip
force of the surface curvature used in the previous trial. Thus, the subjects must have acquired information about surface curvature before
the tilting movement. The curvature could have been determined from
cutaneous afferent information obtained during the initial contact with
the surfaces (Goodwin et al., 1995), in a manner similar to the
acquisition of frictional information by tactile input during initial
contact with a grasped surface (Johansson and Westling, 1984, 1987). It
is also possible that subjects used visual cues. In lifting tasks in
which the shape of the object changed, visual geometric cues modulated
the force coordination in a feed-forward manner (Jenmalm and Johansson,
1997).
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FOOTNOTES |
Received Aug. 3, 1998; revised Sept. 30, 1998; accepted Oct. 2, 1998.
This study was supported by Swedish Medical Research Council Project
08667, Department of Naval Research Grant N00014-92-J-1919, and the
Göran Gustafsson Foundation for Research in Natural Sciences and
Medicine. Antony W. Goodwin was supported by the University of
Melbourne. We thank Dr. G. Westling and L. Näslund for technical support.
Correspondence should be addressed to A. W. Goodwin, Department of
Anatomy and Cell Biology, University of Melbourne, Parkville, Victoria
3052, Australia.
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April 1, 2006;
95(4):
2513 - 2529.
[Abstract]
[Full Text]
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P. A. Chouinard, G. Leonard, and T. Paus
Role of the Primary Motor and Dorsal Premotor Cortices in the Anticipation of Forces during Object Lifting
J. Neurosci.,
March 2, 2005;
25(9):
2277 - 2284.
[Abstract]
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M. A. Smith and J. F. Soechting
Modulation of Grasping Forces During Object Transport
J Neurophysiol,
January 1, 2005;
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137 - 145.
[Abstract]
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H. E. Wheat, L. M. Salo, and A. W. Goodwin
Human Ability to Scale and Discriminate Forces Typical of Those Occurring during Grasp and Manipulation
J. Neurosci.,
March 31, 2004;
24(13):
3394 - 3401.
[Abstract]
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A.-S. Augurelle, A. M. Smith, T. Lejeune, and J.-L. Thonnard
Importance of Cutaneous Feedback in Maintaining a Secure Grip During Manipulation of Hand-Held Objects
J Neurophysiol,
February 1, 2003;
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665 - 671.
[Abstract]
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Y. Ohki, B. B. Edin, and R. S. Johansson
Predictions Specify Reactive Control of Individual Digits in Manipulation
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January 15, 2002;
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600 - 610.
[Abstract]
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I. Birznieks, P. Jenmalm, A. W. Goodwin, and R. S. Johansson
Encoding of Direction of Fingertip Forces by Human Tactile Afferents
J. Neurosci.,
October 15, 2001;
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8222 - 8237.
[Abstract]
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P. Jenmalm, S. Dahlstedt, and R. S. Johansson
Visual and Tactile Information About Object-Curvature Control Fingertip Forces and Grasp Kinematics in Human Dexterous Manipulation
J Neurophysiol,
December 1, 2000;
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M. K. O. Burstedt, J. R. Flanagan, and R. S. Johansson
Control of Grasp Stability in Humans Under Different Frictional Conditions During Multidigit Manipulation
J Neurophysiol,
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J. R. Flanagan, M. K. O. Burstedt, and R. S. Johansson
Control of Fingertip Forces in Multidigit Manipulation
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Top
Abstract
Introduction
