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The Journal of Neuroscience, February 15, 1998, 18(4):1528-1545
Pointing to Kinesthetic Targets in Space
Gabriel
Baud-Bovy1 and
Paolo
Viviani1, 2
1 Department of Psychobiology, Faculty of Psychology
and Educational Sciences, University of Geneva, Carouge, Switzerland,
and 2 Laboratory of Perception, Action and Cognition,
LAPCO, Vita-Salute University HSR, Milan, Italy
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ABSTRACT |
An experiment investigated in human adults the sensorimotor
transformation involved in pointing to a spatial target identified previously by kinesthetic cues. In the "locating phase," a
computer-controlled mechanical arm guided the left [condition LR
(left-right)] or right [condition RR (right-right)] finger of the
blindfolded participant to one of 27 target positions. In the
subsequent "pointing phase," the participant tried to reach the
same position with the right finger. The final finger position and the
posture of the arm were measured in both conditions. Constant errors
were large but consistent and remarkably similar across conditions,
suggesting that, whatever the locating hand, target position is coded
in an extrinsic frame of reference (target position hypothesis). The
main difference between the same-hand (RR) and different-hand (LR)
conditions was a symmetric shift of the pattern of endpoints with
respect to the midsagittal plane. This effect was modeled accurately by assuming a systematic bias in the perception of the postural angles of
the locating arm. The analysis of the variable errors indicated that
target position is represented internally in a spherical coordinate
system centered on the shoulder of the pointing arm and that the main
source of variability is within the planning stage of the pointing
movement. Locating and pointing postures depended systematically on
target position. We tested qualitatively the hypothesis that the
selection of both postures (inverse kinematic problem) is constrained
by a minimum-distance principle. In condition RR, pointing posture
depended also on the locating posture, implying the presence of a
memory trace of the previous movement. A scheme is suggested to
accommodate the results within an extended version of the target
position hypothesis.
Key words:
kinesthetic pointing; frames of reference; arm movements; arm posture; sensorimotor transformations; inverse kinematics; position
sense
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INTRODUCTION |
Reaching out, without looking, for
an object that we had placed nearby is a common action. Thus, a motor
plan can be set up on the basis of postural information acquired during
the previous placing action. However, aside from the earlier (and
somewhat inconclusive) literature on motor memory (Posner, 1967
; see
also Laszlo, 1992), relatively few studies have considered realistic instances in which kinesthetic cues are used to specify the target of a
hand movement (Paillard and Brouchon, 1968; Wallace, 1977; Larish and
Stelmach, 1982; Helms Tillery et al., 1991, 1994). In particular, it is
not known yet whether the neural mechanisms for reaching a position
defined by kinesthetic cues share a common spatial frame of reference
with the mechanisms for reaching memorized visual targets.
When one arm is used both for locating a point in space and for
reaching again that point, the problem of sensorimotor coordination seems to be easier than in the visual case because, in principle, accurate reaching may be achieved by reproducing the same postural angles of the locating phase. This matching strategy, however, is
ineffectual when different arms are involved in locating and reaching
points that do not belong to the midsagittal plane. In this case, a
highly nonlinear (even though well defined) four-dimensional 4D 
4D
mapping would be necessary to transform the locating posture into
the appropriate pointing posture. If so, the computational complexity
in the two situations would be significantly different, and so should
be, one might argue, the resulting accuracy.
The so-called target position hypothesis (MacNeilage, 1970; Russel,
1976) provides a different solution to the problem of sensorimotor
coordination. According to this hypothesis, postural information from
the locating arm is recoded into position information in
three-dimensional (3D) extrinsic space. Thus, the subsequent reaching
would involve a 3D 4D mapping similar to the one contemplated in
the case of visual targets (Soechting and Flanders, 1989a,b; Flanders
and Soechting, 1990; Flanders et al., 1992). The target position
hypothesis applies equally well to the same- and different-hand cases,
the computations involved being similar. In fact, Larish and Stelmach
(1982) argued that if the hypothesis holds true, the pattern of
reaching errors should be fairly equivalent whether the same or
different hands are used.
Recent investigations of kinesthetic pointing in the same-hand (Helms
Tillery et al., 1991) and different-hand (Helms Tillery et al., 1994)
conditions reported that (1) accuracy was much poorer than in the
visual case; (2) the pattern of errors, as well as the relation between
target position and posture, differed from those reported previously by
the same group for visual targets; and (3) in the same-hand condition,
posture matching was the preferred strategy.
We report a further study of kinesthetic pointing in which controlled
arm movements brought the participant's finger to one spatial location
in the absence of vision (locating phase). Then, the participant was
asked to point again to that location, either with the same or with the
opposite arm (pointing phase). With this switched-limb paradigm, we
addressed the following questions: (1) Is hand position coded in a 3D
spatial system of reference analogous to that postulated for visually
directed movements? (2) What is the origin of reaching errors? (3)
Which factors determine arm posture?
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MATERIALS AND METHODS |
Participants. Four right-handed male adults (S1-S4)
volunteered for the study (ages, 29, 31, 34, and 28 years; heights,
175, 168, 176, and 185 cm). Participants were naive about the purpose of the experiment. They gave their informed consent and were paid 15 Swiss Francs. The experimental protocol was approved by the Ethical
Committee of the University of Geneva.
Target positions. Targets (N = 27) were
arranged at the nodes of a three × three orthogonal lattice, each
layer of the lattice including nine targets (Fig.
1). The spacing between horizontal, frontal, and sagittal layers was 150, 100, and 170 mm, respectively, so
that the vertical, frontal, and transversal dimensions of the workspace
were 300, 200, and 340 mm, respectively. The center of the array was
350 mm away and at the same height as the participant's nose. Targets
were numbered consecutively, starting from the furthest frontal layer
and from the lower left to the upper right corner of each layer. The
global frame of reference was provided by a right-handed orthogonal
coordinate system, with the x-axis pointing toward the right
of the subject, the y-axis pointing forward, and the
z-axis pointing upward.
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Figure 1.
The 27 targets arranged at the nodes of a
three × three lattice. The spacing between horizontal, frontal,
and sagittal layers was 150, 100, and 170 mm, respectively.
Dots indicate the average fingertip positions recorded
in the locating phase in one participant. Their perfect alignment
demonstrates the combined accuracy of the guiding and recording
systems. Targets were numbered from the furthest to the
closest frontal layer and from the lower
left to the upper right corner of the layer
[two targets (3 and 25) are identified]. The central (14th) target
was 350 mm from the origin (O) of the indicated
trihedron aligned with the global reference
axis. The origin was at the tip of the nose of the participant (see Materials and Methods).
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Experimental procedure. Participants rested in a tightly
fitting racing-car seat in front of a five-degrees-of-freedom
computer-controlled mechanical arm (the "robot"; SCORBOT ER-VII;
Eshed Robotec LTD). Two adjustable armrests defined a fixed reference
posture for the upper limbs. Except during rest periods, participants
were kept blindfolded. Each trial consisted of two phases. In the
"locating phase," the robot distal segment approached gently either
the right [condition RR (right-right)] or the left [condition LR
(left-right)] index finger of the participant. A switch mounted on
the distal segment checked that contact had been established.
Thereafter, the robot moved to a selected target position. Participants
were instructed to follow the movement, remaining in touch with the switch. The locating movement lasted between 1.3 and 4.5 sec, depending
on the target. One second after the end of the movement, a tone
instructed the participant to move his arm back to the reference
posture. Three seconds after the first tone, a second tone with a
different pitch signaled the beginning of the subsequent "pointing
phase" (the interval between tones was largely sufficient to
reposition the arm). In this phase, the participant had to try and
reach again with his right index finger the target position reached in
the previous phase. When the finger position was judged to be accurate,
he signaled the end of the phase by closing a switch with the right
foot. Then, the arm was brought back to the armrest. Using proximity
sensors mounted on the armrests, the computer checked that, before each
phase, both arms were in the reference posture. In the pointing phase,
accuracy was stressed without explicit time constraints. On every
trial, the hand to be used for locating was indicated by the computer
via a voice synthesizer. The synthesizer also warned the participant
when the initial posture was not correct and whenever contact with the
robot was accidentally interrupted during the locating movement. In
this (very rare) event, the robot stopped until contact was established
again.
Each target was selected 10 times in each condition. The resulting 540 trials [2 (conditions) × 10 (repetitions) × 27 (targets)] were
divided into 20 equal blocks that were administered sequentially, alternating conditions RR and LR. The order of selection of the targets
within each block was randomized under the constraint that no two
consecutive targets belong to the same horizontal, frontal, or sagittal
layer. The presentation of one block required ~12 min, and one
complete experiment (including two calibration phases, see below)
required between 8 and 9 hr divided evenly over three or four sessions.
Short rest periods interrupted the experiment every 25 min.
Data acquisition and processing. Movements were recorded
with a three-camera ELITE system (BTS Technology). The system measures the 3D coordinates of passive markers (diameter, 4 mm) reflecting in
the infrared band (accuracy, 1 mm; sampling rate, 100 Hz). The number
of degrees of freedom of the hand-arm complex was reduced to four by
blocking the wrist with a cast. Movements of the forearm were described
by four markers. Three were mounted on an orthogonal frame fixed to the
cast; one was placed at the tip of the outstretched index finger.
Movements of the arm were described by three markers mounted on a
second orthogonal frame strapped around the biceps. Trunk position was
measured by an additional three-marker frame firmly strapped to the
chest of the participant in correspondence with the sternum. For
certain postures, not all markers remained visible simultaneously to
all three cameras. However, in most cases the tracking software
(ELIPLUS; BTS Technology) filled-in the missing information. In a few
instances, the tracking procedure failed, and it was impossible to
describe fully the posture of the arm.
Both in the locating and pointing phase, data acquisition was triggered
by the opening of the proximity switch on the armrest and was stopped
when the arm came back to rest, closing again the switch. The end of
the forward movement was timed by recording either the control signal
to the robot (in the locating phase) or the foot-operated switch (in
the pointing phase). Shoulder, elbow, and finger positions at this
point were computed by averaging the coordinates of the markers over a
50 msec period.
Calibration. Defining an arm posture from the instantaneous
coordinates of the shoulder and elbow joints in the global coordinate system requires the knowledge of the (invariant) joint positions with
respect to arm- and forearm-based references, respectively. These
anthropometric data were estimated by the following calibration procedure. Let m1,
m2, and m3 be the
instantaneous position of the markers strapped around the arm in the
global coordinate system and [i = m2 
m1,
j = m3 m1, and k = (m2 m1) × (m3 m1)] be
the associated (nonorthogonal) moving reference. The participant was
asked to perform for 5 sec random 3D rotations of the extended arm,
trying to keep the shoulder as still as possible. Any triple of
coordinates
[x,y,z]
in the moving reference corresponds to a point
(p = xi + yj + zk + m1) in the global
coordinate system. The position ps of the
shoulder joint was estimated by finding with a standard minimization
algorithm the triple
[xs,ys,zs]
such that ps has the least variance during
the calibration movements. A similar method was used to compute the
position of the elbow joint. In this case, the calibration movements
involved only the forearm, the elbow being immobilized.
The reliability of the procedure was tested by repeating the
calibration at the end of each experimental session. The data from a
trial were discarded if either of two events occurred: (1) the distance
between any two markers on the same frame differed by >2 mm from the
corresponding session average, or (2) the estimated distance between
shoulder and elbow joints differed by >3 mm from the session
average.
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RESULTS |
Constant errors
The pointing error in a trial is the 3D vector e
joining the true and the estimated target position (endpoint). Overall pointing accuracy was measured by the average absolute error, i.e., by
the average of the length (|e|) of the error vector over
all targets and repetitions. By this criterion, subjects were fairly
inaccurate in both conditions, the average absolute error ranging from
72.6 to 156.0 mm (Table 1).
In a further analysis, we decomposed each error vector into two
components, the constant error across repetitions
(ec) joining the target to the center of
gravity of all endpoints and the variable error
(ev) joining the center of gravity to each
endpoint. The first component measures the systematic-pointing bias.
The second component is a random variable describing the dispersion of
the data with respect to this bias.
Figure 2 contrasts the deformation of the
array of the targets in conditions RR and LR (pooling individual data).
Average endpoints exhibited a rather regular pattern, the orthogonality and orientation of the array being well preserved. Distortions concerned mostly the lateral and vertical dimensions of the array. The
main effect associated with the locating hand was a leftward shift in
condition RR and a rightward shift in condition LR. In condition RR,
the constant error was larger for targets in the left sagittal plane
than for those in the right sagittal plane. The opposite was true in
condition LR.
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Figure 2.
Patterns of constant errors. Perspective view of
the average endpoints for all participants (thick lines)
in conditions RR and LR is shown. The array of targets (thin
lines) is superimposed for comparison.
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The pattern of deformation was consistent across participants (Fig.
3), the main difference being the
position of the array of endpoints with respect to the body. In all
cases, the array of the targets was stretched along the lateral
direction (top view) and compressed along the sagittal
and vertical directions (front and side
views). Table 1 reports the amplitude
(|ec|) and the directional biases
(ecx, ecy,
ecz) averaged over all targets for each
subject and condition separately. In both conditions, participants S1,
S2, and S3 underestimated the distance of the center of the array from
the body (ecy < 0) and overestimated its
elevation (ecz > 0). Individual data confirmed
that the main effect of the experimental manipulation was to shift the
center of gravity toward the left (ecx < 0) in
condition RR and toward the right (ecx > 0) in
condition LR. Similar shifts have been reported by Wallace (1977) and
Larish and Stelmach (1982). The (signed) difference across conditions
between the average directional biases was similar for all
participants. Along the lateral direction, the differences were S1,
60.2 mm; S2, 76.0 mm; S3, 45.3 mm; and S4, 48.5 mm. Both along the
sagittal (S1, 11.4 mm; S2, 3.9 mm; S3, 8.8 mm; and S4, 5.9 mm) and the vertical (S1, 16.7 mm; S2, 31 mm; S3, 13.6 mm; S4,
16.8 mm) directions, differences were less dependent on the arm used
for locating the targets. The systematic error increased only slightly
(S1, 1%; S2, 20%; S3, 24%; and S4, 7%) when different arms were
used for locating and pointing (condition LR) with respect to condition
RR in which the same (right) hand was used.
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Figure 3.
Two-dimensional comparison of the pattern of
constant errors in conditions RR (blue lines) and LR
(red lines). Individual data
(S1-S4) and the population average
(Av) are shown. Each view was obtained by collapsing the
average endpoint positions over one spatial dimension. Top
view, Line crossings are the
[x,y] endpoint coordinates
averaged over one vertical row of targets (e.g., targets 1, 2, and 3).
Front view, Line crossings are the [x,z] endpoint coordinates
averaged over one sagittal row of targets (e.g., targets 1, 10, and
19). Side view, Line crossings are the [y,z] endpoint coordinates averaged
over one transversal row of targets (e.g., targets 1, 4, and 7). The
projections of the array of targets (thin dotted lines)
are superimposed for comparison.
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Variable errors
In this section, we argue that the most significant aspects of the
variable errors are independent of the experimental condition. We
assumed that endpoints are multinormal (3D) Gaussian variates. Thus,
the spatial distribution of the endpoints is characterized by the
confidence ellipsoid defined by the Hotelling
T2 statistics (Morrison, 1976):
where S is the unbiased estimate of the
k-sample covariance matrix of the endpoint coordinates,
m is the center of gravity of the ellipsoid, N is
the total sample size, and p (= 3) is the number of
dimensions. Because of the small number of repetitions, computing the
ellipsoid for each participant, condition, and target separately would
yield unreliable results. To group the data from all participants, we
first computed the average endpoint mh
across repetitions for each participant, condition, and target. Then,
the k-sample covariance matrix:
was computed after subtracting these averages from the endpoint
xih for each trial. Figure
4 shows a perspective view of the
distribution of endpoints that is obtained from collapsing individual
data for each target and condition separately [N = 4 (subjects) × 10 (repetitions) = 40 and k = 4 (subjects) for each of 2 (conditions) × 27 (targets) = 54 ellipsoids]. The direction of the three axes of the ellipsoids is that
of the eigenvectors of the covariance matrix S. The lengths
A, B, and C of the semiaxes were set
to the square root of the corresponding eigenvalues (with this choice,
the probability for the mean endpoint to fall inside the ellipsoid
exceeds 0.99, the corresponding probability for any one endpoint being
0.18).
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Figure 4.
Spatial distribution of the variable error.
Three-dimensional views of the confidence ellipsoids for each target
position in conditions RR and LR are shown. The semiaxes of the
ellipsoids are equal to the square root of the three eigenvalues of the
variance-covariance matrix of the endpoint coordinates. Data from all
participants were pooled.
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From the definition of the confidence ellipsoids, it follows that
comparing the endpoint distributions is equivalent to comparing statistically the underlying covariance matrices. Using Box's (1949)
test of the equality of covariance matrices (see Morrison, 1976, page
252), we compared for each target the endpoint distributions in
conditions RR and LR. None of the 27 pairwise comparisons revealed a
significant difference (p > 0.05). This test,
which compounds all aspects of the variable errors, indicated that
indeed the endpoint distributions were independent of the arm used for
locating. To test more specific differences between the two conditions, we proceeded to analyze independently volume, shape, and orientation of
the ellipsoids.
The volume of the ellipsoid (V = 4/3 
ABC) affords a measure of the dispersion of the endpoints
around their center of gravity. The volume was significantly larger
when different hands were used for locating and pointing (condition LR)
than when the same hands were used (condition RR) [one-way ANOVA;
F(1,52) = 43.46; p < 0.001].
The volume varied across participants (Table 1), but for all of them,
it was significantly larger in condition LR than in condition RR. The
increase (S1, 125%; S2, 87%; S3, 71%; and S4, 44%) was larger than
the increase for systematic errors (see above). A principled
relationship between dispersion and position emerged by averaging the
volume of the ellipsoids across layers. Figure
5 shows the result of collapsing the data across horizontal (top view) and sagittal (side
view) layers. A four-way ANOVA [2 (conditions) × 3 (sagittal layers) × 3 (horizontal layers) × 3 (frontal layers)],
with all interactions except the four-way one, confirmed the condition
effect mentioned above [F(1,8) = 100.7;
p < 0.001] and revealed a significant increase of the variability from the leftmost to the rightmost sagittal layer [F(2,8) = 20.5; p < 0.001].
All other main effects, as well as all interactions, failed to reach
significance (p > 0.05). The sagittal layer
effect was confirmed, for each condition separately, by regressing the
volume against the coordinates of the center of gravity:
V = a0 + a1x + a2y + a3z. In both conditions, the only
significant contribution was that of the x-axis coordinate (RR, a1 = 106.3; t23 = 4.16; p = 0.0004; LR, a1 = 149.2; t23 = 3.64, p = 0.0014;
two-tailed). The positive values of the coefficient a1 indicate that the volume was larger for
ellipsoids belonging to the right sagittal plane that was close to the
starting position of the hand. This result did not confirm the
frequently reported Weber-like increase of variability with movement
extent (however, see Stelmach and Wilson, 1970).
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Figure 5.
Volume of the confidence ellipsoids. Top
view, Data collapsed over the horizontal layers. Side
view, Data collapsed over the sagittal layers. The area of the
symbols is proportional to the corresponding average
volume. Distributions in the margins illustrate the result of
collapsing these averages along the indicated directions. Solid
symbols, Condition RR; empty symbols, condition
LR. Data from all participants were pooled.
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Ellipsoids are cigar-shaped when one semiaxis is significantly longer
than the other two (A > B 
C) and lens-shaped when one semiaxis is significantly
shorter than the other two (A B > C). Figure 6 is a plot of the
ratio A/B against the ratio B/C for all 54 ellipsoids. In this plot,
quasispherical ellipsoids (A/B 1.0;
B/C 1.0) are close to the origin,
whereas lens-shaped and cigar-shaped ellipsoids are close to the upper
left (A/B 1.0;
B/C > 1.0) and lower right
(A/B > 1.0; B/C 1.0) corner, respectively. Using this criterion and Anderson's
(1963) test of the equality of eigenvalues (see Morrison, 1976, page
294), we found that in 30 cases at least one of the ratios was
significantly different from 1 (p < 0.05). By
applying the same criterion to this group of nonspherical shapes, we
identified 21 lens-shaped and 7 cigar-shaped ellipsoids (these figures
correspond to the number of dots within the indicated regions in Fig.
6). Therefore, there was a definite tendency for the ellipsoids to be
lens-shaped. Notice that the distributions of data points in conditions
RR (solid symbols) and LR (empty
symbols) overlapped completely, implying that the shape of
the ellipsoid did not depend on the locating hand.
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Figure 6.
Shape of the confidence ellipsoids. Each data
point corresponds to the ellipsoid for one target (solid
symbols, condition RR; empty symbols, condition
LR). x-axis, Ratio between the first (A) and the second (B)
eigenvalue of the variance-covariance matrix of the endpoint
coordinates; y-axis, ratio between the second (B) and the third (C)
eigenvalue. In this representation, the position of a data point
characterizes the shape of the ellipsoid. Ellipsoids near the
upper left corner are lens-shaped. Ellipsoids near the
lower right corner are cigar-shaped. The
vertical and horizontal lines correspond
to the critical values (p < 0.05) in
testing the hypothesis that either ratio is significantly different from 1. Data from all participants were pooled.
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The dominant orientation of cigar-shaped ellipsoids is that of their
major axis (first eigenvector). Instead, the orientation of lens-shaped
ellipsoids is best characterized by the direction of their smallest
semiaxis (third eigenvector) that is orthogonal to their flattest
surface. The relationship between the center of the ellipsoids and
their orientation was investigated by regressing independently the
azimuth and the elevation of the third eigenvector against the
spherical coordinates of the center of gravity. When all ellipsoids
were included in the analysis, only the azimuth was significantly
dependent on the center of gravity [F(3,50) = 10.65; p < 0.001], the highest correlation being the
one with the azimuth of the center of gravity
(t50 = 5.08; p < 0.001;
two-tailed). Similar results were obtained when each condition was
analyzed separately [RR, F(3,23) = 7.25;
p < 0.002; LR, F(3,23) = 4.89; p < 0.01]. Moreover, all significant coefficients
(p < 0.05) had the same sign and were of the
same order of magnitude, implying that the relationship between
position and dominant direction of the ellipsoids was similar in both
conditions. In condition RR also, the elevation of the third
eigenvector depended on the center of gravity
[F(3,23) = 3.80; p < 0.05].
However, this relationship was not confirmed in condition LR.
To identify the dominant direction of the ellipsoids, we collapsed all
individual data for both conditions along the vertical and transversal
directions (Fig. 7). In the top
view, each ellipsoid pools data corresponding to three
targets with the same [x,y] coordinates (e.g.,
the top left ellipsoid refers to targets 1, 2, and 3). In the
side view, each ellipsoid pools data corresponding to
three targets with the same [y,z]
coordinates (e.g., the bottom right ellipsoid refers to targets 1, 4, and 7). All the resulting ellipsoids exhibited the tendency to be flat
that was detected before pooling. For both views, the dominant
orientation is indicated by lines starting at the center of gravity and
stopping at the intersection with the frontal plane passing through the
shoulders of the participant. The pattern emerging from the top
view shows a clear tendency for the orientations to depend
on the azimuth and distance of the targets, all lines converging
approximately toward a region in front of the shoulder. As seen from
the side view, the orientations were less dependent on
elevation or distance. All lines intersected the frontoparallel plane
at or below the right shoulder.
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Figure 7.
Orientation of the confidence ellipsoids
characterized by the direction of their third eigenvector (see
Results). Top view, Each ellipsoid pools data for one
vertical row of targets. Side view, Analogous results
for each transversal row of targets are shown. Data from all
participants in both conditions were pooled.
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In summary, detailed statistical analysis of the endpoint distribution
showed that all aspects of the variable error but the volume of the
ellipsoids were independent of the locating arm. Moreover, even
differences in volume were not large enough to be detected by the
global comparison between distributions.
Relationship between arm posture and finger position
Target position (three degrees of freedom) does not
determine uniquely the posture of the arm (four degrees of freedom).
Thus, any principled relationship between target position and arm
posture indicates the presence of constraints in the execution of the movement. To verify whether such a relationship did in fact exist, we
took advantage of the fact that all postures compatible with a given
target position are obtained by a rotation 
of the elbow around the
fixed shoulder-hand axis (see ). Thus, the angle (the
posture angle) concentrates the intrinsic indeterminacy of the target-posture relationship. In the following, first we use
this representation of the extra degree of freedom to demonstrate a
relationship between average arm posture and finger position. Then, we
show that this relationship is largely independent of the arm used for
locating.
For the left and right arm, the range of posture angles (in degrees)
was [90,20] and [20,90], respectively. In each phase (locating
and pointing) and each condition, the posture angle was strongly
dependent on the target (one-way ANOVA, target treated as a nominal
variable; p < 0.001 in each phase and condition, for
all participants). For pointing movements, we summarized this dependency by expressing the posture angle as a linear combination of
the spherical coordinates of the finger: = a0 + a
+ a
+ aRR
(Table 2). Across participants and
conditions, the amount of variance accounted for by the regression was
quite high (r12 in Table 2).
Arm posture depended most significantly on the azimuth of the
finger position (p < 0.001), the elbow rotating in the counterclockwise direction as the final position moved toward
the left. The correlation with the elevation was weaker (t values were 10 times smaller than were those for the
azimuth), the elbow rotating in the clockwise direction as the final
position moved upward. Finally, the correlation with the distance
R was highly significant (the value of the coefficient
aR cannot be compared to the other two
because different scales are involved). The dependency of the posture
angle on finger position is consistent with the observation that arm
orientation is a function of azimuth and elevation angles in
straight-arm pointing (Straumman et al. 1991; Miller et al., 1992) and
ball-throwing (Hore et al., 1992, 1994) movements.
Next, we tested whether the relationship between the arm posture and
finger position in the pointing phase depends on the locating arm.
Because the finger position at the end of the pointing phase was not
the same in both conditions and because these variations affected the
posture angle , the test required a preliminary processing. For each
participant, condition, and phase separately, the posture angle was
fitted by third-order polynomial functions of the polar coordinates of
the finger position (accuracy improved only marginally with
higher-degree polynomials). Across subjects and conditions, the amount
of variance accounted for by these fittings varied between 72 and 93%
(r22 in Table 2). The
regression plots in Figure 8 demonstrate
the accuracy of the fitting in one typical subject. The polynomials were then used to interpolate the posture angles at the target positions. Figure 9 shows the
relationship between the interpolated posture angles at the end of the
pointing phase in conditions LR and RR. In participant S4, the average
posture of the right arm was virtually the same, irrespective of the
arm used for locating. The correlation between posture angles was also
very high in participants S2 and S3. In both cases, however, there was
a significant, target-independent tendency to rotate counterclockwise
the right elbow in condition LR with respect to condition RR.
Participant S1 introduced postural variations across targets, resulting
in a weaker correlation. For all participants, the slope of the normal
regression line through the data points was close to 1.
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Figure 8.
Posture angle at the end of the locating
(left panels) and pointing (right panels)
phase in the indicated conditions. Data are for one participant (S4).
Large panels, Dependency of arm posture on target
position. x-axis, Average posture angle for each target;
y-axis, posture angle predicted by a third-order polynomial model at the target position (locating phase)
or at the average endpoint position (pointing
phase). Data points for targets 3 and 25 are identified.
Small panels, Variability around the average posture
angle for each target position. Residuals are the difference between
the actual posture angle in each trial and the corresponding polynomial
prediction.
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Figure 9.
Comparison of the average pointing posture across
conditions. For each participant, panels correlate the
posture angle for pointing in conditions LR (x-axis)
and RR (y-axis). The coordinates of the data
points were computed via a third-order polynomial interpolation of the
relationship between finger position and posture angle. Solid
lines, Normal regressions; dotted lines, 45°
diagonal.
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Postural variability across phases and conditions
The amount of postural variability across repetitions (i.e.,
unaccounted for by target position) was not negligible. This is
illustrated by the plots of the residuals in Figure 8 showing the
difference between the actual posture angle and the value predicted by
the polynomial model for the corresponding endpoint. Here we describe
how the correlation between the residuals was used to gauge the extent
to which arm posture in the pointing phase could be explained by the
corresponding posture in the locating phase. Both panels in
Figure 10 are a plot for each target of
the residual for the locating phase (x-axis) against the
residual for the corresponding pointing (data for all subjects).
Because of the large number of data points, correlation was significant in both conditions. However, the strength of the association was much
higher in condition RR [F(1,815) = 517.03;
p 0; r2 = 0.39] than
in condition LR [F(1,770) = 32.96;
p 0; r2 = 0.04].
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Figure 10.
Partial correlation between posture angles .
In each panel, the residuals for each locating movement
(see Fig. 9) are plotted against the residuals for the corresponding
pointing movements. Data for all participants and all targets were
pooled. The standard (dotted) and normal
(solid) regression lines are also plotted.
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|
The contrast between conditions was confirmed by performing a
correlation analysis for each target and subject separately (Table
3, values based on less than five
measures were omitted). The number of significant
(p < 0.05) correlations was 39 out of 85 (45%)
in condition RR and 13 out of 87 in condition LR (15%), a significant
difference (binomial test; p < 0.001). The same pattern was present in each participant (4 vs 64%, 29 vs 64%, 5 vs
17%, and 30 vs 44% for S1, S2, S3, and S4, respectively). When the
data for all targets was pooled, the correlation for each participant
was significant in both conditions, but again, the strength of the
association was clearly different [for S1; RR,
F(1,221) = 278.15; LR,
F(1,204) = 4.14; for S2; RR,
F(1,148) = 35.31; LR,
F(1,150) = 28.51; for S3; RR,
F(1,194) = 47.41; LR, F(1,181) = 5.39; and for S4; RR,
F(1,246) = 112.45; LR,
F(1,229) = 18.08; p < 0.05 in
all cases]. Whenever the correlation differed significantly from 0, the slope of the regression line was positive in condition RR and
negative in condition LR (a negative slope indicates that the two
elbows rotated in opposite directions). Thus, all participants tended
to adopt a pointing posture similar to that adopted for locating.
Again, this tendency was much stronger when the same arm was used in
both phases.
In the midsagittal plane, pointing in the condition LR could be
achieved accurately by a posture matching strategy (see the introductory remarks). To ascertain whether these targets had a special
status, we considered the strength of the correlation between locating
and pointing postures for targets belonging to different planes. Table
4 reports for both conditions the
proportion of significant correlations for each group of nine targets
lying in one plane (data from all participants). In condition RR, the proportion increased from the distal to the proximal plane and from the
left to the right sagittal plane. There was no significant difference
among the three horizontal planes. In short, the correlation was
highest for targets close to the starting position of the right hand.
In condition LR, the proportion was largest (22%) for targets in the
midsagittal plane, for which pointing posture could be the mirror image
of the locating posture.
Modeling average posture
Arm posture varied somewhat from trial to trial. Yet it depended
strongly on target position. In this section, we argue that the
systematic component of the posture selection process can be accounted
for by a simple kinematic hypothesis. Specifically, we posit that the
selection of one posture (i.e., the solution of the inverse kinematic
problem) complies with an optimum principle. A posture can be described
by a 4D vector P = [,,
,
], where (,)
and (,) are the yaw and elevation angles of the arm and forearm,
respectively (i.e., the so-called orientation angles; Soechting and
Ross, 1984). Suppose that there exists one posture
P* that the motor system construes as a
fixed reference, much like the primary gaze position in oculomotor
behavior. Suppose also that the distance of any one posture
P from this reference is estimated with the Pythagorean
metrics d(P,P*) = |P P*| (scale
factors in the computation of distances may be different for each
component). The motor-planning hypothesis we are entertaining is that
the average posture adopted both in locating and pointing is the only
posture that, at the same time, is compatible with the observed
endpoint and minimizes the distance from the reference (minimum-distance principle).
Testing the hypothesis involved the specification of the reference
posture and of the scale factors. P* was
estimated by the (componentwise) average of the locating postures over
all targets. The scale factors
[c,c,c
,c
] were selected to minimize the difference between actual and predicted posture angles across all targets. The average scale values across subjects were c = 0.289, c = 0.158, c = 0.221, and c = 0.332. The provides
the details of the fitting strategy. We tested the hypothesis for each
subject and each condition separately. Numerical analysis confirmed
that the minimum-distance posture for each endpoint could be identified
unambiguously. Figure 11 summarizes the
results for one participant by plotting for all conditions and each
target the predicted posture angle versus its observed average.
Table 5 reports the actual and predicted correlations for the angles , , , , and in all
conditions and all participants (intercept and slope parameters were
close to 0 and 1, respectively). The high percent of variance accounted for in every case shows the excellent agreement between the data and
the suggested solution to the inverse kinematic problem.
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Figure 11.
Modeling the selection of arm posture. For each
indicated condition and phase, panels correlate the
posture angle averaged over repetitions for each target
(x-axis) with the corresponding prediction
(y-axis) of the minimum-distance model (see
Results). Data are from one typical participant (S4). Solid
lines, Normal regressions; dotted lines, 45°
diagonal.
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Modeling systematic errors
Systematic errors were large, yet regular. The pattern of errors
was similar in conditions LR and RR, being the result of quasisymmetrical shifts with respect to the midsagittal plane. Because
only the arm used for locating changed across conditions, this
symmetric arrangement must reflect some aspect of the locating process.
In fact, it is possible to predict the pattern of error in both
conditions by assuming that the orientation angles of the locating arm
are systematically biased and generate inaccurate target
representations in a 3D spatial reference. Instead, the execution of
the pointing movement toward the targets is not supposed to introduce
further systematic distortions. Such a difference between the locating
and pointing processes is supported by two observations. First, in the
locating phase the target position was not known until the end of the
movement, whereas the participant had a representation of the desired
endpoint even before initiating the pointing movement. Second,
inspection of the trajectories demonstrated the ballistic nature of the
pointing movements. Only very rarely did we observe corrective
movements at the end of this phase. Together, these two observations
point to the conclusion that pointing, unlike locating, movements were
controlled in the feedforward mode and involved very little processing
of sensory cues.
Because the minimum-distance principle translates any configuration of
endpoints into the corresponding configuration of arm postures, we can
estimate the (mis)perceived orientation angles by computing the average
arm posture that the left or right arm would take at the spatial
location reached by the right hand at the end of the pointing movement
[note that the predictions of the minimum-distance model for condition
RR are highly correlated with their actual values (see Table 5)].
Figure 12 summarizes the results for
the same participant of Figure 11. Each panel is a plot of the
perceived versus the actual orientation angles measured in the locating
phase. Collectively, these psychophysical functions convey all the
information concerning the systematic pointing errors. Linear
regressions provided an excellent fitting to the data points in all
cases. To predict the systematic errors, we expressed each perceived
angle as a linear combination of the four real angles. Using again data
from the this same subject (Fig. 13),
we illustrated the adequacy of the model by comparing the actual and
predicted average endpoints with the same format described in Figure
3.
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Figure 12.
Psychophysical functions for perceived
orientation angles. For each indicated orientation angle and both
conditions, panels correlate the experimental values
measured at the end of the locating movement (averaged over repetitions
for each target) and the values that, according to the minimum-distance
model, the locating arm should take to reach the average final
position. Deviations of the normal regressions (solid
lines) from the 45° diagonals (dotted lines)
were interpreted as perceptual biases occurring during the locating
phase.
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Figure 13.
Pattern of constant errors predicted by the
psychophysical functions in conditions RR and LR. The
dotted, interrupted, and solid
lines describe the pattern of the targets, of the average endpoints, and of the predicted endpoints, respectively. The same format of Figure 3 is used.
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Modeling variable errors
By necessity, the diverse postures observed in the locating phase
for any one target identified the same endpoint (i.e., the one set by
the robot). However, because we concluded above that perceived angles
were biased, this many-to-one correspondence between angles and
endpoints was no longer guaranteed at the level of the internal
representation. The idiosyncratic variations of the locating posture
for each target are bound to introduce some variability also in the
perceived hand position and, ultimately, in the planning of the
pointing. We tried to estimate this perceptual contribution to the
pointing variability. First, using the psychophysical functions
described above, we computed the perceived hand position corresponding
to the actual posture in each trial. Then, we computed the associated
confidence ellipsoids with the same procedure that was used for the
pointing data. Shape and orientation of these ellipsoids differed
across conditions, whereas shape and orientation of the ellipsoids for
pointings were similar (see above). However, this apparent
contradiction disappears when one considers the size of the perceptual
contribution. In fact, the volume of the ellipsoids describing the
variability of the perceived hand position was found to be much smaller
(<20%) than that of the pointings. Thus, the perceptual contribution
to the pointing variability was masked by a larger additional source of
noise. The similarity of shape and of orientation of the actual
ellipsoids in both conditions strongly suggests that this additional
contribution pertains to the planning and/or the execution phases.
We reasoned that if planning is the situation in which noise intervenes
most, then it should affect the 3D spatial representation of the target
generated by the locating arm. Alternatively, if noise intervenes
mostly in the execution phase, it should emerge at the level of the
intrinsic coordinates of the pointing arm. The problem of deciding
between the two possibilities was addressed by finding the system of
reference in which the coordinates of the endpoints are most
uncorrelated (Gordon et al., 1994; McIntyre et al., 1997; Vindras and
Viviani, 1998). Five systems of coordinates were considered. For the
extrinsic (3D) reference, these were (1) spherical, centered on the
right shoulder, (2) spherical, centered on the hand initial position,
and (3) orthogonal cartesian. For the intrinsic (4D) reference, these
were (4) orientation angles and (5) joint angles. The amount of
coupling for each coordinate system was gauged using Bartlett's (1954)
statistics (see Morrison, 1976, page 118) that permits one to test
whether a correlation matrix is significantly different from the
identity matrix. For each system, target, and condition separately, we
computed the correlation matrix R = Diag(1/
sii)T
S Diag(1/sii), where
sii values are the diagonal terms of the
covariance matrix S. R and S are
either three × three matrices (for the extrinsic systems) or
four × four matrices (for the intrinsic systems). Finally, the
degree of coupling was defined as the number
Nc of correlation matrices (out of 54) for
which the null hypothesis was rejected at the 0.05 significance level.
In order of increasing coupling, the five systems were ranked as
follows: (1) spherical, shoulder-centered
(Nc = 6); (2) spherical, hand-centered
(Nc = 18); (3) orthogonal cartesian (Nc = 26); (4) orientation angles
(Nc = 41); and (5) joint angles (Nc = 53). Note that the same ranking was
obtained by taking into account separately the 27 ellipsoids for either
condition. In conclusion, the analysis above strongly suggests that
most of the endpoint variability originates within the 3D extrinsic
representation of the target position on which motor planning is
based.
Based on this clear result, we performed a numerical simulation of the
distribution of the variable error. For each condition, uncorrelated
Gaussian noise was added to the coordinates of the individual average
endpoints expressed in a shoulder-centered spherical system. The
variance of the coordinates was estimated from the data for each
condition and participant separately, pooling all targets. The
simulated ellipsoids are shown in Figure
14 with the same format of Figure 7. In
keeping with the correlation analysis above, both the shape and the
direction of these ellipsoids were similar to the experimental ones.
Moreover, the average volume compared favorably with the data, the only
discrepancy being a slight increase in the volume with movement extent
that was not present in the experimental results.
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Figure 14.
Modeling the variable error. Size, shape, and
orientation of the simulated confidence ellipsoids are shown. The same
format of Figure 7 is used.
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DISCUSSION |
We investigated the mechanisms that permit one to point to a
position in space identified previously by kinesthetic cues. Absolute
errors (average across participants and conditions, 119.75 mm) were
larger than those typically observed in experiments on motor memory.
The difference, however, is congruent with the observation that both
constant and variable errors increase with the dimensionality of the
movement. For instance, in the one-dimensional lever-positioning task,
absolute error ranged typically between 30 and 50 mm (Laszlo, 1992).
Comparable values (18.5-40.2 mm) were observed in an experiment using
the same switched-limb technique of this study (Larish and Stelmach,
1982), as well as in a similar study by Wallace (1977). By contrast,
for two-dimensional (2D) movements, Larish and Stelmach (1982) found
that absolute radial errors ranged from 34.9 to 70.0 mm. Also, with a
hand apposition technique, Helms Tillery et al. (1994) found that the
constant error ranged between 36.0 and 67.0 mm for 2D movements and
between 73.0 and 101.0 mm for 3D movements. Finally, errors in our
study were of the same order of magnitude as those reported by
Soechting and Flanders (1989a, page 587) for pointing to (3D,
remembered) visual targets (average absolute error, 116.6 mm). It
should be stressed that the pattern of constant errors (Fig. 3) was
quite similar to that reported for the visual case (see Soechting and
Flanders, 1989b, their Fig. 4).
Direct mapping versus target position hypothesis
Initially, target position must be coded as a set of (at least
four) articular variables. In the introductory remarks, we contrasted
two alternative accounts of how this information is then transformed
between the locating and pointing phases. One possible strategy could
be to establish a direct mapping from the 4D manifold in which the
posture of the locating arm is represented into the 4D manifold in
which the posture of pointing arm is represented (Fig.
15A). Two arguments can be
cited against the 4D 4D mapping hypothesis.
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Figure 15.
Schematic representation of three hypotheses
(A-C) on the sensorimotor transformations
involved in pointing to kinesthetic targets.
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First, in condition RR, direct mapping would reduce simply to
reproducing the same postural angles adopted in the locating phase.
Instead, the mapping would be much more complicated when left arm
posture has to be translated into a right arm posture with the same
endpoint. This asymmetry is difficult to reconcile with the following
facts: (1) errors in both the single- and switched-limb cases were
quite similar, the only distinguishing feature between conditions being
the opposite shifts of the average endpoints; (2) variable errors were
smaller when the right hand was used both for locating and pointing
(Fig. 4), but shape (Fig. 6) and orientation (Fig. 7) of the confidence
ellipsoids were instead similar; and (3) average pointing posture was
fairly independent of which arm had been used for locating (Fig. 9).
Thus, one would have to admit that, although radically different
computations were involved in the two conditions, they nevertheless
resulted in remarkably similar postures and patterns of error.
Second, postural variability (summarized by the angle ) was
considerable, on the order of 40° in both phases (Fig. 8). However, the degree of correlation between left and right postural variability was weaker in condition LR than in condition RR. If indeed target identifications were mediated directly by postural information, one
would expect a much tighter coupling between arms also in condition LR.
A discussion of the coupling in condition RR is deferred until
later.
The fact that kinesthetic information from the left arm could drive the
right arm as reliably as information from the same right arm is much
better in keeping with the alternative account mentioned in the
introductory remarks, namely the one based on the target position
hypothesis that the early intrinsic coding is followed by a stage in
which target position is translated into an extrinsic 3D coordinate
system (MacNeilage, 1970; Wallace, 1977; Larish and Stelmach, 1982). In
fact, as shown schematically in Figure 15B, within this
conceptual framework, the sensorimotor transformation leading to the
pointing movement is similar in both conditions. Moreover, the scheme
is fully compatible with the fact that the average posture of the right
arm was independent of the experimental condition.
The target position hypothesis was tested by Helms Tillery et al.
(1991) by comparing pointing accuracy in two conditions. One was
similar to our RR condition; in the other the target had to be
identified with a stick held by the right hand. Errors with the stick
were larger than were those with the hand. Moreover, they were
unsystematic, subject-specific, and dependent on the location of the
target in the workspace, leading the authors to the conclusion that
"subjects were unable to synthesize a reliable estimate of the
location of their hands in space using only kinesthetic cues" (Helms
Tillery et al., 1991, page 771). In a later study (Helms Tillery et
al., 1994) on the basis of two switched-limb experiments, the same
authors concluded that targeted hand movements subserved by visual and
somesthetic inputs are organized in fundamentally different frames of
reference. The reason for the discrepancy between these results and
ours is not clear. However, two methodological improvements in our
experiment may be cited, the much higher density of points sampled in
the workspace and the availability of repeated measures. Moreover, in
both previous studies, individual performances were highly variable,
making it difficult to compare the results of different participants
tested in the single- and two-hand conditions.
The analysis of the distribution of the variable errors provided useful
clues about the nature of the coordinate system in which target
position is coded. The fact that shape and orientation of the
ellipsoids were similar in both experimental conditions is congruent
with the assumption that one frame of reference is involved
irrespective of the locating hand. Moreover, the analysis of the
coupling among coordinates confirmed that, as postulated by the target
position hypothesis, this frame of reference is extrinsic. It also
suggested that the variable error originates within the early stages of
planning of the pointing rather than within the execution stage.
Finally, correlation analysis permitted us to identify the spherical,
shoulder-centered system of coordinates as the most likely candidate
for the extrinsic reference.
Solving the inverse kinematic problem
Pursuing the hypothesis that target position is eventually coded
in an extrinsic 3D system of reference, one must address the question
of how pointing posture is selected on the basis of this spatial
information. Because >70% of the total variance of the posture angle
was explained by the endpoint position (see Results), we framed the
selection problem as an inverse kinematic problem for the average
posture. Several attempts have been made to account for the dissipation
of the extra degrees of freedom involved in the solution to the inverse
kinematic problem in terms of neural constraints. In particular, it was
suggested that Donders' law that holds for eye movements can be
generalized to arm movements (Hore et al., 1992, 1994; Crawford and
Vilis, 1995). One consequence of this assumption is that arm
orientation should depend only on the azimuth and elevation of the
target. By varying systematically the initial hand position, Soechting
and Flanders (1995) showed that this constraint is not fully respected.
Our results are also at variance with the proposed generalization of
Donders' law. First, we found that arm orientation, as described by
the posture angle , depended strongly on the target radial distance.
Second, the orientation of the pointing arm for any given target was
not constant, being related to the arm orientation during the previous locating movement. Our solution to the problem of dissipating the extra
degrees of freedom takes into account the dependency from the target
radial distance by adopting a biologically plausible cost function
based only on differences between orientation angles. Moreover, by
framing the problem in terms of average posture, this solution leaves
room for accommodating the observed postural variability (see
below).
Admittedly, the minimum-distance principle cannot be construed as a
fully satisfactory solution to the problem, insofar as it does not take
into account the initial position of the hand and the dynamics of the
movement (Soechting and Flanders, 1995). It should also be noted that,
although the principle accounts for the selection of one pointing
posture among the infinite candidates compatible with one final finger
position, it does not address the question of how this set of
candidates is specified by the perceived position of the target.
Despite these limitations, the very accurate predictions of average
posture (Fig. 13) demonstrated that the general notion of optimal
planning, originally developed for point-to-point movements (Hogan,
1984; Flash and Hogan, 1985; Hasan, 1986; Uno et al., 1989; Viviani and
Flash, 1995), leads to sensible predictions also in the case of a
rather complex posture selection problem.
Although the target position hypothesis proved adequate for
interpreting the similarity between the pattern of errors in both conditions, it seems to be too simple to accommodate also the results
of the partial correlation analysis (Fig. 10). Indeed, in condition RR,
a significant component of the postural variability around the mean in
the pointing phase was accounted for by the corresponding variability
of the locating posture. Although much weaker, a similar dependency was
present also in condition LR. Across subjects and targets, all
significant correlations between residuals were negative, indicating a
tendency to raise and lower symmetrically right and left elbows (this
was true also for most of the nonsignificant correlations, see Table
3). Moreover, correlations were stronger for midsagittal targets (Table
4). Because the locating posture cannot be recovered from the
(misperceived) target position, these findings imply that a memory
trace of this postural information is still available during the
pointing phase.
The spontaneous tendency to match postural angles, even in condition LR
in which this strategy was not a viable solution for reaching the
desired position, suggests a functional dissociation between the
processes that control the endpoint and specify the average pointing
posture and those responsible for the trial-by-trial deviations from
this average. The former implement the minimum-distance algorithm,
taking into account only the desired endpoint. The latter, which do not
contribute to the specification of the endpoint, are influenced by the
memory trace of the locating posture. A more complex cost function may
take into account this trace, leading to a model for individual
trials.
A functional scheme
Figure 15C summarizes our view of the sequence of steps
and factors involved in pointing to kinesthetic targets.
(1) The average locating posture adopted in the locating phase is
selected by the minimum-distance principle. The variations around the
average (all compatible with the finger position imposed by the robot)
are primarily idiosyncratic. They do not have a significant influence
on the accuracy of the subsequent pointing.
(2) Systematic biases in the perception of postural angles generate an
erroneous representation of the target. Framing the description of the
constant errors in psychophysical rather than motor terms afforded a
natural explanation of the symmetric lateral shifts of the endpoints in
conditions RR and LR. In fact, the shift would simply be a consequence
of the fact that the pointing movement is made by two arms arranged
symmetrically with respect to the midsagittal plane.
(3) Two processes are active during the pointing phase. The first one
translates back (through the minimum-distance principle) the memorized
(shoulder-centered) 3D representation of the target into an average 4D
postural configuration. The second process is a memory trace
responsible for a portion of the trial-by-trial postural variations
around the average. The extent to which these variations reflect the
locating posture depends on the task condition (RR vs LR). In either
condition, however, the average pointing position is unaffected by the
memory trace.
(4) An additional source of noise blurs the 3D representation of the
target position subserving the planning of the pointing movement.
Ultimately, pointing variability reflects mostly this source of
noise.
In this scheme, the coding of the target position in an extrinsic frame
of reference has the same central role that this coding does in current
theorizing on visuomanual reaching. The fact that the scheme accounted
fairly accurately for the experimental results brings further credence
to the notion of an amodal system of spatial representation shared by
visual and kinesthetic inputs. So far there is no direct
neurophysiological evidence of such a system because the task of
pointing to kinesthetic targets has not been investigated yet in
monkeys. However, the presence in area 5 of cells tuned to the location
of visual targets in a shoulder-centered frame of reference (Lacquaniti
et al., 1995) suggests that the spatial coding postulated by the target
position hypothesis may be implemented in the parietal cortex. Future
work involving a wider range of operating conditions should clarify the
extent and nature of the factors specific to each perceptuomotor
channel.
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FOOTNOTES |
Received Aug. 18, 1997; revised Oct. 31, 1997; accepted Nov. 25, 1997.
This research was supported in part by a grant from the Pôle
Rhône-Alpes de Sciences Cognitives and by Vita-Salute University HSR Research Grant A2876. We thank the two anonymous reviewers for
their critical reading of this article and their suggestions.
Correspondence should be addressed to Dr. Paolo Viviani, Department of
Psychobiology, Faculty of Psychology and Educational Sciences,
University of Geneva, 9, Route de Drize, 1227 Carouge, Switzerland.
| |
APPENDIX |
We describe the characterization of the posture by the Cardan
angles and the application of the minimum-distance principle to the
experimental data. The following notation is adopted:
[x,y,z]: finger
cartesian coordinates with respect to the shoulder.
[xe,ye,ze]:
elbow cartesian coordinates with respect to the shoulder.
La: arm length.
Lf: forearm length.
R
=
:
distance between finger and shoulder.
Posture angle
The Cardan system of reference is illustrated in Figure
16. For any given hand (finger) and
elbow position, an initial arm posture (S,
E0, H0) is
defined in which the finger and the elbow belong to the midsagittal
plane, the finger points forward along the y-axis, and the
elbow flexion is equal to that of the final posture. Then, a unique
sequence of two rotations, 
and 
, moves the finger from the
initial to the final (H2 
H3) position through the intermediate
posture (S, E1,
H1). The same rotations move the elbow to
a position (E2) that corresponds, by
definition, to a zero amount of rotation around the shoulder-finger
axis. Finally, a third rotation, , around this axis brings the elbow to its final (E3) position. The triple
[,,] are the Cardan angles in the zxy
convention.
The Cardan angles can be expressed as a function of the finger and
elbow coordinates by the following formulae:
|
(A.1a)
|
|
(A.1b)
|
|
(A.1c)
|
where and correspond to the azimuth and elevation of the
finger and corresponds to the posture angle used in the Results. The function arctan2(u,v) computes
arctan(v/u), specifying also the quadrant in
which the resulting angle is comprised.
Conversely, finger and elbow coordinates are expressed as a function of
the Cardan angles by the following formulae:
|
(A.2a)
|
|
(A.2b)
|
|
(A.2c)
|
|
(A.3a)
|
|
(A.3b)
|
|
(A.3c)
|
where
is the angle between the shoulder-finger and the shoulder-elbow
axes.
Applying the minimum-distance principle
To any finger position (endpoint) corresponds an infinite set of
postures that are compatible with that endpoint. According to the
minimum-distance principle, the indeterminacy of the inverse kinematic
problem is eliminated by selecting, within this infinite set, the
(unique) quadruple [,,,] (euclidian distance) of orientation angles that minimizes the following cost function:
where
[0,0,0,0]
are the orientation angles of the reference posture
P* and
[c,c,c,c]
are positive scale factors. Any quadruple [,,,] coding
for a given observed endpoint
[x,y,z] must satisfy the geometrical
constraint expressed by the formulae:
|
(A.4a)
|
|
(A.4b)
|
|
(A.4c)
|
This constraint dissipates three of the four postural degrees of
freedom provided by the orientation angles. Thus, the selection of the
minimum-distance posture can be formulated as a one-dimensional problem
by parametrizing the set of compatible postures by any one of the
angles.
Computationally, the most convenient choice is to express the angles
, , and as functions of the coordinates
[x,y,z] and of the arm elevation
angle chosen as the parameter. The angle is provided directly
from Equation A.4c:
|
(A.5a)
|
The other two orientation angles are obtained by solving Equations
A.4a and A.4b, considered
as a nonlinear system in the unknowns sin and sin :
![<UP>sin</UP> &eegr;=<FR><NU>1</NU><DE>2</DE></FR> <FR><NU>x[(x<SUP>2</SUP>+y<SUP>2</SUP>)+(L<SUP>2</SUP><SUB>a</SUB> <UP>sin</UP><SUP>2</SUP>&thgr;−L<SUP>2</SUP><SUB>f</SUB><UP>sin</UP><SUP>2</SUP>&bgr;)]±y<RAD><RCD>&Dgr;</RCD></RAD></NU><DE>L<SUB>a</SUB>(x<SUP>2</SUP>+y<SUP>2</SUP>)<UP>sin</UP>&thgr;</DE></FR>,](/content/vol18/issue4/fulltext/1528/img019.gif)
|
(A.5b)
|
![<UP>sin</UP> &agr;=<FR><NU>1</NU><DE>2</DE></FR> <FR><NU>x[(x<SUP>2</SUP>+y<SUP>2</SUP>)−(L<SUP>2</SUP><SUB>a</SUB><UP>sin</UP><SUP>2</SUP>&thgr;−L<SUP>2</SUP><SUB>f</SUB><UP>sin</UP><SUP>2</SUP>&bgr;)]∓y<RAD><RCD>&Dgr;</RCD></RAD></NU><DE>L<SUB>f</SUB>(x<SUP>2</SUP>+y<SUP>2</SUP>)<UP>sin</UP>&bgr;</DE></FR>,](/content/vol18/issue4/fulltext/1528/img020.gif)
|
(A.5c)
|
where:
The sign indeterminacy in these equations corresponds to the fact
that, for each arm elevation , there are two equivalent postures for
reaching the endpoint that are symmetric with respect to the vertical
plane that contains the endpoint and the shoulder. For a given
endpoint, the range of permissible values of the parameter (i.e.,
the set of compatible postures) is dictated by anthropometric constraints. For instance, when the endpoint distance R is
equal to the length of the fully stretched arm, only one value is permissible. Let:
Then the constraints are expressed as follows:
|
(A.6a)
|
|
(A.6b)
|
|
(A.6c)
|
|
(A.6d)
|
The model fitting was performed by a doubly nested application of
the Simplex algorithm for constrained minimization. At the inner level,
for a fixed set of scale factors, applying the algorithm to the
parameter (constrained by Equations A.6) permitted one
to select, for each target independently, the posture that minimizes
the distance 
from the reference P*. Note that the two
equivalent postures corresponding to a given angle (see above) do
not have necessarily the same distance from the reference configuration. Thus, the procedure at the inner level always tested both solutions. At the outer level, we computed the normal regression across all targets between the actual posture angle and the posture
angle corresponding to each minimum-distance posture. The global cost
function at this level was defined by the quantity 
2 = (1 b0)2 + (1 r)2, where b0 is the
slope of the normal regression line, r = (A2min/A2max 1)/(A2min/A2max + 1) is the so-called normal coefficient of correlation, and A2min and
A2max are the eigenvalues of
covariance matrix. 2 measures the quality of the
fit for the current set of scale factors. The optimal scale factors
were found by minimizing this global cost under the constraint
c + c + c + c = 1.
| |
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Top
Abstract
Introduction
