1Medical Research Council Group for Action and Perception, Centre for Vision Research and Departments of Psychology and Biology, York University, Toronto, Ontario M3J 1P3, Canada; and 2Department of Medical Physics and Biophysics, University of Nijmegen, NL 6525 EZ Nijmegen, The Netherlands
![]() |
ABSTRACT |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Medendorp, W. P.,
J. D. Crawford,
D.Y.P. Henriques,
J.A.M. Van Gisbergen, and
C.C.A.M. Gielen.
Kinematic Strategies for Upper Arm-Forearm Coordination in
Three Dimensions.
J. Neurophysiol. 84: 2302-2316, 2000.
This study addressed the question of how the
three-dimensional (3-D) control strategy for the upper arm depends on
what the forearm is doing. Subjects were instructed to point a
laserattached in line with the upper arm
toward various visual
targets, such that two-dimensional (2-D) pointing directions of the
upper arm were held constant across different tasks. For each such
task, subjects maintained one of several static upper arm-forearm
configurations, i.e., each with a set elbow angle and forearm
orientation. Upper arm, forearm, and eye orientations were measured
with the use of 3-D search coils. The results confirmed that Donders'
law (a behavioral restriction of 3-D orientation vectors to a 2-D
"surface") does not hold across all pointing tasks,
i.e., for a given pointing target, upper arm torsion varied widely.
However, for any one static elbow configuration, torsional variance was
considerably reduced and was independent of previous arm position,
resulting in a thin, Donders-like surface of orientation vectors. More
importantly, the shape of this surface (which describes
upper arm torsion as a function of its 2-D pointing direction) depended
on both elbow angle and forearm orientation. For pointing with the arm
fully extended or with the elbow flexed in the horizontal plane, a
Listing's-law-like strategy was observed, minimizing shoulder
rotations to and from center at the cost of position-dependent tilts in
the forearm. In contrast, when the arm was bent in the vertical plane,
the surface of best fit showed a Fick-like twist that
increased continuously as a function of static elbow flexion, thereby
reducing position-dependent tilts of the forearm with respect to
gravity. In each case, the torsional variance from these surfaces
remained constant, suggesting that Donders' law was obeyed equally
well for each task condition. Further experiments established that
these kinematic rules were independent of gaze direction and eye
orientation, suggesting that Donders' law of the arm does not
coordinate with Listing's law for the eye. These results revive the
idea that Donders' law is an important governing principle for the
control of arm movements but also suggest that its various forms may
only be limited manifestations of a more general set of
context-dependent kinematic rules. We propose that these rules are
implemented by neural velocity commands arising as a function of
initial arm orientation and desired pointing direction, calculated such
that the torsional orientation of the upper arm is implicitly
coordinated with desired forearm posture.
![]() |
INTRODUCTION |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
The purpose of this study was to clarify the rules that govern
the choice between various three-dimensional (3-D) arm
configurations for different motor tasks. The human arm is provided
with multiple degrees of freedom so that a given position of the hand
in space can result from many different joint configurations (e.g.,
Buchanan et al. 1997). For example, one of these
joints
the shoulder
is free to rotate about any axis in 3-D space,
which allows a specific pointing direction of the upper arm to be
obtained in different possible orientations. This poses a degrees
of freedom problem, considered to be one of the most basic, yet
hardest to unravel computational challenges encountered in the area of
neural control (Bernstein 1967
; Turvey
1990
).
In this respect the shoulder is similar to the eye, a structure with 3 dfone more than necessary to specify its two-dimensional (2-D) gaze
direction. It is well established that 3-D orientation of the eye is
uniquely determined by gaze direction (at least when the head is
stationary and the eye is looking far away), effectively reducing the
number of controlled degrees of freedom from three to two
(Donders 1848
). This general principle is now known as
Donders' law. Listing's law further specifies this
constraint as follows: rotation vectors, which describe eye positions
as a rotation relative to some reference position, are confined to a
flat range called Listing's plane (Ferman et al. 1987
;
Tweed and Vilis 1990
). Considering the fact that both
the eye and shoulder have three rotational degrees of freedom, it is
perhaps not surprising that Donders' law also applies to straight-arm
pointing movements (Hore et al. 1992
; Miller et
al. 1992
; Straumann et al. 1991
; Theeuwen
et al. 1993
). In particular, during straight-arm pointing, the
upper arm obeys a rule very similar to Listing's law, leading some to
suggest that the arm-control system might possess a Donders' operator
that takes in desired pointing direction and outputs a command for
desired 3-D arm orientation (Crawford and Vilis 1995
).
In contrast to these observations suggesting a consistent and
reproducible reduction of the number of degrees of freedom, other
authors have reported violations of Donders' law for the arm
(Desmurget et al. 1998; Gielen et al.
1997
; Soechting et al. 1995
). For example,
Soechting et al. (1995)
reported that the orientation of
the upper arm for a given fingertip position in space depends on the
starting position of the targeting arm movement. These results would
seem to suggest that Donders' law has a much more limited application
for understanding the neural control of arm movement, particularly for
bent-arm configurations. Thus at this time the importance of Donders'
law for limb motor control seems tenuous, or at best, controversial.
One possible clue for resolving this controversy comes from recent
experiments that show how 3-D head orientation may depend on
the contribution of the eye versus head position to a particular gaze
direction (Ceylan et al. 2000; Crawford et al.
1999
). During normal gaze shifts where the head acts as a
platform for the eye, head orientations conform to a form of Donders'
law called the Fick strategy (Glenn and Vilis 1992
;
Medendorp et al. 1998
; Radau et al. 1994
;
Theeuwen et al. 1993
). This entails that the
orientations of the head behave qualitatively like those of a Fick
gimbal, which has a horizontal axis nested within a space-fixed
vertical axis. As a result, the rotation vectors representing 3-D head orientations define a twisted saddle-shaped surface with nonzero torsional components at the oblique facing directions. However, when
the head was forced to act like a gaze-pointer (imposed by pin-hole
goggles or a head-mounted laser), its twisted surface flattened out to
become more Listing-like (Ceylan et al. 2000
; Crawford et al. 1999
). Moreover, when head movements
were dissociated from gaze shifts, Donders' law for the head broke
down in favor of a minimum-rotation strategy. Thus if one pooled the
data from all of these conditions, it would appear as though Donders'
law were not obeyed at all, whereas considered individually, different kinematic strategies (of which some obeyed Donders' law) were used to
optimize various motor task constraints.
These results further imply that Donders' law reflects a control
principle for eye and head coordination since the control strategy of the head is dependent on what the eye is doing. In an
attempt to derive general principles from their results, Ceylan et al. (2000) suggested that Listing's law is the optimal
strategy for a system primarily concerned with pointing, whereas the
Fick strategy was thought to be ideal for a weight-bearing inverted pendulum (to minimize torques resulting from gravity).
Just as the head acts as a platform for eye movements, the upper arm acts as a platform for the forearm during normal arm movements. The forearm is sometimes a pointer (like the eye) and sometimes an inverted pendulum with the potential for being used as a weight-bearing pillar, so it could make sense to incorporate elements of different Donders strategies into a control system that accounts for upper arm-forearm coordination. In other words, the choice of strategy for control of upper arm torsion would have to account for the way that it is coordinated with the forearm.
Thus whereas earlier reports suggested that a simple Donders' law is
used in arm control (Hore et al. 1992), more recent
studies (Nishikawa et al. 1999
; Soechting et al.
1995
) show that final arm postures are the result of a complex
combination of kinematic and dynamic factors. The present study pursues
these ideas further, wondering whether there could be a more general
kinematic law that governs the range of arm positions in natural
movement tasks, perhaps choosing different Donders strategies to
optimize different task conditions. In particular, the present study
investigates whether task-dependencies related to coordination with the
forearm could affect the manifestation of Donders' law for
the upper arm. But before proceeding to METHODS,
let us first consider the intimate kinematic linkage between upper arm
orientation and forearm posture and how this might be influenced by
different Donders strategies of the upper arm.
Arm kinematics and theory
Upper arm torsionor rotation of the upper arm around its long
axis
is often equated with the arm's redundant degree of freedom (Hore et al. 1992
). However, this is only true when the
arm is fully extended. In contrast, whenever the elbow is bent, upper arm torsion determines forearm orientation
and thus hand position (Soechting et al. 1995
). Take for example the arm
postures simulated in Fig. 1. This figure
is set up to illustrate two of the main tasks used in the current
study. But more importantly, it shows two different ways in which upper
arm torsion could be used to determine forearm posture in a
kinematically redundant task, and how these different strategies would
be expressed in rotation vector space. Figure 1, left and
middle, shows simulated "stick figures" of the upper arm
and forearm, as viewed from the front of the "subject," whereas
Fig. 1, right, shows the surface of best fit to the
corresponding orientation vectors of the upper arm. In each case, the
task is to point the upper arm toward one of nine targets, with the
elbow angle set at 90°.
|
Let us first consider the difference between the left and
middle columns. If we just look at the central arm position
of each of the Fig. 1, left (A and D),
where the upper arm points at a target straight out of the page, one
can see that the upper arm is bent upward by 90°. We called this the
V90 task. In contrast, for the same target direction and
elbow angle (Fig. 1, B and E, middle
column), the H90 task aligned the forearm horizontally and pointing to the right (subject's left). Thus the upper arm has
been rotating torsionally by 90° between the V90 and H90
tasks (left and middle columns). This is
reminiscent of some of the arm movements described by Soechting
et al. (1995); and there is no question that Donders' law must
be violated to move the arm thus i.e., between these two configurations.
What is at issue here, is that once the baseline
torsion is selected, i.e., within the V90 or H90 task, how it might
further depend on the 2-D pointing direction of the upper arm? In other words, how would upper arm torsion be selected for the other pointing directions shown within each panel (Fig. 1, A, B, D, and
E) and how would this further affect the posture of the
forearm? Let us first suppose that the upper arm follows a Listing's
law strategy (Fig. 1, top row). If the arm orientations
shown in A and B are each allowed their own
reference positionthat is, they are each described relative to the
central arm position of that panel
then the orientation vectors for
the upper arm would have to align in a plane, as shown in Fig.
1C. (But note that if 1 common reference position were used,
these 2 panels would give 2 different planes with a large torsional
shift between them.) In contrast, if in these two tasks the upper arm
followed a Fick strategy (Fig. 1, bottom row), then its
orientation vectors would form a twisted surface (Fig.
1F), i.e., in Cartesian coordinates, its torsion would
depend on pointing direction.
The important thing to note hererelating this back to the stick
figures in the two left columns
is that these different
strategies produce different arm configurations as a function of upper
arm pointing direction. In particular, they would produce different forearm tilts at the oblique arm positions, in the corners of each
panel. For example, note that in the upward-oblique V90 positions the
forearm tilts more inward
as projected onto the page
with the Fick
strategy compared with the Listing strategy. More precisely, it can be
easily shown that with the Fick strategy, the plane containing the
upper arm and forearm remains fixed with respect to the horizon for
every pointing target, whereas the Listing strategy causes this plane
to tilt at the oblique positions. Therefore the choice of 3-D control
strategy for the upper arm
Donders' or otherwise
will have real
consequences for hand-arm posture, and one should bear in mind that
whenever the elbow is bent the representations of upper arm torsion
shown in RESULTS also correspond precisely to tilts in the
forearm plane.
![]() |
METHODS |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Subjects
Experiments were performed on 19 human subjects, who were tested
in nine different task conditions as described in the following text.
The main experiments (Figs. 27, RESULTS) were performed with naive subjects. In some additional control experiments (see Fig.
8, RESULTS), three subjects, who were familiar with the
general purpose of the experiments (but not at that time with the
hypotheses or results), also participated. Their basic results were not
different from those of the other subjects. All subjects signed
informed consent to participate in the experiment. All subjects but one were right-handed, and all were free of any sensory, perceptual, or
motor disorders. All pointing movements were made using the right arm.
Experimental setup
Three-dimensional upper arm orientations were measured using
custom-built 3-D magnetic search coils as described elsewhere (Glenn and Vilis 1992; Henriques et al.
1998
; Tweed et al. 1990
). In 12 subjects, we
also monitored the orientation of the forearm. In addition, the
orientation of the right eye was measured in six subjects using Skalar
search coils. Subjects sat and were tested with the torso rotated 45°
leftward with respect to a frontally placed stimulus array (see
Stimuli) (see also Hore et al. 1992
) so that
the central pointing target was near the center of the arm's
mechanical range. The limb and eye movements were measured using three
mutually orthogonal magnetic fields (frequencies, 90, 124, and 250 kHz)
generated by field coils 2 m across. After demodulation, the three
voltages from each coil were sampled at 100 Hz. Calibration and
accuracy were as described previously (Henriques et al.
1998
; Klier and Crawford 1998
).
Stimuli
Experiments were either done in normal lighting conditions or
with the background in complete darkness. The targets, either 1-cm-diam
white dots (for experiments in the light) or green light-emitting diodes (LEDs; 0.17°; 2.0 cd/m2; for dark
experiments), were mounted on a vertical screen oriented in parallel to
the horizontal-vertical magnetic fields at a distance of 1.1 m
before the subject's eyes. The target array contained a total of nine
targets arranged in a square grid. The four cardinal targets were at
40° right, left, above, and below; the four oblique targets were
48° from the center of the right shoulder. Subjects pointed toward
these targets either with the arm fully extended in the normal way
(Henriques et al. 1998) or with the use of a laser
pointer with the elbow at various configurations described in the next
section. The laser pointer was attached to the distal part of the upper
arm, about 5 cm from the elbow joint above the tendon from the triceps
muscle, and secured in parallel to the upper arm. The central target
was placed so that the upper arm was parallel to the frontal magnetic
field (orthogonal to the target screen) on pointing at it.
Before the experiment began, the subject was familiarized with the
positions of the targets on the screen. At the start of each task, the
subject pointed the upper arm toward and visually fixated the center
target for 3 s to define a reference position for the arm and
right eye, respectively. Thereafter the subject was required to point
toward each of the stimuli in the nine-target array at 2.5-s intervals.
In the light, the stimulus order was determined by verbal commands to
the subject, e.g., up-left, down-right, middle-center (see
Ceylan et al. 2000) (see also Fig. 1), whereas in the
dark, subjects pointed toward the LEDs as they came on (each for
2.5 s with no gap interval in between). In either case, the nine
stimuli were repetitively "presented" in a random sequence of nine
so that subjects pointed toward each target the same number of times
from various initial positions. Experiments in the dark were used to
control for potentially distracting visual feedback from the forearm,
but the results revealed no significant differences for pointing with
or without background lighting (see RESULTS). Sessions were
divided into 50-s blocks, each block including two pointing movements
to each of the nine targets. Each task consisted of three blocks unless
otherwise stated. A brief rest was provided between blocks.
Experimental protocols
The main hypothesis tested in this study is that the control strategy of the upper arm is dependent on the orientation of the forearm relative to the upper arm. To this end, we introduced several task conditions in which we varied the orientation of the forearm relative to the upper arm. The arm-mounted laser paradigm was used to ensure that the upper arm used the same 2-D pointing direction for each target across tasks, without determining the third degree of freedom. Our basic hypothesis was that with straight arm pointing, the upper arm would use a more Listing-like Donders strategy, whereas with the elbow bent and held with the arm in a vertical plane, the upper arm would use the Fick strategy to minimize extraneous torsional torques on the arm resulting from gravity. To test this hypothesis, we used the following tasks.
During the control task, C, subjects made pointing movements (without laser) to the nine targets with the fully extended arm. During the control laser task, CL, the subject again adopted an outstretched arm but now pointed the laser to the target array. In some subjects, there was a slight dissociation of about a few degrees of the pointing direction of the laser and the natural pointing direction of the straight arm. However, by comparing both control tasks we were able to show that the laser pointer did not affect the control strategy of the upper arm (see RESULTS).
During the vertically bent-arm laser tasks, subjects were instructed to first stretch their arm straight out, bring their thumb up, and subsequently rotate their forearm vertically in the direction of the shoulder by three different angles: 45, 90, and 135°, while pointing with the upper arm laser toward the central target. In this way, the initial arm configuration was set with the upper and forearm contained in vertical plane, without providing the subject with any explicit verbal instructions about 3-D arm orientation that might influence their subsequent behavior. We will refer to these task conditions as: the V45 task, the V90 task (as in Fig. 1, left), and the V135 task, respectively. Subsequently, subjects were instructed to point the laser to each LED in the dark and to maintain their initial elbow angle, but no further specific instructions were given regarding the orientation of the forearm in space.
We also performed some additional control experiments. First, we tested whether the configuration, in which the forearm was bent, either horizontally or vertically (as in the preceding text), has implications for the pointing strategy. We examined this by the horizontally bent-arm laser task, the H90 task. During this task, the subjects were instructed to first stretch their arm, point with their thumb to the left, and subsequently rotate their forearm horizontally toward their body over an angle of 90°, meanwhile pointing with the upper arm to the central target (see Fig. 1, middle). Thereafter, subjects pointed the laser to the various targets while preserving the elbow joint angle at 90°.
The next control was designed to see how well subjects would follow a Fick rule when explicitly instructed to maintain the forearm vertically at all times. During this task, the subject initially took the same elbow configuration as during the V90 task but now was explicitly told to keep the forearm vertical with respect to gravity for all pointing directions (as demonstrated by the experimenter). To see his forearm in this task, the subject pointed in dim background lightning to the target array. By definition, this task did not involve a stable elbow angle, but because on average the elbow varied about 90°, depending on target elevation, we called it the V90v task.
Considering our hypothesis that the upper arm might use a Fick strategy to reduce torsional gravitational torques on the forearm, we also wondered whether loading the hand (and thereby increasing these potential torques) would further alter this strategy. Therefore in the V90w task, the subject carried a hand-held 1-kg weight (which was strapped across the hand with the weight nestled in the palm) starting in a 90° vertically rotated forearm position while pointing the laser toward the nine targets. This also acted as a control for inertial effects.
The final control experiment was inspired by the fact that head
movements only obey Donders' law when they are part of a gaze shift
(Ceylan et al. 2000). By using the gaze-fixation
task, GF135, we tested whether there is a similar gaze dependency
for arm movements. During the GF135 task, subjects were instructed to
keep their head still and their eyes on the center target while
pointing the laser to targets in the periphery using the same elbow
configuration as in the V135 task (this angle was used because the
subjects found it to be the least fatiguing). Subjects reported that
gaze-fixation tasks were easy to perform. By measuring movements of the
right eye, we checked whether subjects indeed fixated the center target throughout the task. Except for the gaze fixation task, all experiments took place under head-free conditions. As a variation on this concept
and to test the hypotheses of Straumann et al. (1991)
concerning 3-D eye-arm coordination (discussed later), we also tested
four subjects in the C and V135 task with the head fixed (with the use
of a bite bar) as opposed to moving freely (as in the other experiments).
Data analysis
From the 3-D coil signals, we computed rotation vectors that
represent any instantaneous arm or eye position as the result of a
virtual rotation from a fixed reference position to the current position. In the space-fixed right-handed coordinate system, the rotation vector is defined by
![]() |
(1) |
The rotation vector describing the orientation of the forearm with
respect to the upper arm, FU, was
computed from the rotation vectors characterizing the orientation of
both the upper arm and forearm in space,
US and
FS, respectively, using
FU =
FS
US. In this way, we were able to
check the ability of subjects maintaining a constant elbow angle,
according to the instruction, when pointing for the different bent-arm
configurations. Note that our experimental protocols and hypotheses
were not dependent on a high degree of precision in maintaining the
elbow angle, but we wished to check that subjects did not show any
systematic drift in this angle. In all subjects, we found some
trial-to-trial variation for all elbow angles with largest variation
for the V45 task (about 10° SD). The standard deviations of the elbow angle for the C, CL, V45, V90, V135, and H90 task were 3.3, 4.3, 10.1, 7.2, 3.1, and 7.4°, respectively (averaged across subjects), which we
deemed sufficiently small for the purpose of the present experiments.
The important analysis in this study concerned the 3-D orientation of
the upper arm. Note that we did not analyze the trajectories of the
ongoing movements but rather the range of orientations used during
fixations, as in the previous study by Ceylan et al. (2000). Therefore onset and offset of each arm movement between targets were determined on the basis of an angular velocity criterion (velocity threshold 5°/s) (see Medendorp et al. 1999
).
All onset/offset markings were visually checked and corrected if
necessary. The 3-D pattern of upper arm orientations at the offset
positions were then computed by fitting a second-order surface to the
rotation vector data (Hore et al. 1992
; Miller et
al. 1992
; Straumann et al. 1991
; Theeuwen
et al. 1993
) as follows
![]() |
(2) |
If parameters d, e, and f are zero, the surface
is planar, which means that Listing's law holds perfectly. A negative
twist score (parameter e) indicates that orientations of the
arm are similar to those produced by a Fick gimbal system, which has a horizontal rotation axis nested within a vertical rotation axis. A
perfect Fick gimbal has a twist score of 1. In contrast, for a system
that behaves like a perfect Helmholtz gimbal system, for which the
order of nesting in the rotation axes is reversed compared with the
Fick-system, the twist score would be +1 (Theeuwen et al.
1993
). But in practice, each of these parameters can fall anywhere in the continuum from Fick, to Listing, to Helmholtz, and
beyond (Ceylan et al. 2000
).
The scatter of the data relative to the fitted surface (commonly denoted as the thickness of the surface) is defined by the standard deviation of the distances of all samples in the rx direction to the fitted surface (in degrees). The smaller the thickness, the closer the rotation vectors stay to their surface, and therefore the better Donders' law is obeyed. Unless otherwise specified, an ANOVA was used to determine whether differences in the results between various task conditions were statistically significant (P < 0.05).
![]() |
RESULTS |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Task-dependent manifestations of 3-D constraints
To test the hypothesis that the control strategy of the upper arm can be manipulated by changes in arm kinematics, six subjects performed the following pointing tasks: C (control), CL (straight arm laser), and the three vertically bent-arm laser tasks V45, V90, and V135 (see METHODS). The question to be addressed is whether these various kinematics of the arm affect the way in which the control system of the upper arm deals with its three rotational degrees of freedom. To this end, we analyzed the upper arm data by using rotation vectors, which represent any instantaneous arm position as the result of a virtual rotation from the reference position to the current position. We will start by describing the results within a general framework before presenting each of the various findings in more detail.
Figure 2 presents the data of
subject HH for each of the tasks. Figure 2, left,
shows the horizontal and vertical components of the rotation vectors
for the upper arm in magnetic field coordinates. During the laser tasks
(CL, V45, V90, and V135), the 2-D arm trajectories were generally more
curved than those in the control task (C), but the end points where the
arm is pointing at the targetsthe subject of this study
remained
about the same. These end points are shown as squares (
) in the
"side view" and the "top view" panels (middle and
right), which show their torsional components as a function
of their horizontal and vertical components, respectively. These plots
show that, for all tasks, the subject keeps the torsional components
limited to a restricted range for all movement directions.
|
To quantify and visualize the shape of the 2-D surface defined by these
arm orientations, we fitted Eq. 2 to the data for the
movement end points (). In Fig. 2, the side view and top view of the
fitted 2-D surfaces (represented as vertical-horizontal grids) are
superimposed on the data. At first glance, these surfaces seem to fit
the data. We will provide more detailed quantitative analysis to check
the actual adherence of the data to these surfaces in subsequent
sections (see Figs. 4 and 5, and Table
1). But for the time being, we will focus
on the shape of these surfaces.
|
These surface plots immediately revealed several noteworthy differences
between the straight-arm pointing tasks (C and CL) and the bent-arm
laser tasks. For pointing with the fully extended arm (C and CL), the
surface of rotation vectors was relatively flat (i.e., Listing-like,
see Fig. 1C), meaning that upper arm "torsion" remains
approximately the same, independent of the target/pointing direction.
However, in the bent-arm laser tasks (V45, V90, V135), the surfaces of
best fit were more twisted (as in Fig. 1F), meaning that now
arm torsion depended on pointing direction. Specifically, for
upward-leftward and downward-rightward pointing directions, the upper
arm now took on a clockwise torsion (in space-fixed coordinates),
whereas the opposite corners took on a counterclockwise torsion. In
other words, the direction of this twist was consistent with
the twist observed with the Fick strategy (Hore et al.
1992; Medendorp et al. 1998
).
The observation that the fitted surface becomes more twisted during vertically bent-arm laser pointing was a general finding in all six subjects tested with this experimental protocol. In Fig. 3 we have depicted the side views of the fitted surfaces for each subject during the course of the experiment. As can be seen, the surface was fairly flat across most subjects during the control tasks (C and CL). However, all subjects showed a consistent magnitude and direction of twist during the bent-arm laser tasks (V45, V90, and V135), always in the Fick-like pattern. Moreover, there appeared to be a tendency for the surfaces to become progressively more twisted for larger elbow angles.
|
To substantiate this observation, we averaged the quantitative results of all subjects, and summarized them in Fig. 4. Graphic depictions of the average surface fits (i.e., based on fit parameters averaged across subjects) are shown in A, whereas B plots the average (±SE) twist score for each task. The difference between CL and C was not statistically significant [F(1,5) = 3.1, P = 0.14], indicating that laser pointing does not affect the control strategy of the upper arm. However, there was a systematic relationship between the upper-arm twist score and elbow angle (see Fig. 4B).
|
Although the average twist scores for all tasks were intermediate
between the ideal Listing value (0) and the ideal Fick value (1), the
degree of elbow flexion changed the value of the score along this
continuum. On average the 2-D surface of the upper arm was rather flat
(small twist score) for both the standard control task (C) and the
laser control task (CL), whereas it became progressively more twisted
(more negative twist score) in the Fick direction for pointing with
larger elbow angles (V tasks). A pair-wise comparison among the
different laser tasks revealed the following statistical analyses:
CL-V45 significant [F(1,5) = 11.3, P = 0.02]; V45-V90, significant [F(1,5) = 7.12, P = 0.04]; V90-V135 significant
[F(1,5) = 7.60, P = 0.04]. Moreover,
an ANOVA revealed significant interactions between elbow angle and
twist score [F(4,20) = 31.0; P
0.001],
suggesting that the elbow configuration is an important constraint on
the control strategy of the upper arm across the vertical-arm laser tasks.
Figure 4C shows the average (±SE) torsional thicknesses (SD) of the orientation ranges relative to their fitted surfaces, which quantifies the goodness of fit of our surfaces. In all tasks, the average (across subjects) thickness of the fitted planes was close to 4°, and the differences in thickness for different task conditions were not significant [F(4,20) = 0.47, P = 0.76]. This suggests that, despite the changes in the shape of the fitted surfaces for different elbow angles, adherence of the arm orientations to the fitted planes was equally as good in each of our vertical-arm laser tasks as the controls.
Dependence of arm orientation on previous movement history
So far the results indicate that a 2-D surface can describe the
upper arm orientations adopted during each task reasonably well. The
torsional thickness of the fitted surfaces was about 4°, which is
small considering the large torsional range of shoulder movements.
However, this is still large compared with the thickness of Listing's
plane of the eye (~1°) (Straumann et al. 1991;
Tweed and Vilis 1990
) and comparable to other ranges
that were said to not obey Donders' law (Ceylan et
al. 2000
; Soechting et al. 1995
). So where do we
draw the line between a system that obeys Donders' law and one that
does not?
One possible way is to rely not on torsional thickness per se but
rather on another part of the original definition of Donders' lawthat eye orientation is independent of the previous saccade path
(Donders 1848
). This is not always true for certain
movements of the eyes (Crawford and Vilis 1991
), head
(Ceylan et al. 2000
), and arm (Soechting et al.
1995
). But if it held true here, we could claim that these
movements still obeyed a form of Donders' law, just not as precisely
as the oculomotor system.
To test this, we calculated for each final position the torsional
distance to the fitted surface when starting in one of the eight other
positions. When this distance does not significantly deviate from zero,
there is no dependence on starting position. Figure
5, showing the results for both the
control task and the V90 task, indicates that the torsional distance to
the surface is not significantly different from zero with only a few
minor exceptions (t-test, P > 0.05 for and P < 0.05 for
). This indicates that there are
no systematic trends, producing starting position dependencies in the
scatter of the surfaces. Similar results were found for the CL, V45,
and the V135 task, suggesting that
by the definition outlined in the
preceding text
Donders' law was obeyed within each of the tasks,
albeit with a considerable amount of random scatter.
|
Does Donders' law hold globally
Up until now our analysis revealed that Donders' law for the
upper arm holds in good approximation for any particular elbow configuration. The question to be faced now is whether it is also obeyed across different tasks. Soechting et al.
(1995) emphasized that Donders' law for the arm does not hold
for movements under more general testing conditions. Is there a
discrepancy with their results and the results of the present experiment?
To explore this issue, we recomputed the arm orientations for each elbow configuration by taking one common reference position, which is the particular position when the subject is pointing to the center target in the control task (C). (Note that the previous section focused on the shape of the best-fit surface with a separate reference position in each task, so that torsional shifts between tasks would not be evident). The results for subject HH are shown in Fig. 6A, where each panel illustrates the torsional range of upper arm orientations across all tasks (gray patch) together with the specific set of arm rotation vectors for the indicated task (black subspace). As the figure demonstrates, each specific elbow configuration introduced a mean torsional shift (in addition to the twist effect described in the preceding text), and this shift was different for different elbow angles.
|
Accordingly, the corresponding arm orientations cover a sub-range of the overall range of rotation vectors. The torsional thickness for each particular task condition, ranging from 2.1 and 5.4° in this subject, was much smaller than the thickness of a 2-D surface fitted to all movement endpoints, which was 12.5°. Across all subjects tested, the average scatter of the total set of rotation vectors was 11.9 ± 3.4° (mean ± SD), ranging from 6.5°to 16.1°. This suggests that upper arm orientations do not obey Donders' law globally.
As shown by Fig. 6B, each sub-range of rotation vectors can be characterized by a different surface with a different torsional offset within the overall range. For further clarification, Fig. 6B, bottom, shows the mean shape and shift (relative to the common reference point) of the surfaces of all subjects. This shows that the 2-D surfaces for each different elbow angle can be characterized by a specific twist and torsional offset.
Why a Fick strategy during vertically bent-arm pointing
So far in all our bent-arm experiments, subjects
adopted a certain initial elbow configuration by bending their arm in
vertical direction, which led to the adoption of a Fick-like strategy. This strategy ensures that vertical movements will occur via the shortest path, which would, as argued by Hore et al.
(1992), constitute the most energy-efficient strategy for the
work against gravity. However, since such work would
decrease as the elbow flexed (moving the center of mass
toward the body and thus reducing torque on the shoulder joint), this
argument does not account for the monotonic increase of the
Fick-like twist that we observed with increasing elbow flexion. Another
consequence of Fick behavior is that a particular orientation of the
forearm will remain constant with respect to both the horizon and the
line of the pointing arm, like an earth-fixed telescope (Hore et
al. 1992
).
Based on this argument, for vertically bent-arm pointing where the
forearm is an inverted pendulum, it might be advantageous to use a
Fick-like strategy because it minimizes torques in torsional direction
due to gravity. Note that Nishikawa et al. (1999) showed that the plane of the arm did not remain invariant with
respect to gravity during a reaching task. However, with the static
elbow angles used in the current study, our subjects could have been tapping into a postural control strategy where minimization
of torsional torques with respect to gravity might be expected to be
more important. If so, then 1) one might expect an even
stronger Fick strategy when carrying a load during vertically bent-arm pointing, whereas 2) by contrast, one might expect the
advantages of Listing's law to prevail for horizontal
bent-arm pointing, where joint torques due to gravity are unavoidable,
and thus there is less incentive to optimize movement kinematics
according to the Fick strategy.
We tested these hypotheses on the basis of the following task conditions: control task (C), 90° horizontally bent-arm laser pointing (H90), the standard 90° vertically bent-arm laser pointing (V90), and 90° vertically bent-arm laser pointing by carrying a hand-held 1 kg weight (V90w). For comparison, in a fifth task (V90v), we explicitly instructed subjects to maintain their forearm vertical with respect to gravity during pointing to see if this would produce an even more extreme Fick-like constraint. The mean results of all subjects are given in Fig. 7. Figure 7A illustrates the average shape of the fitted surface for the various task conditions in the same format as Fig. 4. The corresponding twist scores and thickness values of the fitted surfaces (averaged over all subjects) are given in Fig. 7, B and C, respectively.
|
As hypothesized, the plane remained flat during horizontally bent-arm
pointing (H90). The twist score, at a value of 0.12, was even less
negative, although not significantly different from the control task
[F(1,16) = 0.6, P = 0.90]. Thus the
arm configurations observed in the H90 task resembled the Listing
configurations shown in Fig. 1B rather than the Fick
configurations shown in Fig. 1E.
In contrast, for vertically bent-arm pointing tasks (V90, V90v, and
V90w), the twist score became more negative, reaching a value of about
0.46 (average) as it did in the previous experiments. In comparison
with the control task (C), the increase in twist score was highly
significant [F(3,32) = 8.6, P < 0.001]. However, the load task (V90w) did not have any further effect
on the shape of the surface and neither did the V90v task. An ANOVA
indeed revealed no significant differences between the twist scores for all three V90 task conditions [F(2,21) = 0.31, P = 0.52]. So, both the specific instruction task
(V90v) and the loading task (V90w) failed to change the shape of the
surface compared with natural bent-arm pointing (V90). To summarize,
all of the vertical bent-arm tasks produced a more twisted
surface than the H90 and straight-arm controls, and each by the same
amount, tending to show the more Fick-like configurations illustrated
in Fig. 1D rather than the Listing configurations shown in
Fig. 1A.
Finally, in all tasks except the V90v task, the thickness of the fitted planes was about 4°, as can be seen in Fig. 7C. The differences in thickness values across the C, H90, V90, and V90w task conditions were not significant [F(3,32) = 0.86, P = 0.94]. However, statistical analysis suggested that Donders' law was much better obeyed in the V90v task compared with the normal V90 task [F(1,16) = 9.3, P = 0.02]. Thus although the shape of the best-fit surface remained fairly constant in this task (see preceding text), the accuracy of how Donders' law was obeyed could be manipulated by instruction and voluntary intent.
Gaze dependency of arm orientations
Ceylan et al. (2000) showed that head movements
violated Donders' law when they were dissociated from gaze shifts. We
wanted to test whether there is a similar gaze dependency for arm
movements, which are also strongly linked to gaze during pointing tasks
(e.g., Henriques et al. 1998
). To check this idea, we
applied the gaze fixation task (GF135), in which subjects were
instructed to make bent-arm pointing movements (elbow 135°) to
targets while keeping their gaze fixed on the center target. Six
subjects were tested doing the following five tasks in the light: C,
V135, GF135, CL, C (in this order).
In general the fixation task caused no trouble for the subjects. The standard deviations (averaged across subjects) of horizontal and vertical direction of the eye in space (gaze) for the entire duration of the task were only 1.6 and 1.4°, respectively, indicating that during this task arm movements were effectively dissociated from gaze shifts. However, an examination of the 2-D upper arm trajectories in this task showed that subjects were still able to point toward the targets with reasonable accuracy.
The effect of gaze fixation on the 3-D orientations of the upper arm is summarized in Fig. 8, A and B, which shows the twist score (Fig. 8A) and the thickness scores (Fig. 8B) for the various tasks. Although the gaze fixation task tends to slightly increase the scatter of the fitted 2-D surfaces relative to control V135 data (see Fig. 8B), the differences for the thickness between the various task conditions were not statistically significant [F(4,20) = 2.74, P = 0.06]. So in the case of the arm, the accuracy of how well Donders' law is obeyed did not depend on gaze direction. Neither was the shape of the best-fit surface affected by the gaze fixation task (see Fig. 8A). Statistical analysis revealed no significant differences [F(1,5) = 0.04, P = 0.85] between the twist scores in the gaze-free bent-arm task (V135) and the gaze-fixed bent-arm condition (GF135). Thus the implementation of Donders' law for the upper arm appears to be independent of gaze direction.
|
A related question is whether Donders' law of the arm is influenced by
Donders' law of the eye. Straumann et al. (1991)
suggested that the function of Donders' laws of the eye, head, and arm
is to create a synergy between these segments for coordinated action in
any part of their workspace. If such a linkage exists, then one might
expect that a change in eye orientation might affect the way that the
arm is oriented. It is well known that the eye in space obeys
Listing's law when the head is fixed, whereas it obeys the Fick
strategy when the head is free to move (Glenn and Vilis
1992
; Radau et al. 1994
; Tweed et al.
1990
). Therefore we repeated the C and V135 task with both the
head-fixed and -free conditions in four subjects for comparison. The
results are shown in Fig. 8, C and D. As shown in
Fig. 8C, this variable had no effect on the data: the upper
arm best-fit surface continued to be consistently flat with the arm
straight and consistently twisted with the arm bent vertically,
independent of gaze kinematics.
Coefficients of the fitted surfaces
Up to this point, we have quantified the range of arm orientations
on the basis of the twist score (parameter e in Eq. 2) of the fitted surface and the torsional shift (parameter
a) compared with a common reference position (Fig. 6),
thereby ignoring the values of the other coefficients characterizing
the fits. Potentially, these other parameters could be important in
developing a kinematic rule for arm control. To obtain insight in these
parameters, Table 1 lists all six parameters for each of the various
task conditions (C, CL, V45, V90, V90v, V90w, V135, GF135, and H90)
averaged across the number of subjects (n) who performed the
task in head-free conditions (except GF135). By using an ANOVA, we
determined whether there are differences in the parameter values across
tasks. The significance level, P, is given in the
bottom row of Table 1. The average torsional thickness is
given by , which ranged between 2.6 and 4.8° and is not
significantly different among tasks (P = 0.08).
Note that each task had its own reference position in this analysis.
Therefore parameter a, which quantifies the torsional deviation relative to the reference position, was rather small and
never significantly different from zero (t-test,
P < 0.05). Coefficients b and c
characterize the orientation of the plane. The value of parameter
b, which specifies the linear relationship between torsion
and the vertical arm orientation, ranged from 0.01 to 0.24 and was
only significantly different from zero for positive values
(t-test, P < 0.05). This reflects the
arm's tendency to roll clockwise when pointing downward and
counterclockwise when pointing upward. The c scores,
quantifying the relationship between torsion and the horizontal arm
orientation, differed only significantly from zero in the V90v task.
Although an ANOVA revealed that parameters b and
c are significantly different among tasks (see P
value in bottom row of Table 1), each specific value remains close to zero, which indicates that the plane is nearly aligned with
the yz plane of our coordinate system.
Parameters d, e, and f describe the curvature of
the surface. Parameter d specifies the curvature along the
torsional axis with the vertical arm orientation and is in some tasks
significantly different from zero (t-test, P < 0.05). A negative score means that the arm rolls counterclockwise
when pointing upward or downward. An ANOVA showed no significant
differences of the d score among tasks (P = 0.38), indicating that parameter d remains constant for all
task conditions. Similar results were found for parameter f,
which ranged from 0.12 to 0.08. This coefficient, which describes the
curvature along the torsional axis with horizontal arm orientation, was
not significantly different from zero for any of the tasks (t-test, P < 0.05). Also here, an ANOVA
revealed that this coefficient remains fairly constant among the
various task conditions (P = 0.59).
To conclude, as can be seen in Table 1, parameter e,
describing the twist of the surface, was always negative and
significantly different from zero in all tasks but one (H90;
t-test, P < 0.05). From the second-order
terms, it turned out that only the twist score varied highly
significantly among tasks (P 0.0001). This suggests that
most of the change in curvature in the fitted surfaces is captured by
just one parameter (e), expressing the twist of the surface
along a continuum, from Listing to Fick.
![]() |
DISCUSSION |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
This study has concentrated on the question of whether the control
strategy of the upper arm is dependent on its peripheral linkage to the
forearm. When our data are pooled across experiments and different
elbow configurations, the results show (see Fig. 6) that the upper arm
violates Donders' law (or at least does not obey a single Donders'
law) corroborating the findings of Soechting et al.
(1995). But strikingly, when one considers upper arm
orientation when pointing with a specific forearm posture, Donders'
law is consistently obeyed (see Fig. 5). Moreover, it turned out that
the manifestation of this Donders strategy is different for different
forearm postures. In cases where the forearm is fully extended or when
it is horizontally bent, a Listing-type of strategy is used (see Figs.
1, B and C, and 7), whereas in cases where the
forearm acts as an inverted pendulum, the upper arm uses a Fick-like
strategy to position the forearm (see Figs. 1, D and
F, and 2-4). These results suggest that the various forms of Donders' law observed in arm movements may provide glimpses into a
more general set of kinematic rules. Since the control strategy of the
upper arm was dependent on the forearm orientation, these kinematic
rules can be interpreted as a coordination strategy. Furthermore, we
were able to show that Donders' law for the upper arm does not
coordinate with Donders' law of the eye and that its implementation is
independent on gaze direction (Fig. 8).
Purpose of Donders' law for the arm
The kinematic redundancy of the arm has its basis in the number of
joints as well as in the large number of muscles acting across these
joints. Because of the multiple degrees of freedom, the position of the
hand in space can be reached by many joint configurations. Therefore
one of the major problems in motor control is how the upper arm control
system deals with this redundancy problem when controlling the forearm
(Buchanan et al. 1997; Turvey 1990
).
Donders' law is one possible solution to the kinematic redundancy
problem, reflecting a coordination strategy for specific upper
arm-forearm interactions.
However, in cases where the kinematic redundancy is reduced, one could
expect a break-down of Donders' law. For example, for the eye it has
been shown that it violates Listing's law during the vestibuloocular
reflex (Crawford and Vilis 1991). Also for the arm, it
is clear that one can voluntarily rotate the arm about any axis in 3-D
space, resulting in violations of Donders' law.
The present study examined several kinematically redundant
pointing behaviors of the upper arm, and found that for a limited set
of conditions (e.g., any fixed elbow angle) Donders' law held at least
as well as for straight-arm pointing but took on different forms. Could
this be a mechanical effect? Since we have not measured electromyographic (EMG) characteristics and other factors related to
muscle forces and biomechanics, it cannot be ruled out that these have
had an effect (Kamper and Rymer 1999). On the other hand, since we can freely rotate our arms torsionally, the arm is
obviously not mechanically constrained to Donders' law. Two mechanical
parameters were altered in our paradigms: the baseline level of
torsional twist in the shoulder socket (potentially affecting muscle
pulling directions) and the geometry of the arm's inertia. However,
any torsional lag on the upper arm arising from forearm inertia would
be expected to produce a one-dimensional curvature in the fitted
surface
in opposite directions for the H90 and V90 tasks
rather than
the flat or twisted surfaces (respectively) that were actually found.
Moreover, the progressive increase in the twist score observed in the
V45-V90-V135 series (where shoulder torsion did not change) is
incompatible with either mechanical explanation. This suggests that it
is the neural system that is choosing different forms of
Donders' law. Clearly, the neural system cannot ignore muscle force
and dynamic aspects of joint torques (Nishikawa et al.
1999
). To the contrary, it must account in an exquisite fashion
for these factors to optimize some variable.
Contrary to the observation of Straumann et al. (1991)
and compatible with the conclusion of Theeuwen et al.
(1993)
, the 2-D surfaces for the arm that we obtained do not
coordinate with Listing's plane of the eye. The fitted surfaces of the
upper arm were flat when pointing with the arm straight and twisted
with the arm bent vertically, irrespective of 3-D eye orientation. This
indicates that Donders' law does not serve as a synergistic control
principle for eye and arm.
Nor was elbow angle alone the sole determinant of the 2-D surface for
the upper arm because different forms of Donders' law were observed
for the same 90° elbow angle depending on forearm orientation.
Instead the important factor appears to be the interaction between elbow angle and torsional arm posture. In particular Donders' law of the upper arm appears to be influenced by forearm posture against gravity. Note that when the arm is fully extended,
upper arm torsion has little or no effect on the work done to maintain forearm posture against gravity. In this under-constrained condition, the upper arm thus used the best strategy to preserve Donders' law and
still take the shortest route between any of two arm positions: Listing's law (Ceylan et al. 2000). However, when the
elbow is bent, upper arm torsion determines the orientation of the
forearm with respect to gravity (Fig. 1), possibly providing a new
constraint on arm posture. Thus in the case where the forearm was
aligned vertically like an inverted pendulum, the upper arm adopted a Fick-like strategy. From a purely phenomenological viewpoint
without getting into cause and effect
this clearly reduced torques on the
forearm due to gravity.
To understand this point, note that gravity will produce no torsional torques on an inverted pendulum that is held to be perfectly vertical, as in the Fick constraint (Fig. 1D), but will exert growing torques for increasing off-Fick tilts, like those seen in the Listing strategy (Fig. 1A). These torques may seem negligible compared with the overall work-load of the arm, but over time, and particularly when bearing a heavy load, they become energy costly and even mechanically dangerous. Now, it is another matter to speculate that the system is actually designed to account for these factors, but one way to test this idea would be to perform our experiment with the subjects' bodies tilted on their sides to see whether the kinematic strategies for the V90 and H90 conditions reverse.
In critique of these ideas, the question arises why subjects did not
implement a more pronounced Fick strategy (more negative twist score)
when pointing with the 1-kg weight (V90w-task), where the potential
gravitational torques are even larger. Furthermore why did the
Fick-like twist not decrease from the V90 condition to V135 task, where
gravity would have less effect? One has to bear in mind that the
advantages of Listing's law do not go away for the bent armthis just
brings in potential disadvantages. One possibility is that in our
(somewhat unnatural) tasks there was an internal competition between
the factors that weigh the system toward the Listing versus Fick
strategies with elbow angle tending to tip the scales toward the
latter. However, once again this is all just speculation
the important
point, addressed in the next section, is that an upper-arm control
strategy resembling Donders' law was obeyed in these tasks and that
this strategy was systematically modified as a function of arm configuration.
How general is Donders' law as a control principle for the upper arm?
In the case of the eye, where Donders' law has long been studied,
we have seen a progression of accepting Donders' law, rejecting it and
then revising it into more complex forms. For example, Donders assumed
that his law applied to all eye movements. Then it was thought that
Listing's law for the eye does not hold for near vision. Later work,
however, has shown that the Listing plane of each eye rotates as
vergence increases (Minken and Van Gisbergen 1996;
Mok et al. 1992
). Now we know that a higher form of
Donders' law called L2, which incorporates both eye "joints,"
captures the behavior more completely (Tweed 1997
).
Could the same general principle also apply to arm movements?
If so, then one might explain an apparent discrepancy between results
in the literature: some authors found that Donders' law is obeyed
during straight-arm pointing movements (Hore et al.
1992; Miller et al. 1992
; Straumann et
al. 1991
; Theeuwen et al. 1993
), whereas others
found violations of Donders' law under less restricted conditions
(Desmurget et al. 1998
; Gielen et al.
1997
; Soechting et al. 1995
). Even when we
consider only our limited set of tasks, it is clear that Donders' law
does not hold up in general. However, for each specific task, one finds a certain range of arm orientations emerging, dependent on the limited
conditions that were set, like elbow angle and forearm orientation.
Taking these ideas one step further, if one considers an even wider set
of orientations than those in the current study, including all upper
arm and forearm orientations, elbow angle, effects of dynamic and
static forces, and their cross-correlations, they may all be optimized
according to certain learned or preprogrammed rules. These
context-dependent rules could form a lawful set of equations within a
kinematic hyperspace, where they could optimize for both
kinematic and dynamic factors (Kelso et al. 1991;
Soechting et al. 1995
; Turvey 1990
).
Viewed this way, if one looks at all possible kinematically redundant
hand paths (at least where paths are not constrained), there are
arrangementssuch as those explored in the current study
for which
Donders' law can be preserved at least for the movement end points
(Crawford et al. 1999
). But there are also situations that are not kinematically redundant (like turning a door-knob) or
where dynamic factors override kinematics, requiring a violation of
Donders' law. For example, Soechting et al. (1995)
demonstrated situations where large torsional rotations in the upper
arm provide a clear advantage in minimizing kinetic energy. Thus our
various Donders surfaces could be viewed as various slices cut along
iso-Donders surfaces through the kinematic hyperspace, whereas the task
employed by Soechting et al. (1995)
might be viewed as
cutting tangentially (or orthogonally) to these slices.
As shown in our study, the fitted 2-D surfaces resemble a kinematic
strategy for the upper arm related to how the forearm is held. Thus the
idea of a kinematic hyperspace must be incorporated into the idea of
coordination (although the latter also includes temporal
dynamics which were not addressed here) (Buchanan et al.
1997; Turvey 1990
). In this respect, our results
provide a good analogy with the head-movement study of Ceylan et
al. (2000)
, where similar task-dependencies govern the
manifestation of Donders' law for the head when controlling the eye.
Relation to other models of the control system
In the past, many different types of models have been proposed to describe the kinematics of human arm movements. Usually, the validity of these models was tested by comparison of predicted and measured postures of the arm or movements trajectories of the hand. However, hardly any studies have discussed these various models in the context of degrees of freedom of movement control.
EQUILIBRIUM-POINT HYPOTHESES.
One of the best-known models for movement control is the so-called
equilibrium-point hypothesis (Feldman et al. 1998;
Polit and Bizzi 1978
). In the context of Donders' law,
the final equilibrium point for antagonistic muscle forces need
incorporate not only the pointing direction of the arm but also its
orientation. The question is, does the brain determine this final
orientation by explicitly computing the corresponding muscular
equilibrium points? Since it is undeniably true that the mechanical
plant must have some equilibrium point at any one time
that will not
equal current position during motion
it is difficult to argue against
this theory on the basis of behavior alone, but as a control strategy,
it poses certain problems (Gottlieb 1998
). For example,
in the oculomotor system, it is clear that movements are not generated
by equilibrium-point commands but rather by velocity commands that are
sent directly to the plant and to a neural integrator that computes a
tonic 3-D eye orientation signal (Crawford et al. 1991
;
Robinson 1975
). In a sense, the latter provides an
equilibrium point command to the ocular motoneurons, but this only
specifies current eye position and is computed in somewhat
passive fashion in response to movement commands. Likewise, we suggest
that Donders' law of the arm is implemented through similar kinematic
commands (see NONHOLONOMICS AND VELOCITY CONTROL).
TRAJECTORY MINIMIZATION.
As explained by Ceylan et al. (2000), Listing's law is
optimal for pointing in the sense that it guarantees the shortest route between any of two arm positions under the constraint of Donders' law.
But it does not provide the absolute shortest path between joint
positions, for to do so consistently for all paths and positions predicts systematic violations of Donders' law (Ceylan et al. 2000
; Crawford et al. 1999
; Tweed and
Vilis 1990
). Therefore the finding that arm torsion does not
depend on starting position (see Fig. 5) is incompatible with
"minimum-rotation" principles or at least a strict interpretation
of them that would hold across all situations (Rosenbaum et al.
1999
; Soechting et al. 1995
; Uno et al.
1989
). The Fick strategy does provide the shortest path
rotation for vertical movements, but only at the cost of lengthening
the path for most horizontal movements (Hore et al. 1992
).
TRAJECTORY MINIMIZATION AGAINST GRAVITY.
Again, in the present study, a tendency to minimize rotation of the
upper arm was only observed for a subset of movements (vertical) at a
subset of elbow configurations (bent-vertical). Does this mean that the
Fick strategy was primarily being used to minimize angular arm
displacements for work against gravity, as suggested by Hore et
al. (1992)? Unlikely because these displacements became less
optimal (i.e., less Fick and more Listing) as the elbow
extended
increasing the work load against gravity, leading us back to
the postural arguments that we began with. Moreover, it would appear
that the Fick-like behavior first reported by Hore et al.
(1992)
may pertain more closely to the hand (which is what they
measured) and so relate to factors other than those that we are considering.
NONHOLONOMICS AND VELOCITY CONTROL.
Based on our discussion so far, one is left with the idea that arm
movements for pointing to (distant) targets are planned in
unconstrained, kinematically redundant coordinates that must then be
converted into the appropriate points in a kinematic hyperspace. The
various theories of trajectory minimizationin their strictest sense
are not compatible with our data but can be incorporated together with various Donders' strategies into the rules for a kinematic hyperspace, as discussed in the preceding text. So how then
would the brain implement these rules? One possibility is that the
brain could have a little box for each little sub-rule by learning and
maintaining internal models to determine the motor commands required to
perform specific tasks (Kawato 1999
; Kawato and
Wolpert 1998
). But there may be a more parsimonious alternative.
![]() |
ACKNOWLEDGMENTS |
---|
We thank L. Sergio, L. Harris, and E. Klier for critical comments on the manuscript.
This study was supported by a grant (MT-13357) from the Canadian Medical Research Council (MRC). During the period of this study, D.Y.P. Henriques was supported by an E. A. Baker Foundation-Canadian National Institute for the Blind/MRC Doctoral Research Award, and J. D. Crawford was supported by an MRC scholarship.
![]() |
FOOTNOTES |
---|
Address for reprint requests: J. D. Crawford, Dept. of Psychology, York University, 4700 Keele St., Toronto, Ontario M3J 1P3, Canada (E-mail: jdc{at}yorku.ca).
Received 10 March 2000; accepted in final form 25 July 2000.
![]() |
REFERENCES |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|