ECOLOGICAL PSYCHOLOGY $(I),43-69
Copyright O 19%, Lawrence Erlbaum Associates, Inc.
Is Limb Proprioception a
Function of the Limbs'
Inertial Eigenvectors?
Christopher C. Pagano
Clemson University, Clemson, SC and
Centerfor the Ecological Study of Perception and Action,
University of Connecticut, Storrs, CT
Steven R. Garrett
Center for the Ecological Study of Perception and Action,
University of Connecticut, Storrs, CT
M. T. Turvey
Center for the Ecological Study of Perception and Action,
University of Connecticut, Storrs, CT and
Haskins Laboratories, New Haven, CT
The abilityof humans to match the spatial orientation of occluded contralateral limbs
was investigated. We hypothesized that this ability is tied to the inertial eigenvectors
of each limb, which correspond to the limb's axes of rotational symmetry. In two
experiments, the coincidence between the eigenvectors and the spatial axes of a
forearm was broken with the expectation that the matching of occluded forearms
would vary as a function of the eigenvectors. Overall, the angles at which the limbs
were positioned were affected by the direction in which the eigenvectors were oriented
by small appended masses. Discussion focused on the importance of physical invariants for perception, and their relation to hypothesized frames of reference for proprioception and motor concrol.
Research has shown that one's impression of elbow position reflects forearm
orientation in physical space rather than the elbow joint angle. Soechting (1982),
for example, asked participants to point the right arm at a target and then reproduce
-
--
-
Requests for reprints should be sent to Christopher C. Pagano, Department of Psychology, 418
Brackett Hall, Box 34151 1, Clemson, SC 29634-151 1. E-mail: [email protected].
44
PAGANO, GARRETT, TURVEY
with the left arm either the right elbow joint angle or the orientation of the right
forearm. The standard deviation was significantly greater for matching joint angle
than for matching limb orientation. Similar experiments by Worringham, Stelmach,
and Martin (1987) also showed that participants are less accurate at perceiving joint
angles than perceiving forearm inclination, and further demonstrated that errors in
joint angle perception are biased toward matching forearm inclination. These
results provide evidence that limb orientation sense is a function of spatial variables
rather than the registration of joint angles, where these spatial variables may be
defined relative to an absolute frame of reference anchored either in the body (e.g.,
the trunk) or the environment (e.g., gravitational or spatial vertical and horizontal
axes; Soechting, 1982; Soechting & Ross, 1984). In additional experiments, participants were asked to point to a location in space, then reproduce that location
by orienting the arm to the same target endpoint, or by indicating the endpoint with
a handheld pointer (Helms Tillery, Flanders, &Soechting, 1991). The error in the
pointing task was greater than in the limb-orienting task, indicating that the limb
is perceived according to its orientation in space, rather than the location of its
endpoint in space (Helms Tillery et al., 1991; see also Flanders, Helms Tillery, &
Soechting, 1992).' Taken together, these experiments demonstrated that limb
orientation perception is tied to spatially oriented variables independent of the joint
angles, and can be quantified with reference to the environment in which the limb
is embedded. Additionally, the orientation of the limb's longitudinal axis relative
to the vertical has been suggestedas one such variable (Soechting, 1982; Soechting
& Ross, 1984).
Our research is directed at an alternative possibility first suggested by Pagano
and Turvey (1995): that the eigenvectors of the limb's inertia tensor are candidate
spatially oriented parameters for proprioception. The eigenvectors of the limb are
specific to the orientation of the forearm's mass distribution in space, not the
particular joint angles that gave rise to that orientation or the geometric position
of the endpoint in space. It is possible that the eigenvectors of the inertia tensor
account for the spatially oriented nature of proprioception observed in previous
research. A complete exposition of this hypothesis requires a definition of the inertia
tensor and an outline of a recent body of work that has established the sensitivity
of the haptic system to the inertia tensor. This work has focused primarily upon
participants' ability to perceive object properties by dynamic touch.
DYNAMIC TOUCH AND THE INERTIA TENSOR
When one grasps and wields an occluded object conjointly, there is perception of
both aspects of the object and how the body segments are oriented relative to the
'The difference in error cannot be attributed to the use of a handheld pointer. In control conditions,
participants pointed with equal accuracy to a virtual target using either the finger or pointer.
PROPRIOCEPTION AND INERTIAL EIGENVECTORS
45
object held, and vice versa (e.g., Pagano, Carello, & Turvey, in press). Though
different than their visual counterparts (see Burton, Turvey, & Solomon, 1990;
Pagano & Tuney, l993), these concurrent haptic perceptions of holding must play
a significant role in the control of actions involving the held object. Because of the
obvious role of vision in the control and coordination of activity, the role of the
haptic system is likely to go unnoticed. Without touch, however, the scaling of
muscular forces to the dimensions of the limb segments and handheld objects could
not achieve the fluency and autonomy required by these skilled behaviors (e.g.,
Forget & Lamarre, 1987; Ghez, Gordon, Ghilardi, Christakos, & Cooper, 1990;
Ghez, Gordon, Ghilardi, & Sainburg, 1995; Teasdale et al., 1994). The kind of
touching characterized by the wielding and hefting of objects is referred to as
dynamic touch. It relates primarily to the tensile states of muscles and tendons as
these tissues undergo deformation during exploratory and performatory activity
(Gibson, 1966). Gibson explained that dynamic touch "is a perceptual system in
its own right. More than any others, it is perception blended with performance, for
the information comes from muscular effort" (1966, p. 128). Thus, dynamic touch
is different from cutaneous (or tactile) touch in that muscle sensitivity plays a greater
role in information detection than does skin sensitivity. It is the haptic system based
primarily upon the "muscle sense," and is aptly described as kinesthesis employed
in the perception of objects and surfaces.*Recent research has identified the inertia
tensor Iij as the relevant mechanical quantity to which such perception is tied (e.g.,
Solomon & Turvey, 1988; see Turvey & Carello, 1995, for a review).
Rotational motions about a futed point of the kinds characteristic of movements
about a joint follow from
where . is the matrix product and x is the vector cross product (Goldstein, 1980).
In wielding an object (or moving a limb), the torque Ni, angular velocity 4, and
angular acceleration q,vectors are coupled by the inertia tensor Is. Thus I,, is a
parameter (a constant) that couples the varying torques and varying rotations. Ii, is
represented mathematically by a matrix of numbers. The calculations of these
components are done with respect to a rectangular coordinate system Oxyz.
Patently, there are indefinitely many sets of three perpendicular axes xyz that can
be anchored at a point of rotation 0 (located at the wrist, elbow, or shoulder). For
each choice of Oxyz, the components of 4 differ, but the nature in which the tensor
specifies properties of the object does not change. This is a basic property of tensors
(Lovett, l989), inertia measured about one set of axes can be transformed to inertia
measured about a different set of axes. In general, a tensor is a hypernumber-a
2 ~ ePagano,
e
Carello, and Turvey (in press) for a discussion of how dynamic touch relates to other
types of touch (e.g., haptic touch and kinesthesis);see Fitzpatrick, Carello, and Turvey (1994), Pagano
and Turvey (1995), and Turvey (1994) for discussions of the muscle sense and its role in dynamic touch.
46
PAGANO, GARRETT, TURVEY
matrix of numbers that taken together express a physical state of affairs and
transform in a particularly simple way (Moon & Spencer, 1986). Different coordinate systems Oxyz result in different tensorial components (numbers in the matrix
change), but the manner in which the tensor transforms is such that the tensor as
a whole (with all the components considered together) continues to d e h e the
object property it quantifies. As a time-independent and coordinate-independent
quantity, L, is an invariant rendering of the persistent material distribution of the
limb and wielded object. With a given point of rotation, If does not change: It is
constant property of the rigid object. It can, therefore, be used to quantify the
information for perceiving the object's unchanging dimensions (e.g., Solomon,
1988; Solomon & Turvey, 1988; Turvey, 1994). Even when an object's motions
occur about several joints, such as the wrist, elbow, and shoulder taken singly or in
combination, an invariant rendering of I,, can be found that maps onto perceived
object properties (Pagano, Fitzpatrick, & Turvey, 1993). The implication is that
dynamic touch is tuned to the invariant parameters of the object's dynamics, rather
than to the varying states (displacements,velocities) and torques (see Amazeen &
Turvey, 1996).
Represented mathematically by a symmetric 3 x 3 matrix (Goldstein, 1980), I,,'s
diagonal terms (I,, I,, I,,)-referred to as moments of inertia-uantify
the object's
rotational inertia with respect to the three orthogonal axes of rotation. 1,s' off-diagonal terms (I,, G,,I,,, I,,, I,, I,), known as products of inertia, quantify the object's
rotational inertia in directions perpendicular to the axial rotations and reflect the
asymmetrical mass distribution of the object about the axes. 4 is a symmetric tensor;
accordingly, I, = I,, and the nine components reduce to six. The moments and
products of inertia comprising I,, are calculated with respect to a rectangular
coordinate system Oxyz. Clearly, many triplets of perpendicular axes can be anchored at the point of rotation 0.For each choice of Oxyz, the components of IiJ
differ. There is, however, a form of I, that is independent of Oxyz. This invariant
form is with respect to the principal axes or eigenvectors of I,,. If the eigenvectors
are chosen as the axes, then 4 is diagonalized: The principal moments of inertia, or
eigenvalues, are on the diagonal, and all other entries are equal to zero (that is,
there are no products of inertia) (Goldstein, 1980). The eigenvectors are the
directions, with respect to 0, about which the object's resistances to rotation are
distributed evenly; they are the symmetry or body axes with respect to the fixed
point. These principal directions are fixed in the object and rotate with it. The
eigenvalues (11,12, L) are the object's resistances to rotation about the respective
eigenvectors (el, e2, ej), where one eigenvalue is the maximum moment of inertia
for the object (II),one eigenvalue is the minimum moment of inertia (IJ, and the
remaining eigenvalue is intermediate (12). Thus the eigenvectors are the axes of
maximal (el), minimal (e4, and intermediate (ez) resistance to rotational acceleration. For any real object, unique eigenvectors and eigenvalues can be found about
any given point of rotation (except for cases of degeneracy, such as when 12 = b ).
A geometric representation of I,, is provided by the inertia ellipsoid. If all possible
PROPRIOCEP'ITON AND INERTIAL EIGENVECTORS
47
axes p are passed through the point ofrotation 0 , and lengths OA, equal to (1p)-%
are laid off on each axis, the locus of points A is an ellipsoid called the inertia
ellipsoid. It is a quadratic surface with semi-axes of lengths (11)-95, (12)-%, and
(13)-%(e.g., Lovett, 1989). Thus, its radii are inversely proportional to the
eigenvalues of the inertia tensor, and correspond to the eigenvectors. In sum, the
inertia ellipsoid is a geometric representation of the distribution "in the mean" of
the mass of a body with respect to its point of rotation (Goldstein, 1980; Starzhinskii, 1982).
In light of recent research, a useful summary is that Iij provides the domains for
two sets of functions: One consists of the principal moments of inertia or eigenvalues
that map onto perceived object "magnitudes," such as object length (e.g., Fitzpanick, et al., 1994; Solomon & Turvey, 1988), shape (Burton et al., 1990), and
weight (Amazeen &Turvey, 1996); the other consists of the principal axes ofinertia
or eigenvectors that map onto perceived object "directions," such as an object's
orientation in the hand (Pagano & Turvey, 1992; Turvey, Burton, Pagano, Solomon, & Runeson, 1992), and the location of the hand relative to a wielded object
(Pagano, Kinsella-Shaw, Cassidy, &Turvey, 1994). In other words, the magnitudes
of the inertia ellipsoid, quantified by the eigenvalues, map onto perceived object
magnitudes, and the orientations of the inertia ellipsoid, quantified by the eigenvectors, map onto perceived object orientations.
EIGENVECTORS OF THE INERTIA TENSOR AS
CANDIDATE PARAMETERS FOR PROPRIOCEPTION
As demonstrated by Pagano and Turvey (1995), the understanding of dynamic
touch summarized in the preceding may apply not only to how one perceives
"attachments to the skin" (Gibson, 1966) such as tools and instruments, but also
to the very traditional concern of how one perceives the body itself. Because the
body, its limbs, and its limb segments are describable through liis, defined about the
respective joint rotation "points," we can hypothesize that a person's knowledge
about the dimensions and directions of his or her body and its appendages is given
continuously by the eigenvalues and eigenvectors of the respective tensors. Simply
put, this ability of dynamic touch to use the tissue deformation consequences of lii
may have a proprioceptive role.
Of particular importance to our hypothesis is the shape of limbs. Bodies of
animals are composed of cylindrical parts, approximately round or elliptical in
cross-section with a readily identifiable longitudinal axis (Wainwright, 1988). The
spatial orientation of a cylindrical object can be defined by the orientation of the
longitudinal axis. For example, the geometric orientation of two limb segments
can be determined from the angle created by their longitudinal axes; this method
is analogous to specifyingthe orientation of limb through joint angles or inclination
relative to vertical (e.g., Soechting, 1982). An alternative and specifically dynamic
48
PAGANO, GARRETT,TURVEY
definition of a limb's orientation assumes it to be in motion about an end point
corresponding to a relevant joint. The spatial orientation of one of the object's
eigenvectors about that point is coincident with the geometric orientation of the
longitudinal axis. This is because the point of rotation for a limb segment, located
at a joint, is typically on the longitudinal axis of that segment. The crux of the
matter is that although the shape of a limb or object, and thus its longitudinal axis,
is a geometric property, the haptic system is stimulated by mechanical parameters.
The inertial eigenvectors are mechanical parameters, and thus may quantify
properties of tissue deformation patterns. To reiterate, they are the directions of
maximal resistance, minimal resistance, and intermediate resistance, to rotational
acceleration. The coincidence of the longitudinal axis orientation and the orientation of the e3 eigenvector may allow one to know about the former by means of
the latter through dynamic touch (e.g., motions of the limbs). That is, generalizing
from what has been shown to be the case with handheld objects, the hypothesis
is that one can know about the spatial orientation of a limb by detecting its
eigenvectors. By breaking the coincidence between e3 and the longitudinal axis
of the arm, the series of experiments described below is directed at the question:
Is the perception of limb orientation a function of a limb's eigenvectors or of its
geometric orientation? A remaining issue is that for properties of limbs and limb
segments to be perceived according to 4, this argument must be applied to
multiple I,+, corresponding to multiple parts of the body concurrently.
As discussed, our research investigates the hypothesis that the orientation of an
arm is perceived according to its inertial eigenvectors defined at a relevant point of
rotation. This hypothesis should be considered part of a more general one: that
dispositions of limbs and limb segments (including magnitudes and orientations, as
well as their changes and rates of changes) are given by properties of Iij defined at
each point of rotation. Given that any movement of the body and its limbs typically
involves motions about several joints concurrently, a separate Ii, can be defined for
each segment and its corresponding joint. That is, an Iij can be assigned to each
point in joint space (the joints of the limbs and limb segments). Accordingly, there
(Pagano et al., 1993).This 1,)-fieldhas proven useful
is an inertia tensor field or 4-field
to understanding the constancy of object properties perceived by wielding about
varying degrees of freedom (Pagano et al., 1993). In ~rinciple,the hefield is
sufficiently structured to be informative not only about postures, but also about
transformations of postures. Like the contrast between the optic array (at a fixed
point of observation) and the transforming optic array (at a changing point of
observation; see Gibson, 1966, 197911986), the specifying ability of the I,,-field
should be enhanced by transformations over time.
The hypothesized Ii,-fieldfollows from other theories that propose proprioception
is a function of mechanical parameters acting locally at each joint. It has been
suggested, for example, that detection of gravitational torque N,acting at each joint
may account for the perception of limb orientation (Worringham & Stelmach,
1985; Worringham, Stelmach, & Martin, 1987). The torque produced at a joint
PROPRloCEPTlON AND INERTIAL EIGENVECTORS
49
due to gravity is proportional to the limb's angle relative to the gravitational vertical.
N, is minimal when the limb's center of mass (CM) is aligned verticallywith respect
to the joint, and maximal when CM is aligned horizontally. It is possible that
individual joint torques and neighboringjoint torque patterns can contribute to the
limb position sense (Worringham & Stelmach, 1985). That is, proprioception may
be a function of an N,-field. There are several reasons, however, to expect that
proprioception is a function of Iij,rather than N,.The first is that changing the load
on a limb also changes N, at every position; thus, N , cannot provide reliable limb
position information unless an inference process is used to compare N, at one angle
to the N , of the same limb at some other angle (Worringham & Stelmach, 1985).
In contrast, the eigenvectors of I,, are not affected by symmetrical loading, as
described in the methods of Experiment 1. Secondly, N, acts only with respect to
the gravitational vertical and fails, therefore, to address one's ability to perceive
limb orientations in planes perpendicular to gravity. Even within the vertical plane,
N,is not determinate: A mass oriented 45" above the horizontal, for example, has
the same N, as an equal mass 45" below the horizontal. In contrast, Ivis indifferent
to planes of motion. Significantly, Ii, is gravity independent. It is the three-space
expression of the second moments of an object's mass (kg) distribution about a fixed
point; it is not tied to an object's weight (kg m s - ~ ) .
The finding that perceived limb orientation varies as a function of N , has not
held up in all experiments investigating it. While Worringham and Stelmach
(1985) found evidence in favor of N,, Soechting (1982) did not (see also Cohen,
1958). It is possible that the participants in Soechting's experiment detected the
particular N, perturbations used, and may have ignored their impressions of torque
(Worringham & Stelmach, 1985). If such was the case however, then the question
arises as to what parameter governed the successful perception of limb orientation,
if it was not N,. Relatedly, Carello, Fitzpatrick, Domaniewicz, Chan, and Turvey
(1992) investigated participants' abilities to perceive the extent of occluded rods
held steadily at different angles relative to the vertical. They found the first
moment of the object's mass distribution to affect perceived extent more than N,,
which is variable over angle. Like I,,, the first moment is an invariant mechanical
parameter tied to the object's mass distribution. They hypothesized that the
influence of angle on perception is less likely the result of the object's N, than of
muscle deformation specific to the torsion of the limb's tissues needed to position
the object at different inclinations relative to the limb. Following their lead, the
present thesis investigates the possibility that 4, an invariant mechanical parameter
tied to an object's or limb's mass distribution, provides the basis for one's proprioceptive abilities.
In previous experiments directed at the eigenvectors' hypothesis (Pagano &
Turvey, 1995), participants were asked to point to a visual target with an occluded
arm while holding an object. Small masses were appended to the object, so as to
break the coincidence between the arm's eigenvectors and spatial axes. In these
experiments, participants were restricted to movements about the shoulder, with
50
PAGANO, GARRETT, TURVEY
e3 of the arm being manipulated in a horizontal plane. The outcome was that,
relative to participants' pointing in a symmetrical mass condition where e3 remained unaltered, participants pointed farther to the right when the left side was
weighted (e3 rotated to the left), and farther to the left when the right side was
weighted (e3 rotated to the right). This was the expected outcome from the
hypothesis that the perceived orientation of the arm varies with the manipulation
of the arm's eigenvectors. When, for example, the arm was pointed 2" to the left
of a target, and the participant perceived it to be pointed directly at the target,
then the participant perceived the arm in that configuration to be 2" to the right
of its actual orientation. The results of these experiments indicated that the
perception of limb direction was varied as a function of the limb's e3 eigenvector.
In contrast to the Pagano and Turvey (1995) experiments, our experiments
employed a matching procedure in which the perceived orientation of a forearm is
reproduced with the contralateral forearm. Confirming the eigenvectors' hypothesis
in the context of a matching task would be an important generalization. Velay, Roll,
and Paillard (1989), for example, have contrasted the results from pointing tasks
with those from matching tasks, and have concluded that the two manipulations
may involve separate proprioceptive mechanisms.' As discussed earlier, the matching procedure has been employed to demonstrate that position sense at the elbow
reflects forearm orientation, rather than elbow joint angle (Soechting, 1982;
Worringham et al., 1987). In experiments by Soechting (1982) and Worringham
et al. (1987), each forearm's e3 was coincident with the segment's geometric orientation, as specified by the orientation of its longitudinal axis. Thus, possibly their
participants were actively matching the orientation of the forearm eigenvectors. In
our experiments, masses were attached to a "splint" attached to the forearm such
that the inertial eigenvectors were rotated upward, downward, or not at all. It was
expected that participantswould orient the contralateral arm oriented above, below,
or even with the manipulated target arm, respectively, to match the orientation of
the forearm eigenvectors. Alternatively, if participants are sensitive to the geometric
angle at the joint or the spatial orientation of the longitudinal axis, then no difference
in limb matching should be observed as a function of the differential weighting.
Regardless of the particular task in question (e.g., matching instead of pointing), it
is expected that the dependence of perceived limb orientation on lowill remain the
same. Velay et al. (1989) suggested that in both matching and pointing tasks "... the
basic position sense is coded in terms of relative angular position in intrapersonal
space" (p. 191). According to our hypothesis, this angular position may be defined
in terms of the limb's inertial eigenvectors.
'~lthoughimportant differences exist between the pointing task used by Pagano and Tunrey (1995)
and that used by Velay et al. (1989),the fundamental similarity is that both required continuous
knowledge of limb orientation in extrapersonal space.
PROPRIoCEPnONAND INERTIAL EIGENVECTORS
51
EXPERIMENT 1
The eigenvectors of the target arm were manipulated by asking participants to hold
an object with small appended weights. The object was cross-shaped, consisting of
two wooden dowels attached perpendicularly at their midpoints. The object was
held such that one dowel, the stem, extended backwards against the inside of the
forearm coincident with the longitudinal axis of the forearm and forward from
between the middle and ring fingers. The other dowel, the cross-piece, extended
laterally from either side of the closed fist, and was held to remain perpendicular to
the ground plane. When held in this manner, the object's stem points in the same
direction as the forearm. That is, the longitudinal axis of the cyliindrical stem is
parallel to the longitudinal axes of the forearm. Thus, the tasks "point the forearm
in a particular direction" and "point the stem in a particular direction" are identical.
Masses were added to the cross-piece to alter the eigenvectors of the target forearmrod system. The added masses did not alter the matching forearm's eigenvectors.
The handheld objects, along with the three mass conditions used in Experiment
1,are depicted schematicallyin Figure 1. In each example in Figure 1, e3 is indicated
by an arrow, the arm by solid lines, the object by dashed lines, and the masses by
black squares. The el and ez eigenvectors also extend from 0,both being perpendicular to e3 and to each other, with el parallel and e2 perpendicular to the ground
plane. The rotation of e, depicted in Figure 1 also involves an equal rotation of e2,
\
----
FIGURE 1 The e3 eigenvector of the right forearm was manipulated by appending small weights to a handheld object. For the
matchingleft forearm, the weights always were in the symmetrical
arrangement.
---
;: X
I
both rotating in the same direction by the same amount. The masses were attached
to the object to reorient e3 of the limb and the object configuration upwards,
downwards, or not at all (see Figure 1 top, lower left, and lower right, respectively).
The rotation of ez and e3 occur at about 0,so that in effect, they rotate about el,
which remains unaltered by the mass manipulation. In all cases, the geometric
orientation of the forearm (as specified by elbow joint angles) remains unaltered.
For one third of the trials in Experiment 1, a mass was attached to the cross-piece
above the object's midpoint. The effect was to rotate e3 of the forearm an estimated
6.3" upwards (with a point of rotation in the elbow and the wrist joints fixed, and
O" being coincident with the longitudinal axis of the forearm). In another third of
the trials, a mass was attached below the object's midpoint. The effect was to rotate
e3 of the forearm an estimated 6.3"downwards. In the remaining trials, two smaller
masses were attached to the object, one on either side of the midpoint, such that
e3 of the arm remained unaltered by the addition of the symmetrically weighted
object, while the eigenvalues equaled those of the single mass configurations.
Manipulating the eigenvectors in a vertical plane in this manner meant that over
each experimental condition, (a) the limb's eigenvalues and overall mass remained
invariant, and (b) the gravitational torque varied only slightly, thereby ruling out
any possibility of observed effects being due to these parameters. Importantly, the
top versus bottom positioning of the mass changed the arm's eigenvectors without
changing the angle of the arm at the elbow or the orientation of the limb's
longitudinal axis relative to the vertical. The eigenvalues and geometric orientation
of the arm remained unaffected by the manipulation of the eigenvectors, and were
similar in all conditions. In consequence, an effect of mass position would be
contrary to the hypothesis that the perception of the forearm's direction is based
solely upon the arm's geometric orientation.
Methods
Participants. Seven graduate students and staff members, all right-handed,
associated with the University of Connecticut participated in Experiment 1 on a
volunteer basis. Four were women and three were men. No participant had any
foreknowledge of the specific hypothesis in question, nor of the experimental
conditions in use.
Materials. Two cross-shaped objects similar to those used in Pagano and
Turvey (1995) were constructed out of oak dowels 32.5 cm in length with a radius
of .6 cm. The cross-piece extended 16.25 cm to either side of the center of the metal
brace, and the stem extended 8 cm forward and 24.5 cm backward from the center
of the metal brace. The total mass of each object, excluding added weights, was
PROPRIOCEPTION AND INERTIAL EIGENVECTORS
53
TABLE I
Moments and Products of Inertia (g . an /1,000)for the Object and Object + Limb
Combination About the Point of Rotation in the Elbow for Experiment 1
Mass Condition
Object
200gU
100gUand 100gD
200g D
Object + limb combination
200 g U
100gUand 100gD
200gD
298.3
298.3
298.3
350.1
350.1
350.1
52.3
52.3
52.3
93.0
0.0
-93.0
895.7
895.7
895.7
945.3
945.3
945.3
64.4
64.4
64.4
93.0
0.0
-93.0
Note. U = up; D = down.
"L,
=
Irr = 0.
114.8 g. Masses were attached to one or both branches of the handeheld object in
three configurations: 200 g above the midpoint, 100 g above and 100 g below the
midpoint, or 200g below the midpoint. Masses were placed 15cm from the midpoint
of the cross. The moments and products of inertia for the objects, as well as the
objects + arm configuration, are given in Table 1. The I,, values for the the objects
+ arm configuration were diagonalized to arrive at the the limb's e3 orientation for
each mass condition. Calculations revealed that placement of the masses were such
that the target limb's e3 was rotated 6.3" up, 0°, or 6.3" down with respect to the
arm's longitudinal axis when the object was held in the hand. The eigenvalues and
geometric orientation of the arm were similar in all conditions.
Apparatus. The participant was seated in a chair and blindfolded. The
elbows were positioned on the chair's horizontal arms with the participant's upper
arms oriented roughly 65" with respect to the horizontal. The participant's arms
were bare around and below the area of the elbow. The objects were placed one in
each hand, such that the stem extended backward against the inside of the forearm
coincident with the longitudinal axis of the arm and forward from between the
middle and index fingers. The cross-piece extended laterally from either side of the
closed fist, and was held to remain parallel to the sagittal plane. The objects were
secured to the participant's forearms with tape and Velcro straps. Kinematic data
regarding the motions of the participant's arms were collected using a three-dimensional sonic digitizer (Science Accessories Corporation, Shelton, CT) and associated MASS kinematic analysis software (Engineering Solutions, Columbus, OH).
To collect motion data on the oscillating limbs, high-frequency sound emitters (3
cm long x .5 cm wide) were attached to the distal tips of the handheld objects. The
54
PAGANO, GARRETT, TURVEY
emitted sounds were detected by four microphones aligned in a plane perpendicular
to the ground plane and parallel to the frontal plane of the participant. The sonic
digitizer calculated the xyz coordinates of an emitter in three-space, and thus the
distal tip of an arm object configuration, by detecting the distances from each
microphone to the emitter. An emitter's signal was sampled at 91 Hz for the duration
of each trial, passed through an A-D converter, and stored on the hard disk of a
PC. Each individual trial lasted 30 sec. Given that only final positions were of
interest, the data were down-sampled to 15 Hz before analysis.
+
Procedure. Throughout Experiment 1, the right arm served as the target arm
and the left arm as the matching arm. Participants began half the trials with the
target arm in the up (U) position (roughly 75' with respect to horizontal) and the
matching arm in the down (D) position (roughly-30" with respect to horizontal)
and the remaining trials with the target arm in the D position, and the matching
arm in the U position. The experimenter began each trial by switching on a
metronome that repeatedly sounded a triplet of tones, with 2 sec separating the
three tones. This continuous repetitive pattern lasted for the duration of each trial.
When the participant was ready, Tone 1 acted as the signal to move the target arm
about the elbow to some angle intermediate between the positions of the target arm
and matching arm. Tone 2 signaled the participant to move the matching arm to
an angle such that the matching arm was parallel to the target (see Figure 2). Tone
3 signaled the participant to return both arms to their starting positions, completing
the cycle of three tones with their corresponding movements. This cycle was
repeated without interruption for the duration of the trial, with 2-sec intervals
between each tone and each cycle. When one complete cycle of three tones was
FIGURE 2 The configuration used in Experiment 1. Participants intended to match the left
forearm with the right forearm. Participants were blindfolded.
PROPIUoCEPTION AND INERTIAL EIGENVECTORS
55
completed successfully, the participants were asked to indicate if they felt that the
task was being done correctly; if so, the sonic digitizer was switched on by the
experimenter to begin data collection. The participant was instructed to sample
randomly with the target the full range of angles between the target and matching
arm's starting positions. To ensure that the full range of angles was sampled, the
experimenter observed each trial and instructed the participant to use an upper, a
middle, or a lower angle, if the corresponding area was being neglected. Two
baseline trials were performed without the metronome: One with the target arm
started in the U position, and one with the target arm started in the D position.
The symmetrical mass condition was used in these trials. During the baseline trials,
participants moved the forearms in the same manner as in the mals with the
metronome, but did so at any pace they found comfortable. The shoulder and wrist
joints remained fixed by the apparatus during every trial.
At the beginning of each trial, masses were attached to the cross-piece of the
target object in one of the three experimental configurations. Each of six different
conditions (3 mass conditions x 2 target starting positions, U or D) was presented
to the participant twice, in addition to two baseline conditions (one for each target
starting position), for a total of 14 trials per participant. Four participants received
the trials for the U target starting position before the trials for the D target starting
position, and the three remaining participants received the target starting positions
in the reverse order. Each participant received the mass conditions in a different
randomized order. At the beginning of the experiment, the participant was given
two practice trials using the starting positions for the first seven trials: the first with
neither the blindfold nor added weights, and the second with the blindfold and both
objects weighted symmetrically. After the first seven trials, the participant performed a third practice trial using the starting positions for the second seven trials,
with the blindfold worn and both objects weighted symmetrically. Data were not
collected during the practice trials. The participant rested the arms for about two
minutes between trials, but was allowed to take longer breaks as needed to avoid
fatigue. The experiment lasted about 45 min per participant. Participants received
no feedback about their performance during the course of the experiment.
Eigenvector calculations.
The orientation ofes was computed for each arm
+ object combination using regression equations and procedures provided by
Reynolds (1978; see also Chandler et al., 1975; Clauser et al., 1969) applied to the
body dimensions of one representative participant (see Appendix for the regression
equations and body dimensions used; for similar use of these equations; see also
Barac-Cikoja & Turvey, 1993; Pagano & Turvey, 1995). From these regression
equations, the mass, distance of segment CMs from the point of rotation in the
elbow, and principal moments of inertia for the limb segments were computed. The
parallel axis theorem then was used to transform the principal moments of inertia
for each segment about its respective CM to moments and products of inertia about
56
PAGANO, GARRETT, TURVEY
the elbow; these quantities were combined with those for the object and attached
object
masses to get the moments and products of Ii, about 0 for the limb
combination of each mass condition (see Table 1). Each I,, was diagonalized to arrive
at its eigenvalues and eigenvectors. The eigenvectors, expressed as coordinates in
Oxyz, were transformed into angles about 0.In each case, el remained perpendicular to the longitudinal axis of the arm, and ez and e3 were reoriented by the same
amount within a plane perpendicular to e l , where 00 would be coincident with the
longitudinal axis of the arm. From these calculations, it was estimated that the mass
conditions caused e3 of the arm to be oriented 6.3" above the arm's longitudinal
axis, 00, or 6.3" below, or 6.3"' 0°, and -6.3" respectively. As Table 1 shows, the
moments of inertia (I,, I,, I,,) remain essentially constant over the different mass
conditions, while the products of inertia (I,) change considerably. The sign of I,,
indicates in which direction e3 will be rotated to diagonalize the inertia tensor, and
the magnitude of I,,, relative to the magnitudes of the moments of inertia, corresponds to the magnitude of this rotation.
+
Data reduction. The time series for each 30-sec trial consisted of 450 sets
of xyz coordinates for the distal tip of each handheld object, one set every 67 msec.
Each set of points was converted into an angle within the yz plane, with O"
corresponding to the forearm horizontal, and +90° corresponding to the forearm
oriented vertically upwards. The yz plane was parallel to the midsagittal plane, and
passed through the point of rotation in the elbow. The angle for the right arm was
subtracted from the corresponding angle for the left arm to get the relative angle of
the two arms at each of the 450 points in time. This relative angle was 0" when the
two arms were perfectly parallel, negative when the left arm (matching arm) was
positioned lower than the right (target), and positive when the left arm was
positioned higher than the right. The relative angles for only those points in time
when the participant was matching the two arms were averaged to get each
participant's mean relative angle for each condition. The relative angle when both
arms were at their starting positions, or when the target was moved to an intermediate position and the matching arm was still in its starting position, was not
included in this relative angle because the participant did not intend to match the
orientations of the arms at those points in time.
Results
The relative angles for the two target starting positions and three mass conditions
for each of the seven participants in Experiment 1are presented in Table 2. Overall,
and
"
the mean relative angle for the U and D target starting positions were 4.4
-3.2" respectively. The mean relative angle for upwards, zero, and downwards
rotations of e3 were -1.7", -3.9"' and -5.8" respectively. A 2 x 3 analysis of variance
PROPRIOCEPTION AND INERTIAL EIGENVECTORS
57
(ANOVA) with within-subjects factors of Target Starting Position and Mass
Configuration resulted in no main effect for target starting position (F < 1) but a
significant main effectfor mass condition, F(2,12) = 11.5,p < .002. The interaction
was not significant (F < 1). A Tukey honestly significant difference (HSD) test
determined that the mean relative angle for upwards and downwards rotations of
e3 were significantly different (P < .01), whereas other comparisons were not (P >
.05). The lack of a main effect for target starting position suggests that participants
were not biased to matching higher or lower depending on the direction in which
the matching arm was moved to reach the orientation of the target arm. Inspection
of Table 2, however, suggests that some participants displayed such a bias, although
the direction ofbias differed across participants. Participant 1, for example, pointed
higher when the target arm started in the upper position compared to when the
target arm started in the lower position in all three mass conditions, whereas
Participant 2 did just the opposite. The main effect for mass indicates that matching
differed as a function of how the object was weighted. As predicted, compared to
matching in the symmetrical mass condition, the trend was for the participants to
orient the matching arm higher when the upper portion of the target arm was
weighted, and lower when the lower portion of the target arm was weighted. Thus,
the relative matching directions corresponded to the direction in which the eigenvectors were rotated by the added weight. Figure 3 depicts the relative angle for
each of the seven participants as a function of e3. The data in Figure 3 have been
normalized by subtracting the mean relative angle for each participant from that
participant's relative angle values. In this way, Figure 3 represents the matching
bias due to the mass manipulation, unaffected by each participant's individual bias
to place the matching arm higher or lower. The mean relative angles were -1.2",
3.9", 0.2", -2.1°, -17.0°, -1.3", and -10.2", respectively,for Participants 1 through
7. Five of the seven participants tended to place the matching forearm in a more
flexed (vertical) position than the target arm. This result is similar to those observed
in previous experiments (Velay et al., 1989; Worringham and Stelmach, 1985),
where participants also tended to place the matching forearm in a more flexed
position than the target.
A 2 x 2 ANOVA with within-subjects factors of target starting position and
metronome condition performed on the baseline (metronome off)and symmetrical
mass (metronome on) conditions resulted in no main effect for target starting
position (F < l), or metronome condition, F(1,6) = 1.0,p = .35.The interaction
was not significant (F < 1). The lack of main effect indicates that relative angles
with the metronome on, with participants constrained to moving when signaled by
the metronome, did not differ from relative angles with the metronome off, with
the participants setting their own pace.
To rule out the possibility that participants were matching Ng of the forearms,
we investigated whether the relative angle of two limbs in the Above and Below
mass conditions varied as a function of the orientation of the target arm. Because
cross-shaped objects were used to manipulate mass in a direction perpendicular to
TABLE 2
Relative Angle As a Function of the Three Mass Conditions and Two Starting Positions for
the Target Arm Used for Each of the Seven Participants in Experiment 1
Target Mass Confimcrahbn
Participant
Target Starting Positia
1
UP
2
UP
3
UP
4
UP
5
UP
6
UP
7
UP
Overall
UP
Above
Symmetric
Below
Down
Down
Down
Down
Down
Down
Down
Down
FIGURE 3 Mean relative angle as a
-8 -6 -4 -2 0 2 4 6 8
Eigenvector Angle (Deg)
function of the eigenvector angles for
the seven participants in Experiment
16 = .37x + 0 ) .
PROPRIOCEPTION AND INERTIAL EIGENVECTORS
59
the longitudinal axis of the arm, the difference in Ng between the two limbs varied
as a function of arm orientation. Specifically, in both the Above and Below
conditions, the Ng for the two arms were identical when both were oriented
horizontally, and the difference in Ng between the two arms increased as the arms
were moved towards the vertical. The difference in Ng between the two arms can
be predicted from the orientation of the target arm using simple regression (y =
.004x .003, = .98, p c .0001, n = 131). It is thus expected that ifparticipants
were matching Ng,the relative angle between the two limbs would vary as a function
of the target angle. Of the fourteen comparisons tested with simple regression-7
x 2 (Participant X Condition: Above vs. Below)-four resulted in a significant rZ,
but with none matching the expected function. For two of the four significant
regressions, the relative angle between the two arms decreased as the arms became
more vertical instead of increasing, as would be expected by the Ng hypothesis4
Thus it appears that the participants in Experiment 1 were not matching the Ng of
the two forearms, but were biased towards matching the inertial eigenvectors.
-
EXPERIMENT 2
Experiment 2 was designed to replicate the findings of Experiment 1 with two
important differences: First, the left arm served as the target, and the right arm
served as the matching arm. As in Experiment 1,e3 of the right arm was manipulated,
whereas e3 of the left arm remained unaltered. Thus in Experiment 2, e3 of the
matching arm was manipulated, whereas in Experiment I, es of the target arm was
manipulated. Second, the overall amount of mass attached to the target arm was
varied. In half of the trials, two 100-g masses were attached symmetrically to the
object held in the target arm as in Experiment 1, and no mass was attached in the
remaining trials. In all cases, ej of the target (left) arm remained unaltered. This
second manipulation was designed to further test the possibility that participants
are matching Ng.According to the Ng hypothesis, the matching arm should be
positioned more vertically relative to the unweighted target arm than when positioned relative to the weighted target arm.
Method
Participants. Seven graduate students and faculty (two women and five
men) associated with the University of Connecticut participated in Experiment 2
he four significant simple regressions were: Participant 1 Below, y = .113x- 3.59 :( = .39;p <
.01,n = 20);Participant 4 Above, y = .125x- 5.126:( = .21,p < .05,n = 20);Participant 4 Below,
y = -.092x + .003 = .30,p < .05,n = 20);and Participant 7 Below, y = -.105x + 18.236:( =
.26,p< .05,n = 20).Polynomial regressions also were tested, but the basic result remained unchanged.
(t
60
PAGANO, GARRETT, TURVEY
on a volunteer basis. All participants were right-handed, and none had any
foreknowledge of the specific hypothesis in question or of the experimental conditions in use. All participants gave their informed consent prior to their inclusion in
the study.
Materials and apparatus. The cross-shaped objects and experimental arrangement from Experiment 1 were used.
Procedure. The procedure from Experiment 1 was used, with the following
alterations. The left arm was the target arm and the right arm was the matching
arm. The metronome was used during all trials with a 3-sec separation between
tones and with the individual trial duration extended to 45 sec. This allowed the
participants a more comfortable pace. At the beginning of each trial, masses were
attached to the cross-piece of the matching object in one of the three experimental
configurations. During half of the trials, two 100-g masses were placed symmetrically on the cross piece of the target object, and no masses were attached to the
target arm during the remaining trials. Each of the 12 different conditions (3 Mass
Configurations for the Matching Arm x 2 Target Mass Conditions x 2 Target
Starting Positions) was presented to the participant once, for a total of 12 trials per
participant. Four participants received the trials for the U target starting position
before the trials for the D target starting position, and the three remaining participants received the target starting positions in the reverse order. Each participant
received the target arm mass conditions and matching arm mass configurations in
a different randomized order. The experiment lasted about 45 min per participant.
Participants received no feedback about their performance during the course of the
experiment.
Results
As in Experiment 1, the angle for the right arm was subtracted from the corresponding angle for the left arm to arrive at the relative angle between the two forearms.
Thus for Experiment 2, this relative angle was 0" when the two forearms were
perfectly parallel, negative when the right (matching arm) was positioned higher
than the left arm (target), and positive when the right arm was positioned lower
than the left. The relative angles for those points in time when the participant was
matching the two arms were averaged to get each participant's mean relative angle
for each condition.
The relative angles for the two target arm mass conditions and three matching
arm mass conditions for each of the seven participants in Experiment 2 are presented
in Table 3. Overall, the mean relative angle for the U and D target starting positions
PROPRIoCEPnoN AND INERTIAL EIGENVECTORS
61
TABLE 3
Relative Angle As a Function of the Two Target Mass Conditions and
Three Matching Arm Mass Configurations for the Target Arm Used for
Each of the Seven Participants in Experiment 2
Mmhing Arm Mass Confguration
Participant
1
2
3
4
5
6
7
Overall
Target Mass Condith
Above
Symmetric
Below
Weighted
Unweighted
Weighted
Unweighted
Weighted
Unweighted
Weighted
Unweighted
Weighted
Unweighted
Weighted
Unweighted
Weighted
Unweighted
Weighted
Unweighted
were -4.0" and -1.3" respectively. The mean relative angle for the weighted and
unweighted target arm conditions were -3.4" and -1.9" respectively. The mean
relative angle for upward, zero, and downward rotations of e3 were -1.7', -3.9", and
-5.a0, respectively. A 2 x 2 x 3 ANOVA with within-subjects factors of Target
Starting Position, Target Mass Condition, and Matching Arm Mass Configuration
resulted in no main effect for target starting position, F(l, 6) = 1.5, p > .05, a
significant main effect for target mass condition, F(1, 6) = 7.0, p < .05, and a
significant main effect for matching arm mass configuration, F(2, 12) = 17.9, p <
.001. No interactions were significant: Target Mass Condition x Matching Arm
Mass Configuration F(2, 12) = 1.2, p > .05 (all other Fs < 1). A Tukey HSD test
determined that the relative angles for the downward rotation ofe3 were significantly
different from those for the upward rotation of e3 (p < .01) as well as those for the
symmetrical weighting (P < .05), whereas the difference between the upwards and
symmetric conditions did not reach significance (P > .O5). The lack of a main effect
for target starting position indicates that, as in Experiment 1, participants were not
biased to matching higher or lower depending on the direction in which the
matching arm was moved to reach the target arm orientation. The main effect for
target mass condition indicates that target matching differed as a function of
whether the target arm was weighted. The overall trend was for participants to place
the matching arm more vertically than the target arm, and to do so to a greater
62
PAGANO, GARRETT, TURVEY
degree when the matching arm was symmetrically weighted than when the target
arm was unweighted. Thus, participants tended to match the limbs more closely
when the mass (and thus the N,) of the two limbs differed. This result is contrary
to the N, hypothesis, which predicted the opposite result. Note, however, that the
eigenvectors hypothesis predicted no difference as a function of this manipulation.
Tne main effect for the target mass configuration indicates that matching
differed as a function of how the object was weighted. As predicted, compared to
matching in the symmetrical mass condition, the trend was for participants to orient
the matching arm lower when the upper portion of the matching arm was weighted,
and higher when the lower portion of the matching arm was weighted. Thus, as in
Experiment 1, the relative matching directions corresponded to the direction in
which the eigenvectors were rotated by the added weight. Figure 4 depicts the
relative angle for each of the seven participants as a function of e3. The data in
Figure 4 have been normalized by subtracting the mean relative angle for each
participant from that participant's relative angle values. The mean relative angles
were -4.3", -9.0", -2.6", 2.6", -7.1°, -4.1°, and 5.9" respectively for participants 1
through 7.
The mean relative angles indicate that five of the seven participants in Experiment
2 tended to place the matching forearm in a less flexed (more horizontal) position
than the target arm. This result appears as contrary to that of Experiment 1, where
five out of seven participants tended to place the matching forearm in a more flexed
position than the left target arm. The results are similar, however, in that in the
matching configuration, the right forearm tended to be more flexed than the left for
10 out of the 14 total participants, regardless of whether the right forearm was
employed as the target arm (Experiment 1) or the matching arm (Experiment 2).
FIGURE 4 Mean relative angle as a
-8-6-4-2 0 2 4 6 8
Eigenvector Angle (Deg)
hnction of the eigenvector angles for
the seven participants in Experiment
2 0, = .28x 0).
+
PROPRIoCEPTION AND INERTIAL EIGENVECTORS
63
DISCUSSION
The two experiments demonstrate that forearm matching is reliably biased by
manipulations of the eigenvectors of the limb segment's inertia t e n ~ o rIn
. ~ agreement with the research of Pagano and Turvey (1995), our results ~rovidefurther
confirmation of the general hypothesis that the inertial properties of the limbs have
proprioceptive consequences. This hypothesis also has been evidenced by the
findings of Ghez, Gordon, and colleagues (Ghez et al., 1990, 1995; Gordon,
Ghilardi, & Ghez, 1994) in respect to the guidance of limb movements, and is
implicit in Hogan's "mobility tensor" theory of multijoint posture and movement
control (Hogan, 1985; see also Mussa-Ivaldi et al., 1985).
In both Experiments 1 and 2, the shift in perceived limb matching due to
eigenvector manipulations was consistently less than predicted inboth experiments,
as demonstrated by the regression slopes of .37 and .28 obtained in Experiments 1
and 2 respectively (see Figures 3 and 4). These slopes were similar to those observed
in Pagano and Turvey's (1995) Experiment 3. As in the present experiments,
participants in Pagano and Turvey's Experiment 3 were restricted to motions
occurring primarily within a single plane. In experiments allowing free movement
of the limb, the observed slope predicting pointing bias from e3 tended to be two to
three times steeper (Pagano & Turvey, 1995, Experiments 1 and 2). The low slope
observed in the present data may be due, in part, to restrictions placed on
participants' exploratory movements during the experiments. A low slope obtained
in cases of restricted movements, compared to those allowing participants to move
more freely, may underscore the dynamic nature of parameters for proprioception.
lois a parameter of rotational dynamics about a fixed point that is revealed
during active exploration. Greater accuracy in position sense with active as
opposed to passive movements has been demonstrated (e.g., Paillard & Brouchon,
1968, 1974), implicating a dynamic component originating within the muscles
over and above signals from joint receptors. In studies by Paillard and Brouchon
(1968, 1974), blindfolded participants were asked to move one arm to a position
along a vertically fixed rod, then to match its elevation with the other arm.
Positioning error was greater when the target arm was passively moved into
position by the experimenter compared to when it was actively positioned by the
participant. It was concluded that proprioceptive information originating in the
discharge of muscle spindle receptors, brought into play by self-induced movement
and absent during stabilized position of the limb, may contribute to the perception
of a limb's spatial position (Paillard & Brouchon, 1974). The major distinction
between the active and passive case is that the former is a case of exploratory
behavior; the latter is not. In the experiments by Pagano and Turvey (1995) for
-
-
' ~ o t ethat this result means experimenters should be aware that any apparatus applied directly to the
limb of their participanc (e.g., manipulandum, potentiometers, splints, etc.) may bias matching or
pointing performance if the device alters the eigenvectors of the limb.
64
PAGANO, GARRETT, TURVEY
example, the active exploration (i.e., the dynamics) occurred when the participant
freely moved the limb in three-space before pointing, or when the participant moved
the limb in a restricted manner within a single plane. The pointing behavior was
found to be more strongly determined by the dynamic parameter I,, when less
restricted exploration was allowed: the condition that better reflected natural
movements. The less restricted wielding of Pagano and Turvey's Experiments 1 and
2 allowed participants to more fully explore the dynamics of the limb; consequently,
Ii, was better revealed in that case. Thus, the dependency of limb proprioception on
I,, observed in the present experiments already has been found to be stronger in tasks
allowing a fuller range of movement. This finding may reveal a major shortcoming
of the often used matching procedure: The limited range of motion allowed in the
matching task is not adequate to reveal the full power of the dynamic parameters
used in proprioception.
Much work has been directed at identifying the coordinate system in which limbs
are controlled (e.g., Soechting, Helms Tillery, &Flanders, 1990).A proposed model
of the sensorimotor transformations involved in targeted arm movements assumes
that the preferred coordinate system is shoulder centered (Flanders et al., 1992).
The possible arbitrariness of the origin of this coordinate system has been commented upon, with different locations both within and outside the body advanced
as competing candidates (e.g., Alexander, 1992; Blouin et al., 1992). In our view
(see also Hayward, 1992), the paramount issue should not be the particular
coordinate system in which parameters for proprioception are rendered (e.g., head
centered vs. shoulder centered), but rather the nature of the parameters themselves
(e.g., Iijvs.joint angles or arm inclination; or more generally, dynamic vs. geometric).
In light of the transformation law of tensors discussed in our introduction (see also
Lovett, 1989; Moon &Spencer, 1986), the origin for Ii, may be translated from the
actual point of rotation to some other location. In doing so, no argument need be
made with respect to the preferred origin used by participants in the perception and
control of limbs and objects, because each origin provides a rendering of the same
physical facts. With such transformations, Is computed about this new point will
remain specific to the aspects of a limb or object's mass distribution of relevance to
dynamic touch (Pagano & Turvey, 1992; Pagano et al., 1993). The origin of the
coordinate system may be chosen by the scientist on the basis of which origin offers
the best description of the phenomenon in question. The numbers from one
coordinate system, for example, may make it easier to obtain quantities that enter
a regression with, and thus better predict, a relevant dependent variable. The use
of one coordinate system over another does not change the physical facts at issue;
thus, it does not automatically imply a preferred coordinate system used by the
nervous system (only the scientist's preferred coordinate system). In research by
Pagano and Turvey (1992), for example, participants were required to perceive the
orientation of an object in the hand by wielding. I,,about the actual point of rotation
was transformed to I,, about the proximal end of the object. This transformation
allowed for a rendering of 4 as an angular position in a coordinate system centered
PROPRIOCEPTIONAND INERTIAL EIGENVECTORS
65
around the portion of the handheld object, permitting the experimenter to use lii
to predict perceived object orientation. This does not require the conclusion that
the same coordinate system and origin are used by the participants. Rather, it allows
for the conclusion that participants are sensitive to tissue deformations specific to
the eigenvectors of 4 when perceiving the orientation of handheld objects by
wielding. These tissue deformations, like all physical facts, remain invariant to the
particular coordinates used to measure them. Participants capitalize upon such
physical facts during the act of perception; they need not heed the articular
coordinates required by the experimenter for their discovery and quantification.
Similarly, Pagano et al. (1993) found that when the same object is wielded about
one of several joints, taken singly or in combination, the perceived magnitude of
that object remains invariant. To best predict the performance of the participants,
the liis computed about the actual points of rotation were translated to an 0
common to all wielding conditions. This, again, does not suggest a coordinate
system within which the participants act, but rather an invariant property of the
wielding dynamics to which the perceptual constancy is tied. In sum, the use of a
particular coordinate system to describe behavior does not necessarily imply the use
of that coordinate system by the participant to produce the behavior. To the
contrary, the participant's perception may be specific to coordinate-free quantities.
The types of parameters found to be relevant to perception (e.g., tensors) may set
aside questions regarding coordinate systems.
The eigenvectors of I,, provide an intrinsic frame of reference for any object or
limb in rotation about a point of rotation. Although based upon the distribution of
mass and revealed through the patterning of forces, this reference frame is spatial
and thus, is commensurate with spatial frames used concurrently by vision and the
sensorimotor system (Pagano & Bingham, 1995; for a discussion of transformations
between such reference frames, see also Soechting & Flanders, 1992). The inertial
eigenvectors are the object's axes of rotational symmetry, the axes about which all
bearing forces are zero (i.e., there are no products of inertia). For example, e3 is the
axis of minimal resistance to rotational acceleration for a given object. Thus,
rotations about e3 require the least work. Soechting, Buneo, Herrmann, and
Flanders (1995) demonstrated that in pointing tasks requiring multijoint limb
movements, participants tend to use limb movements involving rotation about axes
chosen to minimize work. They identified the longitudinal axis ofeach limb segment
as candidate axes for the minimization of inertia-thus, the minimization of work.
However, a complete analysis may be in terms of e3 for the entire multisegmented
limb taken as a whole. Although e3 for a single limb segment typically is coincident
with the longitudinal axis of that segment, this may not be the case for a multisegmented limb. With the forearm and upper arm taken together, e3 for the entire arm
rotated about the shoulder is coincident with the longitudinal axis of the upper arm
and forearm only when the elbow is at full extension (i.e. when the angle between
the forearm and upper arm is 180"). As the forearm is rotated about the elbow
towards a position of 90" relative to the upper arm, the e3 for the entire arm rotates
66
PAGANO, GARRETT, TURVEY
away from the longitudinal axis of upper arm, and extends from the shoulder
through a point along the forearm (see Turvey et al., 1992, for a related description
of the eigenvectors for L-shaped rods). Future research should be directed at
uncovering the extent to which the eigenvectors of multisegmented limbs may be
used to perceive limb positions and to constrain the movement trajectories of both
limbs and handheld objects.
ACKNOWLEDGMENTS
Experiment 1 formed part of a doctoral dissertation presented by Christopher C.
Pagano to the University of Connecticut.
This work was supported in part by National Science Foundation Grant SBR
93-0937 1, awarded to M. T. Turvey and Claudia Carello; a University of Connecticut Doctoral Dissertation Fellowship; an American Psychological Association
Doctoral Dissertation Research Award; and U.S. Public Health Service Individual
National Research Service Award lFS32NS09575-01, awarded to Christopher C.
Pagano.
We thank Claudia Carello and Endre Kadar for their contributions to this
research, as well as Geoffrey Bingham, Michael Stassen, Dragana Barac-Cikoja, and
an anonymous reviewer for comments made on an earlier version of this article.
Figure 2 was prepared by Claudia Carello.
REFERENCES
Alexander, G. E. (1992). For effective sensorimotor processing must there be explicit representations
and reconciliation of differing frames of reference?Behavioral Brain Sciences, 15,321-322.
Amazeen, E. L., & Turvey, M. T. (1996). Weight perception and the haptic "size-weight illusion" are
functions of the inertia tensor.Joumd of E x p k m t a l Psychologv: Human Perception and PerfOTrnaTlCe,
22,2 13-222.
Barac-Cikqa, D., &Turvey, M. T. (1993).Hapticallyperceiving size at a distance.Joudof Experimental
Psychology: General, 122,347-370.
Blouin, J., Teasdale, N., Bard, C., Fleury, M. (1992) The mapping of visual space is a function of the
structure of the visual field. Behavioral Brain Sciences, 15,326327.
Burton, G., Turvey, M. T., & Solomon, H. Y. (1990) Can shape be perceived by dynamic touch?
Perception Et Psydwphysics, 48,477-487
Carello, C., Fitzpatrick, P., Domaniewicz, I., Chan, T. C., &Turvey, M. T. (1992). Effortful touch with
minimal movement. Journal of Expenmental Psychdogv: Human Perception and Petformance, 18,
290-302.
Chandler, R. F., Clauser, C. E., McContville,J. P., Reynolds, H. M., &Young, J. W. (1975). Inuestigation
of inertial pmpertaes of the human body, Find report, Apr 1, 1972 - Dec. 1974 (AMRL-TR-74-137).
Dayton, OH: Aerospace Medical Research Laboratories,Wright-Patterson Air Force Base.
Clauser, C. E., McContville, J. P., &Young, J, W. (1969). Weight, volume and center of macs of segments
of the human body (AMRL-TR-69-70). NASA CR-11262. Dayton, OH: Aerospace Medical Research Laboratories, Wright-Patterson Air Force Base.
PROPRIoCEPTION AND INERTIAL EIGENVECTORS
67
Cohen, L. A. (1958). Contributions of tactile, musculo-tendinous and joint mechanisms to position
sense in human shoulder. Jacmal of Neurophysldogy,21,563-568.
Fitzpatrick, P., Carello, C., &Turvey, M.T. (1994). Eigenvalues of the inertia tensor and exreroception
by the "muscular sense." Neuroscience, 60,551-568.
Flanders, M., Helms Tillery, S. I., & Soechting, J. F. (1992). Early stages in a sensorimotor transformation. Behavioral Brain Sciences, 15,309-362
Forget, R, & Lamarre, Y. (1987). Rapid elbow flexion in the absence of proprioceptive and cutaneous
feedback. Human Neumbtdogy, 6,27-37
Ghez, C., Gordon, J., Ghilardi, M. F., Christakos, C. N., &Cooper, S. E. (1990). Roles ofproprioceptive
input in the programming of arm trajectories. Cold Spring Harbor Symposia on Quantitative Bldogy,
LV, 837-847
Ghez, C., Gordon, J., Ghilardi, M. F., &Sainburg, R. (1995).Contributionsof vision and proprioception
to accuracy in limb movements. In M. S. Gazzaniga (Ed.), The cognitive neurosciences. Cambridge,
MA: MIT Press.
Gibson, J. J. (1966). The senses considered US perceptual systems. Boston: Houghton Miftlin
Gibson, J. 1. (1986). The e c o w apprwh to visual perception. Hillsdale, NJ: Lawrence Erlbaum
Associates, Inc. (Original work published 1979)
Goldstein, H. (1980). Clussical mechanics. Reading, MA: Addison-Wesley
Gordon, I., Ghilardi, M., & Ghez, C. (1994). Accuracy of planar reaching movements 11. Systematic
extent errors resulting from inertial anisotropy. Experimental Brain Research, 99, 112-130.
Hayward, V. (1992). Physical modeling applies to physiology, too. Behauioral Brain Sciences, 15,
342-343.
Helms Tillery, S. I., Flanders, M., & Soechting, J. F. (1991). A coordinate system for the synthesis of
visual and kinesthetic information. TheJournal of Neuroscience, 11, 770-778,
Hogan, N. (1985). The mechanics of multi-joint posture and movement control. Bidogrcal Cybernetics,
52,315-331.
Lovett, D. R. (1989). Tenrorpr@ertier of q s t n l s . Philadelphia: Adam Hilger.
Moon, P., &Spencer, D. E. (1986). Theory of holm. Cambridge, England: Cambridge University Press.
Mussa-Ivaldi, F. A., Hogan, N., Bizzi, E. (1985) Neural, mechanical, and geometric factors subserving
arm posture. TheJoumal of Neuroscience, 5,2732-2743.
Pagano, C. C., & Bingham, G. P. (1995). Spatial frames for motor control would be commensurate with
spatial frames for vision and proprioception, but what of the control of energyflows? Behavioral and
Brain Sciences, 18,773.
Pagano, C. C., Carello,C. & Turvey, M. T. (in press). Exteroception and exproprioception by dynamic
touch are different functions of the inertia tensor. Perceptwn 19Psychophysics.
Pagano, C. C., Fitzpatrick, P., & Turvey, M. T. (1993). Tensorial basis to the constancy of perceived
object extent over variations in dynamic touch. Perception E ) Psychophysics, 54,43-54.
Pagano, C. C., Kinsella-Shaw, J. M., Cassidy, P. E., &Turvey, M. T. (1994). Role of the inertia tensor
in haptically perceiving where an object is grasped. Joumal of Exprmentnl Psychology: Human
Perception and Performance, 20, 276-285.
Pagano, C. C., &Turvey, M. T. (1992). Eigenvectors of the inertia tensor and perceiving the orientation
of a hand-held object by dynamic touch. Perception E ) Psychophysics,52,617424.
Pagano, C. C., & Turvey, M. T. (1993). Perceiving by dynamic touch the distances reachable with
irregular objects. Ecdogrcal Psychdogy, 5, 125-15 1.
Pagano, C. C., & Turvey, M. T. (1995). The inertia tensor as a basis for the perception of limb
orientation.Joumal of E q e m e d Psychology: Human Perception and PedDrmrrnCe, 21, 1070-1087.
Paillard,J., &Brouchon, M. (1968). Active and passive movements in the calibration of position sense.
In S. J. Freedman (Ed.), The neur@sychology of spatially oriented behavior (pp. 37-55). Homewood,
IL: Dorsey.
Paillard, J., & Brouchon, M. (1974). A proprioceptive contribution to the spatial encoding of position
cues for ballistic movements. Brain Research, 71, 273-284.
68
PAGANO, GARRETT, TURVEY
Reynolds, H. M. (1978). The inertial properties of the body and its segments. In E. Churchill, L. L.
Laubach,]. T. McContville, &I. Tebbetts (Eds.),Anthropometric source bmk, Volume I: Anthropomrtry fw designers. (NASA Reference Publication 1024, pp. 1V-1-N-76). Washington, DC: NASA
Scientific and Technical Information Office.
Soechting, 1. F. (1982). Does position sense at the elbow reflect a sense of elbow joint angle or one of
limb orientation? Brain Research, 248,392-395.
Soechting, J. F., Buneo, C. A., Herrmann, U., & Flanders, M. (1995). Moving effortlessly in three
dimensions: Does Donder's law apply to arm movements? The Journal of Neuroscience, 15,
6271-6280.
Soechting, J. F., & Flanders, M. (1992). Moving in three-dimensional space: Frames of reference,
vectors, and coordinate systems. Annud Review of Neumscience, 15, 167-191.
Soechting, J. F., Helms Tillery, S. I., & Flanders, M. (1990). Transformation from head- to shouldercentered representation of target direction in arm movements. Journal of Cognitive Neumscience, 2,
32-43.
Soechting,J.F., &Ross, B. (1984). Psychophysicaldetermination ofcoordinate representationof human
arm orientation. Neuroscience, 13, 595404.
Solomon, H. Y. (1988). Movement-produced invariants in haptic explorations: An example of a
self-organizing, information-driven, intentional system. Human Movement Science, 7,201-224.
Solomon, H. Y., &Turvey, M. T. (1988). Haptically perceiving the distances reachable with hand-held
objects. Journal ofEx@imental Psychology: Human Perceptiun and Perjomrance, 14, 404-427.
Starzhinskii, V. M. (1982) An advanced course oftheoretical m e c h s . Moscow: MIR.
Teasdale, N., Bard, C., Fleury, M., Paillard, J., Forget, R., &Lamarre, Y. (1994). Bimanual interference
in a deafferented patient and control subjects. In S. P. Swinnen, H. Heur, J. Massion, & P. Casaer
(Eds.), Interlimb cmrdination: Neural, dynnmical, and cognitive constraints. San hego, CA: Academic.
Turvey, M. T. (1994). From Borelli (1680) and Bell (1826) to the dynamics of action and perception.
Journal of Spot and Exercise Psychology, 16, S128S157.
Turvey, M. T., Burton, G., Pagano, C. C., Solomon, H. Y., & Runeson, S. (1992). Role of the inertia
tensor in perceiving object orientation by dynamic touch.Journal of Etperimental Psychology: Human
Perception and Performance, 18, 714-727.
Turvey, M. T., & Carello, C. (1995). Eynamic touch. In Epstein W, Rogers S (Eds.), Handbook of
perception and cognition, Vol. 5. Perception of space and motion (pp 401-490). New York: Academic.
Velay, 1. L., Roll, R., Paillard, J. (1989). Elbow pasition sense in man: Contrasting results in matching
and pointing. Human Movement Science, 8, 177-193
Wainwright, S. A. (1988).AXISand circumference: The cylindrical shape ofplants and animals. Cambridge,
MA: Harvard University Press.
Womngham, C. I., &Stelmach, G. E. (1985). The contributionof gravitational torques to limb position
sense. Experimental Brain Research, 61,3842.
Worringham, C. J., Stelmach, G. E., & Martin, Z. E. (1987). Limb segment inclination sense in
proprioception. Experimental Brain Research, 66,653-658.
APPENDIX: CALCULATING 1 j j FOR LIMB SEGMENTS
Body dimensions used in the calculation of mass, center of mass, and moment of
inertia of the limb segments (for a precise definition of all dimensions, see Clauser
et al., 1969):
Total body weight: 70 kg
Elbow breadth: 6.5 cm
PROPRIOCEPTION AND INERTIAL EIGENVECTORS
69
-
Forearm length (radiale stylion): 25 cm
Forearm circumference: 23 cm
Wrist breadth: 6.0 cm
Wrist circumference: 18 cm
Fist diameter: 7.0 cm
Regression equations used to calculate (a) the mass (kg) of the limb segments, (b)
the distance (cm) of the limb segment and object centers of mass from the point of
rotation in the elbow, and (c) the principal moments of inertia (g cm2)of the limb
segments about CM (using total body weight in g; Chandler et al., 1975; Clauser
et al., 1969; Reynolds, 1978):
a. Forearm: .081 (wrist circumference) + .052 (forearm circumference) 1.650 = 1.004
Hand: .029 (wrist circumference) + .075(wristbreadth) + .031 (fist diameter) - .746 = .443
b. Forearm: .440 (forearm length) + .761 (wrist breadth) - 5.645 + upper arm
length = 9.9
Hand: Forearm length + 6.0 = 3 1.0
c. Forearm:l, = 1.O84 (total body weight) - 4,8 12 = 7 1,068
I,, = 1.062 (total body weight) - 5,444 = 68,896
I,, = ,271 (total body weight) - 9,020 = 9,950
Hand& = I, = I,, = 215 (mass of hand) x [(fist diameter)/212= 2,17 1