Motor Control, 1998, 2, 352-378
O 1998 Human Kinetics Publishers, Inc
Grasp Stiffness as a Function of Grasp
Force and Finger Span
Clayton L. Van Doren
The purpose of this study was to determine whether direct measurements of
grasp stiffness agreed with stiffness inferred from the slopes of isovolitional
force-span characteristics derived from previous grasp-effort matching data.
Grasp stiffness for three-finger pinch was measured as a function of initial
force and finger span using step displacements applied in a do-not-intervene
paradigm. Subjects pinched a free-floating, motorized manipulandum in each
hand and squeezed both with equal effort; one of the hands was perturbed at
random. Stiffness was calculated from the initial and final steady-state values
of force and span. The effects of step amplitude, rise-time, and initial load
stiffness were investigated; grasp stiffness decreased significantly for larger
steps, increased slightly for longer rise-times, and was unaffected by load stiffness. Grasp stiffness then was measured as a function of initial force and span
using a single set of step parameters. Stiffness increased significantly in proportion to force but was changed only slightly by span. It was concluded that
the perturbation and effort-matching measures of stiffness are not equivalent
and represent different components of motor behavior.
Equilibrium-point (EP) models of motor control hypothesize that posture
results from the equilibrium between external forces acting on a limb and internal,
muscular forces that are generated as a function of the difference between the
limb's actual position and a centrally programmed "virtual" position or endpoint
(for reviews, see Bizzi, Hogan, Mussa-Ivaldi, & Giszter, 1992; Latash, 1993).
Movement occurs when the equilibrium is disturbed by a voluntary shift in the
virtual endpoint along a virtual trajectory (Flash, 1987; Gribble, Ostry, Sanguineti,
& Leboissiere, 1998; Hogan, 1985; Latash & Gottlieb, 1991, 1992) or by changes
in the external load. The requirements of mechanical equilibrium ensure that motor behavior depends strongly on the relationships between force and position engendered by the neuromuscular system. In Feldman's version of the EP model
(Asatryan & Feldman, 1965; Feldman, 1966, 1986), for example, the central motor command shifts the position of a force-position characteristic by changing the
threshold position of the tonic stretch reflex (denoted by the variable A). Muscle
force is recruited if the limb or joint position exceeds this threshold. The A-model
is controversial (see the target article by Feldman & Levin, 1995, and the accomClayton L. Van Doren is with the Department of Orthopedics, Case Western Reserve
University, Cleveland, OH 44106.
352
Grasp Stiffness
353
panying commentaries for representative samples of support and criticism) but
provides an excellent account for the steady-state forces and finger spans produced when equal voluntary motor commands (efforts) for three-finger pinch are
exerted simultaneously and bilaterally against equal or unequal compliant loads
(Van Doren, 1998). The results from that study showed that (a) the set of steadystate grasp forces characteristic of a constant voluntary motor command are described well by a linear function of finger span, (b) the slope of this "isovolitional
force-span characteristic"' (IFSC) did not appear to change significantly with changing levels of grasp effort, but (c) the location of the characteristic shifted systematically to shorter spans for greater efforts. As shown in Figure 1 (data from Figure
3, Van Doren, 1998), the intercept of the IFSC with the span axis behaves like a
"virtual span" determined by the motor command, that is, the span that would be
produced if the fingers were not loaded. The actual force and span for a given
motor command and load are given by the intersection (equilibrium) of the IFSC
and the load characteristic.
The role of the IFSC as described above is to generate an initial, steadystate grasp output in response to a given motor command. However, the springlike
properties of hand grasp also determine the response to perturbations away from
the initial state and are responsible for maintaining a stable grip. The initial states
and perturbation responses may be related simply if the force-span characteristic
represents all of the possible output states for a given motor command. That is,
we can start at a grasp state (x, j) and then externally impose a perturbation by
4
5
6
7
8
9
Span (cm)
-
Figure 1 Representative isovolitional force-span characteristics (IFSCs, solid lines)
derived from effort matching data for three-finger pinch applied bilaterally to equal
or unequal compliant loads. Data from Van Doren (1998). Each characteristic represents
the forces and spans that result when the same motor command (effort) is applied to a
variety of compliant loads. IFSCs at shorter spans were produced by greater efforts.
One of the loads is represented by the dashed line for a spring with a stiffness of 6 N/
cm. The equilibrium force and span for a given effort-load combination are given by
the intersection of the appropriate load line and IFSC. An example is marked with the
arrow. According to the A-model of Feldman (e.g., Feldman, 1986), the shape of the
IFSC is determined by passive, active intrinsic, and reflex components, and its location
along the position axis (span, in this case) corresponds to the threshold of the tonic
stretch reflex set by the motor command.
354
Van Doren
changing the load to produce a new state (x + k c , f + Af). The new state, by supposition, will also be a member of the characteristic if the subject does not intervene
and change the motor command in response to the perturbation. The grasp stiffness,
then, is equal to the ratio of the force increment to the position increment, A P h ,
which is also the characteristic's slope. The linear isovolitional force-span characteristics that we derived from effort matching data had constant slopes and suggest,
therefore, that grasp stiffness should be constant, independent of force and span.
A constant stiffness is at odds, though, with expectations based on positionand force-induced changes in muscle activation and mechanics. Grasp force changes
with grip size or finger span (Bechtol 1954), as does the distribution of forces
among contributing fingers (Kinoshita, Murase, & Bandou, 1996), due to changes
in the moment arms of different joints and shifts in the operating point on the
length-tension curves of relevant muscles (e.g., An, Ubea, Chao, Cooney, &
Linscheid, 1983; Lee & Rim, 1990). It seems likely, then, that stiffness should
vary with finger span as weH, though such measurements have not been made
previously. Results found at otherjoints have been mixed. Stiffness about the ankle
is known to change with angle (Weiss, Kearney, & Hunter, 1986a, 1986b), but not
so at the elbow (Gottlieb & Agarwal, 1988).
All studies, in contrast, have found that stiffness increases with the initial force,
whether at the ankle (Allum & Mauritz, 1984; Hunter & Kearney, 1982; Sinkjaer,
Toft, Andreassen, & Hornemann, 1988),elbow (Gottlieb & Agarwal, 1988),or wrist
(Gielen & Houk, 1984). Similar results have been obtained from single finger joints
(Becker & Mote, 1990; Capaday, Forget, & Milner, 1994; Carter, Crago, & Gorman,
1993; Carter, Crago, & Keith, 1990),and endpoint stiffnesshas been measured for the
index finger in one (Hajian & Howe, 1997) and two (Milner & Franklin, in press)
dimensions. Data from individual fingers may not be relevant for pinch or grasp,
however, since the net internal stiffness measured between the thumb and fingers is
produced by multiplejoints and the forces are wholly supplied by the fingers and are
not balanced by the arm or an external support (i.e., a mechanical ground).
Stiffness during ungrounded pinch has been measured only in four studies
(Hajian, 1997; Hermsdorfer, Wessel, Mai, & Marquardt, 1994; Karason &
Srinivasan, 1995; Van Doren, 1996), and none of them provide data comparable to
the IFSCs derived from effort matches. The results of Hermsdorfer et al. (1994)
were obtained for diagnosing cerebellar ataxia or Freidreich's ataxia, and the instructions called for subjects to intervene purposefully as quickly as possible, violating the isovolitional requisite. Hajian (1997) and Karason and Srinivasan (1995) did
use a "do-not-intervene" paradigm but were interested only in the initial transient
prior to the onset of the stretch reflex. Van Doren (1996) made preliminary measurements of steady-state grasp stiffness using a do-not-intervene paradigm, but the data
were sparse. Also, the perturbations were applied by a pneumatic manipulandum that
provided little control over initial force and span and produced rather sluggish force
perturbations (rise-time >200 ms). Because the pneumatic manipulandum had a
relatively low mechanical impedance, the displacement and velocity of each step
covaried with the initial conditions,providing inconsistent input to stretch reflexes.
As we shall see below, step size has a very significant effect on stiffness, so the
sketchy results from Van Doren (1996) must be interpreted with caution.
The experiments described here provide measurements of grasp stiffness
using mechanically ungrounded electromechanical manipulanda to apply step
changes in finger span over a variety of initial conditions (spans and forces) and
Grasp Stiffness
355
for a range of step parameters (amplitude, rise-time, and initial load stiffness), all
under instructions to the subject to not intervene. The manipulanda and conditions
were chosen to reproduce the previous effort-matchingparadigm (Van Doren, 1998)
as closely as possible, so that a more direct comparison can be made between the
grasp stiffness calculated from the slope of the IFSCs and grasp stiffness measured
directly. If the values agree, then steady-state grasp generated by exerting equal
efforts and the response to perturbations applied while effort is kept constant can
be described by a single mechanism represented by the force-span characteristic.
Methods
Subjects
Five subjects (3 male, 2 female, age 27-39 years) completed three control experiments, and 8 subjects (5 male, 3 female, age 23-39) completed the main experiment. EMG measurements were made on 5 of the latter subjects in a separate
session. Three subjects (including the author) participated in all components of the
study. One of these subjects had Raynaud's syndrome but was free of symptoms at
the time of the experiments. All subjects but one were right-handed (self-report),
and the other subject was ambidextrous. All subjects gave verbal informed consent prior to participation in the study, and the procedures were approved by the
Institutional Review Board at MetroHealth Medical Center.
Manipulanda
The manipulanda were custom built (Sensable Devices, Cambridge, MA), each
consisting of a single motor (Maxon RE 025) with a threaded capstan connected to a
pair of linear slide bearings via a cable transmission, as drawn in Figure 2A. Rotation
of the motor drove the slide bearings in opposite directions with equal force. Finger
and thumb contact plates were mounted to the bearings, and the thumb plate was
instrumented with eight strain gauges configured as a pair of Wheatstone bridges.
The gauges measured differential strain that was proportional to the applied force
independent of the point of contact. Even so, loose-fitting clips for the thumb and
index finger were fashioned from wrap-around guitar picks and were mounted on
each contact plate to provide repeatable contact locations. Strain gauge output was
amplified using a low-gain preamp (Analog Devices AD620) mounted on the thumb
plate followed by a remote, secondary amplifier (Entran Sensors, Inc., Fairfield, NJ).
The overall sensitivity of the system was roughly 25 mV/N. Finger span was defined
operationally as the separation of the contact plates ind was measured via a rotary
encoder mounted on the motor shaft (span resolution = 0.00164 cm). The analog
signals for force and span were sampled at 1342 Hz2with 12-bit AID converters in
one computer while the controllers ran independently on an integer DSP board in a
second computer. The latter sampled the encoder output with 16-bit resolution using
a custom-built decoder circuit and a pair of 12-bit AID converters.
The position signal was used to implement a simple stiffness control law,
i = k(x - x,), where i is the motor current (proportional to torque), x is the measured span, and k and xo are the programmed stiffness and rest length, respectively. The stiffness k was modest (3-12 Nlcm, see below) while the subject
squeezed the manipulandum to achieve the target force but was set to the maximum stable stiffness of the system (32 Nlcm) during and after the perturbation
Van Doren
A
SIDE VlEW
Enc
Motor
Slide Bearing
Q
I I
@
8
Index Finger
Clip g
F
Strain
Gauges
Thumb Clip
Finger Contact
Plate
Thumb Contact
Plate
BOTTOM VlEW
D
Slide Bearing
Figure 2A - Side and bottom views of the motorized manipulanda. The motor drives
a pair of slide bearings in opposite directions via a cable transmission. Thumb and
finger contact plates (with clips to minimize variationsin contact locations) are mounted
on the slide bearings. The thumb contact plate is also instrumented with 8 strain gauges
to measure force. A low-gain preamp is mounted on the thumb contact plate. The
separation of the contact plates is defined as the span and is measured via a rotary
encoder mounted on the motor shaft.
Grasp Stiffness
Manipulandum
I
Figure 2B - Posture of experimental subject and sketch of manipulandum support.
The latter allowed each manipulandum to move freely in three directions and rotate
about three axes.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (sec)
Figure 2C - Representative data (force, top trace; span, bottom trace) from a single
trial in the main experiment. Forces and spans are expressed relative to their initial or
target values (12 N and 6.4 cm in this trial), as averaged over a 0.1 s interval just prior
to the perturbation onset. The traces are unfiltered. Vertical lines denote 40 ms step
rise-time.
Time (sec)
Figure 2D - Ensemble median force (top) and span (bottom)traces from 16 replicate
trials in the rise-timeexperiment.Traces are unfiltered and expressed relative to their
initial or target values (6 N and 6.4 cm, respectively). Vertical lines denote different
step rise-times (40,100, and 250 ms), as labeled.
in all experiments. At the instant just before the perturbation was applied, the
controller rest length was changed simultaneously with the stiffness so that the motor current was unchanged. Then, the rest length was shifted by an amount Ax,, with
a sigmoidal trajectory over time. A stiff controller was used during the perturbation
so that the actual displacement would be nearly equal to the nominal value, Axo,
independent of the load presented by the hand. Representative force and span trajectories during a perturbation are shown in Figure 2C for a single t i a l and in
Figure 2D for ensemble medians across sets of 16 replicate trials.
Each manipulandum was mounted in a gimbal attached to a counterweighted
parallelogram frame, allowing the manipulandum to "float" with free translation
in three directions and rotation about three axes (Figure 2B). The suspension guaranteed that the fingers and thumb exerted equal and opposite forces and did not
apply any forces counterbalanced by the arm or external supports. Subjects rested
their forearms on the chair armrests or on a padded table with their wrists and
hands extending past the end of the supports. A screen was used to block the subjects' view of their hands.
Timing
Subjects were prompted to begin each trial by a computer beep, and they squeezed
both manipulanda equally using a three-finger pinch until they reached the de-
Grasp Stiffness
359
sired target force, as indicated by a null meter. The null meter was controlled by
one of the hands selected to be the reference at random and unknown to the subject
as per an effort-matching protocol used previously (Van Doren, 1995, 1998). The
centraI green LED of the meter was illuminated if the force applied by the reference
hand was within k5% of the nominal target force, and yellow and red LEDs flanking the center were illuminated for progressively larger errors. The null meter measured the error between the actual force and the desired force but did not indicate
reference magnitude. Subjects were instructed to squeeze as quickly and smoothly
as possible. A step perturbation was applied to either of the hands (selected at random) at a random time up to 1 s after the subject had achieved and held the target
for 0.5 s. Subjects were instructed not to intervene in any way when the perturbation was applied. The randomization of the perturbed hand and the timing helped
minimize interventions. Subjects were also instructed to count backward or recite
lyrics in their head to provide additional distraction.
Step Parameters
The first series of three control experiments was used to determine if the grasp
stiffness measured using displacement steps was affected by three step parameters: rise-time, amplitude, and initial stiffness. "Initial stiffness" was the programmed stiffness of the manipulandum maintained while the subject made the
initial squeeze to the target force prior to the perturbation. The programmed rest
length of the manipulandurn was covaried with its stiffness in each trial so that the
target force and span were the same in all trials (6.0 N and 6.4 cm). Rise-time was
varied in the first control experiment (40, 100,250 ms) with constant amplitude (1
cm) and initial stiffness (6 Nlcm). Step direction (increasing or decreasing span),
perturbed hand (left or right), and reference hand (left or right) were used with
equal frequency, in all combinations, and in random order. Step amplitude was
varied in the second experiment (0.2, 0.5, and 1.0 cm), with initial stiffness and
rise-time held constant at 6.0 Nlcm and 40 ms, respectively. The initial stiffness
was varied in the third experiment (3.0,6.0, 12 Nlcm) with constant rise-time (40
ms) and step amplitude (1.0 cm).
One particular combination of parameters was common in all three control
experiments to assess repeatability (40 ms rise-time, k1.0 cm amplitude, 6.0 Nlcm
initial load stiffness). Each session consisted of 96 trials (3 levels of the varied
parameter x 2 step directions x 2 reference hands x 2 perturbed hands x 4 replications) and lasted about an hour. Subjects were given a few seconds rest between
trials and a 30 s rest between blocks of six trials. Fatigue was not reported by any
subject. Each control experiment was completed in a separate session, and the
order was fixed since the parameters chosen for later sessions depended on the
results of previous sessions (e.g., the results of the rise-time experiment led to
selection of 40 ms as the standard used in later sessions). The fixed order is unlikely to confound the interpretation of the results since most comparisons were
made across parameter values used within a session.
Initial Force and Span
The first three control experiments led to the selection of a single set of step parameters (40 ms rise-time, 0.5 cm amplitude, 6.0 Nlcm initial stiffness; see Re-
360
Van Doren
sults) that were used in the main experiment to measure grasp stiffness as a function of the initial (target) force (2.12, 3.00, 4.24, 6.00, 8.48, 12.0 N) and finger
span (2.4,4.4, 6.4 cm). Subjects first completed one or two practice sessions using the same protocol as the step amplitude control experiment. All 8 subjects
proceeded to the main experiment when their ensemble median responses appeared
free from triggered reactions or interventions (see Results). Each subject completed 288 trials (6 target forces x 3 target spans x 2 step directions x 2 hands
perturbed x 2 reference hands x 2 replications) over three sessions, each with 96
trials. As before, the reference hand, perturbed hand, and step direction were randomized. All combinations of the six target forces and three spans were used in
each session in approximately equal numbers and in pseudorandom order. The
trials with the 12 N target force were purposefully distributed throughout each
session to prevent motor overheating.
EMGs and Muscle Excursions
Two additional sets of measurements were completed to identify potential active
and passive contributions to the observed grasp stiffness. First, muscle excursions
were calculated approximately for the extrinsic finger flexors (flexor digitorum
superficialis [FDS], flexor digitorum profundus [FDP]) from finger joint angles
measured while 4 additional subjects (2 male, 2 female) held blocks with widths
from 1.0 cm to 9.0 cm. The blocks were held using three-finger pinch in a posture
similar to that used with the manipulanda.The metacarpal-phalangeal (MP), proximal interphalangeal (PIP), and distal interphalangeal (DIP) angles were measured
from the index finger, and the carpal-metacarpal (CMC) and interphalangeal (IP)
angles were measured from the thumb. Wrist angles were not measured, although
subjects voluntarily maintained a fixed posture throughout the measurements. The
angles were measured with a goniometer once for each of six block widths and
were averaged across subjects for each width. Excursions e were calculated using
the following formula, which assumes that the effects of rotations about multiple
finger joints are additive:
where A@,is the average change in angle relative to the middle block size (4.5 cm)
for the ith joint, and cq is the moment arm (tendon excursion measured per degree
of rotation for the ith joint) either for the FDP or FDS (from An et al., 1983). The
absolute excursions were converted to relative excursions by dividing by muscle
lengths (FDS: 20.7 cm, index finger belly, distal and proximal components combined; FDP: index finger belly; from Lieber, Jacobson, Fazeli, Abrams, & Botte,
1992) and assuming that tendons did not stretch significantly. The potential effects
of undetected wrist movements were approximated using a moment arm of 1.5 cm
for both the FDS and FDP (see Gonzalez, Buchanan, & Delp, 1997), which yields
relative excursions of 0.13% and 0.18% per degree of wrist movement, respectively. The calculated excursions were then compared to previously observed criteria for evoking phenomena such as muscle yield versus elastic deformation of
cross-bridges.
Second, EMGs were measured from a separate group of 5 subjects to verify
that the perturbations evoked changes in the steady-state EMG from three hand
Grasp Stiffness
361
muscles involved in three-finger pinch: first dorsal interosseus (IDI), flexor
digitorum superficialis (FDS), and extensor digitorum cornmunis (EDC). EMGs
were measured from the left arm only using pairs of self-adhesive, AglAgC1 surface electrodes placed over locations determined by palpation. EMG signals were
band-pass filtered from 10 to 200 Hz and amplified by gains of 9004,500 (MMRS
amplifiers, BAK Inc.). The conditioned signals were sampled at 2000 Hz via 12bit AID converters controlled by a third computer. Data were collected while subjects completed the step-size experiment (16 trials for three step amplitudes in
both directions) with the left hand using a target force of 6 N and steps with a 40
ms rise-time and 6.0 Nlcm initial stiffness.
Data Analysis
Data from each trial were analyzed from 0.5 s prior to the onset of the perturbation
until 1 s after the onset. Force, span, and EMG data from like trials in a given
experiment (e.g., the 16 trials with the same rise-time but with either left or right
hand perturbed and/or serving as the reference) were processed by subtracting the
initial value from each trial (averaged over 0.1 s just prior to the step onset) and
then taking the point-by-point median across trials in the ensemble. The median
served to reject outlying responses from trials in which subjects intervened. Unfiltered median data were used in all analyses, but data were filtered for the accompanying plots using a fourth-order Butterworth filter with a cutoff frequency at 67
Hz (1120th of the force and position sampling frequency), applied forward and
backward to avoid introducing time shifts.
Calculating the median rejected the occasional voluntary intervention or
triggered reaction that produced relatively large increases or decreases in force
at random, but the median could not correct for the subtle and consistent interventions exhibited by some subjects. Three examples are shown in the left half
of Figure 3, which plots the median forces from the perturbed hand from the step
amplitude control experiment. As was typical for all subjects and conditions, the
responses had three components (see also Figure 2C & D). The force during the
perturbation was dominated by the passive mechanics of the hand and
manipulandum. The force peak during the middle of the step was probably due
to viscosity (because the steps were sigmoidal, the peak velocity occurred
halfway through the step), and the force peak near the end of the step (i.e.,
at the time of peak acceleration) was likely due to inertia. The force then declined (toward zero) until roughly 55-75 ms after step onset when the force
increased again due to the short-latency stretch reflex. The reflex force peaked
at roughly 130-140 ms and diminished thereafter until the response reached a
plateau.
All of these features are consistent with previous studies that imposed position perturbations on the digits (e.g., Akazawa, Milner, & Stein, 1983; Carter et al.
1990, 1993). Subject 5 (Figure 3, top) also showed a slow, steady decline in force
starting about 200 ms after the step onset on nearly every trial regardless of step
amplitude or direction. Subject 4 (Figure 3, middle), in contrast, consistently produced an increasing force, and Subject 3 (Figure 3, bottom) showed no systematic
change. The timing and consistency of these late changes within subjects but not
across subjects suggest that the variations were due to voluntary or triggered reactions and not slow intrinsic or reflex effects. If so, then the responses were due to
362
Van Doren
..
.
. . .. .. . . . .. .............. . ..
.
. ... ........ .... ..
_.
L
I
X
_
/
0
0.1 0.2 0.3
-------
0.4 0.5 0.6 0.7
0.8 0.9
1.0
0
0.1
0.2 0.3 0.4 0.5
0.6 0.7 0.8 0.9 1.0
Time (sec)
Figure 3 - (left) Plots of force (relative to the 6.0 N target force prior to the
perturbation) measured from the perturbed hand as a function of time in response to
perturbations with three amplitudes (0.2,0.5, and 1.0 em) and two directions (F, flexion
or decreasing span, and E, extension or increasing span). Each trace is the low-pass
filtered, ensemble median of 16 replicate trials.Vertica1 dotted lines show the rise-time
of the step perturbation. Subject 5 (top) tended to relax following the perturbation
regardless of the size or direction of the step. Subject 4 (middle) exhibited the opposite
tendency, and Subject 3 (bottom) had flat tong-term responses. (right) The consistent
trends or interventions were accounted for by separating the responses into symmetric
S and antisymmetricA components, as described in the text. The former were used to
calculate the stiffness, and the latter indicated the drift in the motor command over
time.
363
Grasp Stiffness
a drift in the motor command or effort from the preperturbation value. The drift
would corrupt estimates of stiffness if the final, steady-stateforces and spans were
compared directly to the preperturbation values. Therefore, the median responses
from equal-amplitude pairs of extension E (increasing span) and flexion F (decreasing span) steps were partitioned into symmetric S and antisymmetricA components (Figure 3, right) as follows:
S=-
E - F A = -E + F
2 '
2
All four components are labeled for the I .O-cm steps for Subject 5 in Figure 3 (top,
heavy lines).
The symmetric component is that part of the response which has the same
amplitude but opposite directions for extension and Bexion steps. Stiffness was
calculated from the symmetric response by averaging the forces and spans for 0.1
s prior to the step onset and over the interval between 0.5 and 1.0 s after the step
onset, and then dividing the change in force by the change in span. The antisymmetric component is that part of the response which changes in the same direction
regardless of the step direction. The antisymmetric component will be attributed
here to a change in motor command, and it will change the force to which the
stiffness estimate is referenced but will not affect the stiffness estimate.
The EMGs were rectified, but not filtered, prior to taking ensemble medians across replicate trials. The ensembles were averaged over two time windows, 0.1 s prior to the step onset and over the interval from 0.5 to 1.0 s after
step onset, but were not parsed into symmetric and asymmetric components.
Rather, relative changes in the EMG were calculated for each step size (flexion
and extension) by taking the logarithm of the ratio of the final, steady-state EMG
to the initial, preperturbation EMG. The ratio obviates the need for explicit scaling of the EMGs.
The stiffnesses calculated from the symmetric component were subjected to
repeated-measures ANOVAs to test for significant effects of step parameters (risetime, amplitude, initial stiffness, initial span, and initial force). A critical value of
a = .05 was used in all cases. Because stiffnesses are essentially slopes, they do
not add arithmetically and their distributionsbecome highly skewed near extreme
values of It...Therefore, stiffnesses were converted to angles via an arctangent
transformation prior to performing the ANOVA (Colebatch & McCloskey, 1987;
Van Doren, 1995). Logarithms of forces and EMG ratios were used similarly to
equalize the variance over their ranges. The results from the main experiment were
also regressed to derive an expression relating stihess to initial force. In this case,
the variables were transformed (arctangent of stiffness and logarithm of force) so
that their variances were approximately equal over their respective ranges. The
data were fit with a line, K = mf, + b, but the objective function was defined in the
transformed coordinates.
Results
Step Parameters
The effect of step rise-time (40,100, and 250 r n ~ is) ~shown in Figure 4A for two
subjects representing the smallest (left) and largest (right) variation. Step size and
Van Doren
364
initial stiffness were constant (1.0 cm and 6.0 Nlcm, respectively). On average,
grasp stiffness calculated from the symmetric component increased with increasing rise-time, as shown in Figure 5A. The effect of rise-time was relatively small
(16% change on average) but was statistically significant ( p = .01, one-way
repeated-measuresANOVA). It seemed advantageous to use the longer rise-times
at first since they felt much smoother than the abrupt, 40 ms steps and seemed
less likely to evoke interventions (Feldman, 1986). However, it was very difficult
to detect intervention when it happened because forces often continued to increase
or decrease smoothly after the step ended. The shortest steps were used in subsequent experiments to facilitate discrimination of the passive, short-latency reflex
and long-latency responses.
k==:
Subject 2
Subject 1
4
Time (sec)
Figure 4 - Forces from the perturbed hand from the three control experiments,
plotted as in Figure 3. (A) Responses for three rise-times (40,100, and 250 ms;indicated
by the vertical dotted lines) with constant step size (33.0 cm) and initial load stiffness
(6.0 Nlcm) prior to the perturbation. Subject 4 (left) had the most variation in grasp
stiffness measured for different rise-times; Subject 2 (right) had the least. Overall, the
measured grasp stiffness increased modestly with rise-time. (B) Similar results for
steps with constant initial stiffness (6.0 N/cm) and rise-time (40 ms) but varying step
amplitudes (40.2,0.5,1.0 cm), showing greatest (left, Subject 1) and least (right, Subject
2) variation in grasp stiffness across step amplitudes. The arrows show the steadystate forces that would have resulted if grasp stiffness had remained constant at the
value measured for the 1.0 cm step.
Grasp Stiffness
3
2
1
0
g
2
-1
-2
-3
?!
0)
2
9
1
0
-1
-2
-3
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Time (sec)
Figure 4 - (C) Responses for steps with different initial stiffness (3.0, 6.0, and
12 Nlcm) but constant amplitude (1.0 cm) and rise-time (40 ms), showing the greatest
(left, Subject 3) and least (right, Subject 2) variation in grasp stiffness with changes in
initial load stiffness. Grasp stiffness was unaffected by load stiffness. @) Comparison
of responses across control experiment sessions for steps with constant parameters:
6.0 Nlcm load stiffness, 40 ms rise-time, and 1.0 cm amplitude. Subject 5 (left) had the
greatest variation in grasp stiffness, due largely to the responses from the initial session
(arrow) contaminated by large and consistent interventions. Subject 2 (right) had the
least variability in responses across sessions.
Step size had a large and significant effect on grasp stiffness. Figure 4B
shows the results from two subjects with the largest and smallest variations (left
and right, respectively), and Figure 5B plots stiffness as a function of step size for
all subjects. It is difficult to judge whether stiffness changed from the force traces
alone since the step size changed, so the arrowheads at the right of each graph in
Figure 4B show the forces that would have resulted if the stiffness was the same
for all steps and equal to that measured for the 1 cm step. The force produced by
the 0.2 cm step was much larger than expected relative to the 1 cm step, indicating
that the stiffness increased as the step size decreased. The increase was large (56%
on average) and statistically significant ( p < .0001, one-way repeated-measures
ANOVA). An amplitude of 0.5 cm was used throughout the main experiment, but
caution must be used in interpreting those results in view of the dependence of
stiffness on step size.
Van Doren
366
0
30
0
0
100
Rise Time (ms)
300
2
0
0.2
1
Step Size (cm)
10
20
Initial Stiffness (Nlcm)
Rise Time Amplitude Initial k
Experiment Parameter
Figure 5 - Stiffnesses calculated from symmetricresponse componentsfor all subjects
(A Subject 1, Subject 2, A Subject 3, 0 Subject 4, W Subject 5) and each of the
conditions tested in the control experiments, showing effects of (A) step rise-time, (B)
step amplitude, (C)initial load stiffness, and @) session number or type. Grasp stiffness
increased modestly with rise-time (p = .01), decreased markedly as step size increased
@ < .0001), did not change with initial load stiffness (p = .39), and decreased slightly
with session number (p = .03). The latter effect was due to the relatively high stiffnesses
exhibited in the first sessionsof Subjects 3 and 5, probably resulting from intervention.
The median responses from two subjects for each of the three initiai load
stiffnesses (i.e., the stiffness of the manipulandum prior to the perturbation) are
shown in Figure 4C. Again, the subject on the left exhibited the most variation
over the range of initial stiffnesses, and the subject on the right exhibited the least.
The step size and rise-time were constant (f1 cm and 40 ms, respectively). Variations in the initial load stiffness (3.0, 6.0, 12 Nlcm) had almost no effect on the
measured grasp stiffness as shown in Figure 5C. The grasp stiffness increased very
slightly (by about 1%) with initial load stiffness but not significantly ( p =.49, oneway repeated-measures ANOVA). Since load stiffness had no effect, the middle
value (6 Nlcm) was used in the main experiment.
Last, it is useful to compare the responses to the same step applied across the
three different control experiments (load stiffness 6.0 Nlcm, rise-time 40 ms, amplitude k1.0 cm). The sessions were conducted in the same order for each subject
Grasp Stiffness
367
(variable rise-time, step size, and load stiffness) so the effect of order was redundant with the effect of experiment type. There was a modest order/type effect on
the measured grasp stiffness (p = .03, one-way repeated-measures ANOVA) with
an overall decrease in stiffness of 22% from the first to the last session. It is likely
that the reduction reflects changes in triggered reactions or interventions exhibited
by Subjects 3 and 5. The arrowhead in Figure 4D points to the extension step for
Subject 5 from the first experiment (effect of step rise-time). The response is characterized by the lack of a clear reflex peak since it was swamped by the subsequent
intervention. The intervention disappeared in the later two sessions. This adaptation or learning effect was present, though smaller, for Subject 3 (Figure 5D) and
absent completely in the 3 other subjects. At worst, the intervention corrupts some
of the results of the first control experiment (rise-time effect) but does not invalidate the choice of rise-time for the remainder of the study. As a precaution before
the main experiment, though, subjects completed one or two practice sessions until their responses were free of such large interventions and displayed a clear reflex
peak.
Initial Force and Span
The results of the main experiment are summarized in Figure 6, which plots grasp
stiffness as a function of the initial force f,, with the initial span x, varied parametrically. The results for each subject are offset for clarity in Figure 6A and are
plotted in transformed coordinates (arctangent of stiffness, logarithm of force).
Stiffnesses were calculated using the symmetric component of flexion and extension step pairs, and the initial force was amended by subtracting the antisymmetric
component from the force prior to the perturbation. Overall, span had a small but
significant effect on stiffness (p = .05, two-way repeated-measuresANOVA), such
that the stiffness at x,= 4.4 cm was 5% lower than at either 2.4 or 6.4 cm. The
effect of initial force was large and highly significant ( p c .0001), accounting for
93% of the variance not due to the subject effect. The (transformed)stiffnesses and
forces were averaged across subjects and spans, and fit via linear regression (see
Data Analysis), as shown in Figure 6B (linear coordinates). Both the slope (m =
0.40 NIcmfN) and intercept (b = 0.82 Nlcm) were significant @ < .0001).
Muscle Excursions
The average angles measured across 4 subjects for six block widths (finger spans)
are plotted in Figure 7A. The index finger MP joint exhibited the largest angular
displacements, extending markedly with increasing span. The thumb IP joint and
the index DIP joint flexed modestly for larger blocks. Index PIP and thumb CMC
angles extended slightly for increasing span. The index finger angles were used to
calculate rough estimates of the relative excursions of the FDS and FDP muscles
according to Equation 1 and as plotted in Figure 7B. The relative excursions for
both muscles were nearly the same, showing a 0.54% change in muscle length per
centimeter change in finger span (linear regression slope). As a result, flexor lengths
would change only 2.2% over the 4 cm range of initial spans used in the main
experiment, and the 0.5 cm perturbations would produce muscle excursions of
only 0.27%. Wrist movement could also contribute to muscle length changes at
rates of 0.13% per degree and 0.18% per degree for the FDS and FDP, respectively
(see Methods). Therefore, changes in wrist position between trials on the order of
0
5
10
Initial Force (N)
15
Figure 6 - Grasp stiffness (expressed as arctangent) plotted as a function of initial
force (expressed as logarithm) for each subject, identified by the different symbols.
Three subjects (asterisks) also participated in the control experiments. Data for each
subject offset by 0.1 rad from the next-lower subject for clarity. Each point is the
stiffness calculated from symmetric component of the ensemble median of 8 trials in
each loading direction. Each set of connected points corresponds to a different initial
span (2.4,4.4,6.4 cm), which had a small effect on grasp stiffness. Stiffness did increase
significantly with grasp force. (B)Average stiffnesstaken across subjects plotted versus
average force in linear coordinates. Error bars show k l standard error. The linear fit
had a slope of 0.40 N/cm/N and an intercept of 0.82 Nlcm. Averages, standard errors,
and the objective function of the fit were calculated in transformed coordinates.
512-15" or changes during a perturbation of k1.5-2.0" could have produced muscle
excursions comparable to those due to finger movement. Informal observations
suggested that wrist movements were not this large, but the possibility cannot be
rejected rigorously.
Grasp Stiffness
Finger Span (cm)
Figure 7 - (A) Index finger and thumb joint angles measured with a goniometer
while subjects pinched blocks of different width (index finger: 0 DIP, A PIP, MP;
thumb: IP, A CMC). Each point is the average of one measurement for each block
width in each of 4 subjects. Finger span is equated operationally to the block width.
Standard errors of the angles ranged from 3" to 10'. (B) Relative changes in muscle
length for the extrinsic finger flexors (0 FDP, PDS) calculated using Equation 1.
Muscle lengths at different spans are expressed as a percentage difference relative to
the muscle length corresponding to a finger span of 4.5 cm (e.g., changing span from
4.5 cm to 9.0 cm increased the muscle length by about 2.4%). The data pooled for both
muscles are fit well by the function: e% = m(x - b), where m = 0.54%/cm, b = 4.1 cm,
and rZ = .99. That is, the muscle length changed about 0.5% per centimeter change in
span.
Logarithms of the final-to-initial EMG ratios are plotted versus step size in Figure
8. Each point is the median taken across 5 subjects. All of the muscles (FDS, EDC,
and 1DI) showed a significant effect of step size (one-way repeated-measures
ANOVA for each muscle, p < .05). Extension steps stretched the flexors (FDS,
1DI) and increased their tonic EMG, as expected. Flexion steps shortened these
muscles and produced little change in their EMG. The EMG from the EDC, curiously, increased for extension steps (which shortened the muscle) and decreased
for flexion steps (which stretched the muscle).
Van Doren
FLEXION
EXTENSION
Step Size (cm)
Fiyre 8 - Change in EMGs of three muscles (0 1D1, A FDS, and EDC) as a
function of step size. The change in the EMG is plotted as the Iogarithm of the ratio of
the average rectified EMG taken over an interval 0.5-1.0 s following the step onset
(finai) to the average taken over an interval 0.1 s prior to the step (initial).
Discussion
The results of this study and their immediate interpretation are straightforward,
but they pose some difficulty in interpreting the significance of isovolitional forcespan characteristics derived from previous effort matching experiments. The main
observations were that the stiffness of three-finger pinch increased markedly with
the pinch force at a rate of 0.40 N/crn/N (on average) over the range of roughly 2.0
to 12 N (up to 10-20% of typical MVCs; see Van Doren, 1993). Stiffness varied
srightly (by about 5%)for finger spans from 2.4 to 6.4 cm, corresponding roughly
to a 2% change in the lengths of the extrinsic finger flexor muscles (FDP and FDS,
assuming no wrist movement). These results must be tempered, however, by acknowledging that stiffness was measured with a particular method (step displacements) and with particular parameters (f0.5 cm amplitude, 40 ms rise-time, 6.0 NI
cm initial stiffness). The control experiments showed that a different choice of
initial load stiffness would have had no effect on the grasp stiffness measurements,
and grasp stiffness might have increased slightly for longer rise-times. Grasp stiffness certainly would have increased or decreased significantly for smaller or larger
steps, respectively.
There may be some doubt also regarding the calculation of stiffness from the
symmetric components of the force and span trajectories and the suggestion that
the antisymmetric components result from changes in the motor command or effort drift. That interpretation is supported by the typical time course of the antisymmetric component, which was near zero around 200 ms (approximately equal
to the voluntary reaction time) and then steadily drifted in one direction or the
other. The direction and amplitude of the drift were consistent across conditions
within a subject but were not consistent across subjects. This behavior suggests
that, after the short-latency reflex response and before voluntary reactions could
appear, the asymmetry was small but increased at later times. The consistency
within a subject and the variability across subjects suggest that the slow changes
Grasp Stiffness
371
were due to idiosyncratic interventions, perhaps triggered reactions. W h t snch
reactions, the passive, intrinsic, and reflex components wmld imm%&-re to gmrdvce
a nearly symmetric springlike response to extension and flexion steps (see Becker
& Mote, 1990; Crago, Houk, & Hasan, 1976; Nichols & Houk, 1976). Also, the
use of the symmetric components of span and force to calculate stiffness is equivalent to the empirical procedure of "averaging7' the stiffness fsm opposing pairs of
steps by simply dividing the total change in force by the total Change in span. The
use of such an averaging technique as a data-analysis tool is independent of the
interpretation of the antisymmetriccomponent.
Qualitatively,the present measurements of grasp stiffness are similar to many
previous measvrements of joint and limb stiffness. First, stiffness varied slightly
(5%)across finger spans, though it is not clear why the stifkess s b m d be smaller
for the middle span (4.4 cm) compared to larger and smaller spans. The decline
could be due to a particular juxtaposition of joint angles and muscle lengths, but
the present data are too inconclusive to identify a mechanism with confidence.
The marginal effect of finger span seems to reflect the variable results reported in
previous studies, where joint angle may (e-g., Weiss et al., 1986b) or may not
(Gottlieb & Agarwal, 1988) influence stiffness. In contrast, the general trend that
total stiffness and its components (intrinsic and reflex) increase with initial force
has been observed many times, for isolated muscles (e.g., Kirsch & Kemey, 1997;
Rack & Westbury, 1974), single muscles acting at a joint (Akazawa et al., 1983;
Carter et al., 1990, 1993), and multiple muscles acting at a single joint such as the
elbow (Bennett, 1993; Gottlieb & Agarwal, 1988; Latash & Gottlieb, 19901, the
ankle (Gottlieb and Agarwal, 1979; Hunter & Kearney, 1982; Sinkjaer et al., 1988;
Weiss, Hunter, & Kearney, 1988), or the MCPjoint of the finger (Hajian & Howe,
1997). Planar stiffness (or, more generally, impedance) of the entire arm has also
been measured by two-dimensionalperturbations applied to the hand (Dolan, Friedman, & Nagurka, 1993; Gomi & Kawato, 1996; Mussa-Ivaldi, Hogan, & Bizzi,
1985; Tsuji, Morasso, Goto, & Ito, 19951, and stiffness again increases with the
load force (Perreault, Krisch, & Acosta, 1997).
The observation that stiffnessincreased slightly with increasing rise-time, in
contrast, differs from the results of Latash and Gottlieb (19901, who found that
stiffness was unaffected by changes in the risetime of torque steps from 20 to 800
ms for 7 of 9 subjects. However, the imposed torque steps produced steps in displacement that lasted from roughly 500 to over 1,000 ms-all longer than any of
the displacement steps applied here (40-250 ms). Stiffness increased roughly 20%
on average over the latter range, but the results ((Figure5A) hint that the effect may
be insignificant at the longer rise-times. In fact, mean comparisons {contrasts)
show that the stiffness measured with 40 ms steps is significantly lower than that
measured with either 100 or 250 ms steps (g I .01) and that the latter did not differ
significantly O,= .7). Carter et al. (1990) found that the stiffness of the electrically
stimulated (but voluntarily relaxed) human flexor pollicis longus muscle decreased
abruptly (yielded) for rotations of the thumb IP joint larger than 3" for 100 ms,
linear, ramp perturbations. Slow (500 ms) perturbations did not produce yielding.
Rack and Westbury (1974) found likewise that yielding was more severe at faster
velocities for a fixed amplitude stretch. Perhaps the 40 ms steps used in the present
study produced yielding as well (though, see below), reducing the measured stiffness. Carter et al. (1990) also showed that yielding is diminished in muscle with
intact reflexes, consistent with the modest effect measured in this study.
372
Van Doren
Yielding also increases and stiffness decreases with step size (Carter et al.,
1990; Gottlieb & Agarwal, 1988; Nichols & Houk, 1976; cf. Sinkjaer et al., 1988)
or the amplitude of sinusoidal or stochastic perturbations (Kearney & Hunter, 1982;
Kirsch, Boskov, & Rymer, 1994; Rack & Westbury, 1974), consistent with the
current results (Figure 5B). Yielding typically occurs for muscle stretches greater
than 1-2% of muscle length (in cat soleus, at least; Rack & Westbury 1974)due to
a decline in the number of attached cross-bridges (e.g., Joyce, Rack, & Westbury,
1969; see also Kirsch et a]., 1994, for a discussion). The largest steps applied here
elongated the extrinsic finger flexors roughly 0.5% (Figure 7B), which should be
in the range of elastic deformation of the cross-bridges rather than their disruption.
It is possible, then, that the change in stiffness with step size reflects a change in
the force generated by reflexes rather than changes in the intrinsic muscle stiffness. Changes in step size did produce graded changes in the tonic stretch reflex
response as reflected in the EMGs measured from the FDS, 1DI and EDC (Figure
8). The measurements show that a reflex contribution is possible, at least, but its
magnitude cannot be calculated since the relative contribution of these muscles
versus others is uncertain (see Maier & Hepp-Reymond, 1995). Moreover, some
of the muscle activity was probably necessary to stabilize the finger and wrist
joints as grasp force changed. The EDC, for example, became more active for
steps that shortened it, perhaps to offset an increasing wrist flexion moment generated by the long finger flexors (e.g., Gonzalez et al., 1997).
Quantitatively, it is more relevant to compare the current results to previous
studies that measured grasp stiffness directly via perturbations (Hajian, 1997;
Hermsdiirfer, Mai, & Marquardt, 1992; Hermsdorfer et al., 1994;Van Doren, 1996)
or inferred grasp stiffness indirectly from effort-matching data (Van Doren, 1995,
1998).All of the direct measurements used pneumatic actuators to apply perturbations to a two-finger pinch (Hajian, 1997; Hermsdorfer et al., 1994; Karason &
Srinivasan, 1995; see also Hermsdorfer et al., 1992) or three-finger pinch (Van
Doren, 1996). The two-finger studies all measured stiffness only during the step
transition prior to the influence of reflexes and so are not directly comparable to
the present study. The prereflex stiffness (e.g., average 3.0 Nlcm, two-finger pinch,
Hermsdorfer et al., 1994), however, is not dramatically different from the steadystate stiffness observed here. Van Doren (1996) used a three-finger pinch and measured stiffnessduring the steady state, but the initial forces exceeded most of those
used in the present study. The earlier measurements also used force steps that produced relatively slow changes in span (>200 ms) and variable step sizes on the
order of a few millimeters. The resulting stiffnesses, as a consequence, may be
higher than those measured with 40 ms, 0.5 cm displacement steps, based on the
trends measured in the present control experiments. Nonetheless, the data from the
present study and those from Van Doren (1996) seem to form a contiguous set
(Figure 9A, open and filled points) and show that the stiffness of a three-finger
pinch is roughly proportional to grasp force from 2 to 50 N.
The "grasp stiffness" inferred from effort-matchingexperiments (Van Doren,
1998) behaves quite differently. The IFSCs constructed from the matching functions were nearly linear over a wide range of initial conditions (x, 4.7-8.2 cm, f,
2.2-34 N, load stiffness 1.5-12.6 Nlcm), and the slopes were nearly constant at
Grasp Stiffness
1
1
10
initial Force (N)
60
Figure 9 - Comparison of stiffnesses measured from the present study (open symbols,
replotted from Figure 6 but without offsets) to previous measurements made with a
pneumatic manipulandum (filled points, each from a single trial; from Van Doren,
1996) and to slopes of compliance characteristics derived from effort matches (crosses;
from Van Doren, 1998). The two sets of perturbation data appear to form a contiguous
set. In contrast, the slopes of the compliance characteristics were comparatively large
and nearly independent of force.
turbation method, the fingers and thumb supposedly start at some initial grasp
state corresponding to a point (f,, x,) on a particular compliance characteristic.
Then, if the load is changed without changing the motor command, the grasp output is supposed to shift to another point on the same characteristic. If so, then the
stiffness calculated from the initial and final steady states will equal the slope of
compliance characteristic (Feldman, 1986;Gottlieb & Agarwal, 1988). In the matching paradigm, in contrast, equal efforts are exerted simultaneouslyor serially against
the different loads, which should also yield points on a single compliance characteristic. The results of the present study (perturbations) and the previous study
(effort matching, Van Doren, 1998) suggest that the two methods are not actually
equivalent. The stiffnesses calculated from the pemubations are clearly too small
compared to the slope of the compliance characteristicderived from effort matches.
It is possible, though not likely, that the discrepancy is due to differences in
the loads. The initial conditions in the present study were always achieved against
a spring load with a constant stiffness (6.0 Nlcm), and perturbations were applied
by simultaneously increasing the load stiffness (32 Nlcm) and shifting the effective rest length (see Methods). The matching experiment used fixed springs with
stiffnesses from 1.55 to 12 Nlcm. The grasp stiffness may have been altered automatically in response to differences in the load stiffness, but no such effect was
observed in the third control experiment described here in which the effects of
initial load stiffness were investigated explicitly (albeit over a narrower range).
Alternatively, the narrower range of forces encountered by subjects in the present
experiment (-2-12 N) compared to the previous experiment (-2-34 N) may have
prompted subjects to use more compliant grasps. Context effects are common in
sensory scaling (e.g., Foley, Cross, & O'Reilly, 1990; Gescheider & Hughson,
1991; for a review see Poulton, 1989), have been o b s e ~ e din effort scaling (Burgess, Cooper, Gottleib, & Latash, 1995), and merit further study.
-
-
374
Van Doren
It is more likely that perturbations imposed by transient changes in the load
simply do not represent ''pertwbations7' produced by grasping different loads from
the outset. That correspondence is Iikely to fail for large perturbations since yielding will occur when a perturbation is imposed but not when two different loads
are grasped. For very small displacements, the response to imposed perturbations
will be due to the passive and intrinsic mechanics of muscle rather than to different levels of reflexive recruitment. The response to any perturbations may be affected by the extent, force, duration, and direction of voluntary movements that
precede it as a result of thixotropic (history-dependent) properties of muscle (e.g.,
Hagbarth, Nordin, & Bongiovanni, 1995; Wiegner, 1987). Last, it seems unlikely
that the detailed dynamics of cross-bridge binding and motor unit recruitment will
be the same for both paradigms at all levels. The situation is similar to that encountered when measuring the Iength-tension properties of isolated muscle. The
classical approach (Rack & Westbury, 1969) is to stimulate a muscle electrically
at a fixed rate while it is held at a series of different lengths and measure the
resulting force. In a sense, the paradigm applies a repeatable motor command
against different loads and is similar in structure to effort matching. Perturbations
(muscle stretches) produce force and length responses that fail, in general, to return to the static length-tension curve in the steady state (e.g., Joyce et al., 1969).
That is, perturbations do not represent transitions between points on the lengthtension curve, and the slope of the length-tension curve does not represent muscle
stiffness.
So, which paradigm measures stiffness correctly-perturbations or effort
matching? Both methods are isovolitional (Burgess et al., 1995)but may best represent two modes of behavior. The isovolitional force-span characteristic (or other
such force-position characteristic) is a coIlection of output states that are the results of exerting the same motor command against different loads, but it is not
equivalent to a physical compliance. Even so, the motor behavior represented by
the IFSC may allow different objects to be grasped successful~ywithout particularly accurate "tuning" of the motor command. Imposed perturbations, on the other
hand, may adequately represent the motor response to changes in the load caused,
for example, by limb movements while an object is grasped. In other words, perturbations describe the behavior of the system about an operating point which is,
itself, an equilibrium point on the compliance characteristic (or, an equilibrium
state belonging to the set that comprises the characteristic)for a given motor command. Regardless of the specific interpretation, however, it is unlikely that any
one measure of stiffness can describe the whole spectrum of motor behavior, and
any particular measure should be interpreted judiciously and within an appropriate
context.
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Notes
'The IFSC, as defined here, is analogous to the "invariant characteristic" defined by
Feldman (Asatryan & Feldman, 1965), and the "joint compliance characteristic" defined by
Latash and Gottlieb (1990). "Isovolitional" is used as per Burgess et al. (1995) to make it
Van Doren
378
clear that the voluntary motor command is fixed for a given IFSC. "Invariant" in Feldman's
sense also refers to a fixed motor command. The generic term "force-span" is used in favor
of "compliance" since it is not clear whether the IFSC represents a bona fide physical compliance. It is more likely that the F S C is a set of states that is characteristic of, but not
formally equivalent to, a particular motor command.
2Theproximity of the strain gauges to the motors caused severe pickup of the motor
drive signal: a pulse-width modulated (PWM) square wave at 22 kHz. To reject this noise,
the clock signal generated within the PWM circuit in the motor amplifiers (Model 303,
Copley Controls, Westwood, MA) was monitored and used to trigger data acquisition so
that the force was sampled away from the square-wave transitions.
3The actual step rise times were slightly smaller than the nominal values (typically
30, 95, and 245 ms) due to the finite stiffness of the manipulanda. Likewise, actual step
amplitudes were about 10-20% below their nominal values. For simplicity, the nominal
values are used as labels in the plots and analyses since exact values are not critical.
Acknowledgments
This research was funded by grant NS-27958 from the National Institutes of Health.
I thank Mr. Scott Heavner for excellent technical assistance and appreciate the helpful comments and suggestions from Drs. Robert F. Kirsch, Wendy M. Murray, T. Richard Nichols,
and two anonymous reviewers,