Exp Brain Res (2003) 152:17–28
DOI 10.1007/s00221-003-1507-4
RESEARCH ARTICLE
Florin Popescu · Joseph M. Hidler · W. Zev Rymer
Elbow impedance during goal-directed movements
Received: 12 November 2001 / Accepted: 14 April 2003 / Published online: 23 July 2003
Springer-Verlag 2003
Abstract The mechanical properties and reflex actions of
muscles crossing the elbow joint were examined during a
60-deg voluntary elbow extension movement. Brief
unexpected torque pulses of identical magnitude and
time-course (20-Nm extension switching to 20-Nm flexion within 30 ms) were introduced at various points of a
movement in randomly selected trials. Single pulses were
injected in different trials, some before movement onset
and some either during early, mid, late or ending stages of
the movement. Changes in movement trajectory induced
by a torque pulse were determined over the first 50 ms by
a nearest-neighbor prediction algorithm, and then a
modified K-B-I (stiffness-damping-inertia) model was
fit to the responses. The stiffness and damping coefficients estimated during voluntary movements were compared to values recorded during trials in which subjects
were instructed to strongly co-contract while maintaining
a static posture. This latter protocol was designed to help
determine the maximum impedance a subject could
generate. We determined that co-contraction increased
joint stiffness greatly, well beyond that recorded under
control conditions. In contrast, the stiffness magnitudes
were quite small during routine voluntary movements, or
when the subjects relaxed their limb. Furthermore, the
damping coefficients were always significant and increased measurably at the end of movement. Reflex
activity, as measured by EMG responses in biceps and
triceps brachii, showed highly variable responses at
latencies of 160 ms or greater. These reflexes tended to
activate both elbow flexors and extensors simultaneously.
These findings suggest that very low intrinsic muscle
stiffness values recorded during point-to-point motion
render an equilibrium point or impedance control approach implausible as a means to regulate movement
trajectories. In particular, muscle that is shortening
against inertial loads seems to exhibit much smaller
stiffness than similarly active isometric muscle, although
some degree of damping is always present and does not
simply co-vary with stiffness. Although the limb muscles
can be co-contracted statically or during movement with
an observable increase in stiffness and even task performance, this control strategy is rarely utilized, presumably
due to the greater energetic cost.
Keywords Elbow · Impedance · Stiffness · Ballistic
Part of this work was presented earlier in abstract form (Popescu
and Rymer 1999)
F. Popescu ())
Laboratorio di Tecnologia Medica, Istituti Ortopedici Rizzoli,
Via di Barbiano 1/10, 40136 Bologna, Italy
e-mail: [email protected]
Tel.: +39-051-6366865
Fax: +39-051-6366863
F. Popescu
Department of Mechanical Engineering, Northwestern University,
2145 Sheridan Road, Evanston, IL 60208, USA
J. M. Hidler · W. Z. Rymer
Department of Biomedical Engineering, Northwestern University,
2145 Sheridan Road, Evanston, IL 60208, USA
W. Z. Rymer
Department of Physical Medicine and Rehabilitation,
Northwestern University Medical School,
303 East Chicago Ave., Chicago, IL 60611, USA
Introduction
The dynamic behavior of the human limb performing a
voluntary task is the result of a complex interaction
between neuromuscular mechanics, passive joint properties and the mechanical environment. Generally, the
mechanical behavior of the limb has been modeled using
the most simple and plausible mathematical representations of muscle visco-elasticity, based on the idea of local
linearization of the force-length (FL) and force-velocity
(FV) curves characteristic of skeletal muscle. This
representation is commonly known as K-B-I, indicating
stiffness (K), viscous damping (B) and inertia (I),
analogous to that of an equivalent second order massspring damper system. While there have been multiple
efforts directed towards understanding the role of intrinsic
18
mechanical and reflex muscle properties during reflex
muscle activation in upper and lower extremity muscles
(Gottlieb and Agarwal 1972; Bennett et al. 1992; Bennett
1993, 1994; Gomi and Kawato 1997), there have been
relatively few attempts to estimate K, B and I during
voluntary movements that are under the subject’s control.
This is primarily because of the technical difficulties
associated with imposing controlled force perturbations
during unconstrained voluntary motion.
While such investigations are inherently difficult, the
availability of accurate estimates of muscle mechanical
properties during limb motion would be of great value in
understanding the strategies available for controlling limb
trajectory. For example if mechanical impedance during
limb motion remains high, then impedance control could
provide a legitimate strategy for trajectory control (Hogan
1985). Conversely, if mechanical impedance is very low,
then other control strategies may be more appealing and
appropriate. Accordingly, our objective is to estimate the
mechanical impedance of muscle during voluntary limb
movements to help us assess the potential impact of limb
mechanical impedance on neuromuscular control strategies.
One approach to investigating how muscle properties
(such as stiffness and damping) influence the limbs’
capacity to resist external disturbances has been to study
the limb kinematic response to small, random perturbations applied during simple movements (Bennett et al.
1992; Gomi and Kawato 1997; Xu and Hollerbach 1999).
For example, Bennett et al. (1992) attempted to characterize limb mechanics by using an air-jet actuator to
deliver small random disturbances in force during elbow
movements. However, the voluntary limb movements
were continuous and cyclical rather than point-to-point,
and the imposed air jet perturbations small (2 Nm peakto-peak) and limited in bandwidth. Other estimates
recorded limb force responses during sinusoidal positional perturbations, imposed during point-to-point positioncontrolled movements (Bennett 1994). These studies
relied on a subject’s capacity to repeat joint motion so
that there was a precise match between the intended limb
motion and the actual movement imposed by the device.
This is clearly a difficult task, and one whose performance proved difficult to assess. In spite of these
uncertainties, stiffness was reported to drop significantly
during movement and limb motion was also underdamped, while the magnitude of the stretch reflex
increased at later stages of movement
In a more recent study, Gomi and Kawato (1997)
attempted to estimate stiffness and damping of the limb
by applying unexpected force pulses to the hand at
various stages of a planar movement. In these experiments, the stiffness was shown to increase during motion,
albeit slightly. Changes in trajectory were fit by K and B
matrices for 300 ms post perturbation, so that stretch and
potentially even so-called “long-loop” reflexes may have
had a major effect on K and B estimates.
In fact, none of the existing studies of limb mechanical
impedance during limb motion made an attempt to
identify the mechanical properties of the plant itself
(i.e., the intrinsic visco-elasticity). Rather, they quantified
the net visco-elasticity of the limb, which includes both
the intrinsic mechanical properties of the limb and the
effects of reflex action without separating the relative
contributions of each constituent. Furthermore, no rigorous review of the actual applicability of the K-B-I
formulation in the face of complex dynamics has yet been
performed, although efforts to overcome the difficulties
outlined above are underway (Burdet et al. 2000).
In the present study, we sought to estimate the
mechanical properties of the elbow musculature during
voluntary motion and to model these properties mathematically. The model we used was one that relates
changes in torque to changes in trajectory. The intent of
this work is to use these data to gain insight on the
constraints and to characterize the control strategies used
for goal-directed voluntary limb movements.
Materials and methods
Subjects
Five subjects (3 male, 2 female) participated in the study. Each
participant was instructed on the intent and protocol of the study
and gave informed consent. The protocol was approved by
Northwestern University’s Institutional Review Board.
Instrumentation
Torque-controlled pulses were delivered to the subject’s elbow
using a DC motor (Cleveland Machine Controls, F563), digitally
controlled with a Pentium PC (see Fig. 1). A torque transducer
Fig. 1 Apparatus and protocol for elbow perturbation study. The
subject is shown from above. Note the fiberglass cast on the
forearm attached to an aluminum beam under the arm by means of
a light vise pressed on as the cast solidified. The motor is seen as a
box under the elbow, and the torque transducer-co-axial with the
motor shaft and the elbow-as a circle. The fan-out lines shown
correspond to the onsets of the various perturbations given. The
trapezoids around the ‘PRE’ and ‘ENDING’ lines indicate the
relative widths of the starting and target regions as seen by the
subject on the monitor
19
Table 1 Description of different perturbation types. See
Fig. 1 for a graphical description
Perturbation condition
Pulse onset
PRE, STIFF
EARLY
MID
LATE
ENDING
About 100 ms before the green light signal
The point at which the cursor crossed –17.5
The point at which the cursor crossed 0
The point at which the cursor crossed 17.5
80 ms after the 17.5 crossing (roughly at the end of motion).
(Himmelstein, 2030) mounted between the motor shaft and the
subject was used to measure the amount of elbow torque throughout
each trial. This torque signal was also used as feedback to the motor
controller such that the system operated under closed-loop torque
control. This ensured the elbow torque was well controlled,
accounting for actuator effects such as friction. A precision
potentiometer and a tachometer mounted on the motor shaft were
used to measure elbow joint angle and angular velocity, respectively. Surface EMGs were recorded differentially from the triceps
and biceps using a DelSys Bagnoli-4 EMG system. All signals were
anti-alias filtered at 500 Hz prior to sampling at 1000 Hz using a
16-bit data acquisition board (Keithley Metrabyte, DAS 1802
HRDA).
Protocol
Each subject was seated in a BIODEX chair, with the torso
restrained using a lap belt and shoulder straps. The height of the
chair was adjusted such that the right arm was kept in the horizontal
plane of the shoulder. The subject’s arm was first cast from the
wrist to the elbow using Delta-Lite fiberglass. The cast was wound
tightly to reduce wobble in the forearm soft mass, but not so too
tight as to induce discomfort or ischemia. After hardening, the arm
cast was then attached to a rigid aluminum beam extending from
the motor shaft using additional casting material. A shallow vshaped aluminum piece was mounted to the aluminum beam,
providing protrusions upon which the humeral condyles made rigid
contact. A steel hose clamp was tightened around the wrist area and
aluminum beam, further tightening the interface between the
subject’s forearm skeletal structure and the motor.
The inertia of the subject’s forearm-hand was first estimated
using a 20-s transient sinusoidal position input with 4 successive
periods of increasing frequency (2-12 Hz) and 0.5-deg amplitude.
During these trials, the subject was asked to not intervene but
instead relax. The same procedure was performed post-experiment
to the cast and aluminum beam only to estimate fixture inertia.
The main experimental protocol consisted of 250 repeated
extension movements, with a schematic of the protocol shown in
Fig. 1. The subject’s position was represented by a 1.51.5 cm box
on a computer monitor, and the start and end targets represented by
1-cm-diameter dots, each of which were center aligned on a
horizontal axis with a 15-cm distance between start and target dots.
The start position was chosen to be about 20 deg from maximal
elbow flexion, and the target was placed 60 deg towards extension.
The subject was instructed to keep the cursor box over the start dot,
wait until it turned green, then move and stop in a single motion
such that the target dot would be contained inside the position box
within 300 ms. The graphical display was such that the subject had
an effective €2.5 mm margin of error on the screen, which
corresponded to €1.67% of the movement distance and €1 deg of
elbow motion. After completing the movement, the target circles
turned white and the subject returned to the start position to await
the onset of the next trial. Movement time was displayed following
each trial so that the subject could adjust their speed accordingly in
order to complete the movement within the instructed time.
The initial 10 trials of the sequence were ’premovement’
bumps, in that the subjects relaxed at the start target, which was
followed by a brief, bi-polar and symmetric torque pulse. These
torque perturbations consisted of bi-directional pulses, each with a
15-ms width (30 ms total perturbation time) and approximately 20Nm amplitude. The subject was then instructed to move to the
target position and hold until instructed to return to the start. The
second 10 trials were the same as the first 10, except that the
subjects were asked to co-contract to a comfortable max with no
visible tremor. This condition was deemed STIFF so that between
the relaxed and stiffened paradigms, upper and lower impedance
bounds could be estimated.
The remaining movements had a 50% chance of having visual
position feedback (targets always remained on the screen during
each trial), and a 10% chance of occurrence for each type of
perturbation (see Fig. 1 and Table 1). The disturbances were
randomly and independently selected, with only one perturbation
per movement. As such, roughly half of the trials the subject
performed did not contain any perturbations. During each trial, the
motor ran in closed-loop torque control using the torque signal from
the torque cell as feedback. This process eliminated any resistance
due to friction in the motor and provided an extremely smooth
environment in which the movements were made.
Data processing
Since the applied perturbations were quite fast, the resulting
motions had high frequency content. This effectively led to limb
inertia being the dominant part of the response (Rack 1981). Two
independent methods were used to estimate inertia: the sinusoidal
input frequency response protocol described above, and a regression model based on anthropometric data taken from the subjects
(Zatsiorsky and Seluyanov 1985). The former method led to much
tighter bounds on the inertia value derived. The other was used to
validate this value (see Results). The inertial value used to fit the
perturbed trials naturally included that of the fixture coupled to
arm.
Inertial estimates were derived by first decomposing each
constant frequency period into its sinusoidal frequency-amplitude
characteristics by a Finite Fourier Transform and extracting the
main frequency and amplitude within that period (there were four
such periods). For a purely inertial system, the magnitude of torque
is equal to the product of the inertia, the magnitude of the position,
and the frequency squared. Plotting the ratio of the torque
amplitude to the position amplitude versus the frequency squared
results in a straight line, with the slope equal to the inertia. The
amplitude and frequency range chosen were such that other joint
components such as reflexes and visco-elasticity did not have a
significant contribution to the dynamic response recorded. This was
confirmed by the EMG traces and the local linearity of the inertial
plot described above.
Traditionally, the dynamics of the joints of the human body in
reaction to perturbations have been approximated by a simple
visco-elastic model: the K-B-I model (stiffness, damping and
inertia in parallel). During preliminary investigations, it became
clear that a linear second order K-B-I model did not adequately fit
perturbations to the human forearm, despite fitting artificially
constructed K-B-I systems using the same apparatus and analysis
methods. The problem arose from independent motion of forearm
soft tissue within the cast. This soft tissue comprises approximately
80% of the forearm mass (Clarys and Marfell-Jones 1986) and has
also been reported earlier as a confounding factor in the determination of the moment of inertia of limb segments (Allum and
Young 1976). For our study, we found that as the cast was applied
tighter to the arm, particularly around the upper forearm, the limb
appeared to become more rigid, although there was a limit to the
20
cast tightness that a subject could tolerate, due to ischemia and
discomfort.
As such, modifications were made to the model to incorporate
these added dynamics, such that the relationship between position
and elbow torque was represented by:
t ¼ I€
a þ Bq_ þ Kq
T a_ þ a ¼ q
ð1Þ
This second order model relates applied torque (t) to joint angle
(q) through the familiar parameters K, B and I. To address the
separate motion of limb mass, a time delay (T seconds) was added
as well as an angle, a, which references the inertial component,
where a represents the angle between the joint axis and the center
of mass (as opposed to the location of the wrist). The concept
behind this change is that the center of mass generally tends to lag
the movement of the bone by a small amount of time, and the
simplest dynamic model of such behavior is a first-order filter. The
transfer function of the limb impedance in the Laplace domain
reveals two poles and one minimum phase zero:
q ðsÞ
Ts þ 1
¼
tðsÞ ðI þ BT Þs2 þ ðB þ KT Þs þ K
ð2Þ
As the time lag, T, approaches zero, the transfer function
approaches that of a standard K-B-I system. The time delay, T, was
chosen to be 5 ms for all subjects, by considering the initial 15 ms
of all responses. The inertial parameter, I, was fixed for each
subject using the procedure described above.
Stiffness and damping parameters were adjusted to fit the mean
position response to the mean torque input for each perturbation
condition in the following manner. For a given perturbation
condition, we collected torque and position in N trials:
ti ðtÞ ¼ recorded torque; i ¼ 1::N
bi ðtÞ ¼ recorded position; i ¼ 1::N
bi ðtÞ ¼ bA;i ðtÞ bU;i ðtÞ
qðti ðtÞÞ ¼ qðK; B; I; T; ti ðtÞÞ
90% and looking to see if data and model curves had similar
structure) it is decided that Dmin is acceptable, then the region of
best fit in parameter space, given measurement uncertainty, is one
for which:
DðK; BÞ Dmin þ
ð3Þ
The third definition in Eq. 3 denotes that a prediction of the
underlying trajectory, bU, i (t), is subtracted from the measured
position. The last definition simply establishes q(ti (t)) as the model
prediction, based on the torque recorded in the i-th trial using what
we will call the K-B-I-T model (so as to distinguish it from K-B-I).
Stiffness, K, and damping, B, were adjusted such that for 50 ms
following the perturbation onset, the mean discrepancy between the
simulated and recorded position traces was minimized:
Dmin ¼ min
ð4Þ
E qðtÞ b
K;B
Fig. 2 Trajectory matching. Shown are two trajectories, which
match well up to the 0-deg crossing point where the MID
perturbation condition occurs (here t0=121), and their subsequent
growing deviation, indicating an underlying limit to the predictability of ballistic motion for fixed-task parameters
t¼0::50
The Euler method of differential equation simulation with time
steps of 1 ms proved to be numerically stable for the values of K
and B tested. Although we fit the average position trace to the
average torque trace, we took into account that the uncertainty in
these averages was related to their standard error:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Vari ðqðti ðtÞÞÞ
qSE ðtÞ ¼
N1
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Vari ðbi ðtÞÞ
bSE ðtÞ ¼
ð5Þ
N1
Traditional statistics of model fit such as chi-square probability
or Bayesian likelihood are based on the concept that deviations
from model predictions to data are random and due primarily to
measurement errors. As there are very few statistical methods
available to deal with the case when the errors in a model fit are
much greater than attributable to measurement error (Sakamoto et
al. 1989), we consider K-B values equivalent if the variability of
the predictions of different K-B values is less than the inter-trial
variability. Thus, if a model is only capable of fitting some
physiological process to within Dmin, and by a general measure such
as VAF or the ’eye’ test (we used both, checking for VAF’s above
E jqSE j þ
t¼0::50
E jbSE j
t¼0::50
ð6Þ
Prediction of arm trajectory
Whenever experiments were performed that altered voluntary
motion, the expected trajectory of the limb at the time of the
perturbation must be predicted. The effect of a perturbation is
dependent on what the movement would have been if the
perturbation was absent.
A nearest neighbor approach to extrapolation was used, for
which a large database of unperturbed trajectories was collected.
The database could be used to train or construct a predictor function
(Burdet et al. 2000) that is small and efficient. Since this function is
not used in real-time to control the motor coupled to the arm,
computational efficiency is not needed, but merely accuracy.
For this purpose we can simply construct a metric (norm) which
measures the similarity between 2 trajectories up to, or from, a
particular point in time with a forgetting factor, w. This metric can
be used to pick the most similar trajectory history from the
database, and its form is:
Z t0
d ðy1 ; y2 ; t0 ; wÞ ¼
ðexp t=wÞ ðy1 ðtÞ y2 ðt þ DtÞ DyÞ2 dt ð7Þ
1
Note that we introduce two parameters, Dt and Dy, which can be
adjusted to minimize this norm. The same norm can be used to
compare trajectory values in the future by the coordinate transformation t’=t. The value for w chosen was 100 ms, such that it
would match twice the approximate bandwidth of human voluntary
movement. An example of a trajectory match for to consistent with
a MID perturbation is shown in Fig. 2.
It follows that one can collect all predictions of unperturbed
trajectories for each choice of to corresponding to the EARLY, MID
and LATE perturbation types, and construct respective functions of
time called “unpredictability functions” which represent the
growing uncertainty in prediction versus time. This function is
zero for the PRE and POST conditions since they are quasi-static.
21
The variability in perturbed trajectories, perturbed minus predicted,
will be expected to be greater than the unpredictability functions.
The remaining variability can be attributed to trial-to-trial changes
in the mechanical properties of the lower arm and the small changes
in the perturbing torques applied.
EMG analysis
Surface EMG data was pre-filtered, rectified and integrated over
successive 40-ms time windows, which began at the onset of the
perturbation (with the exception of EMG profiles of the entire
movement, where the windows began at the onset of trial
recording). EMG activity was scaled to the EMG activity collected
during the time period when subjects voluntarily co-contracted (the
STIFF condition); in this way, biceps activity could be compared to
triceps activity, as agonist and antagonist forces are equal during
co-contraction.
Results
Moment of inertia
Accurate identification of inertia is essential, because the
inertia dominates the response to fast perturbations and it
potentially affects the values of K and B we obtain. That
is, small errors in inertial estimates may propagate
erroneous estimates of K and B. Table 2 shows the
inertial estimates for each subject, along with the
estimates derived using the Zatsiorsky regression model.
Although there is a general agreement between the two
estimates of inertia, and therefore validation of the
frequency response estimation technique, it is clear from
the size of error bars that the frequency response method
provides the far more accurate measurements, with about
3% accuracy.
Fig. 3 Torque perturbations (bottom) and resulting position traces
(top). The thick dotted trace is a sample reaction for the STIFF
condition; the rest are all the reactions of a subject to the PRE
condition (both are isometric). The gray region (<50 ms) indicates
the portion of the response used for the K-B-I-T
The first aim of this study was to characterize the
mechanical state of the limb during voluntary movement
by applying brief perturbations to the subject’s arm. An
example perturbation for the relaxed, PRE motion
condition as well as for the contracted, STIFF condition
is shown in Fig. 3. The resulting torque profiles of the
perturbations and the resulting position traces are also
plotted in the figure, where it can be seen that the torque
perturbation changed little from condition to condition.
The position responses indicate the normal trial-to-trial
variability because of both subtle changes in the torque
input as well as changes in the mechanical state of the
limb. Note that for the PRE case shown, there is no
variability associated with predicting the intended movement trajectory since the subject is attempting to keep the
limb stationary. Note also the effect a large degree of cocontraction has on position. The difference seen in Fig. 3
between the co-contracted and relaxed conditions is about
as large as we can expect since strong co-contraction
induces near maximal stiffness.
The dynamic response to perturbations at the various
positions during a subject’s movement is shown in Fig. 4.
There is clearly an increase in variability of the response
in the moving perturbation conditions (EARLY, MID,
LATE and ENDING) when compared with the PRE
perturbation condition. Even with an accurate prediction
method such as nearest neighbor with a large training set
(see Materials and methods), trajectories can only be
predicted with up to 0.5-deg accuracy for 50 ms into the
future. Torques are also different between actual and
Table 2 Inertial estimates for each subject. The 95% confidence
interval of frequency response estimate is from the linear regression
fit between squared frequency and amplitude. To the linear
regression predictors of forearm/hand inertia (Zatsiorsky and
Seluyanov 1985) from limb geometry data, we added to the
sinusoid estimates of fixture inertia, confidence interval being
derived from the published r-values of the prediction and do not
include measurement errors
Effects of perturbations on joint position
I, frequency response
(kg m2/rad)
95% confidence interval
(kg m2/rad)
I, Zatsiorsky+cast
(kg m2/rad)
95% confidence interval
(kg m2/rad)
I. fixture
(kg m2/rad)
104.5
95.3
110.9
48.4
48.8
5.4
6.7
6.7
2.6
2.6
130.3
91.1
110.2
57.9
64.9
43.0
25.2
34.7
16.3
19.1
25.2
25.2
26.7
16.8
20.2
22
Fig. 4 Actual trajectories minus
unperturbed trajectories. Shown
are position traces (black) and
torque profiles (light gray,
scaled, shown in 10-Nm units)
for each perturbation type. The
inertial response derived from
the average torque profile and
the inertial estimate (if K and B
were zero) is also shown (dotted)
predicted values, but they are much less variable because
torque is the controlled variable by the motor. All traces
of the same condition were aligned in time to provide the
best match among torque profiles.
For modeling purposes, we used the averaged response
for each condition in order to reduce the effect of
prediction uncertainty. This is because the prediction
error associated with the intended trajectory is not due to
random noise but is a slowly growing function. Fitting
each trial independently would therefore result in distorted values of K and B values. To assess the robustness of
limb trajectory to perturbations, we calculated the ratio of
the maximum deflection of the limb to the maximum
deflection attributable solely to the inertia, over the
interval ending 50 ms after the perturbation onset.
We used this measure to gain insight into the
mechanical state of the limb (i.e., the intrinsic stiffness)
without immediately confronting the mathematical difficulties and statistical uncertainty in fitting K-B-I to the
responses. This ratio has limited meaning outside the
paradigm shown, and there is no linear relationship
between it and stiffness. The results are shown in Fig. 5,
where it is illustrated that the only noticeable increases in
effective stiffness within the time period for which
reflexes are not yet active occurs during voluntary cocontraction.
Modeling of forearm impedance
The second aim of this study was to establish the
usefulness of the K-B-I model. Although it proved
necessary to modify the structure of the model in order
to account for the movement of soft mass on the upper
forearm, the K-B-I-T model is similar enough so that the
term may be used interchangeably with K-B-I (see
Materials and methods for formulation and optimization
procedures). The fundamental results of this paper are
shown in Fig. 6.
Fig. 5 Limb robustness to perturbations. Ratios (M) of the
maximum deflection of the arm within 50 ms to similar torque
perturbations to the deflections we would expect for a purely
inertial system, by perturbation condition. Five bars in each
grouping represent 5 subjects. Subject #4 shows a substantial
deviation from this trend, as it was observed from the noisy EMG
traces that she did not substantially co-contract despite the
instruction
The regions of possible K and B values (see Materials
and methods) which achieve fits within the standard error
bounds are shown in the lower trace of Fig. 6. The shape
of these regions indicates the relative sensitivity of the fit
to B and K, respectively. By varying the inertial
parameter within €3% of our inertial estimate, we found
that these shapes were minimally affected. The extrema of
the regions can be used as lower and upper bounds on our
B and K estimates for each condition, summarized in
Figs. 7 and 8, respectively. For the perturbation condition
during motion (EARLY, MID, LATE and ENDING), the
uncertainty region is determined by our prediction
algorithm (see Materials and methods) and the variability
23
Fig. 6 Best fits and parameter uncertainty. Top: Average changes
in position bounded by standard error curve (light gray), mean best
model fit bounded by standard error (dark gray), position change
attributable to inertia (top trace-dotted) and mean torque profile in
10-Nm units (gray saw-tooth trace). Bottom: Regions of K-B space
which are within standard error bounds of the best fit value (see
text). Arcs delineate line of critical damping. Only 1 subject is
shown in this figure
Fig. 8 K values obtained for the best fit for the K-B-I-T model,
with lower and upper bounds and mean of best fit across subjects
(horizontal line). Note that bounds are not independent of other
parameters, notably B
co-contraction, the region of uncertainty produced by our
analysis allows an increase in B during co-contraction;
however, the only noticeable increase observed is in K,
which increases by more than an order of magnitude.
While the uncertainty in our measurement does not
establish the modulation of intrinsic K with precision,
there is no observable change in our upper bound for K
during movement, and the value allowed by the best
estimate during relaxed or movement conditions is very
slight compared to K during co-contraction.
EMG response to perturbations
Fig. 7 B values obtained for the best fit for the K-B-I-T model,
with lower and upper bounds and mean of best fit across subjects
(horizontal line). Note that bounds are not independent of other
parameters, notably K
in the mechanical properties. For the isometric trials
(STIFF and PRE) the uncertainty regions are determined
by the variability in mechanical properties only, hence the
more precise knowledge of K and B.
Figures 7 and 8 show that the model fits are much
more sensitive to B than to K because the bounds, relative
to the mean, are tighter. There is a slight increase in B at
the ENDING phase of movement when compared to the
relaxed state, and a slight drop during movement. During
Our observations indicate that the moving arm displays
very low K values when evaluated over the first 50 ms
following a brief perturbation. But when the response is
assessed over longer time periods, as in Fig. 3 (on the
order of 200 ms), we observed that the trajectories began
to return towards the intended trajectories, as in the fully
relaxed limb, but only after some significant changes in
EMG activity were seen, suggesting the presence of
reflexes and other neurally mediated reactions to the
perturbation.
The typical EMG profiles, along with typical movements in 3 subjects, are shown in Fig. 9. The EMG
profiles are typically tri-phasic, with an agonist burst, an
antagonist burst, and finally mild co-activation. The
figure shows that EMG changes from subject to subject
and scales with the level of muscle activity. Figure 9
shows that outside of the PRE condition, all other
perturbations occur during periods of distinct muscle
activity rather than coasting periods of low muscle
activity during a ballistic motion. This indicates that the
lack of stiffness increase is not due to perturbations being
applied during periods of muscle inactivity.
24
Discussion
The finding of low values of intrinsic limb stiffness
during movement is surprising in that we had expected
stiffness to increase significantly above that present under
passive conditions. We discuss the potential impact of
uncertainties in measurement, the relation of the reported
estimates of K and B to known muscle physiology,
alternative explanations for the kinematic behavior of the
perturbed limb, and the impact of changes in mechanical
properties on movement generation.
Mechanical states of the limb during movement
as measured by K and B
Fig. 9 EMG profiles for 3 subjects and typical trajectories. The
main agonist (triceps) and the antagonist (biceps) integrated
rectified EMG profiles are shown as percentages of EMG activity
during forceful voluntary co-contraction (collected during the
STIFF protocol condition). Also shown are typical (most predictable) trajectories in position for each subject, along with a
pictorial representation of perturbation onsets for each protocol
condition
Fig. 10 Illustration of the significance of B. Top: Sample velocity
profile (smooth curve) and necessary torque (leading trace) for an IB system with parameters derived for one subject with a relaxed
limb (PRE). Also shown is the velocity profile resulting from a
torque profile like the one shown but with the negative (braking)
portion removed. Bottom: Same information as Top but with only
an inertial system assumed. Notice in the absence of a braking pulse
the limb reaches a high steady state velocity and would eventually
ram into the elbow joint lock
It is usually believed that mechanical properties of the
passive forearm are mainly inertial. Using sinusoidal
position perturbations of varying frequency and with
moderate levels of preload, Rack (1981) characterized
both physiological properties of muscle as well as
dynamic behavior of the elbow joint. The numerical
results, along with those of other studies, are given in
Table 3.
The first values in Table 3 are derived from Nyquist
plots in the Rack studies, and do not necessarily emulate a
linear second order K-B-I linear model because of
delayed reflex contributions. Interestingly, K values in
Rack are lower than those obtained by Cannon and
Zahalak (1982), presumably because of the low level of
pre-load. The damping values recorded in our current
study lie within the range found for isometric muscle
contraction reported in the earlier study (Cannon and
Zahalak 1982).
If we assume that the torques needed to generate
motion are primarily determined by limb inertia, we can
estimate these values readily, as shown in the upper trace
of Fig. 10. The Cannon and Zahalak study suggests that
for an extensor moment of about 8 Nm, the stiffness is
around 60 Nm/rad. This estimate is a much higher value
than the best fit values in the present study (where the
extensor driving torque is about 8 Nm around the time of
the EARLY perturbation) and is higher than the upper
bound obtained for K during movement (Fig. 8). This
discrepancy suggests that stiffness during movement is
lower than what we might expect from predictions using
the quasi-static slope of the Force-Length relationship in
muscle, presumably measured by Cannon and Zahalak.
Table 3 Numerical values of stiffness (K), damping (B) and inertia (I) obtained at the elbow joint in different studies
K
Nm/rad
B
Nm/rad/s
I
kg m2/rad
Condition
Study
3–5
30 (before) 20 (during)
10
5–15, higher while moving
5€2 + x 240, 0<x<1
.2
1 to 2
–
–
0.4 to 1
–
.1
–
–
.08 to 1.2
Isometric, 0.75 Nm preload
Cyclical 0.6 rad, 1 s, voluntary motion
Slow (1 mvmt/sec)
2d ballistic, voluntary motion
Isometric, Relaxed, x level of Preload
(Rack 1981)
(Xu 1999)
(Bennett 1993)
(Gomi 1996)
(Cannon 1982)
25
This difference can be attributed to a variety of physiological mechanisms, but static force-length slopes are
nevertheless used in many simplified muscle models.
Implications of low stiffness during movement
In contrast to Bennett et al. (1992) who used an air-jet to
study limb mechanics, the results from our present study
suggest that stiffness does not really drop measurably
during movement, but is essentially negligible before and
throughout the movement. The modest decrease in
stiffness seen by Bennett may have been an artifact
resulting from the use of large sinusoid-like movements
of the elbow in which there was essentially continuous
muscle activity. In the present study, a ‘rest’ meant a
period of muscle relaxation prior to the subject’s instruction to move.
Furthermore, the results of our study also do not
support the assertion that intrinsic muscle stiffness
increases significantly during movement, as suggested
by Gomi and Kawato (1997). The discrepancy in this case
may be attributed to reflex contributions on the effective
stiffness they measured. The addition of reflex action over
a latency on the order of hundreds of milliseconds (Gomi
and Kawato 1997), which is significant and potentially
includes components other than the stretch reflex, could
lead to a significant increase in estimated stiffness.
We found that stiffness modulation is low for the
relaxed limb and remains low throughout the movement,
despite significant activation of muscle. Unlike other
studies of stiffness, the present study calibrated estimates
against those induced by co-contraction, which was found
to be two orders of magnitude larger than those recorded
during relaxation or during movement.
Implications for equilibrium point (EP) control
We believe that our findings do not support the role of
equilibrium point approaches towards the control of
voluntary motion. To illustrate, we will begin by assuming, as in Fig. 10, that there is an inertial driving torque of
8 Nm at the EARLY stage of movement and an intrinsic
mechanical stiffness of about 5 Nm/rad (from Figs. 6 and
8). With the driving torque being the product of stiffness
and the difference between virtual and actual trajectories,
as claimed by the EPH hypothesis (Astrayan and Feldman
1965; Bizzi et al. 1978), this error signal would have to be
approximately 1.6 radians (90 deg), an error of impractical size for realistic control.
In the event that reflexes play an integral role in the
forward dynamics of the limb as an autonomous motor
servo functioning at the spinal level, we may expect the
reflex to enhance effective impedance magnitude. Such
reflex actions develop force only after some delay and are
not a component of stiffness, as estimated here. Reflex
actions would have an effect on impedance magnitude if
we assume that impedance is a linear transfer function. In
the quasi-static case, Rack (1981) has shown that the
increase in impedance magnitude due to reflexes at the
elbow is quite limited by stability considerations (to about
twice that of intrinsic stiffness, transiently), similar to
reports of ankle impedance (Kearney et al. 1997). In
another quasi-static study (Bennett 1993) the contribution
of the stretch reflex to disturbance rejection itself was
shown to be very small, which we also noted in our EMG
data. Hence, we believe that the reflex could not increase
effective stiffness in the forward dynamics enough to
allow it to be part of a simplified control calculation, as
discussed above, even if it was part of an automatic
positioning servo.
Even if the stretch reflex could result in quadrupled
effective stiffness relative to intrinsic, a value exceeding
that supported by previous studies (Bennett et al. 1992),
we would still need about 23 deg of difference between
virtual and actual trajectory. At about 160 deg of elbow
extension, or shortly after velocity peak in our study, the
’equilibrium point’ would have to be outside the joint
range of motion.
Because of the small stiffness values, our findings
argue against the ability of spring-like intrinsic properties
of muscle to generate movement, a key postulate of one
variant EP Hypothesis (Bizzi et al. 1978). They also raise
doubts about the ability of the stretch reflex to sufficiently
increase stiffness via a servo-loop like mechanism, as
implied by another EP Hypothesis variant (Astrayan and
Feldman 1965). Had the elbow muscles been more
strongly co-contracted during movement, sufficient force
generation from joint elasticity might have been possible.
Mechanisms of joint stiffness
The main factor determining intrinsic muscle stiffness and
damping are the dynamics of actin and myosin crossbridges within the muscle fiber (Huxley and Simmons
1972). Tonic increases in activation increase the number
of active cross-bridges. Stiffness of muscle rises proportionately to that number (Huxley and Simmons 1972),
whereas damping is proportional to the number of crossbridges in transition to and from non-force generating
states. This mechanism predicts a coupling between K
and B under constant activation without any history
dependence. Moving muscle exhibits lower stiffness than
isometric muscle because a lower number of crossbridges are attached while movement occurs (Zahalak
1986).
Controlled experiments with single rat and frog muscle
fibers reveal that static stiffness behaves differently and is
higher than dynamic stiffness (Cecchi et al. 1986; Ettema
and Huijing 1994; Bagni et al. 1998) and that damping is
always present, increasing with activation. Also possibly
responsible for the surprising lack of stiffness observed is
that common muscle models were derived for and assume
tonic activation conditions (Hill 1922; Zahalak 1981,
1990). However, phasic changes in activation as well as
activation history (Sandercock and Heckman 1997) have
26
been observed to alter the development of stiffness and
damping.
Reflex contributions to disturbance rejection
during movement
The behavior of isolated muscle preparations reveals
different mechanical properties for reflexive vs. areflexive muscle. Reflexively active muscle is stiffer and is
more linear during stretch, while exhibiting less yield
(Nichols and Houk 1976; Houk et al. 1981). Concurrently,
areflexive (yet active) muscle shows more effective
damping in isolated muscle as well as in humans
(Rothwell et al. 1982b; Lin and Rymer 1993, 1998;
Milner 1993, 1998). This could explain many of the
discrepancies between this study (which reports low
intrinsic K and B) and its predecessors (which measured
effective K and B, including reflex contributions).
One interesting question remaining is, given low
intrinsic stiffness and also servo stiffness at the spinal
cord level, how do fast movements remain robust in the
face of external perturbations? Our previous results
showed this robustness to be limited without visual
correction (Popescu and Rymer 2000), that the stretch
reflex appears in the muscle being stretched while the
subsequent reactions result in the transient but significant
co-contraction of agonist and antagonist muscles, and that
the integrated EMG response profiles were not strongly
correlated across latency (Popescu and Rymer 1999) and
are hence likely to imply functionally independendent
control loops. Earlier studies (Desmedt and Godaux 1978)
also pointed to long-loop reactions as essential to
disturbance rejection during movement.
If the longer latency co-contraction reactions observed
have the same effect as observed in the trials where cocontraction was asked of the subjects before movement,
the plant would become stiffer and also increase in
damping factor. This would aid in bringing the suddenly
perturbed motion to a halt close to the intended endpoint.
In preliminary work, we have observed a diminished long
latency response to a second perturbation within a
movement with respect to the initial one (Popescu and
Rymer 2000). The non-automatic nature of longer loop
responses, as well as their likely changes to plant
properties via co-contraction, allows high effective gain
without compromising stability, unlike increasing the
gain of the stretch reflex. This type of strategy is
consistent with the idea that longer-loop reactions compensate for inadequacies in stretch reflex action (Cooke
1980; Bennett 1993) and that disturbance rejection in the
limb is much more complex than would be expected of a
position servo (Button et al. 2000).
To sum up our observations, the low stiffness of the
joint during movement makes it unlikely that the difference between actual and desired (or ’virtual’) limb
trajectory is used as an error signal to generate movement,
and the primarily inertial nature of limb mechanics do not
pose a control problem, either in complexity or stability
(see below for a discussion on movement termination).
Low joint stiffness may compromise robustness to
external perturbations; however, a limited stretch reflex
combined with more powerful longer latency reactions
are available to maintain the limb on track. Although
increasing stiffness is a strategy which is sometimes used
during movement, via co-contraction, increasing both
robustness and movement accuracy (Kawato 2000), the
CNS seems to prefer maintaining the limb at a very low
stiffness level, a strategy which is potentially energy
efficient and relies on multiple reflex pathways and visual
correction for stability in the face of perturbations.
Significance of damping
The final aim of this study (see Introduction) was to
determine the role of observed viscosity in ending
movement. At first glance, it may seem that the amount
of damping observed was small throughout the movement, and did not increase significantly. To show that the
level of damping was indeed significant, some simple
simulations were run to see if the value of B found by
identification of K-B-I-T is functionally significant. In
Fig. 10, we see the torque profile required for the values
of I and B identified for the first subject in order to
produce a sample unperturbed movement. These values
were obtained by using the simple equation F = I a + B v,
where a (acceleration) and v (velocity, shown in red) are
measured. The torque profile shows a large agonist
‘pulse’ and a small antagonist braking pulse.
Let us posit, initially, what would happen to the
movement trajectory if the CNS omitted the braking force
entirely. As seen in the upper trace of Fig. 10, the effect is
hardly noticeable. The damping enables motion to be
halted. But if there was no damping in the system (Fig. 10
lower trace) the results are striking-the limb coasts into a
joint-locked position. In fact, in the horizontal plane,
where gravity does not have an effect, any failure to
properly balance the accelerating and braking mechanical
energy will fail to bring the limb to rest. Adding stiffness
to a system which is very lightly damped will merely
result in large oscillations whenever we do not accurately
balance the agonist and antagonist pulses. This arrangement would require substantial feedback corrective
activity to bring the limb to rest, not to mention to
achieve endpoint accuracy.
The results of this study showed that stretch reflex
activity appeared at the end of movement. Since stretch
reflex activity is thought to decrease effective damping
(Lin and Rymer 1993, 1998; Milner and Cloutier 1998), it
may seem that intrinsic damping may not aid in stopping
movement, as shown in Fig. 10. However, as was
repeatedly pointed out in this discussion, voluntary
control of movement against known loads is likely
determined by intrinsic rather than reflexive limb mechanics. The values of K and B measured in this study
suggest that intrinsic muscle properties are more advan-
27
tageous to control of point-to-point movement than
reflexive properties.
Impedance modeling
The values of K and B estimated in this study are
comparable to parameters derived in other studies, but
only in a general sense. Each study is different in the
details of the functional form of the impedance model and
in subtraction of the underlying trajectory, details which
can affect the values of parameters obtained greatly. In
fact, stiffness and damping are not directly measurable
properties of muscle, like length or duration. These
attributes apply to a particular mathematical form of an
impedance model (in our case K-B-I-T) and to a specific
situation (the posture, movement amplitude and speed of
the protocol).
In the case of unknown loads, removing afferent input
severely impairs the ability of the human to counteract
these loads (Rothwell et al. 1982a). Our results support
the idea that this behavior requires special reflexes and
reactions, which are not automatic in the manner of spinal
reflexes (Dufresne et al. 1978), are of higher latency and
magnitude, and therefore are likely mediated by supraspinal pathways. In the case of known loads, the intrinsic
elasticity of the moving non-cocontracted limb is so low,
and the stretch reflex may be so limited in its ability to
increase limb impedance magnitude, both due to loop
delay related stability margins and phasic depression of
servo loop gain (Gottlieb and Agarwal 1972), that the
brain must plan movements considering limb properties
which are essentially of negligible elasticity and mild
damping in comparison with inertial forces. Under this
scenario, computational simplicity, endpoint accuracy and
robustness of control are likely traded off for gains in
metabolic efficiency, agility and dexterity.
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