 |
INTRODUCTION |
The concept of a simple muscle
stiffness control of balance during quiet standing was introduced by
Winter et al. (1998)
. The crux of their arguments and
experimental evidence was that the controlled variable, center of mass
(COM), was virtually in phase with the controlling motor variable,
center of pressure (COP). The spring stiffness was estimated indirectly
from the tuned frequency of the mass, spring, damper mechanical system. If a normal reactive control were present, the efferent and afferent latencies (Horak and Nashner 1986
) combined with the
biomechanical delays of recruitment of muscle twitches would result in
a COP delay of over 100 ms after the COM (Rietdyk et al.
1999
). Morasso and Schieppati (1999)
have argued
that a simple spring stiffness control does exist but that "there
must be something in the control circuitry that compensates for the
original delays" and that "the phase-lock between COP and COM is a
necessary consequence of physical laws." In the presence of such
criticism the purpose of this paper is to provide further experimental
evidence of a simple spring control at the ankles using direct measures
of ankle moments and sway angles. Also, evidence is presented from the
stiffness characteristics of the ankle plantarflexors whose
nonlinearities provide a simple and stable operating point for control
of upright posture.
 |
METHODS |
Direct measures of stiffness control in quiet standing can be
done in two separate ways. The first way requires a full
three-dimensional (3D) kinematic and kinetic analysis using two
forceplatforms (cf. Rietdyk et al. 1999
). Such a
technique in the sagittal plane yields the ankle moments and angles for
both limbs. A regression of these two variables yields a plot of ankle
moment versus ankle angle and the slope of the curve yields the ankle
stiffness (N · m/rad) for each ankle. A sum of the left and
right ankle stiffness is a direct measure of the anterior/posterior
(A/P) stiffness constant that was previously estimated
indirectly (Winter et al. 1998
).
An alternate and less cumbersome direct estimate of either the A/P and
medial/lateral (M/L) stiffness can be made from the readily
measured time records of COP and COM (cf. Winter et al. 1998
). Figure 1 presents the
common inverted pendulum model in the sagittal plane. Here the COM and
COP are measured relative to the ankle joint, and the COM is located at
a distance h above the ankle joint. The sway is measured by
the angle of the line joining the ankle to the center of mass. Body
weight (mg) is the weight of the body above the ankle joint, and the
vertical reaction force R does not include the reaction
force of the feet that are essentially stable in quiet standing, and is
equal to body weight. The sum of the left and right ankle moments,
Ma, is
|
(1)
|
|
(2)
|
Or, if we plot a regression of
Ma versus
sw,
the slope of that linear regression is
Ka, and the closeness of that
regression to a straight line (its R2
score) will be a measure of how closely our estimate of
Ka resembles a pure spring. For a pure
spring the R2 would be 1 reflecting a
perfectly straight line.

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Fig. 1.
Inverted pendulum model showing the variables center of mass (COM),
center of pressure (COP), body weight (mg), and height,
h of COM, from which direct measure of muscle stiffness
can be estimated.
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|
In balance control studies the latter technique is less complex because
it requires only one force platform and has less kinematic markers than
a full 3D inverse dynamic biomechanical analysis (a minimum of 3 non-colinear markers are required to define each segment of the body).
Thus the direct stiffness estimates reported here will be confined to
this latter technique that requires precise and accurate measures of
COM. Contrary to the claims of Morasso et al. (1999)
,
the direct estimate from our 14 segment COM model is not
"cumbersome" and does not "require a very critical
calibration." The 23-marker system in our 14-segment model takes less
time to prepare subjects and patients than is needed for 3D clinical
gait analysis that have been routinely done for the past decade (cf. Ounpuu et al. 1991
). This direct measure of COM is also
applicable to clinical studies of balance that routinely require
analysis of responses to external perturbations (cf. Horak et
al. 1992
).
 |
RESULTS |
In a separate set of analyses on 10 adult subjects standing
quietly, a full 3D COM and COP biomechanical analysis was conducted. Using Eqs. 1 and 2,
Ma(t) and
sw(t) over the 10 s were
analyzed. A sample regression of Ma
versus
sw for one subject is presented in Fig.
2. The slope of this line is 13.04 N
· m/deg or 747.0 N · m/rad. The
R2 for this 10-s trial was 0.954. For
all 10 subjects the A/P stiffness estimates are reported with their mgh
(gravitational spring) in Table 1.

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Fig. 2.
Regression of net ankle moment Ma vs. sway
angle sw. Slope of this regression (N · m/deg) is
the net ankle stiffness Ka. A perfect spring
would yield a perfectly straight line regression with an
R2 = 1. Thus the
R2 score here (0.954) is a measure of how
close the ankle spring is to a perfect spring.
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|
Across all the 10 subjects, the ankle stiffness during quiet standing
averaged 858.9 N · m/rad, which was 8.8% higher than the
gravitational spring stiffness (mgh), which averaged 789.4 N · m/rad. To achieve this stiffness these subjects had to sway forward an
average of 3.67 ± 0.28°. The
R2 score averaged 0.918, which is a
measure of how close the ankle stiffness approached that of an ideal
spring. Because the subjects in this study were assessed for only
10 s, the magnitude of their changes in sway and ankle moments
were somewhat less that if they were assessed over longer periods
(Winter et al. 1998
).
 |
DISCUSSION |
None of the studies of ankle stiffness reported in the literature
replicated the quiet standing conditions in our study. Hof (1998)
had his subjects standing with the ankle in a special
measuring device, but the subjects balanced themselves with their hands on a rod; he reported muscle stiffnesses of 250-400 N · m/rad. When we consider what the mgh threshold value was for both limbs, his
study would predict 500-800 N · m/rad, which is close to the threshold of mgh = 674 N · m/rad that we predicted for a
68-kg subject. All the other studies had their subjects lying prone with varying degrees of knee flexion. An examination of the series elastic characteristics of the plantar-flexor muscles yields an estimate of the muscle stiffness. A replication from such a study by
Winters and Stark (1988)
, including the summation of all
series elastic components of muscles involved, is presented in Fig.
3. For an 80-kg adult standing with his
COP 5 cm anterior to the ankle joint, the total ankle moment would be
about 40 N · m. Thus with a muscle tone of 20 N · m per
ankle, the ankle stiffness can be estimated from the slope of this
nonlinear summation curve at the operating "a" point in Fig. 3. Two
summation curves are plotted: 1) sum of soleus and
gastrocnemii and remaining plantarflexors and 2) sum of
soleus and two times gastrocnemius and remaining plantarflexors. This
second summation is based on the assumption that the gastrocnemii
(which were shortened in this experiment because the subject was lying
prone with knee flexed) to contribute proportionally to their
physiological cross-section area, which is just over half of that of
the soleus. The slope of this final summation curve is 6.05 N · m/deg. Thus for two ankles the combined stiffness would be predicted to
be ~700 N · m/rad. This again is just below the values we
estimated for subjects who were standing with their ankles slightly
dorsiflexed and supporting themselves entirely by their plantarflexors.
The functional importance of the nonlinear characteristics of these
postural muscles must be emphasized. A stable operating point results
from this nonlinearity. If the subject were to sway forward, the slope
increases, and if he were to sway backward, the slope decreases. Thus
the stable operating point is when the subject sways sufficiently to an
operating point where the slope safely exceeds the mgh gravitational
threshold.

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Fig. 3.
Series elastic curves for the plantarflexor muscles replicated from
Fig. 5 in Winters and Stark (1988) . Two new summation
curves are plotted: 1) sum of soleus and gastrocnemii
and remaining plantarflexors and 2) a 2nd summation to
account for the fact that the subject was lying prone with knee flexed,
here the gastrocnemii contributed proportionally to their physiological
cross-section area. For this subject with a muscle tone of 20 N
· m, the operating point "a" is predicted.
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Morasso and Schieppati (1999)
developed an
inverted pendulum model that incorporated a spring control that was
similar to the model developed by Winter et al. (1998)
.
Morasso and Schieppati in their model claim that the in-phase
relationship between COP and COM that we found was not a function of a
simple spring control. In the direct method reported here, the in-phase
relationship is evident between Ma and
sw, which, from Eqs. 1 and 2, reinforces the in-phase relationship between COP and COM.
Morasso and Schieppati presented a theoretical reactive control model
that also result in COP and COM to be in phase, but their reactive
feedback model erroneously contained no afferent or efferent
transmission delays.
Both reports showed that the borderline stiffness required to overcome
gravity was mgh, where h is the height of the COM above the
ankle joints. Thus a spring of stiffness mgh N · m/rad was the
minimum required at the ankle joints to overcome the unbalancing gravitational forces. However, Morasso and Schieppati
(1999)
assumed the presence of noise (white noise plus
quasi-periodic spike noise) but gave no reference as to where this
noise is assumed to be located. However, if we assume that they are
referring to the neural spike train of action potentials seen at the
motor endpoint, we can agree that these impulses will result in some
noise due to the resultant train of twitches seen at the muscle tendon. The noise in the muscle stiffness controller is very low frequency and
low in amplitude. Muscle stiffness is controlled by muscle tone, which
is a summation of recruited muscle twitches in the balance control
muscles. Milner-Brown et al. (1973)
showed
that the twitches were a critically damped impulse response. The
soleus, for example, with twitch times, T, can be
represented by a low-pass mechanical filter. For a critically damped
second-order system, the cutoff frequency of the filter is related to
the time-to-peak of the impulse response, T, by
|
(3)
|
For the soleus, with a twitch time over 100 ms,
fc = 1.5 Hz. Thus the COP frequency
response would be predicted to have negligible power above 5 Hz, and
this has been reported in the literature (Powell and Dzendolet
1984
; Soames and Atha 1982
; Tokumasu et al. 1983
). An estimate of this ripple can be made from the
COP-COM signal from Winter et al. (1998)
, and this was
<0.1 cm on 40 trials for 10 subjects. The amplitude of COP in quiet
standing is about 5 cm, thus the noise in the stiffness controller is
not only low frequency but is also <2%. Therefore the safe value of
the stiffness does not have to exceed the gravitational spring (mgh) by
a large amount. Thus the predicted safe stiffness to be 1,050 N
· m/rad in excess of mgh proposed by Morasso and Schieppati
(1999)
is not justified, and our direct estimate of ankle
stiffness is sufficient to control posture during quiet standing.
Estimates of static ankle muscle stiffness in the literature
(Hof 1998
; Winters and Stark
1988
) reinforce the results of our indirect method
(Winter et al. 1998
) and our direct method reported here. The in-phase relationship between COP and COM that would occur in
a simple spring control cannot be explained by the Morasso and
Schieppati model because their reactive model failed to include afferent and efferent delays. The excess by which Morasso and Schieppati claim stiffness would have to exceed the gravitational load
is not justified by their erroneous assumption of spike noise in the
motor system. Rather, the noise in the ankle spring is minimal as
predicted by the ripple due to summation of twitches in the ankle plantarflexors.
Address for reprint requests: D. A. Winter.