Biomedical Engineering Reference
In-Depth Information
to solve the problem. According to this theory, the intervention of the CNS is lim-
ited to the selection of an appropriate tonus for the muscles of the ankle joint, in
order to establish an ankle stiffness that stabilises an otherwise unstable mechani-
cal system. Thus, in this view the stabilisation of quiet standing is a fundamentally
passive process without any significant active or reactive component, except for the
background setting of the stiffness parameters. The system equation of the human
inverted pendulum (see
Figure 17.13
,
left) is as follows:
I
p
r
=
mgh
sin
(
r
)+
u
ankle
(17.5)
where r is the sway angle,
m
and
I
p
are the mass and moment of inertia of the body,
h
is the distance of the COM (Center of Mass) from the ankle, g is the acceleration of
gravity, and u
ankle
is the total ankle-torque. Under the assumption that the stabilising
ankle torque is only determined by the ankle stiffness
K
ankle
and that sway angles are
small, then from Equation 17.5 we can derive a critical value of the stiffness:
K
critical
=
mgh
(17.6)
=
≈
In quiet standing the sway angle (and the horizontal shift of the COM
y
h
r)
oscillates with a frequency bandwidth below 1 Hz and such oscillations are asso-
ciated with shifts (
u
) of the center of pressure (COP) on the support surface that
have slightly bigger amplitude and substantially larger frequency band (Figure 17.13,
right; [3]). It is easy to demonstrate [48] that
u
is proportional to the ankle torque
and that
u
and
y
are linked by the following equation:
g
h
e
(
y
=
y
−
u
)
(17.7)
where
h
e
is an
effective distance
which is quite close to the ankle-COM distance and
takes into account the distribution of masses along the body axis. In this equation,
which only expresses biomechanical relations and is independent of modalities of
control,
y
is the controlled variable and
u
the control variable. It can also be de-
scribed by saying that the COM-COP difference is proportional to the acceleration
of the COM.
Figure 17.13
(right) shows the relationship between the two curves.
The COP, which mirrors the time-course of the ankle torque, has a higher frequency
band but with good approximation the two curves are in phase (the peak of the cross-
correlation occurs at a null delay). Moreover, it has been shown [21] that the emg
activity of the ankle muscles anticipates the COM-COP pattern and thus it cannot be
determined by segmental reflexes.
A simulation of the human inverted pendulum [50], activated by realistic muscle
models compatible with experimentally measured muscle properties [30] has shown
that the system is unstable. Moreover, direct estimates of the ankle stiffness have
been carried out [40, 51] with different experimental methods. Both agree that the an-
kle stiffness during standing consistently is below the critical level mentioned above
and thus is unable to stabilise the body by alone.
Figure 17.14
illustrates the latter
approach, based on a motorized platform mounted on top of a force platform. The
motor generates small and rapid angular perturbations (1
o
in less than 150 ms): the
related COP shifts (see
Figure 17.11,
right) are proportional to the ankle stiffness.
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