Biomedical Engineering Reference
In-Depth Information
insufficient to stabilise the unstable plant and thus the only feasible solution is an-
ticipatory control. In other cases, involving the upper extremity and with required
levels of muscle stiffness which fall inside the physiological range of values, the so-
lution adopted by the brain appears to depend upon the task: a) stiffness modulation
in the case of the arm movements in a divergent force field, b) anticipatory compen-
sation in the case of the manual stabilisation of an inverted pendulum. Which kind
of circumstances might explain such difference of implementation?
Let us first consider, in general, the dynamics of
fall
which underlies all these
paradigms. Equation 17.7 was derived in the specific case of the standing posture
but, with a suitable abstraction, it is applicable to all the unstable loads characterized
by an equilibrium state and a divergent force field:
x
u
,wherebis a (positive)
parameter depending on the structure of the unstable plant and
u
is the compensatory
control variable. If we consider the transfer function corresponding to the equation
above, we realise that the system has two real poles:
p
=
b
x
−
=
±
√
b. The positive pole
is the source of instability and, in absence of an appropriate corrective action, de-
termines the exponential
fall
from the equilibrium state (
x
=
0) with the following
/
√
b. In the case of the standing posture b
time constant:
T
=
1
=
g
/
h
e
and if we
assume that
h
e
=
1mweget
T
=
320 ms; in the case of the manually controlled
inverted pendulum of
Figure 17.15
b
=
g
/
h
,
h
=
1
.
8 m and thus
T
=
430 ms; in
the case of the reaching movements in the divergent force field [14] b
M
,
where
K
field
is the elastic constant of the field (about 200 N/m) and
M
is the apparent
mass of the hand in the direction collinear with the field (about 1 Kg), thus giving
T
=
K
field
/
70 ms. Therefore we can say that in the two cases in which there is evidence of
anticipatory compensation the characteristic time constant of the fall is much longer
than in the case in which stiffness modulation is the adopted strategy. This result
might be explained by considering that anticipatory compensation must have enough
time to recover the intrinsic delays of the reafferent pathways (which come close to
100 ms) in order to generate functionally useful anticipatory commands. Therefore,
anticipatory compensation in unstable tasks is only feasible if the critical time hori-
zon before a catastrophic fall is significantly longer than 100 ms. As a matter of
fact, if we consider a variety of stabilisation tasks such as standing on stilts, rope-
walking, balancing a stick etc., it is easy to recognize the fact that the purpose of
common tricks, like spreading out the arms or holding a long balancing rod is just to
increase the natural
falling time
, thus giving time to the internal model to generate
an appropriate stabilisation action.
≈
17.5.4
Implementing anticipatory compensation
In the case of the upright posture, the underlying control process sketched in the
previous section can be made evident by a particular type of analysis of the posturo-
graphic data (the time course of the COP coordinates provided by a force platform)
which is based on sway density plots (SDP, [3]). A SDP is simply constructed by
counting the number of consecutive samples of the COP trajectory that, for each time
instant, fall inside a circle of given radius (typically 2.5 mm). It is worth noting that
if such trajectories were generated by a random-walk process the sway density curve
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