Biomedical Engineering Reference
In-Depth Information
priate feedforward control. In the initial part of the movement, in which acceleration
is high but velocity is low, the first inertial element of Equation 17.1 dominates the
dynamical behavior: u
actuator
(
J
T
F
ext
. Therefore, the apparent iner-
tia applied by the arm to an external load is given by the following relationships, in
the case of non-redundant arms:
t
)
≈
I
(
q
)
q
+
(
q
)
J
T
J
T
))
−
1
I
))
−
1
(
q
)
F
load
≈
I
(
q
)(
q
)
⇒
F
load
≈
[
(
q
(
q
)(
J
(
q
]
x
=
I
(
x
)(
x
)
where
x
is the vector which identifies the position/orientation of the end-effector.
The end-effector inertia, represented by the matrix
I
, is not isotropic, as shown by
the inertia ellipse of
Figure 17.12
(top). It is worth noting that the principal axis is
approximately aligned with the forearm: only in this direction (and the correspond-
ing orthogonal direction) the force and acceleration vectors are collinear. This means
that, if a force vector is generated in a given direction, the corresponding accelera-
tion vector has a sideway component which tends to deviate the movement from its
intended path, with the exception of the principal directions. Figure 17.12 (bottom)
displays, for each experimental subject, the relationship between the aiming error
and the movement direction
relative to the principal direction of the inertia ellipse
:
we can see that the error tends to vanish in the principal direction and is charac-
terized by sideways deviations in the other directions which are consistent with the
directional characteristics of the inertia. Therefore we may conclude that the pattern
of aiming errors can be attributed to a defective cerebellar feedforward controller.
Such biomechanical explanation in terms of unaccounted interaction forces is also
consistent with the fact that both patients and controls exhibit a similar pattern of
aiming errors: the difference is that the feedforward compensation in the controls is
more effective than in the cerebellar patients, although it is not perfect.
In cerebellar patients, the analysis of smoothness has shown that the final part of
the movements, which is typically sensitive to feedback corrections, is not specifi-
cally affected and this suggest that the cerebellar impairment is primarily related to
feedforward rather than feedback control. It is also worth mentioning that in another
pathological syndrome with a relevant impairment of motor coordination (Hunting-
ton's chorea), the observed brief, small amplitude, and involuntary dancelike move-
ments cannot be interpreted as a disruption of feedforward control but rather as a
selective impairment of the feedback mechanisms, active in the terminal part of the
movement [62]. Since the Huntington's chorea, a genetic disease, is known to be
associated with a malfunction of the caudate nuclei which are part of the basal gan-
glia, one might conclude that internal models for anticipatory feedforward control
are likely to be localized in the cerebellum whereas internal models for anticipatory
feedback control are likely to involve the basal ganglia and the cortico-thalamic loop.
(
x
)
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