Biomedical Engineering Reference
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However this solution is not numerically robust, close to the singularities of
the kinematic transformation, and is
non-integrable
in the sense that, if we
command a robot by means of this algorithm in relation with a desired closed
trajectory in task space repeated several times (as in crank turning), we get a
trajectory in joint space which is not closed and tends to drift towards the joint
limits of the kinematic chain.
4.
Inverse dynamics and interaction forces:
computing inverse dynamics corre-
sponds to the solution of the following equation
J
T
u
actuator
(
t
)=
I
(
q
)
q
+
C
(
q
,
q
)
q
+
G
(
q
)+
(
q
)
F
ext
(17.3)
where
q
(
t
)
is the planned/desired trajectory in the joint space,
I
(
q
)
q
identifies
the inertial forces proportional to the acceleration,
C
(
q
,
q
)
q
the Coriolis forces
quadratically dependent upon speed,
G
the gravitational forces indepen-
dent of time, and
F
ext
the external disturbance/load applied to the end-effector
(
Figure 17.3
).
This equation is highly nonlinear and acts as a sort of
internal
disturbance
which tends to induce deformations of the planned trajectories.
In particular, the inertial terms is predominant during the initiation and ter-
mination of the movements whereas the Coriolis terms is more important in
the intermediate, high-speed parts of the movements.
Figure 17.4
shows a
simulation involving a planar arm with two degrees of freedom. A set of 8
different trajectories were generated starting from the same initial point. For
each of them Equation 17.1 was applied and the computed torque vectors were
re-mapped as end-effector force vectors in the following way:
(
q
)
J
T
u
actuator
(
t
)=
(
q
)
F
end-effector
(17.4)
J
T
))
−
1
u
actuator
(
⇒
F
end-effector
=(
(
q
t
)
The figure shows that the patterns of end-effector forces are quite variable in
relation with the movement direction. In particular, each force vector can be
decomposed into a longitudinal component (oriented as the intended move-
ment direction) and a transversal component. While the longitudinal compo-
nent corresponds to the usual inertial resistance that would be present even
if the degrees of freedom were controlled separately, one after the other, the
transversal component is related to the non-linear dynamics determined by
the interaction among degrees of freedom and, if unaccounted for, will tend
to deviate laterally the planned trajectory. As can be seen from the figure,
the order of magnitude of such lateral disturbances or interaction forces is the
same as the main inertial components. Moreover it can be seen that the initial
and final parts of the movements (characterized by low velocity but high ac-
celeration) tend to be more affected than the intermediate high-velocity part.
Obviously interaction torques only occur in multi-joint movements: if move-
ments are broken down as sequences of single-joint rotations, then interaction
forces disappear and this might explain why in several neuromotor pathologies
movement segmentation is a typical adaptation to the impairment.
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