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# Grasp and Manipulation of Objects with a Multi-Fingered Hand ...

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4.4 Kinematic motion control with redundancy resolution
4.4
Kinematic motion control with redundancy
resolution
In the case of kinematically determinate and, possibly, redundant grasp, a suitable
inverse kinematics algorithm can be adopted to compute the fingers and contact
variables corresponding to a desired motion of the object. Hence, this framework
can act as a kinematic motion controller to perform the desired object manipula-
In detail, in view of (4.9), the CLIK algorithm with redundancy resolution,
based on the preudo-inverse of the Jacobian matrix, is given by equation:
q
i
i
= ~
J

i
(q
i
,
i
)G
T
i
(
i
)(
d
+ K
i
e
o
i
) + N
o
i
(q
i
,
i
)
i
,
(4.10)
where ~
J

i
is a right (weighted) pseudo-inverse of ~
J
i
, e
o
i
is a pose error between the
desired and the current pose of the object, K
i
is a (6 × 6) symmetric and positive
definite matrix,
i
is a suitable velocity vector corresponding to a secondary task,
and
N
o
i
= I - ~
J

i
~
J
i
(4.11)
is a projector in the null space of the Jacobian matrix. The asymptotic stability
of the equilibrium e
o
i
= 0 for system (4.10) can be easily proven (see [89] for the
case of a standard CLIK algorithm).
The computation of the Jacobian pseudo-inverse can be avoided by adopting
an alternative CLIK algorithm based on the transpose of the Jacobian matrix,
given by equation:
q
i
i
= ~
J
T
i
(q
i
,
i
)G
T
i
(
i
)K
i
e
o
i
+ N
o
i
(q
i
,
i
)
i
.
(4.12)
The asymptotic stability of the equilibrium e
o
i
= 0 for system (4.12) can be easily
proven in the case v
d
= 0 (see [89] for the case of a standard CLIK algorithm). No-
tice that, if N
oi
is computed according to (4.11), the computation of the Jacobian
pseudo-inverse is not avoided with algorithm (4.12).
In principle, n
f
independent CLIK algorithms can be used, one for each finger,
all with the same input, namely, the desired object's pose T
d
and velocity
d
.
However, some secondary tasks may involve all the variables of the system at the
same time, e.g., those related to quality of the grasp.
Hence, the complete CLIK scheme with redundancy resolution includes n
f
decentralized feedback loops, one for each finger, and a centralized feedforward
action depending on the whole configuration of the system. This is shown in the
block diagram of Figure 4.2 for the scheme based on the pseudo-inverse of the
Jacobian matrix, in which ~
q
i
= q
T
i
T
i
T
.
47

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Summary :

## In detail, in view of (4.9), the CLIK algorithm with redundancy resolution, based on the preudo-inverse of the Jacobian matrix, is given by equation: q i i = ~ J i (q i , i )G T i ( i )( d + K i e o i ) + N o i (q i , i ) i , (4.10) where ~ J i is a right (weighted) pseudo-inverse of ~ J i , e o i is a pose error between the desired and the current pose of the object, K i is a (6 × 6) symmetric and positive definite matrix, i is a suitable velocity vector corresponding to a secondary task, and N o i = I - ~ J i ~ J i (4.11) is a projector in the null space of the Jacobian matrix.

Tags : clik,jacobian,algorithm,matrix,motion,system,desired,case,redundancy,resolution,pseudoinerse,pose,object
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