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(
J
)
⊥
, respectively. These have the following relationships with the
pseudo-inverse of
J
.
N
(
J
) by
N
J
+
J
)
N
R
−
(
J
)=
(
I
(14.20)
(
J
)
⊥
=
(
J
+
)
N
R
.
(14.21)
The range of
J
+
and the range of
I
J
+
J
are orthogonal, and their projections are in
different spaces. Therefore, the implementation of the second task does not disturb
the fist manipulation variable.
Let us consider the problem:
−
k
min
V
(
q
)
,
q
∈
R
,
(14.22)
where
k
is the number of robot joints. The classical solution is to move the robot
according to the gradient of the cost function, computed in the articular space
q
=
κ
g
(
q
)=
−
κ
Q
∇
q
V
,
(14.23)
where
is a positive scalar, used as a gain, and
Q
is a constant positive matrix.
Pre-multiplying (14.23) by
κ
q
V
,weget
∇
d
dt
V
(
q
)=
q
VQ
−
κ∇
∇
q
V
≤
0
.
(14.24)
Thus,
V
decreases with time as long as
∇
q
V
= 0, and remains constant when
∇
q
V
=
0. It is not uncommon to choose
Q
as the identity matrix
I
.
Consider now a potential field
V
Φ
=
V
(
Φ
(
q
)). Using the chain rule, we have
=
∂Φ
∂
T
˙
Φ
q
,
q
and combining this with (14.23) we obtain
∂Φ
∂
T
∂Φ
∂
T
Q
∂Φ
∂
=
∂Φ
∂
˙
Φ
q
=
−
κ
Q
∇
q
V
=
−
κ
∇
Φ
V
.
(14.25)
q
q
q
q
If we choose
Q
=
∂Φ
∂
†
∂Φ
∂
+
,
(14.26)
q
q
in which the superscript † denotes the transpose of the pseudo-inverse, (14.24) is
verified because
Q
is a positive matrix. Substituting (14.26) into (14.25), we obtain
˙
Φ
=
−
κ∇
Φ
V
.
(14.27)
From (14.23) and (14.26), we obtain
∂Φ
∂
†
q
=
−
∇
Φ
V
Φ
,
(14.28)
q
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