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q
d
=
J
1
y
1
d
+
J
2
(
y
2
d
−
J
2
J
1
y
1
d
)+(
I
J
1
J
1
−
J
2
J
2
)
k
2
−
(14.9)
as the desired joint velocity that first realizes trajectory
y
1
d
and then realizes
y
2
d
as closely as possible using the remaining redundancy. This approach iteratively
constructs the equations used to perform more subtasks, in this case using
k
2
to
perform the third task, if
I
J
2
J
2
is not zero.
In the case when the second subtask is specified by a criterion function, we select
the vector
k
1
to make the criterion
p
as large as possible. One natural approach is to
determine
k
1
by the following equations:
J
1
−
J
1
−
k
1
=
κ
g
(14.10)
g
=
∇
q
V
,
(14.11)
R
n
,and
where
g
∈
κ
is an appropriate positive constant. The desired joint velocity,
q
d
,isgivenby
q
d
=
J
1
y
1
d
+
J
1
J
1
)
g
κ
(
I
−
.
(14.12)
Since
p
is the criterion function, from (14.12), we have
p
=
V
(
q
)=
∂
V
q
=
g
T
J
1
y
1
d
+
g
T
(
I
J
1
J
1
)
g
κ
−
.
(14.13)
∂
q
J
1
J
1
) is nonnegative definite, the second term on the right-hand side of
(14.13) is always nonnegative, causing the value of criterion
p
to increase.
With this approach,
k
1
is the gradient of the function
V
(
q
) and the vector
Since (
I
−
−
J
1
J
1
)
g
corresponds to the orthogonal projection of
k
1
on the null space of
J
1
.The
constant
κ
(
I
is chosen so as to make
p
increase as quickly as possible under the
condition that
q
d
does not become excessively large.
κ
14.2.2
The Gradient Projection Method Applied to Robotic
Control
The optimization problem is typically represented by a constraint set
X
and a cost
function
f
that maps elements of
X
into real numbers [13, 1]. A vector
x
∗
∈
X
is a
local minimum of
f
over the set
X
if it is no worse than its feasible neighbors; that
is, if there exists an
ε
>
0suchthat
f
(
x
∗
)
x
∗
<
ε
.
≤
,∀
∈
−
f
(
x
)
x
Xw th
x
(14.14)
A vector
x
∗
∈
X
is a global minimum of
f
over the set
X
if it is no worse than all
other feasible vectors, that is,
f
(
x
∗
)
≤
f
(
x
)
,∀
x
∈
X
.
(14.15)
The local or global minimum
x
∗
is said to be strict if the corresponding inequality
above is strict for
x
=
x
∗
.
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