Information Technology Reference
In-Depth Information
Assume also that the desired trajectory for
y
1
is given by
y
1
d
(
t
). A vector which is
suitable for describing a manipulation task is called a manipulation vector.
Subsequent subtasks of lower priority could be specified in a variety of ways.
The two most common approaches are as follows:
1. an
m
2
-dimensional manipulation vector
y
2
is given by
y
2
=
f
2
(
q
) and the second
subtask is specified by the desired trajectory
y
2
d
(
t
);
2. a criterion function
p
=
V
(
q
) is given, and the second subtask is to keep this
criterion as large as possible.
In either of these cases, the typical way to address lower-priority tasks is to ex-
ploit the null space associated to the primary task. Differentiating (14.1) with respect
to time yields
y
1
=
J
1
q
,
(14.2)
where
J
1
is the Jacobian matrix of
y
1
respect to
q
. When the desired trajectory
y
1
d
is given, the general solution for
q
is
q
=
J
1
y
1
d
+(
I
J
1
J
1
)
k
1
,
−
(14.3)
where
k
1
is an
n
-dimensional arbitrary constant vector. The first term on the right-
hand side is the joint velocity to achieve the desired trajectory,
y
1
d
(
t
). When there
are multiple solutions for
q
satisfying the equation, this term gives a solution that
minimizes
, the Euclidean norm of
q
. The second term on the right-hand side
reflects the redundancy remaining after performing the first subtask.
Consider the case when the second subtask is specified by the desired trajectory
y
2
d
(
t
) of the manipulation variable
y
2
. Differentiating
y
2
=
f
2
(
q
),wehave
q
y
2
=
J
2
q
.
(14.4)
Substituting
y
2
=
y
2
d
and (14.3) into (14.4), we obtain
J
2
J
1
J
1
y
2
d
−
y
1
d
=
J
2
(
I
−
J
1
)
k
1
.
(14.5)
For the linear
Ax
=
b
, the general solution is given by
x
=
A
+
b
+(
I
A
+
A
)
k
−
,
(14.6)
J
2
=
J
2
(
I
J
1
J
1
), from (14.5) and
where
A
+
is the pseudo-inverse of
A
. Letting
−
(14.6), we obtain
k
1
=
J
2
(
y
2
d
−
J
2
J
1
y
1
d
)+(
I
J
2
J
2
)
k
2
,
−
(14.7)
where
k
2
is an
n
-dimensional arbitrary constant vector. Note that the relation
J
1
J
1
)
J
2
=
J
2
(
I
−
(14.8)
holds. Therefore, substituting (14.7), (14.8) into (14.3) we obtain
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