Biomedical Engineering Reference
In-Depth Information
Joint space
Work space
J
R (J)
N (J)
O
Figure 16.3
Nonlinear and redundant mapping.
T
, x
¼
x
1
,
x
2
, ...,
x
n
T
,
m
>
n
, and
where
u ¼ u
1
,
u
2
, ...,
u
m
½
½
dx
¼
J d
u
(16
:
2)
where
J
is a
n
m
matrix. As shown in Figure 16.3, the range and null spaces of
J
are
,
N
(
J
)
¼
R
(
J
)
¼
x
2
R
n
: x
¼
J
(
u
)
u
for
8
u 2
R
m
:
J
(
u
)
u 2
R
m
u ¼
0
(16
:
3)
and dim
R
(
J
)
þ
dim
N
(
J
)
¼
m
.
Assuming the Jacobian
J
is known, we summarize five typical inverse kinematics approaches:
1.
By using the transpose of the matrix
J
, we calculate
u ¼
J
T
(x
d
x)
(16
:
4)
where x
d
is the desired end-effector position (Chiacchio et al., 1991).
For the case when rank (
J
)
¼
n
, we use
J
þ
, the pseudo-inverse of
J
, to obtain
2.
u ¼
J
þ
x
u ¼
J
þ
x
þ
(
I
J
þ
J
)
h
or
(16
:
5)
where
JJ
þ
¼
I
and vector (
I
J
þ
J
)
h 2
N
(
J
). When rank(
J
(
u
))
<
n
, then
J
is singular, the joint
u
is
the singular configuration (Klein and Huang, 1983).
By specifying additional task constraints to extend
J
as a full rank square matrix
J
e
, we have
(Baillieul, 1985)
3.
u ¼
J
1
e
x
(16
:
6)
4.
The regularization method to minimize the cost function
k
dx
J
d
u kþl k
d
u k
.
5.
Based on compliance control, by using the relations:
t ¼
J
T
F
;
t ¼
K
u
d
u ,
F
¼
K
x
dx;
dx
¼
J
d
u
(16
:
7)
and
then we have
K
u
¼
J
T
K
x
J
;
and therefore d
u ¼
(
J
T
K
x
J
)
1
J
T
K
x
dx.
In approach 3, the specification of the additional task constraints may be closely related to the
Bernstein's concept of synergy. However, from the biological point of view, the main problem
inherent in all the above approaches is the assumption that the system's Jacobian is known
a priori, which seems unlikely in biological system. In addition, the cost functions and task
Search WWH ::
Custom Search