Biomedical Engineering Reference
In-Depth Information
4.5.1
Forward recursive kinematics
We can define 4
4 matrices A
j
, B
j
, C
j
as recursive position, velocity, and accel-
eration transformation matrices, respectively, for the
j
th
joint. Given the link trans-
formation matrix (T
j
) and the kinematics state variables for each joint (
q
j
,
3
q
j
,
and
q
j
), we have for
j
5
1to
n
:
A
j
5
T
1
T
2
T
3
?
T
j
5
A
j
2
1
T
j
(4.29)
A
j
2
1
@
T
j
@
B
j
5
A
j
5
B
j
2
1
T
j
1
q
j
_
q
j
(4.30)
2
T
j
@
2B
j
2
1
@
T
j
@
A
j
2
1
@
A
j
2
1
@
T
j
@
C
j
5
B
j
5
A
j
5
q
j
2
C
j
2
1
T
j
1
q
j
_
q
j
1
_
q
j
_
q
j
(4.31)
1
q
j
2
where A
0
5
½
I
and B
0
5
C
0
5
½
0
.
After obtaining all the transformation matrices A
j
, B
j
, C
j
, the global position,
velocity, and acceleration of a point in Cartesian coordinates can be calculated as
0
r
j
5
A
j
r
j
;
0
0
r
j
5
B
j
r
j
;
r
j
5
C
j
r
j
(4.32)
where r
j
contains the augmented local coordinates of the point in
j
th
coordinate
system.
4.5.2
Backward recursive dynamics
Based on forward recursive kinematics,
the backward recursion for dynamic
analysis is accomplished by defining 4
3
4 transformation matrix D
i
and 4
3
1
transformation matrices E
i
, F
i
, and G
i
as follows.
Given the mass and inertia properties of each link, and the external force
f
k
T
k
h
x
k
h
y
k
h
z
0
for the link
k
defined in the global coordinate system, then the joint actuation torques
k
f
x
k
f
y
k
f
z
0
and the moment h
k
T
5
½
5
½
τ
i
are
computed for
i
5
n
to 1 as (
Hollerbach, 1980
):
@
A
i
@
2
g
T
@
A
i
@
q
i
E
i
2
f
k
@
A
i
@
q
i
F
i
2
G
i
A
i
2
1
z
0
τ
i
5
tr
q
i
D
i
(4.33)
where
D
i
5
I
i
C
i
T
1
T
i
1
1
D
i
1
1
(4.34)
E
i
5
m
i
i
r
i
1
T
i
1
1
E
i
1
1
(4.35)
k
r
f
δ
ik
1
F
i
5
T
i
1
1
F
i
1
1
(4.36)
h
k
δ
ik
1
G
i
5
G
i
1
1
(4.37)
with D
n
1
1
5
G
n
1
1
5
½
0
; I
i
is the inertia matrix for link
i
;
m
i
is
the mass of link
i
; g is the gravity vector;
E
n
1
1
5
F
n
1
1
5
i
r
i
is the location of center of mass of
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