Biomedical Engineering Reference
In-Depth Information
4 matrices denoted by A
j
, B
j
, and C
j
to represent recursive posi-
tion, velocity, and acceleration transformation matrices for the
j
th
joint, respec-
tively. Given the link transformation matrix (T
j
) and the kinematics state
variables for each joint, angular displacement, velocity and acceleration (
q
j
,
We use 4
3
q
j
,
_
and
q
j
), we have for
j
€
1to
n
:
5
A
j
5
T
1
T
2
T
3
?
T
j
5
A
j
2
1
T
j
(7.1)
A
j
2
1
@
T
j
@q
j
_
B
j
5
A
j
5
q
j
B
j
2
1
T
j
1
(7.2)
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
1
C
j
2
1
T
j
1
q
j
_
q
j
1
_
q
j
€
q
j
(7.3)
q
j
where A
0
5
1 and B
0
5
C
0
5
0. Then, the global position, velocity, and accelera-
tion of a point in the Cartesian coordinate system can be calculated using the fol-
lowing formulas:
0
r
j
5
A
j
r
j
;
0
0
r
j
5
B
j
r
j
;
r
j
5
C
j
r
j
(7.4)
where
0
r
j
and r
j
are global and local augmented coordinates, respectively.
7.5
Dynamics formulation
7.5.1
Backward recursive dynamics
Based on forward recursive kinematics, the backward recursion for the dynamic
analysis is accomplished by defining a 4
1
transformation vectors E
i
, F
i
, and G
i
as follows. Given the mass and inertia proper-
ties of each link,
4 transformation matrix D
i
and 4
3
3
force f
k
5
k
f
x
0
and the moment
k
f
y
k
f
z
the external
h
k
5
k
h
x
k
h
y
k
h
z
0
for the link
k
, defined in the global coordinate system, the
joint actuation torques derived in Chapter 4 (Equation 4.33) are
τ
i
for
i
n
to 1 as:
5
tr
@
A
i
@
g
T
@
A
i
@
f
k
@
A
i
@
G
i
A
i
2
1
z
0
τ
i
5
q
i
D
i
q
i
E
i
2
q
i
F
i
2
(7.5)
2
I
i
C
i
1
D
i
5
T
i
1
1
D
i
1
1
(7.6)
m
i
r
i
1
E
i
5
T
i
1
1
E
i
1
1
(7.7)
k
r
f
δ
ik
1
F
i
5
T
i
1
1
F
i
1
1
(7.8)
G
i
5
h
k
δ
ik
1
G
i
1
1
(7.9)
where D
n
1
1
5
0 and E
n
1
1
5
F
n
1
1
5
G
n
1
1
5
0; I
i
is the inertia matrix for link
i
;
m
i
is the mass of link
i
; g is the gravity vector;
i
r
i
is the location of center of
k
r
f
mass of link
i
in the local frame
i
;
is the position of the external force in the
Search WWH ::
Custom Search