Biomedical Engineering Reference
In-Depth Information
The kinetic energy
K
i
of link
i
is calculated from the kinetic energy of a dif-
ferential mass element as
ð
1
2
Trð
v
i
v
i
Þdm
K
i
5
(4.13)
m
where,
Tr
is the trace of a matrix, i.e., the sum of the diagonal elements of a
matrix.
Therefore, the total kinetic energy
K
of the human-link system is
ð
X
2
X
n
n
1
Trð
v
i
v
i
Þdm
K
5
K
i
5
(4.14)
m
i
5
1
i
5
1
q
j
using
Equation (4.12)
, the total kinetic energy can be expressed in terms of joint
space as
Expanding the velocities and expressing them in terms of joint velocities
_
2
X
X
n
n
1
2
q
T
M
ð
q
Þ
q
1
K
5
5
M
ik
_
q
i
_
q
k
(4.15)
i
1
k
1
5
5
_
where
q is the joint velocity vector and
M
ik
is the (
i, k
) element of the mass-
inertia symmetric matrix M
ð
q
Þ
such that
"
#
T
!
X
n
0
T
j
ð
q
Þ
@
0
T
j
ð
q
Þ
@
Tr
@
I
j
@
M
ik
ð
q
Þ
5
i
;
k
1
;
2
; ...;
n
(4.16)
5
q
k
q
i
j
5
max
ði;kÞ
and I
j
is the inertia matrix as below:
2
3
I
xx
1
I
yy
1
I
zz
2
I
xy
I
xz
m
i
x
i
2
2
4
5
2
I
xx
2
I
yy
1
I
zz
I
xy
I
yz
m
i
y
i
2
2
2
I
i
5
(4.17)
I
xx
1
I
yy
2
I
zz
I
xz
I
yz
m
i
z
i
2
2
2
m
i
x
i
m
i
y
i
m
i
z
i
m
i
where
m
i
is the mass of link
i
,(
x
i
,
y
i
,
z
i
) is the location vector of center
of mass of link
i
, expressed in terms of
i
th
coordinate frame,
I
xx
,
...
,
I
xy
,
...
to
i
th
are the moments/products of inertia of link
i
with respect
coordinate
frame.
The potential energy
P
i
of each link
i
is
m
i
g
T
0
r
i
52
m
i
g
T
0
T
i
i
r
i
Þ;
n
(4.18)
where
i
r
i
is the center of mass vector of link
i
with respect to the
i
th
local coordi-
nate frame and g is the augmented 4
P
i
52
ð
i
1
;
2
; ...;
5
1 gravity vector.
3
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