Biomedical Engineering Reference
In-Depth Information
Table 4.1 shows the ratio of the foot center of mass location relative to its proximal end to be
0.5, so
d
¼
0
:
5
ð
l
Þ¼
0
:
032 m. Therefore,
foot
2
10
4
kg m
2
I
¼ð
7
:
12
Þð
0
:
365 kg
Þð
0
:
032 m
Þ
foot
=
cm
10
4
kg m
2
¼
3
:
38
I
foot=cm
represents the centroidal mass moment of inertia about the transverse principal axes
of the foot (
y
0
and
z
0
in Figure 4.30). Consequently,
10
4
kg m
2
I
y
0
y
0
¼
3
:
38
10
4
kg m
2
I
z
0
z
0
¼
3
:
38
The foot is approximated as a cylinder with a length to radius ratio of 6. The ratio of trans-
verse to longitudinal (
x
0
) mass moments of inertia can be shown to be approximately 6.5.
Then the longitudinal mass moment of inertia (about
x
0
in Figure 4.30) may be estimated as
10
5
kg m
2
I
x
0
x
0
¼
5
:
20
Having estimated the anthropomorphic values for the foot, the kinetic analysis may now
begin. The unknown ankle reaction force,
F
A
, is found by using Newton's Second Law, or
P
F
¼
m
a
:
F
g
þ
F
A
m
foot
g
¼
m
foot
a
foot
F
A
¼
m
foot
a
foot
k
F
g
þ
m
foot
g
k
¼ð
0
:
365 kg
Þ½
2
:
09
i
0
:
357
j
0
:
266
k
m
=
s
ð
3
:
94
i
15
:
21
j
þ
242
:
4
k
Þ
N
s
2
þð
0
:
365 kg
Þð
9
:
81 m
=
Þ
k
¼
3
:
18
i
þ
15
:
08
j
238
:
9
k
N
Euler's equations of motion (Eqs. (4.44)-(4.46)) are then applied to determine the unknown
ankle moment reaction
M
A
. Euler's equations are defined relative to the principal axes fixed
to the segment—that is,
z
0
fixed to the foot. It is noted, however, that the data
required for the solution presented previously—for example,
x
0
,
y
0
,and
v
foot
and
a
foot
—are expressed
relative to the laboratory coordinate system (
). Consequently, vectors required for the
solution of Euler's equations must first be transformed into the foot coordinate system. In
the preceding data set, the foot anatomical coordinate system was given as
x, y, z
e
fax
¼
0
:
977
i
0
:
0624
j
0
:
202
k
e
fay
¼
0
:
0815
i
þ
0
:
993
j
þ
0
:
0877
k
e
faz
¼
0
:
195
i
0
:
102
j
þ
0
:
975
k
z
0
,or
i
0
,
j
0
,and
k
0
. Recall from the discussion
in Section 4.2.2 that coefficients in the expression for
e
fax
represent the cosines of the angles
between
x
0
,
y
0
,and
where
e
fax
,
e
fay
,and
e
faz
correspond to
x
0
and
x
0
and
x
0
and
, respectively. Similarly, the coefficients in the expres-
sion for
e
fay
represent the cosines of the angles between
x
,
y
,and
z
y
0
and
y
0
and
y
0
and
x
,
y
,and
z
,and
