Biomedical Engineering Reference
In-Depth Information
each other,
z
an
must be corrected:
z
an
=
(x
an
×
y
an
)
:
x
z
=−
=
176
.
872,
y
z
=
446
.
42,
z
z
7841
.
27. Note that none of these vectors are unity length, and
they are normally reported as a unit vector. For example, the length of
x
an
is
245.423 cm; thus, dividing each coordinate by the length of
x
an
yields a unity
vector for
x
an
:
x
x
=
=−
=
0
.
9925,
y
x
0
.
1190,
z
x
0
.
290. Similarly for
y
an
as
a unit vector,
x
y
=
0
.
1204,
y
y
=
0
.
9913,
z
y
=−
0
.
0537. Similarly as a unit
=−
=
=
vector, the corrected
z
an
is:
x
z
0
.
9981. We
can now construct the leg anatomical-to-global matrix [LA to G] and it is:
⎡
0
.
0225,
y
z
0
.
0568,
z
z
⎤
0
.
9925
0
.
1204
−
0
.
0225
⎣
⎦
−
0
.
1190
0
.
9913
0
.
0568
0
.
0290
−
0
.
0537
0
.
9981
Note that the diagonal values are almost
=
1, indicating that the subject
stood in the calibration position with his three anatomical axes almost per-
fectly lined up with the global axes. A more useful transformation matrix is
the leg global-to-anatomical [LG to A], which is the transpose of [LA to G]:
⎡
⎤
0
.
9925
−
0
.
1190
0
.
0290
⎣
⎦
0
.
1204
0
.
9913
−
0
.
0537
−
0
.
0225
0
.
0568
0
.
9981
These anatomical axes have their origins at the ankle joint. However, from
our inverse dynamics, it is more convenient that the origin of the anatomical
axes (see Figure 7.2) be located at the COM of the leg segment with coordi-
nates 0, 0, 0. We must now establish the local coordinates of the ankle and
knee joints and the three tracking markers relative to this new origin at the
COM.
From the COM the ankle vector
=
(global ankle - global COM),
x
al
=
X
a
−
X
c
=
2
.
815
−
5
.
001
=−
2
.
186,
y
al
=
Y
a
−
Y
c
=
10
.
16
−
28
.
151
=−
17
.
991,
z
al
=
Z
a
−
Z
c
=
22
.
685
−
21
.
710
=
0
.
975
.
The anatomical ankle vector is the product of [LG to A][ankle vector]:
⎡
⎤
⎡
⎤
⎡
⎤
−
−
≈
0
.
9925
0
.
1190
0
.
0290
2
.
186
0
⎣
⎦
⎣
⎦
=
⎣
⎦
0
.
1204
0
.
9913
−
0
.
0537
−
17
.
991
0
.
975
−
18
.
15
≈
−
0
.
0225
0
.
0568
0
.
9981
0
This anatomical ankle vector lies along the line joining the ankle to knee
and the ankle is 18.15 cm distal of the COM. The
x
and
z
components of this
ankle vector are theoretically 0 but were calculated
0 because of the limited
number of decimal points in our numbers. If we were to repeat this procedure
≈
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