Biomedical Engineering Reference
In-Depth Information
53
.
27
◦
or
53
.
27
◦
, and
s
3
=
=
0
.
5981,
∴
θ
3
=
−
0
.
8014 or
−
0
.
8014. The only
53
.
27
◦
because
valid solution is
θ
3
=−
0
.
8000. To summarize the
results of these three rotations (see Figure 7.1), to bring the global axes in line
with the anatomical axes requires an initial rotation about the global
X
axis of
−
−
c
2
s
3
≈
8
.
92
◦
. This will create new
Y
and
Z
axes and will be followed by a rota-
2
.
71
◦
about the
Y
axis. This rotation creates new
X
and
Z
axes.
The final rotation is the largest (because we are analyzing the leg segment
during swing), and it is
tion of
−
53
.
27
◦
, which creates the final
X
,
Y
, and
Z
axes. These final axes are the anatomical
x
−
z
axes shown in Figure 7.2.
Finally, to get the COM of the segment, we must calculate
c
in GRS coor-
dinates. We have
c
in the leg anatomical reference, and it is
−
y
−
=
- [Anatomical
m
T
2
vector]
=
[0.0000, 17.991, 3.833]. In the GRS,
c
=
[AtoG][0
.
0000, 17
.
991, 3
.
833]
⎡
⎤
⎡
⎤
⎡
⎤
0
.
5974
0
.
8000
−
0
.
0472
0
.
000
17
.
991
3
.
833
14
.
212
11
.
346
2
.
793
⎣
⎦
⎣
⎦
=
⎣
⎦
=
−
0
.
7873
0
.
5969
0
.
1544
0
.
1515
−
0
.
0550
0
.
9868
From
Figure
7.2,
the
global
vector
R
c
=
R
m
+
c
=
[20
.
812, 36
.
646,
34
.
653].
As an exercise students can repeat these calculations for frames 5 and 7
with the answers:
θ
1
=−
8
.
97
◦
,
θ
2
=−
1
.
31
◦
,
θ
3
=−
56
.
02
◦
,
Frame 5 :
R
c
=
[16
.
120, 36
.
325, 34
.
697]
8
.
56
◦
,
θ
2
=−
4
.
08
◦
,
49
.
89
◦
,
θ
1
=−
θ
3
=−
Frame 7 :
R
c
=
[25
.
429, 36
.
818, 34
.
623]
7.3 DETERMINATION OF SEGMENT ANGULAR VELOCITIES
AND ACCELERATIONS
Recall from Section 7.1.2 and Figure 7.2 that we had to determine three
time-varying rotation angles,
θ
1
,
θ
2
, and
θ
3
, prior to transforming from the
GRS to the anatomical axes. The first time derivative of these transformation
angles yields the components of the segment angular velocities:
ω
=
d θ
1
/dt
·
e
x
+
d θ
2
/dt
·
e
y
+
d θ
3
/dt
·
e
z
(7.7
a
)
where
e
x
,
e
y
and
e
z
denote the unit vectors of the three rotation axes
x
,
y
, and
z
shown in Figure 7.1. Consider an angular velocity,
ω
, about axis
x
; here,
ω
=
d θ
1
/dt
·
e
x
and there is no rotation of
θ
2
or
θ
3
. This angular
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