Biomedical Engineering Reference
In-Depth Information
velocity can be expressed as:
⎡
⎤
θ
1
0
0
⎣
⎦
ω
=
The second angular velocity,
ω
=
e
y
, plus the component of
ω
that is transformed by [
2
] in Equation (7.2), can be expressed as:
d θ
2
/dt
·
⎡
⎤
⎡
⎤
⎡
⎤
⎡
⎤
⎡
⎤
⎡
⎤
c
2
θ
1
0
s
2
θ
1
c
2
θ
1
θ
2
s
2
θ
1
θ
1
0
0
0
θ
2
0
0
θ
2
0
c
2
0
−
s
2
01 0
s
2
ω
=
⎣
⎦
+
⎣
⎦
⎣
⎦
=
⎣
⎦
+
⎣
⎦
=
⎣
⎦
0
c
2
Similarly, the third angular velocity,
ω
=
d θ
3
/dt
·
e
z
, plus the contribution
from
ω
that is transformed by [
3
] in Equation (7.3), gives us:
⎡
⎤
⎡
⎤
⎡
⎤
⎡
⎤
⎡
⎤
c
2
θ
1
0
s
2
θ
1
s
3
θ
2
−
s
3
c
2
θ
1
+
c
3
θ
2
s
2
θ
1
c
3
c
2
θ
1
+
0
0
θ
3
c
3
s
3
0
0
0
θ
3
⎣
⎦
+
⎣
⎦
⎣
⎦
=
⎣
⎦
+
⎣
⎦
ω
=
−
s
3
c
3
0
001
⎡
⎤
c
3
c
2
θ
1
+
s
3
θ
2
⎣
s
3
c
2
θ
1
+
c
3
θ
2
s
2
θ
1
+
θ
3
⎦
=
−
Decomposing
ω
into its three components along the three anatomical axes:
⎡
⎤
⎡
⎤
⎡
⎤
θ
1
θ
2
θ
3
ω
x
ω
y
ω
z
c
2
c
3
s
3
0
⎣
⎦
=
⎣
⎦
⎣
⎦
−
c
2
s
3
c
3
0
ω
=
(7.7
b
)
s
2
01
We can now calculate the three segment angular velocities,
ω
x
,
ω
y
, and
ω
z
,
that are necessary to solve the 3D inverse dynamics equations developed in
the next section. Recall that the time varying
θ
1
,
θ
2
, and
θ
3
are calculated
from Equation (7.5), and the time derivatives of these angles are individually
calculated using the same finite difference technique used in two dimensions;
see Equation (3.15). The three segment angular accelerations,
α
x
,
α
y
, and
α
z
,
can now be calculated using either of the finite difference Equations (3.17)
or (3.18c). We now have all the kinematic variables necessary for our 3D
kinetic analyses.
7.4 KINETIC ANALYSIS OF REACTION FORCES
AND MOMENTS
Having developed the transformation matrices from global to anatomical and
from anatomical to global, we are now in a position to begin calculating
the reaction forces and moments at each of the joints. Because the ground
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