Biomedical Engineering Reference
In-Depth Information
(b)
T
1
−
T
2
−
Fl
1
sin
(q
2
)
=−¨
q
1
sin
(q
2
)(m
1
r
1
+
m
2
l
1
+
m
3
l
1
)
q
2
m
1
(r
1
)
2
m
3
(l
1
)
2
m
2
(l
1
)
2
+¨
+
I
1
+
+
+
q
3
cos
(q
3
−
q
2
)
+
( q
3
)
2
·
sin
(q
3
−
q
2
)
(m
2
r
2
l
1
+
m
3
l
1
l
2
)
+
¨
q
4
)
2
q
4
cos
(q
4
−
q
2
)
+
(
˙
q
2
)
(m
3
r
3
l
1
)
·
sin
(q
4
−
+
g(m
1
r
1
+
m
2
l
1
+
m
3
l
1
)
cos
(q
2
)
(c)
T
2
−
T
3
−
Fl
2
sin
(q
3
)
=−¨
q
1
sin
(q
3
)(m
2
r
2
+
m
3
l
2
)
q
2
)
2
+
[
q
2
cos
(q
3
−
¨
q
2
)
−
(
¨
·
sin
(q
3
−
q
2
)
]
(m
2
r
2
l
1
+
m
3
l
1
l
2
)
+
q
3
m
2
(
r
2
)
2
+
I
2
+
m
3
(
l
2
)
2
+
¨
q
3
)
−
q
4
)
2
q
4
cos
(q
4
−
(
˙
·
sin
(q
4
−
q
3
)
]
(m
3
r
3
l
2
)
+
g(m
2
r
2
+
m
3
l
2
)
cos
(q
3
)
(d)
T
3
=−¨
q
1
m
3
r
3
sin
(q
4
)
+¨
q
2
cos
(q
4
−
q
2
)
q
2
)
(m
3
r
3
l
1
)
+
¨
q
2
)
2
sin
(q
4
−
+
(
˙
q
3
cos
(q
4
−
q
3
)
sin
(
q
4
−
q
3
)
(
m
3
r
3
l
2
)
+
(
q
3
)
2
·
q
4
m
3
(r
3
)
2
I
3
+
+¨
+
m
3
gr
3
cos
(q
4
)
These four equations are the minimum number of equations that describe
the model. They are highly nonlinear, tightly coupled, lengthy, and prone
to errors. By breaking up the model at joints and introducing quasi-DOF in
the form of additional generalized coordinates and constraint equations that
counteract these superfluous coordinates, a larger set of equations is obtained
with equations that are shorter, manageable, and contain more information
about the model.
The tight coupling between these four equations demonstrates the potential
errors that were mentioned in Section 8.0.2. An error in the mass of any
segment, for example, will result in errors in all four coordinates. Mass
m
3
,the
mass of the HAT, appears in all four equations and will affect the calculation
of
q
1
,
q
2
,
q
3
, and
q
4
. The errors will also accumulate over time. Thus, it is
essential that, before the model is used to answer a research question, all the
anthropometrics and internal joint constraints be almost perfect such that an
internal validation is achieved.
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