Biomedical Engineering Reference
In-Depth Information
reaction forces are measured in the GRS and the moments of inertia are known
in the anatomical axes, these previously determined transformation matrices
are used extensively in the kinetic calculations. All joint reaction forces are
initially calculated in the GRS, and all joint moments are calculated in the
anatomical axes.
7.4.1 Newtonian Three-Dimensional Equations of Motion
for a Segment
All reaction forces are calculated in the GRS and, because the gravitational
forces and the segment COM accelerations are readily available in the GRS,
it is convenient to calculate all segment joint reaction forces in the GRS.
Students are referred to the 2D link-segment equations and free-body diagram
equations in Section 5.1. Figure 7.3 is now presented to demonstrate the steps
required to calculate kinetics for this segment. The only addition is the third
dimension,
z
. We are given the three distal reaction forces either as measures
from a force plate or from the analysis of the adjacent distal segment. It
should be noted that the reaction forces and moments at the distal end are in
the reverse direction from those at the proximal end, the same convention as
was used in Section 5.1.
Step 1:
Calculate the reaction forces at the proximal end of the segment in
the GRS:
F
X
=
ma
X
or
R
XP
−
R
XD
=
ma
X
(7.8
a
)
F
Y
=
ma
Y
or
R
YP
−
R
YD
−
mg
=
ma
Y
(7.8
b
)
=
ma
Z
R
ZP
−
R
ZD
=
ma
Z
F
Z
or
(7.8
c
)
where
a
X
,
a
Y
,
a
Z
are the segment COM accelerations in the
X
,
Y
,
Z
GRS
directions and
R
XP
,
R
XD
,
R
YP
,
R
YD
,
R
ZP
,
R
ZD
are the proximal and distal
reaction forces in the
X
,
Y
, and
Z
axes.
Step 2:
Transform both proximal and distal reaction forces into the anatomical
axes using the [G to A] matrix transformation based on
θ
1
,
θ
2
, and
θ
3
[see
Equation (7.5)]. We will now have the proximal and distal reaction forces
in the anatomical axes
x
,
y
,
z
:
R
xp
,
R
xd
,
R
yp
,
R
yd
,
R
zp
,
R
zd
.
Step 3:
Transform the distal moments from those previously calculated in the
GRS using the [G to A] matrix to the anatomical axes, as before, based
on
θ
1
,
θ
2
, and
θ
3
:
M
xd
,
M
yd
,
M
zd
. We now have all the variables necessary
to calculate the proximal moments in the anatomical axes.
7.4.2 Euler's Three-Dimensional Equations of Motion for a Segment
The equations of motion for the 3D kinetic analyses are the Euler equations.
Considerable simplification can be made in the rotational equations of motion
if these equations are written with respect to the principal (anatomical) axes
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