Biomedical Engineering Reference
In-Depth Information
Local
coordinate
system
Segment
center of
gravity
r
i
Zero
potential
energy
Δ
h
0'
ii
Tr
Y
0
Tr
ii
X
Z
FIGURE 3.5
Potential energy.
same vertical baseline where the potential is considered zero. Consequently, we
introduce the idea of minimizing the change in potential energy. It is calculated
as follows. Each segment in the human model has a specified center of mass as
depicted in
Figure 3.5
.
The vector from the origin of a link's local coordinate system to its center of
mass is given by r
i
, where the subscript indicates the relevant local coordinate sys-
tem. In order to determine the position of any part of the body, we use the DH
transformation matrices
ð
i
2
1
Þ
T
i
. Note that r
i
is actually an augmented 4
3
1 vector
with respect to local coordinate system-i, rather than a 3
1 vector typically used
with Cartesian space, as discussed in subsection 10.2.1; g
5
½
0
3
T
is
the augmented gravity vector. When the avatar moves from one configuration to
another, P
i
represents the potential energy of the initial configuration, and P
i
repre-
sents the potential energy of the current configuration. The potential energy terms
for the i
th
body part are P
i
5
2
g 00
T
0
i
2
1
T
0
1
?
T
0
i
r
i
and P
i
5
T 0
i
2
1
m
i
g
m
i
g
T
1
?
T
i
r
i
.In
0
i
1
T
0
1
?
2
T
0
i
r
i
2
0
i
2
1
Figure 3.5
,
h
i
is the y-component of the vector
T
1
?
T
i
r
i
.
Δ
The final performance measure, which is minimized, is defined as follows:
X
κ
i
5
1
ð
P
i
2
2
P
i
Þ
f
Delta
2
potential
2
energy
ð
q
Þ
5
(3.10)
Note that
Equation (3.10)
can be written in the form of a weighted sum as
follows:
X
κ
i
5
1
ð
m
i
g
Þ
2
2
f
Delta
2
potential
2
energy
ð
q
Þ
5
ð
Δ
h
i
Þ
(3.11)
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