Biomedical Engineering Reference
In-Depth Information
4.2.2 Coordinate Transformations
3-D Direction Cosines
When studying the kinematics of human motion, it is often necessary to transform body
or body segment coordinates from one coordinate system to another. For example, coordi-
nates corresponding to a coordinate system determined by markers on the body, a moving
coordinate system, must be translated to coordinates with respect to the fixed laboratory, an
inertial coordinate system. These three-dimensional transformations use direction cosines
that are computed as follows.
Consider the vector
A
measured in terms of the uppercase coordinate system
XYZ
,
shown in Figure 4.8 in terms of the unit vectors
I
,
J
,
K
:
A
¼
A
x
I
þ
A
y
J
þ
A
z
K
ð
4
:
15
Þ
The unit vectors
I
,
J
,
K
can be written in terms of
i
,
j
,
k
in the
xyz
system
I
¼
cos y
xX
i
þ
cos y
yX
j
þ
cos y
zX
k
ð
4
:
16
Þ
J
¼
cos y
xY
i
þ
cos y
yY
j
þ
cos y
zY
k
ð
4
:
17
Þ
K
¼
cos y
xZ
i
þ
cos y
yZ
j
þ
cos y
zZ
k
ð
4
:
18
Þ
where y
xX
is the angle between
i
and
I
, and similarly for the other angles.
Substituting Eqs. (4.16)-(4.18) into Eq. (4.15) gives
A
¼
A
x
cos y
xX
i
þ
cos y
yX
j
þ
cos y
zX
k
þ
A
y
cos y
xY
i
þ
cos y
yY
j
þ
cos y
zY
k
ð
4
:
19
Þ
þ
A
z
cos y
xZ
i
þ
cos y
yZ
j
þ
cos y
zZ
k
or
i
þ
A
x
cos y
yX
þ
A
y
cos y
yY
þ
A
z
cos y
yZ
A
¼
A
x
cos y
xX
þ
A
y
cos y
xY
þ
A
z
cos y
xZ
j
ð
4
:
20
Þ
k
þ
A
x
cos y
zX
þ
A
y
cos y
zY
þ
A
z
cos y
zZ
Consequently,
A
may be represented in terms of
I
,
J
,
K
or
i
,
j
,
k
.
Z
z
A
x
X
Y
y
FIGURE 4.8
Vector
A
, measured with respect to coordinate system
XYZ
is related to coordinate system
xyz
via
the nine direction cosines of Eq. (4.20).
