Biomedical Engineering Reference
In-Depth Information
be the global reference frame. Let X
i
,
Y
i
and Z
i
(resp. X
j
,
Y
j
and Z
j
)
Let
be
the three axis of reference F
i
(resp. F
j
). Let us define:
T
i
→
=
X
i
Y
i
Z
i
T
j
→
=
X
j
Y
j
Z
j
where T
i
→
(resp. T
j
→
)
stands for the transformation matrix from the local frame
F
i
(resp. F
j
)
. Thesematrixes can be computed according to the external
markers and joint centers for each time:
to the world
T
j
→
=
T
j
→
i
×
T
i
→
where T
j
→
i
stands for the transformation matrix from F
j
to F
i
, i.e. the transformation
due to the action of the joint between F
i
and F
j
.
T
j
→
i
is thus given by:
⎛
⎞
t
11
t
12
t
13
t
21
t
22
t
23
t
31
t
32
t
33
T
T
i
→
=
⎝
⎠
T
j
→
i
=
T
j
→
×
In the numerical model, this matrix is the product of three elementary symbolic
matrixes associated with each degree of freedom and which depends on the chosen
sequence of Euler angles. Let T
x
,
T
y
and T
z
be the elementary matrixes for a rotation
along the X, Y and Z axes respectively. From the theoretical point of view, for a ZYX
sequence (recommended by the International Society of Biomechanics), T
j
→
i
could
also be expressed as the product:
T
j
→
i
=
T
x
×
T
y
×
T
z
The resulting matrix has the following shape:
⎛
cos
θ
y
cos
⎞
(θ
z
)
−
cos
(θ
y
)
∗
sin
(θ
z
)
sin
(θ
y
)
cos
θ
y
sin
T
j
→
i
⎝
⎠
=
···
···
−
(θ
x
)
···
···
cos
(θ
x
)
cos
(θ
y
)
As T
j
→
i
should be equal to T
j
→
i
:
atan
t
12
t
11
θ
z
=−
atan
t
23
t
33
θ
x
=−
atan
t
13
×
cos
(θ
z
)
θ
y
=
t
11
