Biomedical Engineering Reference
In-Depth Information
⊡
⊣
⊤
⊦
=
⊡
⊣
⊤
⊦
⊡
⊣
⊤
⊦
⊡
⊣
⊤
⊦
,
(
c
2
c
3
−
s
1
s
2
s
3
)
−
s
3
c
1
(
s
2
c
3
+
s
1
s
3
c
2
)
c
3
s
3
0
s
3
c
3
0
00 1
−
10 0
0
c
1
c
2
0
s
2
010
−
s
2
0
c
2
R
=
(
s
3
c
2
+
s
1
s
2
c
3
)
c
1
c
3
(
s
2
s
3
−
s
1
c
2
c
3
)
−
s
1
−
c
1
s
2
s
1
c
1
c
2
0
s
1
c
1
where
s
i
3.
From the known joint rotation matrix
R
, angles of rotation can be derived as:
=
sin
(ʵ
i
),
c
i
=
cos
(ʵ
i
),
i
=
1
,
2
,
⊜
R
12
+
R
22
,ʵ
1
=
c
1
=
arctan
(
R
22
/
c
1
),
ʵ
2
=
arctan
(
−
R
31
/
R
33
), ʵ
3
=
arctan
(
−
R
12
/
R
22
).
Finite Helical Axis Representation
In the finite helical axis (FHA) or screw approach, joint motion is represented at each
instant by a translation
s
along a line of space and a rotation
ʵ
about the same line,
T
and the position
T
of one of its points [
71
]. The relation between
u
and the rotation matrix R is given
by:
R
determined by a unit direction attitude vector
u
=
(
u
x
,
u
y
,
u
z
)
V
ʵ
uu
T
, where:
S
ʵ
=
=
C
ʵ
E
+
S
ʵ
A
(
u
)
+
sin
(ʵ),
C
ʵ
=
cos
(ʵ),
V
ʵ
=
1
−
C
ʵ
,
E
-(3,3) is the identity matrix and
A
(
u
)
is the screw symmetric matrix, defined as:
⊡
⊤
0
−
u
z
u
y
⊣
⊦
A
(
u
)
=
u
z
0
−
u
x
−
u
y
u
x
0
When the joint ro
tation matrix
R
is known, the sine of the rotation
angle can be found
from 2
S
⊟
2
2
2
ʵ
=±
(
R
32
−
R
23
)
+
(
R
13
−
R
31
)
+
(
R
21
−
R
12
)
,
while the angle of
rotation
C
ʵ
).
The orientation of the FHA, i.e. the components of the unit vector u, is given by:
ʵ
about the axis
u
is given by the relation
ʵ
=
Ar c
tan
(
S
ʵ
/
2
u
x
=
(
R
32
−
R
23
)/
S
ʵ
,
2
u
y
=
(
R
13
−
R
31
)/
S
ʵ
,
2
u
z
=
(
R
21
−
R
12
)/
S
ʵ
.
Note that for very small rotation angles(
, the axis of rotation
u
is not well-defined
due to the small magnitude of both numerator and denominator.
Following a previously-advised joint rotational motion representation [
60
], orien-
tation vector
ʵ)
θ
=
ʵ
·
u
projections (also called OVP convention) was adopted because
of its robustness [
71
]. The distal bone frame has been projected in this work on the
proximal bone frame to represent the distal joint segment displacements.
Linear Multiple Regression
To analyze the available motions, the dependent DoFs, i.e. clavicle and scapula
motions, were plotted using the humerus displacement as the independent DoF. Raw
