Biomedical Engineering Reference
In-Depth Information
The DCM is then obtained by combining the effect of the three rotations in
the same order, [
φ
]
=
[
φ
3
][
φ
2
][
φ
1
]. In expanded form,
⎡
⎤
c
1
c
3
−
s
1
c
2
s
3
−
c
3
s
1
+
c
1
c
2
c
3
s
2
s
3
⎣
⎦
[
φ
]
=
s
1
s
3
+
c
1
c
2
s
3
−
s
1
s
3
+
c
1
c
2
c
3
c
3
c
2
(8.19)
s
2
s
3
−
c
1
s
2
c
2
The first time derivatives
(ω
1
,
ω
2
,
ω
3
)
of Euler angles are in fact vectors in
the directions
Z
,
x
1
, and
z
2
, respectively. These components expressed in the
xyz
directions are more useful for segment energy calculation. They are given
by the transformation:
⎡
⎤
⎡
⎤
⎡
⎤
ω
x
ω
y
ω
z
s
2
s
3
c
3
0
ω
1
ω
2
ω
3
⎣
⎦
=
⎣
⎦
⎣
⎦
s
2
c
3
−
s
3
0
(8.20)
c
3
01
[
ω
x
,
ω
y
,
ω
z
]
t
Finally, the skew-symmetric matrix associated with
ω
=
is in
the form:
⎡
⎤
0
−
ω
2
ω
y
ω
=
⎣
⎦
ω
z
0
−
ω
x
(8.21)
−
ω
y
ω
x
0
It should be noted that the externally applied torques
τ
(or
T
) are always
expressed as components in the directions
xyz
. The relations between
these torque components and the generalized torques required by Lagrange
equations are given by:
⎡
⎤
⎡
⎤
⎡
⎤
Q
θ
1
Q
θ
2
Q
θ
3
s
2
s
2
s
2
c
3
c
3
τ
x
τ
y
τ
z
⎣
⎦
=
⎣
⎦
⎣
⎦
c
3
−
s
3
0
(8.22)
0
0
1
8.3
SYSTEM ENERGY
At any given time, the system energy may consist of the energy of segments
due to their motion and position in the system, the energy stored in springs
because of their elastic deformation, and the dissipation energy due to fric-
tion in the system. The first and second types of energy are included in the
Lagrangian
L
of the system. The third type can be treated as an external force
applied to the system at the proper points rather than energy, and hence it is
covered under external forces. In the case of dampers, it is possible to write
an energy expression that may be included in Lagrangian equations. The three
types of energy are covered in more detail in the sections that follow.
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