Biomedical Engineering Reference
In-Depth Information
Combining Equations (7.1), (7.2), and (7.3), we get:
⎡
⎤
⎡
⎤
x
3
y
3
z
3
x
0
y
0
z
0
⎣
⎦
=
⎣
⎦
[
3
][
2
][
1
]
(7.4)
Note that the matrix multiplication as shown in Equation (7.4) is not
commutative, which means that the order of the transformations must be
such that [
1
] is done first, [
2
] second, and [
3
] last. In other words,
[
1
][
2
]
=
[
2
][
1
]. An expansion of Equation (7.4) yields:
⎡
⎤
⎡
⎤
⎡
⎤
s
3
c
1
+
s
1
s
2
c
3
s
1
s
3
−
c
1
s
2
c
3
x
3
y
3
z
3
c
2
c
3
x
0
y
0
z
0
⎣
⎦
=
⎣
⎦
⎣
⎦
−
c
2
s
3
c
1
c
3
−
s
1
s
2
s
3
s
1
c
3
+
c
1
s
2
s
3
(7.5)
s
2
−
s
1
c
2
c
1
c
2
7.1.3 Other Possible Rotation Sequences
In theory, there are 12 possible correct rotation sequences; all were introduced
by the Swiss mathematician, Leonhard Euler (1707 - 1783) . The list that
follows gives all possible valid rotation sequences. The example explained
previously is generally referred to as the Cardan system, which is commonly
used in biomechanics, while the
z
-
x
-
z
rotation sequence, generally referred
to as the Euler system, is commonly used in mechanical engineering.
y
−
x
y
−
z
(Cardan)
z
−
x
z
−
y
x
−
x
−
x
−
x
−
x
−
y
x
−
z
z
−
x
z
−
y
y
−
y
−
y
−
y
−
x
−
y
x
−
z
(Euler)
y
−
x
y
−
z
z
−
z
−
z
−
z
−
7.1.4 Dot and Cross Products
In 3D we are dealing almost exclusively with vectors and when vectors are
multiplied we must compute the mathematical function called the
dot
or
cross
product
. The dot product is also called the scalar product because the result
is a scalar while the cross product is also called the vector product because
the result is a vector. Dot product was first introduced in Section 6.08 in
the calculation of the mechanical power associated with a force and velocity
vector. Only the component of the force, F, and the velocity, V, that are
parallel result in the power, P
cos
θ
where
θ
is the angle
between F and V in the FV plane. In 3D the power P
=
F
·
V
=|
F
V
|
=
F
x
V
x
+
F
y
V
y
+
F
z
V
z
.
Cross products are used to find the product of two vectors in one plane
where the product is a vector nomal to that plane. Suppose we have vectors
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