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30 Multibody Systems Approach to Vehicle Dynamics
Applying all three rotations in the sequence , and would result in the
three rotation matrices being multiplied through as follows:
⎡Ax'
⎤ ⎡cos sin 0⎤
⎢
Ay'
⎥
⎢
sin cos 0
⎥
⎢ ⎥ ⎢
⎥
⎣⎢
Az' ⎦⎥
⎣⎢
0 0 1⎦⎥
⎡ cos
0 sin⎤
⎢
0 1 0
⎥
⎢
⎥
⎣⎢
sin 0 cos⎦⎥
⎡1 0 0 ⎤
⎢
0 cos sin
⎥
⎢
⎥
⎣⎢
0 sin cos
⎦⎥
⎡Ax⎤
⎢
Ay
⎥
⎢ ⎥
⎣⎢
Az⎦⎥
(2.24)
⎡Ax'
⎤ ⎡cos sin 0⎤
⎢
Ay'
⎥
⎢
sin cos 0
⎥
⎢ ⎥ ⎢
⎥
⎣⎢
Az' ⎦⎥
⎣⎢
0 0 1⎦⎥
⎡ cos sinsin sincos⎤
⎢
0 cos sin
⎥
⎢
⎥
⎣⎢
sin cossin coscos⎦⎥
⎡Ax⎤
⎢
Ay
⎥
⎢ ⎥
⎣⎢
Az⎦⎥
(2.25)
⎡Ax'
⎤ ⎡cos cos cos sinsin sin cos cos sincossin
sin
⎤
⎡Ax⎤
⎢
Ay'
⎥
⎢
sin cos sin sinsin cos cos sin sincoscos
sin
⎥ ⎢
Ay
⎥
⎢ ⎥ ⎢
⎥ ⎢ ⎥
⎣⎢
Az' ⎦⎥
⎣⎢
sin cossin coscos
⎦⎥
⎣⎢
Az⎦⎥
(2.26)
It should be noted that large rotations such as these are not commutative
and therefore the angles , and cannot be considered to be the components
of a vector. The order in which the rotations are applied is important. As
can be seen in Figure 2.10 applying equal rotations of 90° but in a different
sequence will not result in the same final orientation of the vector.
An understanding that large rotations are not a vector is an important aspect
of multibody systems analysis. Sets of rotations may be required as inputs
to define the orientation of a rigid body or a joint. They will also form the
output when the relative orientation of one body to another is requested.
Z
Rotate α about X Rotate β about Y Rotate γ about Z
Y Y X
β
α
Y
Z
γ
X
Y
X
X
Z
Z
Z
Rotate β about Y Rotate α about X Rotate γ about Z
Y
Y
Y
Z
γ
Z
X
β
Z
α
Y
X
Fig. 2.10
X
X
The effect of rotation sequence on vector orientation