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To introduce different types of description for orientations and to explain related transformations, we start with a simple description of rotations with respect to the axes of a coordinates transformation.
It is assumed, that the body frame
is
congruent to a universal base frame 
,
which means that the orientation of the body
with
respect to the external base frame is
the unity matrix:
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(2.5) |
Any rotation of this body around the axes of the coordinates base frame
with
rotation angle 
leads
to the following rotation matrices:
Rotation around the 
axis:
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(2.6) |
Rotation around the 
axis:
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(2.7) |
Rotation around the 
axis:
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(2.8) |
Figure 2.3 illustrates these rotations for a cubic body.
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Figure 2.3: Base rotations of angle |
Rotation matrices can be multiplied with each others, which leads to a series of rotations of a body around the axes of the base frame. The order of rotations is absolutely important, which means it is not commutative.
For example, let us assume an initial body frame which is congruent with the
universal base frame at the very beginning. First a rotation of angle
around
the -axis
is applied, followed by a rotation of angle
around
the -axis.
This results in the given rotation matrix for the final orientation of the body:
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(2.9) |
Assuming the same initial conditions, but now starting with a rotation of the
same angle around
the -axis
as before, followed by a rotation with same
around
the -axis,
now leads to the following rotation matrix:
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(2.10) |
Figure
2.4 explains
this issue for two selected angles
:
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Figure 2.4: Series of two base rotations |
Consequently any series of rotations with respect to the universal base frame
axes (,,
)
can be described mathematically by a series of multiplications of rotation
matrices. Assuming congruent frames for body
and
base at
the beginning, the sequence of matrices starts with the unity matrix first. In
the order of rotation operations, the related rotation matrices are included
from the left. So, the last rotation is described by the left-most matrix in
this sequence.
Instead of referencing the rotations to the axes of the base frame,
alternatively the axes of the body frame (
,
,
)
can be chosen for this purpose. It turns out, that the previous rotation could
have been achieved with respect to the body frame by rotation of the body around
the -axis
of its own body frame with angle first,
and then by a subsequent rotation of angle 
around
its new -axis
(which has been moved to its new position during the first rotation).
In general, a rotation around all the intermediate locations of body frame axes during a sequence of rotations is described mathematically by multiplying the current orientation matrix with appropriate rotation matrices from the right. However, the matrices themselves remain unchanged as in the case of base frame descriptions.
