Maths - Euler Angles - Sample Orientations

In order to try to explain things and give some examples we can try I thought it might help to show the rotations for a finite subset of the rotation group. We will use the set of rotations of a cube onto itself, this is a permutation group which gives 24 possible rotations as explaned on this page.

heading applied first giving 4 possible orientations:

 

right up

reference orientation

heading = 0
attitude = 0
bank = 0

back up

rotate by 90 degrees about y axis

heading = 90 degrees
attitude = 0
bank = 0

left up

rotate by 180 degrees about y axis

heading = 180 degrees
attitude = 0
bank = 0

forward up

rotate by 270 degrees about y axis

heading = -90 degrees
attitude = 0
bank = 0

Then apply attitude +90 degrees for each of the above: (note: that if we went on to apply bank to these it would just rotate between these values, the straight up and streight down orientations are known as singularities because they can be fully defined without using the bank value)

up left

heading = 0
attitude = 90 degrees
bank = 0

up forward

heading = 90 degrees
attitude = 90 degrees
bank = 0

up right

heading = 180 degrees
attitude = 90 degrees
bank = 0

up back

heading = -90 degrees
attitude = 90 degrees
bank = 0

Or instead apply attitude -90 degrees (also a singularity):

down right

heading = 0
attitude = -90 degrees
bank = 0

down back

heading = 90 degrees
attitude = -90 degrees
bank = 0

down left

heading = 180 degrees
attitude = -90 degrees
bank = 0

down forward

heading = -90 degrees
attitude = -90 degrees
bank = 0

Normally we dont go beond attitude + or - 90 degrees because thes are singularities, instead apply bank +90 degrees:

right forward

heading = 0
attitude = 0
bank = 90 degrees

back right

heading = 90 degrees
attitude = 0
bank = 90 degrees

left back

heading = 180 degrees
attitude = 0
bank = 90 degrees

forward left

heading = -90 degrees
attitude = 0
bank = 90 degrees

Apply bank +180 degrees:

right down

heading = 0
attitude = 0
bank = 180 degrees

back down

heading = 90 degrees
attitude = 0
bank = 180 degrees

left down

heading = 180 degrees
attitude = 0
bank = 180 degrees

forward down

heading = -90 degrees
attitude = 0
bank = 180 degrees

Apply bank -90 degrees:

right back

heading = 0
attitude = 0
bank = 90 degrees

back left

heading = 90 degrees
attitude = 0
bank = 90 degrees

left foward

heading = 180 degrees
attitude = 0
bank = 90 degrees

forward right

heading = -90 degrees
attitude = 0
bank = 90 degrees

encoding of these rotations in quaternions is shown here.
encoding of these rotations in matricies is shown here.
encoding of these rotations in axis-angle is shown here.

 

The working to convert each of these to matrix is shown here.

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