Maths - Euler Angles Examples 90 degree steps- Martin Baker

Maths - Euler Angles - Sample Orientations

In order to try to explain things I thought it might help to work out a simple case where rotations are only allowed in mutiples of 90 degrees. This should make it easier to illustrate the orientation with a simple aeroplane figure, we can rotate this either about the x,y or z axis as shown here:

When we combine these rotations about the x,y and z axies in 90 degree multiples there are 24 possible orientations as shown here:

heading applied first giving 4 possible orientations:

reference orientation

heading = 0
attitude = 0
bank = 0

rotate by 90 degrees about y axis

heading = 90 degrees
attitude = 0
bank = 0

rotate by 180 degrees about y axis

heading = 180 degrees
attitude = 0
bank = 0

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)

heading = 0
attitude = 90 degrees
bank = 0

up forward

heading = 90 degrees
attitude = 90 degrees
bank = 0

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):

heading = 0
attitude = -90 degrees
bank = 0

down back

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

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

heading = 90 degrees
attitude = 0
bank = 90 degrees

left back

heading = 180 degrees
attitude = 0
bank = 90 degrees

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

Apply bank +180 degrees:

heading = 0
attitude = 0
bank = 180 degrees

heading = 90 degrees
attitude = 0
bank = 180 degrees

heading = 180 degrees
attitude = 0
bank = 180 degrees

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

Apply bank -90 degrees:

right back

heading = 0
attitude = 0
bank = 90 degrees

heading = 90 degrees
attitude = 0
bank = 90 degrees

left foward

heading = 180 degrees
attitude = 0
bank = 90 degrees

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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see also:

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