Maths - Matrix algebra - Examples 90 degree steps (original) (raw)

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:

aeroplane reference orientation heading = 0 attitude = 0 bank = 0 ch*ca -ch*sa*cb + sh*sb ch*sa*sb + sh*cb sa ca*cb -ca*sb -sh*ca sh*sa*cb + ch*sb - sh*sa*sb + ch*cb ch=1 ca=1 cb=1 sh=0 sa=0 sb=0 aeroplane rotate by 90 degrees about y axis heading = 90 degrees attitude = 0 bank = 0 ch*ca -ch*sa*cb + sh*sb ch*sa*sb + sh*cb sa ca*cb -ca*sb -sh*ca sh*sa*cb + ch*sb - sh*sa*sb + ch*cb ch=0 ca=1 cb=1 sh=1 sa=0 sb=0 aeroplane rotate by 180 degrees about y axis heading = 180 degrees attitude = 0 bank = 0 ch*ca -ch*sa*cb + sh*sb ch*sa*sb + sh*cb sa ca*cb -ca*sb -sh*ca sh*sa*cb + ch*sb - sh*sa*sb + ch*cb ch=-1 ca=1 cb=1 sh=0 sa=0 sb=0 aeroplane rotate by 270 degrees about y axis heading = -90 degrees attitude = 0 bank = 0 ch*ca -ch*sa*cb + sh*sb ch*sa*sb + sh*cb sa ca*cb -ca*sb -sh*ca sh*sa*cb + ch*sb - sh*sa*sb + ch*cb ch=0 ca=1 cb=1 sh=-1 sa=0 sb=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)

aeroplane heading = 0 attitude = 90 degrees bank = 0 ch*ca -ch*sa*cb + sh*sb ch*sa*sb + sh*cb sa ca*cb -ca*sb -sh*ca sh*sa*cb + ch*sb - sh*sa*sb + ch*cb ch=1 ca=0 cb=1 sh=0 sa=1 sb=0 aeroplane heading = 90 degrees attitude = 90 degrees bank = 0 ch*ca -ch*sa*cb + sh*sb ch*sa*sb + sh*cb sa ca*cb -ca*sb -sh*ca sh*sa*cb + ch*sb - sh*sa*sb + ch*cb ch=0 ca=0 cb=1 sh=1 sa=1 sb=0 aeroplane heading = 180 degrees attitude = 90 degrees bank = 0 ch*ca -ch*sa*cb + sh*sb ch*sa*sb + sh*cb sa ca*cb -ca*sb -sh*ca sh*sa*cb + ch*sb - sh*sa*sb + ch*cb ch=-1 ca=0 cb=1 sh=0 sa=1 sb=0 aeroplane heading = -90 degrees attitude = 90 degrees bank = 0 ch*ca -ch*sa*cb + sh*sb ch*sa*sb + sh*cb sa ca*cb -ca*sb -sh*ca sh*sa*cb + ch*sb - sh*sa*sb + ch*cb ch=0 ca=0 cb=1 sh=-1 sa=1 sb=0

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

aeroplane heading = 0 attitude = -90 degrees bank = 0 ch*ca -ch*sa*cb + sh*sb ch*sa*sb + sh*cb sa ca*cb -ca*sb -sh*ca sh*sa*cb + ch*sb - sh*sa*sb + ch*cb ch=1 ca=0 cb=1 sh=0 sa=-1 sb=0 aeroplane heading = 90 degrees attitude = -90 degrees bank = 0 ch*ca -ch*sa*cb + sh*sb ch*sa*sb + sh*cb sa ca*cb -ca*sb -sh*ca sh*sa*cb + ch*sb - sh*sa*sb + ch*cb ch=0 ca=0 cb=1 sh=1 sa=-1 sb=0 aeroplane heading = 180 degrees attitude = -90 degrees bank = 0 ch*ca -ch*sa*cb + sh*sb ch*sa*sb + sh*cb sa ca*cb -ca*sb -sh*ca sh*sa*cb + ch*sb - sh*sa*sb + ch*cb ch=-1 ca=0 cb=1 sh=0 sa=-1 sb=0 aeroplane heading = -90 degrees attitude = -90 degrees bank = 0 ch*ca -ch*sa*cb + sh*sb ch*sa*sb + sh*cb sa ca*cb -ca*sb -sh*ca sh*sa*cb + ch*sb - sh*sa*sb + ch*cb ch=0 ca=0 cb=1 sh=-1 sa=-1 sb=0

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

aeroplane heading = 0 attitude = 0 bank = 90 degrees ch*ca -ch*sa*cb + sh*sb ch*sa*sb + sh*cb sa ca*cb -ca*sb -sh*ca sh*sa*cb + ch*sb - sh*sa*sb + ch*cb ch=1 ca=1 cb=0 sh=0 sa=0 sb=1 aeroplane heading = 90 degrees attitude = 0 bank = 90 degrees ch*ca -ch*sa*cb + sh*sb ch*sa*sb + sh*cb sa ca*cb -ca*sb -sh*ca sh*sa*cb + ch*sb - sh*sa*sb + ch*cb ch=0 ca=1 cb=0 sh=1 sa=0 sb=1 aeroplane heading = 180 degrees attitude = 0 bank = 90 degrees ch*ca -ch*sa*cb + sh*sb ch*sa*sb + sh*cb sa ca*cb -ca*sb -sh*ca sh*sa*cb + ch*sb - sh*sa*sb + ch*cb ch=-1 ca=1 cb=0 sh=0 sa=0 sb=1 aeroplane heading = -90 degrees attitude = 0 bank = 90 degrees ch*ca -ch*sa*cb + sh*sb ch*sa*sb + sh*cb sa ca*cb -ca*sb -sh*ca sh*sa*cb + ch*sb - sh*sa*sb + ch*cb ch=0 ca=1 cb=0 sh=-1 sa=0 sb=1

Apply bank +180 degrees:

aeroplane heading = 0 attitude = 0 bank = 180 degrees ch*ca -ch*sa*cb + sh*sb ch*sa*sb + sh*cb sa ca*cb -ca*sb -sh*ca sh*sa*cb + ch*sb - sh*sa*sb + ch*cb ch=1 ca=1 cb=-1 sh=0 sa=0 sb=0 aeroplane heading = 90 degrees attitude = 0 bank = 180 degrees ch*ca -ch*sa*cb + sh*sb ch*sa*sb + sh*cb sa ca*cb -ca*sb -sh*ca sh*sa*cb + ch*sb - sh*sa*sb + ch*cb ch=0 ca=1 cb=-1 sh=1 sa=0 sb=0 aeroplane heading = 180 degrees attitude = 0 bank = 180 degrees ch*ca -ch*sa*cb + sh*sb ch*sa*sb + sh*cb sa ca*cb -ca*sb -sh*ca sh*sa*cb + ch*sb - sh*sa*sb + ch*cb ch=-1 ca=1 cb=-1 sh=0 sa=0 sb=0 aeroplane heading = -90 degrees attitude = 0 bank = 180 degrees ch*ca -ch*sa*cb + sh*sb ch*sa*sb + sh*cb sa ca*cb -ca*sb -sh*ca sh*sa*cb + ch*sb - sh*sa*sb + ch*cb ch=0 ca=1 cb=-1 sh=-1 sa=0 sb=0

Apply bank -90 degrees:

aeroplane heading = 0 attitude = 0 bank = -90 degrees ch*ca -ch*sa*cb + sh*sb ch*sa*sb + sh*cb sa ca*cb -ca*sb -sh*ca sh*sa*cb + ch*sb - sh*sa*sb + ch*cb ch=1 ca=1 cb=0 sh=0 sa=0 sb=-1 aeroplane heading = 90 degrees attitude = 0 bank = -90 degrees ch*ca -ch*sa*cb + sh*sb ch*sa*sb + sh*cb sa ca*cb -ca*sb -sh*ca sh*sa*cb + ch*sb - sh*sa*sb + ch*cb ch=0 ca=1 cb=0 sh=1 sa=0 sb=-1 aeroplane heading = 180 degrees attitude = 0 bank = -90 degrees ch*ca -ch*sa*cb + sh*sb ch*sa*sb + sh*cb sa ca*cb -ca*sb -sh*ca sh*sa*cb + ch*sb - sh*sa*sb + ch*cb ch=-1 ca=1 cb=0 sh=0 sa=0 sb=-1 aeroplane heading = -90 degrees attitude = 0 bank = -90 degrees ch*ca -ch*sa*cb + sh*sb ch*sa*sb + sh*cb sa ca*cb -ca*sb -sh*ca sh*sa*cb + ch*sb - sh*sa*sb + ch*cb ch=0 ca=1 cb=0 sh=-1 sa=0 sb=-1

| | | | aeroplane reference orientation | aeroplane rotate by 90 degrees about x axis | aeroplane rotate by 180 degrees about x axis | aeroplane rotate by 270 degrees about x axis | | | --------------------------------------------------------------------------------------------------------------------------------------- | --------------------------------------------------------------------------------------------------------------------------------- | ---------------------------------------------------------------------------------------------------------------------------------- | --------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------- | | | | | aeroplane rotate by 90 degrees about z axis | aeroplane rotate by 90 degrees about y axis | | | | | | | aeroplane rotate by 180 degrees about z axis | | aeroplane rotate by 180 degrees about y axis | | | | | | aeroplane rotate by 270 degrees about z axis | | | aeroplane rotate by 270 degrees about y axis | | | |

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

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

This site may have errors. Don't use for critical systems.

Copyright (c) 1998-2023 Martin John Baker - All rights reserved - privacy policy.