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Maths - Rotations using Matrix - Samples in 90 degree steps

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:

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

 

1 0 0
0 1 0
0 0 1

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

 

0 0 1
0 1 0
-1 0 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

 

-1 0 0
0 1 0
0 0 -1

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

 

0 0 -1
0 1 0
1 0 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

 

0 -1 0
1 0 0
0 0 1

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

 

0 0 1
1 0 0
0 1 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

0 1 0
1 0 0
0 0 -1

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

 

0 0 -1
1 0 0
0 -1 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

 

0 1 0
-1 0 0
0 0 1

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

 

0 0 1
-1 0 0
0 -1 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

 

0 -1 0
-1 0 0
0 0 -1

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

 

0 0 -1
-1 0 0
0 1 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

 

1 0 0
0 0 -1
0 1 0

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

 

0 1 0
0 0 -1
-1 0 0

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

-1 0 0
0 0 -1
0 -1 0

 

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

 

0 -1 0
0 0 -1
1 0 0

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

 

1 0 0
0 -1 0
0 0 -1

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

 

0 0 -1
0 -1 0
-1 0 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

 

-1 0 0
0 -1 0
0 0 1

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

 

0 0 1
0 -1 0
1 0 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

 

1 0 0
0 0 1
0 -1 0

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

 

0 -1 0
0 0 1
-1 0 0

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

-1 0 0
0 0 1
0 1 0

 

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

 

0 1 0
0 0 1
1 0 0

 

 

     

aeroplane

reference orientation

1 0 0
0 1 0
0 0 1

 

aeroplane

rotate by 90 degrees about x axis

1 0 0
0 0 -1
0 1 0

 

aeroplane

rotate by 180 degrees about x axis

1 0 0
0 -1 0
0 0 -1

 

aeroplane

rotate by 270 degrees about x axis

1 0 0
0 0 1
0 -1 0

 
   

aeroplane

rotate by 90 degrees about z axis

0 1 0
-1 0 0
0 0 1

 

aeroplane

rotate by 90 degrees about y axis

0 0 -1
0 1 0
1 0 0

 

      
 

aeroplane

rotate by 180 degrees about z axis

-1 0 0
0 -1 0
0 0 1

 

  

aeroplane

rotate by 180 degrees about y axis

-1 0 0
0 1 0
0 0 -1

 

      

aeroplane

rotate by 270 degrees about z axis

0 -1 0
1 0 0
0 0 1

 

    

aeroplane

rotate by 270 degrees about y axis

0 0 1
0 1 0
-1 0 0

 

      

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

aeroplane

1 0 0
0 1 0
0 0 1

 

aeroplane

1 0 0
0 0 -1
0 1 0

 

aeroplane

1 0 0
0 -1 0
0 0 -1

 

aeroplane

1 0 0
0 0 1
0 -1 0

 

aeroplane

0 -1 0
1 0 0
0 0 1

 

aeroplane

0 0 1
1 0 0
0 1 0

 

aeroplane

0 1 0
1 0 0
0 0 -1

aeroplane

0 0 -1
1 0 0
0 -1 0

 

aeroplane

-1 0 0
0 -1 0
0 0 1

 

aeroplane

-1 0 0
0 0 -1
0 -1 0

 

aeroplane

-1 0 0
0 1 0
0 0 -1

 

aeroplane

-1 0 0
0 0 1
0 1 0

 

aeroplane

0 1 0
-1 0 0
0 0 1

 

aeroplane

0 0 1
-1 0 0
0 -1 0

 

aeroplane

0 -1 0
-1 0 0
0 0 -1

 

aeroplane

0 0 -1
-1 0 0
0 1 0

 

aeroplane

0 0 -1
0 1 0
1 0 0

 

 aeroplane
0 1 0
0 0 1
1 0 0

 

 aeroplane
0 0 1
0 -1 0
1 0 0

 

 aeroplane
0 -1 0
0 0 -1
1 0 0

 

aeroplane

0 0 -1
0 -1 0
-1 0 0

 

 aeroplane
0 -1 0
0 0 1
-1 0 0

 

aeroplane

0 0 1
0 1 0
-1 0 0

 

 

 aeroplane
0 1 0
0 0 -1
-1 0 0

 

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.

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