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:
rightUp
reference orientation
heading = 0
attitude = 0
bank = 0
ch=1
ca=1
cb=1
sh=0
sa=0
sb=0
backUp
rotate by 90 degrees about y axis
heading = 90 degrees
attitude = 0
bank = 0
ch=0
ca=1
cb=1
sh=1
sa=0
sb=0
leftUp
rotate by 180 degrees about y axis
heading = 180 degrees
attitude = 0
bank = 0
ch=-1
ca=1
cb=1
sh=0
sa=0
sb=0
forwardUp
rotate by 270 degrees about y axis
heading = -90 degrees
attitude = 0
bank = 0
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)
upLeft
heading = 0
attitude = 90 degrees
bank = 0
ch=1
ca=0
cb=1
sh=0
sa=1
sb=0
heading = 90 degrees
attitude = 90 degrees
bank = 0
ch=0
ca=0
cb=1
sh=1
sa=1
sb=0
upRight
heading = 180 degrees
attitude = 90 degrees
bank = 0
ch=-1
ca=0
cb=1
sh=0
sa=1
sb=0
heading = -90 degrees
attitude = 90 degrees
bank = 0
ch=0
ca=0
cb=1
sh=-1
sa=1
sb=0
Or instead apply attitude -90 degrees (also a singularity):
downRight
heading = 0
attitude = -90 degrees
bank = 0
ch=1
ca=0
cb=1
sh=0
sa=-1
sb=0
heading = 90 degrees
attitude = -90 degrees
bank = 0
ch=0
ca=0
cb=1
sh=1
sa=-1
sb=0
downLeft
heading = 180 degrees
attitude = -90 degrees
bank = 0
ch=-1
ca=0
cb=1
sh=0
sa=-1
sb=0
heading = -90 degrees
attitude = -90 degrees
bank = 0
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:
heading = 0
attitude = 0
bank = 90 degrees
ch=1
ca=1
cb=0
sh=0
sa=0
sb=1
backRight
heading = 90 degrees
attitude = 0
bank = 90 degrees
ch=0
ca=1
cb=0
sh=1
sa=0
sb=1
heading = 180 degrees
attitude = 0
bank = 90 degrees
ch=-1
ca=1
cb=0
sh=0
sa=0
sb=1
forwardLeft
heading = -90 degrees
attitude = 0
bank = 90 degrees
ch=0
ca=1
cb=0
sh=-1
sa=0
sb=1
Apply bank +180 degrees:
rightDown
heading = 0
attitude = 0
bank = 180 degrees
ch=1
ca=1
cb=-1
sh=0
sa=0
sb=0
backDown
heading = 90 degrees
attitude = 0
bank = 180 degrees
ch=0
ca=1
cb=-1
sh=1
sa=0
sb=0
leftDown
heading = 180 degrees
attitude = 0
bank = 180 degrees
ch=-1
ca=1
cb=-1
sh=0
sa=0
sb=0
forwardDown
heading = -90 degrees
attitude = 0
bank = 180 degrees
ch=0
ca=1
cb=-1
sh=-1
sa=0
sb=0
Apply bank -90 degrees:
heading = 0
attitude = 0
bank = -90 degrees
ch=1
ca=1
cb=0
sh=0
sa=0
sb=-1
backLeft
heading = 90 degrees
attitude = 0
bank = -90 degrees
ch=0
ca=1
cb=0
sh=1
sa=0
sb=-1
heading = 180 degrees
attitude = 0
bank = -90 degrees
ch=-1
ca=1
cb=0
sh=0
sa=0
sb=-1
forwardRight
heading = -90 degrees
attitude = 0
bank = -90 degrees
ch=0
ca=1
cb=0
sh=-1
sa=0
sb=-1
Book Shop - Further reading.
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cover us uk de jp fr caVisualizing Quaternions by Andrew J. Hanson
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