Sample Rotations
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.
In the following table we will need to know what quadrant the results are in,
so I have taken some sample results from Math.atan2
heading applied first giving 4 possible orientations:
reference orientation
angle = 0 degrees
axis = 1,0,0
qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)
q = 1 
rotate by 90 degrees about y axis
angle = 90 degrees
axis = 0,1,0
qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)
q = 0.7071 + j 0.7071

rotate by 180 degrees about y axis
angle = 180 degrees
axis = 0,1,0
qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)
q = j

rotate by 270 degrees about y axis
angle = 90 degrees
axis = 0,1,0
or
angle = 90 degrees
axis = 0,1,0
qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)
q = 0.7071  j 0.7071
(equivilant rotation to:
0.7071 + j 0.7071)

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)
angle = 90 degrees
axis = 0,0,1
qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)
q = 0.7071 + k 0.7071

angle = 120 degrees
axis = 0.5774,0.5774,0.5774
qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)
q = 0.5 + i 0.5 + j 0.5 + k 0.5

angle = 180 degrees
axis = 0.7071,0.7071,0
qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)
q = i 0.7071 +j 0.7071

angle = 120 degrees
axis = 0.5774,0.5774,0.5774
qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)
q = 0.5  i 0.5  j 0.5 + k 0.5

Or instead apply attitude 90 degrees (also a singularity):
angle = 90 degrees
axis = 0,0,1
(equivilant rotation to:
angle = 90 degrees
axis = 0,0,1)
qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)
q = 0.7071  k 0.7071
(equivilant rotation to:
0.7071 + k 0.7071) 
angle = 120 degrees
axis = 0.5774,0.5774,0.5774
qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)
q = 0.5  i 0.5 + j 0.5  k 0.5 
angle = 180 degrees
axis = 0.7071,0.7071,0
qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)
q = i 0.7071 + j 0.7071
(equivilant rotation to:
i 0.7071  j 0.7071) 
angle = 120 degrees
axis = 0.5774,0.5774,0.5774
qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)
q = 0.5 + i 0.5  j 0.5  k 0.5 
Normally we dont go beond attitude + or  90 degrees because thes are singularities,
instead apply bank +90 degrees:
angle = 90 degrees
axis = 1,0,0
qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)
q = 0.7071 + i 0.7071 
angle = 120 degrees
axis = 0.5774,0.5774,0.5774
qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)
q = 0.5 + i 0.5 + j 0.5  k 0.5 
angle = 180 degrees
axis = 0,0.7071,0.7071
qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)
q = j 0.7071  k 0.7071 
angle = 120 degrees
axis = 0.5774,0.5774,0.5774
qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)
q = 0.5 + i 0.5  j 0.5 + k 0.5 
Apply bank +180 degrees:
angle = 180 degrees
axis = 1,0,0
qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)
q = i

angle = 180 degrees
axis = 0.7071,0,0.7071
qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)
q = i 0.7071  k 0.7071

angle = 180 degrees
axis = 0,0,1
qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)
q = k 
angle = 180 degrees
axis = 0.7071,0,0.7071
qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)
q = i 0.7071 + k 0.7071 
Apply bank 90 degrees:
angle = 90 degrees
axis = 1,0,0
(equivilant rotation to:
angle = 90 degrees
axis = 1,0,0)
qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)
q = 0.7071  i 0.7071
(equivilant rotation to:
0.7071 + i 0.7071) 
angle = 120 degrees
axis = 0.5774,0.5774,0.5774
qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)
q = 0.5  i 0.5 + j 0.5 + k 0.5 
angle = 180 degrees
axis = 0,0.7071,0.7071
qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)
q = j 0.7071 + k 0.7071 
angle = 120 degrees
axis = 0.5774,0.5774,0.5774
qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)
q = 0.5  i 0.5  j 0.5  k 0.5 
This site may have errors. Don't use for critical systems.