Maths - Axis-Angle to Quaternion - Sample Orientations

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:

rightUp

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

backUp

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

 

leftUp

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

 

forwardUp

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)

upLeft

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

 

upForward
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

 

upRight

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

 

upBack
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):

downRight

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)

downBack

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

downLeft

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)

downForward

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:

rightForward


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

backRight
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

leftBack

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

forwardLeft

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:

rightDown

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

 

backDown

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

 

leftDown

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

forwardDown

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:

rightBack


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)

backLeft

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

leftForward

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

forwardRight

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
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