Maths - Quaternion to Axis-Angle - 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

q = 1

sqrt(1-qw*qw) = 0

to avoid divide by zero just set axis to arbitary value (does not matter as angle is zero):

angle =0
x = 1
y = 0
z = 0

angle = 0 degrees
axis = 1,0,0

backUp

rotate by 90 degrees about y axis

q = 0.7071 + j 0.7071

sqrt(1-qw*qw) = 0.7071

angle = 2 * acos(qw) = 45*2=90degrees
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

angle = 90 degrees
axis = 0,1,0

 

leftUp

rotate by 180 degrees about y axis

q = j

sqrt(1-qw*qw) = 1

angle = 2 * acos(qw) = 2*90=180deg
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

angle = 180 degrees
axis = 0,1,0

forwardUp

rotate by 270 degrees about y axis

 

q = 0.7071 - j 0.7071

(equivilant rotation to:
-0.7071 + j 0.7071)

sqrt(1-qw*qw) = 0.7071

angle = 2 * acos(qw) = 45*2=90degrees
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

angle = 90 degrees
axis = 0,-1,0

or

angle = -90 degrees
axis = 0,1,0

 

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

q = 0.7071 + k 0.7071

sqrt(1-qw*qw) = 0.7071

angle = 2 * acos(qw) = 45*2=90degrees
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

angle = 90 degrees
axis = 0,0,1

upForward

q = 0.5 + i 0.5 + j 0.5 + k 0.5

sqrt(1-qw*qw) = 0.866

angle = 2 * acos(qw) = 2*60=120deg
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

angle = 120 degrees
axis = 0.5774,0.5774,0.5774

upRight

q = i 0.7071 +j 0.7071

sqrt(1-qw*qw) = 1

angle = 2 * acos(qw) = 2*90=180deg
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

angle = 180 degrees
axis = 0.7071,0.7071,0

upBack

q = 0.5 - i 0.5 - j 0.5 + k 0.5

sqrt(1-qw*qw) = 0.866

angle = 2 * acos(qw) = 2*60=120deg
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

angle = 120 degrees
axis = -0.5774,-0.5774,0.5774

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

downRight

q = 0.7071 - k 0.7071

(equivilant rotation to:
-0.7071 + k 0.7071)

sqrt(1-qw*qw) = 0.7071

angle = 2 * acos(qw) = 45*2=90degrees
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

angle = 90 degrees
axis = 0,0,-1

(equivilant rotation to:
angle = -90 degrees
axis = 0,0,1)

downBack

q = 0.5 - i 0.5 + j 0.5 - k 0.5

sqrt(1-qw*qw) = 0.866

angle = 2 * acos(qw) = 2*60=120deg
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

angle = 120 degrees
axis = -0.5774,0.5774,-0.5774

downLeft

q = -i 0.7071 + j 0.7071

(equivilant rotation to:
i 0.7071 - j 0.7071)

sqrt(1-qw*qw) = 1

angle = 2 * acos(qw) = 2*90=180deg
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

angle = 180 degrees
axis = -0.7071,0.7071,0

downForward

q = 0.5 + i 0.5 - j 0.5 - k 0.5

sqrt(1-qw*qw) = 0.866

angle = 2 * acos(qw) = 2*60=120deg
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

 

angle = 120 degrees
axis = 0.5774,-0.5774,-0.5774

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

rightForward

q = 0.7071 + i 0.7071

sqrt(1-qw*qw) = 0.7071

angle = 2 * acos(qw) = 45*2=90degrees
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

angle = 90 degrees
axis = 1,0,0

backRight

q = 0.5 + i 0.5 + j 0.5 - k 0.5

sqrt(1-qw*qw) = 0.866

angle = 2 * acos(qw) = 2*60=120deg
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

angle = 120 degrees
axis = 0.5774,0.5774,-0.5774

leftBack

q = j 0.7071 - k 0.7071

sqrt(1-qw*qw) = 1

angle = 2 * acos(qw) = 2*90=180deg
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

angle = 180 degrees
axis = 0,0.7071,-0.7071

forwardLeft

q = 0.5 + i 0.5 - j 0.5 + k 0.5

sqrt(1-qw*qw) = 0.866

angle = 2 * acos(qw) = 2*60=120deg
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

angle = 120 degrees
axis = 0.5774,-0.5774,0.5774

Apply bank +180 degrees:

rightDown

q = i

sqrt(1-qw*qw) = 1

angle = 2 * acos(qw) = 2*90=180deg
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

angle = 180 degrees
axis = 1,0,0

 

backDown

q = i 0.7071 - k 0.7071

sqrt(1-qw*qw) = 1

angle = 2 * acos(qw) = 2*90=180deg
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

angle = 180 degrees
axis = 0.7071,0,-0.7071

leftDown

q = k

sqrt(1-qw*qw) = 1

angle = 2 * acos(qw) = 2*90=180deg
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

angle = 180 degrees
axis = 0,0,1

forwardDown

q = i 0.7071 + k 0.7071

sqrt(1-qw*qw) = 1

angle = 2 * acos(qw) = 2*90=180deg
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

angle = 180 degrees
axis = 0.7071,0,0.7071

Apply bank -90 degrees:

rightBack

q = 0.7071 - i 0.7071

(equivilant rotation to:
-0.7071 + i 0.7071)

sqrt(1-qw*qw) = 0.7071

angle = 2 * acos(qw) = 45*2=90degrees
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

angle = 90 degrees
axis = -1,0,0

(equivilant rotation to:
angle = -90 degrees
axis = 1,0,0)

backLeft

q = 0.5 - i 0.5 + j 0.5 + k 0.5

sqrt(1-qw*qw) = 0.866

angle = 2 * acos(qw) = 2*60=120deg
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

angle = 120 degrees
axis = -0.5774,0.5774,0.5774

leftForward

q = j 0.7071 + k 0.7071

sqrt(1-qw*qw) = 1

angle = 2 * acos(qw) = 2*90=180deg
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

angle = 180 degrees
axis = 0,0.7071,0.7071

forwardRight

q = 0.5 - i 0.5 - j 0.5 - k 0.5

sqrt(1-qw*qw) = 0.866

angle = 2 * acos(qw) = 2*60=120deg
x = qx / sqrt(1-qw*qw) =
y = qy / sqrt(1-qw*qw) =
z = qz / sqrt(1-qw*qw) =

angle = 120 degrees
axis = -0.5774,-0.5774,-0.5774


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