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Maths - Matrix to Euler - Sample Orientations

Sample Rotations

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:

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

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

atan2(0.0,0.0)=0.0 (although Math.atan2 returns 0 this is abitary and if both x and y is zero we need to find some other way to get to value)
atan2(0.0,1.0)=0.0
atan2(1.0,1.0)=0.7853981633974483
atan2(1.0,0.0)=1.5707963267948966
atan2(1.0,-1.0)=2.356194490192345
atan2(0.0,-1.0)=3.141592653589793
atan2(-1.0,-1.0)=-2.356194490192345
atan2(-1.0,0.0)=-1.5707963267948966
atan2(-1.0,1.0)=-0.7853981633974483
heading applied first giving 4 possible orientations:

rightUp

reference orientation

1 0 0
0 1 0
0 0 1

heading = atan2(-m20,m00)
attitude = asin(m10)

bank = atan2(-m12,m11)

heading = atan2(0,1)
attitude = asin(0)
bank = atan2(0,1)

heading = 0
attitude = 0
bank = 0

backUp

rotate by 90 degrees about y axis

0 0 1
0 1 0
-1 0 0

heading = atan2(-m20,m00)
attitude = asin(m10)
bank = atan2(-m12,m11)

heading = atan2(1,0)
attitude = asin(0)
bank = atan2(0,1)

heading = 90 degrees
attitude = 0
bank = 0

leftUp

rotate by 180 degrees about y axis

-1 0 0
0 1 0
0 0 -1

heading = atan2(-m20,m00)
attitude = asin(m10)
bank = atan2(-m12,m11)

heading = atan2(0,-1)
attitude = asin(0)
bank = atan2(0,1)

heading = 180 degrees
attitude = 0
bank = 0

forwardUp

rotate by 270 degrees about y axis

0 0 -1
0 1 0
1 0 0

heading = atan2(-m20,m00)
attitude = asin(m10)
bank = atan2(-m12,m11)

heading = atan2(-1,0)
attitude = asin(0)
bank = atan2(0,1)

heading = -90 degrees
attitude = 0
bank = 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

0 -1 0
1 0 0
0 0 1

heading = atan2(-m20,m00)
attitude = asin(m10)
bank = atan2(-m12,m11)

heading = atan2(0,0)
attitude = asin(1)
bank = atan2(0,0)

since M10=1 (north pole)
we use this instead:
heading = atan2(M02,M22)
= atan2(0,1)
bank = 0

heading = 0
attitude = 90 degrees
bank = 0

 

 

upForward

0 0 1
1 0 0
0 1 0

heading = atan2(-m20,m00)
attitude = asin(m10)
bank = atan2(-m12,m11)

heading = atan2(0,0)
attitude = asin(1)
bank = atan2(0,0)

since M10=1 (north pole)
we use this instead:
heading = atan2(M02,M22)
= atan2(1,0)
bank = 0

heading = 90 degrees
attitude = 90 degrees
bank = 0

 

 

upRight

0 1 0
1 0 0
0 0 -1

heading = atan2(-m20,m00)
attitude = asin(m10)
bank = atan2(-m12,m11)

heading = atan2(0,0)
attitude = asin(1)
bank = atan2(0,0)

since M10=1 (north pole)
we use this instead:
heading = atan2(M02,M22)
= atan2(0,-1)
bank = 0

heading = 180 degrees
attitude = 90 degrees
bank = 0

 

upBack

0 0 -1
1 0 0
0 -1 0

heading = atan2(-m20,m00)
attitude = asin(m10)
bank = atan2(-m12,m11)

heading = atan2(0,0)
attitude = asin(1)
bank = atan2(0,0)

since M10=1 (north pole)
we use this instead:
heading = atan2(M02,M22)
= atan2(-1,0)
bank = 0

heading = -90 degrees
attitude = 90 degrees
bank = 0

 

 

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

downRight

0 1 0
-1 0 0
0 0 1

heading = atan2(-m20,m00)
attitude = asin(m10)
bank = atan2(-m12,m11)

heading = atan2(0,0)
attitude = asin(-1)
bank = atan2(0,0)

since M10=-1 (south pole)
which gives:
heading = atan2(M02,M22)
= atan2(0,1)
bank = 0

heading = 0
attitude = -90 degrees
bank = 0

 

 

downBack

0 0 1
-1 0 0
0 -1 0

heading = atan2(-m20,m00)
attitude = asin(m10)
bank = atan2(-m12,m11)

heading = atan2(0,0)
attitude = asin(-1)
bank = atan2(0,0)

since M10=-1 (south pole)
we use this instead:
heading = atan2(M02,M22)
= atan2(1,0)
bank = 0


heading = 90 degrees
attitude = -90 degrees
bank = 0

 

 

downLeft

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

heading = atan2(-m20,m00)
attitude = asin(m10)
bank = atan2(-m12,m11)

heading = atan2(0,0)
attitude = asin(-1)
bank = atan2(0,0)

since M10=-1 (south pole)
we use this instead:
heading = atan2(M02,M22)
= atan2(0,-1)
bank = 0


heading = 180 degrees
attitude = -90 degrees
bank = 0

 

 

downForward

0 0 -1
-1 0 0
0 1 0

heading = atan2(-m20,m00)
attitude = asin(m10)
bank = atan2(-m12,m11)

heading = atan2(0,0)
attitude = asin(-1)
bank = atan2(0,0)

since M10=-1 (south pole)
we use this instead:
heading = atan2(M02,M22)
= atan2(-1,0)
bank = 0


heading = -90 degrees
attitude = -90 degrees
bank = 0

 

 

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

rightForward

1 0 0
0 0 -1
0 1 0

heading = atan2(-m20,m00)
attitude = asin(m10)
bank = atan2(-m12,m11)

heading = atan2(0,1)
attitude = asin(0)
bank = atan2(1,0)

heading = 0
attitude = 0
bank = 90 degrees

 

 

backRight

0 1 0
0 0 -1
-1 0 0

heading = atan2(-m20,m00)
attitude = asin(m10)
bank = atan2(-m12,m11)

heading = atan2(1,0)
attitude = asin(0)
bank = atan2(1,0)

heading = 90 degrees
attitude = 0
bank = 90 degrees

 

 

leftBack

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

heading = atan2(-m20,m00)
attitude = asin(m10)
bank = atan2(-m12,m11)

heading = atan2(0,-1)
attitude = asin(0)
bank = atan2(1,0)

heading = 180 degrees
attitude = 0
bank = 90 degrees

 

 

forwardLeft

0 -1 0
0 0 -1
1 0 0

heading = atan2(-m20,m00)
attitude = asin(m10)
bank = atan2(-m12,m11)

heading = atan2(-1,0)
attitude = asin(0)
bank = atan2(1,0)

heading = -90 degrees
attitude = 0
bank = 90 degrees

 

 

Apply bank +180 degrees:

rightDown

1 0 0
0 -1 0
0 0 -1

heading = atan2(-m20,m00)
attitude = asin(m10)
bank = atan2(-m12,m11)

heading = atan2(0,1)
attitude = asin(0)
bank = atan2(0,-1)

heading = 0
attitude = 0
bank = 180 degrees

 

 

backDown

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

heading = atan2(-m20,m00)
attitude = asin(m10)
bank = atan2(-m12,m11)

heading = atan2(1,0)
attitude = asin(0)
bank = atan2(0,-1)

heading = 90 degrees
attitude = 0
bank = 180 degrees

 

 

leftDown

-1 0 0
0 -1 0
0 0 1

heading = atan2(-m20,m00)
attitude = asin(m10)
bank = atan2(-m12,m11)

heading = atan2(0,-1)
attitude = asin(0)
bank = atan2(0,-1)

heading = 180 degrees
attitude = 0
bank = 180 degrees

 

 

forwardDown

0 0 1
0 -1 0
1 0 0

heading = atan2(-m20,m00)
attitude = asin(m10)
bank = atan2(-m12,m11)

heading = atan2(-1,0)
attitude = asin(0)
bank = atan2(0,-1)

heading = -90 degrees
attitude = 0
bank = 180 degrees

 

 

Apply bank -90 degrees:

rightBack

1 0 0
0 0 1
0 -1 0

 

heading = atan2(-m20,m00)
attitude = asin(m10)
b ank = atan2(-m12,m11)

heading = atan2(0,1)
attitude = asin(0)
bank = atan2(-1,0)

heading = 0
attitude = 0
bank = -90 degrees

 

 

backLeft

0 -1 0
0 0 1
-1 0 0

heading = atan2(-m20,m00)
attitude = asin(m10)
bank = atan2(-m12,m11)

heading = atan2(1,0)
attitude = asin(0)
bank = atan2(-1,0)

heading = 90 degrees
attitude = 0
bank = -90 degrees

 

 

leftForward

-1 0 0
0 0 1
0 1 0

heading = atan2(-m20,m00)
attitude = asin(m10)
bank = atan2(-m12,m11)

heading = atan2(0,-1)
attitude = asin(0)
bank = atan2(-1,0)

heading = 180 degrees
attitude = 0
bank = -90 degrees

 

 

forwardRight

0 1 0
0 0 1
1 0 0

heading = atan2(-m20,m00)
attitude = asin(m10)
bank = atan2(-m12,m11)

heading = atan2(-1,0)
attitude = asin(0)
bank = atan2(-1,0)

heading = -90 degrees
attitude = 0
bank = -90 degrees

 

 
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