How to use Scipy Rotation for transforming Orientations between Coordinate Systems with the help of euler angles?

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So lets assume my body has the following extrinsic orientation in the Coordinate System A:

A = [20,30,40] # extrinsic xyz in degrees

And the following Orientation in the Coordinate System B:

B = [10, 25, 50]

So the transformation from A to B is:

T = [-10, -5, 10]

So that:

B = A + T

Now I want to do the same using scipy.Rotation:

from scipy.spatial.transform import Rotation

A = Rotation.from_euler('xyz' ,[20, 30, 40], degrees=True)
B = Rotation.from_euler('xyz', [10, 25, 50], degrees=True)
T = Rotation.from_euler('xyz', [-10, -5, 10], degrees=True)
result =  A  * T # This seems to be wrong?

print(result.as_euler('xyz', degrees=True)) # Output: [14.02609598 21.61478378 48.20912092]

Where is my mistake here? What am I doint wrong. I need to use scipy rotation because, I will apply that same rotation given in euler angles on quaternions too.

scipy python
1 Answers

The transformation from A to B is incorrect. You need to be careful when considering rotations in 3D as rotations about different axes do not commute with each other.

According to your understanding, the T in the following code should get the object to [0, 0, 0]. But it doesn't.

A = Rotation.from_euler('xyz' ,[20, 30, 40], degrees=True)
T = Rotation.from_euler('xyz', [-20, -30, -40], degrees=True)
result =  A * T
print(result.as_euler('xyz', degrees=True)) 
# output: [-25.4441408    5.14593816  -4.1802616 ]

However, if you reverse the order of the rotations, you go to [0, 0, 0] as expected.

A = Rotation.from_euler('xyz' ,[20, 30, 40], degrees=True)
T = Rotation.from_euler('zyx', [-40, -30, -20], degrees=True)
result =  A * T
print(result.as_euler('xyz', degrees=True)) 

# output: [ 4.77083202e-15  0.00000000e+00 -1.98784668e-15] practically [0,0,0]

The correct transformation from A to B will be T = [-14.74053552, -1.237896, 10.10094351]. Refer to the following.

A = Rotation.from_euler('xyz' ,[20, 30, 40], degrees=True)
AToZero = Rotation.from_euler('zyx', [-40, -30, -20], degrees=True)
ZeroToB = Rotation.from_euler('xyz', [10, 25, 50], degrees=True)
T = AToZero*ZeroToB
print(T.as_euler('xyz', degrees=True))
# output: [-14.74053552  -1.237896    10.10094351]
result =  A * T
print(result.as_euler('xyz', degrees=True)) 
# ouput: [10. 25. 50.]
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