How to convert Euler angles to Quaternions and get the same Euler angles back from Quaternions?

Question

I am rotating n 3D shape using Euler angles in the order of XYZ meaning that the object is first rotated along the X axis, then Y and then Z. I want to convert the Euler angle to Quaternion and then get the same Euler angles back from the Quaternion using some [preferably] Python code or just some pseudocode or algorithm. Below, I have some code that converts Euler angle to Quaternion and then converts the Quaternion to get Euler angles. However, this does not give me the same Euler angles.

I think the problem is I don't know how to associate yaw, pitch and roll to X, Y an Z axes. Also, I don't know how to change order of conversions in the code to correctly convert the Euler angles to Quaternion and then convert the Quaternion to Euler angle so that I am able to get the same Euler angle back. Can someone help me with this?

And here's the code I used:

This function converts Euler angles to Quaternions:

def euler_to_quaternion(yaw, pitch, roll):

        qx = np.sin(roll/2) * np.cos(pitch/2) * np.cos(yaw/2) - np.cos(roll/2) * np.sin(pitch/2) * np.sin(yaw/2)
        qy = np.cos(roll/2) * np.sin(pitch/2) * np.cos(yaw/2) + np.sin(roll/2) * np.cos(pitch/2) * np.sin(yaw/2)
        qz = np.cos(roll/2) * np.cos(pitch/2) * np.sin(yaw/2) - np.sin(roll/2) * np.sin(pitch/2) * np.cos(yaw/2)
        qw = np.cos(roll/2) * np.cos(pitch/2) * np.cos(yaw/2) + np.sin(roll/2) * np.sin(pitch/2) * np.sin(yaw/2)

        return [qx, qy, qz, qw]

And this converts Quaternions to Euler angles:

def quaternion_to_euler(x, y, z, w):

        import math
        t0 = +2.0 * (w * x + y * z)
        t1 = +1.0 - 2.0 * (x * x + y * y)
        X = math.degrees(math.atan2(t0, t1))

        t2 = +2.0 * (w * y - z * x)
        t2 = +1.0 if t2 > +1.0 else t2
        t2 = -1.0 if t2 < -1.0 else t2
        Y = math.degrees(math.asin(t2))

        t3 = +2.0 * (w * z + x * y)
        t4 = +1.0 - 2.0 * (y * y + z * z)
        Z = math.degrees(math.atan2(t3, t4))

        return X, Y, Z

And I use them as follow:

import numpy as np
euler_Original = np.random.random(3) * 360).tolist() # Generate random rotation angles for XYZ within the range [0, 360)
quat = euler_to_quaternion(euler_Original[0], euler_Original[1], euler_Original[2]) # Convert to Quaternion
newEulerRot = quaternion_to_euler(quat[0], quat[1], quat[2], quat[3]) #Convert the Quaternion to Euler angles

print (euler_Original)
print (newEulerRot)

The print statements print different numbers for euler_Original and newEulerRot which I don't want to be the case. For example if euler_original contains numbers like (0.2, 1.12, 2.31) in radians I get this Quaternion --> [0.749, 0.290, -0.449, 0.389] and converting the Quaternion to Euler angles gives me this --> (132.35, 64.17, 11.45) which is pretty wrong. I wonder how I can fix this?

Although I'm interested in getting the above code to work by making changes to it but, I would rather learn how to set up the equations correctly. This way I would know how I can get the correct Quaternions even if the order of rotations (XYZ --> YZX etc) for applying Euler angles is changed.

Answer 1

We can use Rotation from scipy.spatial.transform.

from scipy.spatial.transform import Rotation

# Create a rotation object from Euler angles specifying axes of rotation
rot = Rotation.from_euler('xyz', [90, 45, 30], degrees=True)

# Convert to quaternions and print
rot_quat = rot.as_quat()
print(rot_quat)

The result would be:

[ 0.56098553  0.43045933 -0.09229596  0.70105738]

Then, you can also get it back in Euler angles:

print(rot.as_euler('xyz', degrees=True))

Which results in:

[90. 45. 30.]

As a final check, create a rotation object from the quaternions calculated above and get it as Euler angles:

rot = Rotation.from_quat(rot_quat)

# Convert the rotation to Euler angles given the axes of rotation
print(rot.as_euler('xyz', degrees=True))

Which results in:

[90. 45. 30.]