Fixing Inconsistent Euler to Quaternion Conversion in Python with SciPy

Learn why Euler to quaternion conversions might be inconsistent in Python's SciPy and how to achieve consistent results by using consistent rotation conventions.
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This video is based on the question https://stackoverflow.com/q/79420684/ asked by the user 'Kong' ( https://stackoverflow.com/u/6467567/ ) and on the answer https://stackoverflow.com/a/79420725/ provided by the user 'Upwind Tech' ( https://stackoverflow.com/u/23298550/ ) at 'Stack Overflow' website. Thanks to these great users and Stackexchange community for their contributions.

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Introduction

Converting between Euler angles and quaternions is a common task in 3D rotations, robotics, and computer graphics. However, a frequent source of confusion is why converting Euler angles to quaternions and then back doesn't yield the original Euler angles.

This brief guide addresses that confusion using Python's SciPy library.

The Problem

Here's a common scenario:

You start with Euler angles (e.g., in degrees).

Convert them to quaternions.

Convert those quaternions back to Euler angles.

Unexpectedly, the output Euler angles are not the same as the input.

This happens because Euler angle rotations depend on the order and type of rotations:

Intrinsic rotations: axes move with the object.

Extrinsic rotations: axes fixed in the original reference frame.

Mixing these conventions leads to inconsistent results.

Understanding SciPy's Behavior

In SciPy's Rotation module:

Using uppercase axes labels like 'ZYX' implies intrinsic rotations.

Lowercase labels like 'zyx' imply extrinsic rotations.

If you convert Euler to quaternion with an intrinsic convention but try to get Euler angles back with an extrinsic one, the results vary unexpectedly.

The Solution

To get consistent round-trip conversions:

Use the same case and order in both conversions. For example, if you convert Euler angles to quaternions using 'ZYX', then convert back using 'ZYX' as well.

Corrected code snippet example:

[[See Video to Reveal this Text or Code Snippet]]

Additional Tips

Note that quaternions from SciPy are returned in [x, y, z, w] format.

Use R.from_quat() accordingly and avoid manual rolling unless necessary.

Minor numerical differences after round-trip conversions are expected due to floating-point precision.

Beware of gimbal lock situations where Euler angles may not be unique.

Summary

Inconsistent Euler to quaternion and back conversions typically stem from mixing intrinsic and extrinsic rotations by using different case strings (uppercase vs lowercase) for axes in SciPy's Rotation functions.

Always ensure consistent rotation conventions when converting both ways to avoid unexpected angle changes.

Using this approach guarantees logical, reproducible rotations and simplifies debugging.