Quaternion Order Conventions

Quaternions are expressions of the form w + x i + y j + z k, with x i + y j + z k being the vector/imaginary part and w the scalar/real part. Math libraries differ in the way they order the vector and real part of a quaternion. This can lead to bothersome debugging sessions.

The following table shows quaternion implementations and their chosen order:
X-Y-Z-W 🟥 (scalar-last format)   vs.   W-X-Y-Z 🟦

Library / Framework	Quaternion Format
autolab_core	W-X-Y-Z 🟦
Blender.mathutils	W-X-Y-Z 🟦
Boost	W-X-Y-Z 🟦
Bullet / PyBullet	X-Y-Z-W 🟥
DART (uses Eigen::Quaternion)	W-X-Y-Z 🟦
Drake (uses Eigen::Quaternion)	W-X-Y-Z 🟦
differentiable-robot-model	X-Y-Z-W 🟥
dm_robotics.transformations	W-X-Y-Z 🟦
Eigen - API	W-X-Y-Z 🟦
Eigen - Internal memory ordering	X-Y-Z-W 🟥
FleX	X-Y-Z-W 🟥
iDynTree	W-X-Y-Z 🟦
Isaac Gym	X-Y-Z-W 🟥
Isaac Lab	W-X-Y-Z 🟦
Isaac Sim	W-X-Y-Z 🟦
lietorch	X-Y-Z-W 🟥
jaxlie	W-X-Y-Z 🟦
Klampt	W-X-Y-Z 🟦
MATLAB	W-X-Y-Z 🟦
MuJoCo	W-X-Y-Z 🟦
Nuklei	W-X-Y-Z 🟦
numpy-quaternions	W-X-Y-Z 🟦
OpenRAVE 0.9.0	X-Y-Z-W 🟥
Orocos KDL	X-Y-Z-W 🟥
PhysX	X-Y-Z-W 🟥
PyMesh	W-X-Y-Z 🟦
pyquaternion	W-X-Y-Z 🟦
pyrr	X-Y-Z-W 🟥
pytorch3d	W-X-Y-Z 🟦
pytransform3d	W-X-Y-Z 🟦
quaternions	W-X-Y-Z 🟦
RoMa	X-Y-Z-W 🟥
ROS	X-Y-Z-W 🟥
ROS2 (tf2)	X-Y-Z-W 🟥
scipy.spatial.transforms	X-Y-Z-W 🟥
spatialmath-python	W-X-Y-Z 🟦
squaternion	W-X-Y-Z 🟦
The Library for Uniform Deterministic Sequences and Sets of Samples over 2-sphere and SO(3)	X-Y-Z-W 🟥
Theseus	W-X-Y-Z 🟦
transformations.py (<=2009.04.18)	X-Y-Z-W 🟥
transformations.py (>=2010.05.10, includes trimesh.transformations)	W-X-Y-Z 🟦
Transforms3d	W-X-Y-Z 🟦
tinyquaternion	W-X-Y-Z 🟦
USD (Pixar) - API	W-X-Y-Z 🟦
USD (Pixar) - Internal memory ordering	X-Y-Z-W 🟥
ViSII - A VIrtual Scene Imaging Interface	W-X-Y-Z 🟦
NVIDIA Warp	X-Y-Z-W 🟥
YARP - API	X-Y-Z-W 🟥
YARP - Internal memory ordering	W-X-Y-Z 🟦