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
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