Conversion between quaternions and Euler angles

Spatial rotations in three dimensions can be parametrized using both Euler angles and unit quaternions. This article explains how to convert between the two representations. Actually this simple use of "quaternions" was first presented by Euler some seventy years earlier than Hamilton to solve the problem of magic squares. For this reason the dynamics community commonly refers to quaternions in this application as "Euler parameters".
Contents
Definition
A unit quaternion can be described as:
We can associate a quaternion with a rotation around an axis by the following expression
where α is a simple rotation angle (the value in radians of the angle of rotation) and cos(β_{x}), cos(β_{y}) and cos(β_{z}) are the "direction cosines" locating the axis of rotation (Euler's Theorem).
Rotation matrices
The orthogonal matrix (postmultiplying a column vector) corresponding to a clockwise/lefthanded rotation by the unit quaternion q = q_{0} + iq_{1} + jq_{2} + kq_{3} is given by the inhomogeneous expression
or equivalently, by the homogeneous expression
If q_{0} + iq_{1} + jq_{2} + kq_{3} is not a unit quaternion then the homogeneous form is still a scalar multiple of a rotation matrix, while the inhomogeneous form is in general no longer an orthogonal matrix. This is why in numerical work the homogeneous form is to be preferred if distortion is to be avoided.
The orthogonal matrix (postmultiplying a column vector) corresponding to a clockwise/lefthanded rotation with Euler angles φ, θ, ψ, with xyz convention, is given by:
Conversion
By combining the quaternion representations of the Euler rotations we get
For Euler angles we get:
arctan and arcsin have a result between −π/2 and π/2. With three rotation between −π/2 and π/2 you can't have all possible orientations. You need to replace the arctan by atan2 to generate all the orientation.
Relationship with Tait–Bryan angles
Similarly for Euler angles, we use the Tait–Bryan angles (in terms of flight dynamics):
 Roll – ϕ: rotation about the Xaxis
 Pitch – θ: rotation about the Yaxis
 Yaw – ψ: rotation about the Zaxis
where the Xaxis points forward, Yaxis to the right and Zaxis downward and in the example to follow the rotation occurs in the order yaw, pitch, roll (about bodyfixed axes).
Singularities
One must be aware of singularities in the Euler angle parametrization when the pitch approaches ±90° (north/south pole). These cases must be handled specially. The common name for this situation is gimbal lock.
Code to handle the singularities is derived on this site: www.euclideanspace.com
See also
 Rotation operator (vector space)
 Quaternions and spatial rotation
 Euler Angles
 Rotation matrix
 Rotation representation (mathematics)
External links
 Q60. How do I convert Euler rotation angles to a quaternion? and related questions at The Matrix and Quaternions FAQ
Categories: Rotation in three dimensions
 Euclidean symmetries
 3D computer graphics
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