Dual Quaternion

An 8-number algebraic tool that represents a 3D rotation and translation together in a single object.

A dual quaternion is a quaternion whose coefficients are dual numbers, written q = r + εd, where r and d are ordinary quaternions and ε satisfies ε² = 0, for 8 real components in total. Study pointed out in 1891 that this algebra is well suited to describing rigid-body motion in 3D space, and Kotelnikov independently proposed it in 1895. Just as a unit quaternion represents a rotation, a unit dual quaternion can represent a complete rigid-body transformation, or pose: r encodes the rotation, and d = ½·t·r encodes the translation t. Composing two transformations is just multiplying their dual quaternions, which is more compact than a 4×4 homogeneous transformation matrix and also makes interpolating between two poses convenient. In robotics, open-source libraries such as DQ Robotics use dual quaternions for manipulator kinematic modeling and control; they're also used in computer graphics.

ExampleTo rotate 90° about the z-axis and then translate by (1, 0, 0): r = (cos45°, 0, 0, sin45°), the translation is written as the pure quaternion t = (0, 1, 0, 0), and d = ½·t·r is computed; r and d together give the 8 numbers representing this pose.

Also called

Dual Quaternions, Unit Dual Quaternion

Related

Quaternion · Homogeneous Transformation Matrix · Screw Theory · Pose · Lie Group · Spherical Linear Interpolation (SLERP)

Sources

Wikipedia: Dual quaternion
DQ Robotics（dual quaternion robot modelling and control library）