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Research Papers

Polar Decomposition of Unit Dual Quaternions

[+] Author and Article Information
Anurag Purwar

e-mail: Anurag.Purwar@stonybrook.edu

Q. J. Ge

e-mail: Qiaode.Ge@stonybrook.edu
Computational Design Kinematics Laboratory,
Department of Mechanical Engineering,
Stony Brook University,
Stony Brook, NY 11794-2300

More precisely, a unit quaternion Q corresponds to a special equal-angle double rotation in 4D. It is the composition of Q and its conjugate Q* that results in a 3D rotation in a subspace of 4D with rotation angle θ.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received February 20, 2012; final manuscript received March 23, 2013; published online June 10, 2013. Assoc. Editor: Andrew P. Murray.

J. Mechanisms Robotics 5(3), 031001 (Jun 24, 2013) (6 pages) Paper No: JMR-12-1016; doi: 10.1115/1.4024236 History: Received February 20, 2012; Revised March 23, 2013

This paper seeks to extend the notion of polar decomposition (PD) from matrix algebra to dual-quaternion algebra. The goal is to obtain a simple, efficient and explicit method for determining the PD of spatial displacements in Euclidean three-space that belong to a special Euclidean Group known as SE(3). It has been known that such a decomposition is equivalent to the projection of an element of SE(3) onto SO(4) that yields hyper spherical displacements that best approximate rigid-body displacements. It is shown in this paper that a dual quaternion representing an element of SE(3) can be decomposed into a pair of unit quaternions, called double quaternion, that represents an element of SO(4). Furthermore, this decomposition process may be interpreted as the projection of a point in four-dimensional space onto a unit hypersphere. An example is provided to illustrate that the results obtained from this dual-quaternion based polar decomposition are same as those obtained from the matrix based polar decomposition.

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Figures

Grahic Jump Location
Fig. 1

A double quaternion (G, H) that approximates a dual quaternion (Q,ɛQ0) is obtained by projecting points Q + ɛQ0 and Q-ɛQ0 onto the unit hypersphere

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