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Polar Decomposition of Unit Dual Quaternions

[+] Author Affiliations
Anurag Purwar, Q. J. Ge

Stony Brook University, Stony Brook, NY

Paper No. DETC2012-70882, pp. 1571-1578; 8 pages
  • ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
  • Volume 4: 36th Mechanisms and Robotics Conference, Parts A and B
  • Chicago, Illinois, USA, August 12–15, 2012
  • Conference Sponsors: Design Engineering Division, Computers and Information in Engineering Division
  • ISBN: 978-0-7918-4503-5
  • Copyright © 2012 by ASME


This paper seeks to extend the notion of polar decomposition from matrix algebra to dual quaternion algebra. The goal is to obtain a simple, efficient and explicit method for determining the polar decompositions (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. Examples are 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.

Copyright © 2012 by ASME



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