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.
Polar Decomposition of Unit Dual Quaternions
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.
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Purwar, A., and Ge, Q. J. (June 24, 2013). "Polar Decomposition of Unit Dual Quaternions." ASME. J. Mechanisms Robotics. August 2013; 5(3): 031001. https://doi.org/10.1115/1.4024236
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