An open research question is how to define a useful metric on the special Euclidean group SE with respect to: (1) the choice of coordinate frames and (2) the units used to measure linear and angular distances that is useful for the synthesis and analysis of mechanical systems. We discuss a technique for approximating elements of with elements of the special orthogonal group . This technique is based on using the singular value decomposition (SVD) and the polar decompositions (PD) of the homogeneous transform representation of the elements of . The embedding of the elements of into yields hyperdimensional rotations that approximate the rigid-body displacements. The bi-invariant metric on is then used to measure the distance between any two displacements. The result is a left invariant PD based metric on .
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e-mail: pierrel@fit.edu
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August 2007
Technical Briefs
A Distance Metric for Finite Sets of Rigid-Body Displacements via the Polar Decomposition
Pierre M. Larochelle,
Pierre M. Larochelle
Mechanical & Aerospace Engineering Department,
e-mail: pierrel@fit.edu
Florida Institute of Technology
, Melbourne, FL 32901-6975
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Andrew P. Murray,
Andrew P. Murray
Mechanical & Aerospace Engineering Department,
University of Dayton
, Dayton, OH 45469-0238
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Jorge Angeles
Jorge Angeles
Department of Mechanical Engineering,
McGill University
, Montreal, Quebec, H3A 2A7 Canada
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Pierre M. Larochelle
Mechanical & Aerospace Engineering Department,
Florida Institute of Technology
, Melbourne, FL 32901-6975e-mail: pierrel@fit.edu
Andrew P. Murray
Mechanical & Aerospace Engineering Department,
University of Dayton
, Dayton, OH 45469-0238
Jorge Angeles
Department of Mechanical Engineering,
McGill University
, Montreal, Quebec, H3A 2A7 CanadaJ. Mech. Des. Aug 2007, 129(8): 883-886 (4 pages)
Published Online: July 13, 2006
Article history
Received:
September 10, 2005
Revised:
July 13, 2006
Citation
Larochelle, P. M., Murray, A. P., and Angeles, J. (July 13, 2006). "A Distance Metric for Finite Sets of Rigid-Body Displacements via the Polar Decomposition." ASME. J. Mech. Des. August 2007; 129(8): 883–886. https://doi.org/10.1115/1.2735640
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