A Distance Metric for Finite Sets of Rigid-Body Displacements via the Polar Decomposition

[+] Author and Article Information
Pierre M. Larochelle

Mechanical & Aerospace Engineering Department, Florida Institute of Technology, Melbourne, FL 32901-6975pierrel@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 Canada

J. Mech. Des 129(8), 883-886 (Jul 13, 2006) (4 pages) doi:10.1115/1.2735640 History: Received September 10, 2005; Revised July 13, 2006

An open research question is how to define a useful metric on the special Euclidean group SE(n) 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 SE(n) with elements of the special orthogonal group SO(n+1). 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 SE(n). The embedding of the elements of SE(n) into SO(n+1) yields hyperdimensional rotations that approximate the rigid-body displacements. The bi-invariant metric on SO(n+1) is then used to measure the distance between any two displacements. The result is a left invariant PD based metric on SE(n).

Copyright © 2007 by American Society of Mechanical Engineers
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Grahic Jump Location
Figure 1

The embedding of elements of SE(n−1) in SO(n)

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Figure 2

The 11 planar locations and PF

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Figure 3

The fixed frame and four locations equidistant to location No. 1




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