In a flurry of articles in the mid to late 1990s, various metrics for the group of rigid-body motions, SE(3), were introduced for measuring distance between any two reference frames or rigid-body motions. During this time, it was shown that one can choose a smooth distance function that is invariant under either all left shifts or all right shifts, but not both. For example, if one defines the distance between two reference frames to be an appropriately weighted Frobenius norm of the difference of the corresponding homogeneous transformation matrices, this will be invariant under left shifts by arbitrary rigid-body motions. However, this is not the full picture—other invariance properties exist. Though the Frobenius norm is not invariant under right shifts by arbitrary rigid-body motions, for an appropriate weighting it is invariant under right shifts by pure rotations. This is also true for metrics based on the Lie-theoretic logarithm. This paper goes further to investigate the full invariance properties of distance functions on SE(3), clarifying the full subsets of motions under which both left and right invariance is possible.
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March 2015
Research-Article
Partial Bi-Invariance of SE(3) Metrics1
Gregory S. Chirikjian
Gregory S. Chirikjian
Robot and Protein Kinematics Lab,
Department of Mechanical Engineering,
e-mail: gregc@jhu.edu
Department of Mechanical Engineering,
Johns Hopkins University
,Baltimore, MD 21218
e-mail: gregc@jhu.edu
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Gregory S. Chirikjian
Robot and Protein Kinematics Lab,
Department of Mechanical Engineering,
e-mail: gregc@jhu.edu
Department of Mechanical Engineering,
Johns Hopkins University
,Baltimore, MD 21218
e-mail: gregc@jhu.edu
1This paper was originally presented at the ASME 2014 International Design Engineering Technical Conferences as Paper No. DETC2014-DETC2014-34276.
2Here, Ri is a 3 × 3 rotation matrix and ti is a three-dimensional translation vector.
Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTING AND INFORMATION SCIENCE IN ENGINEERING. Manuscript received August 23, 2014; final manuscript received October 26, 2014; published online January 13, 2015. Editor: Bahram Ravani.
J. Comput. Inf. Sci. Eng. Mar 2015, 15(1): 011008 (7 pages)
Published Online: March 1, 2015
Article history
Received:
August 23, 2014
Revision Received:
October 26, 2014
Online:
January 13, 2015
Citation
Chirikjian, G. S. (March 1, 2015). "Partial Bi-Invariance of SE(3) Metrics." ASME. J. Comput. Inf. Sci. Eng. March 2015; 15(1): 011008. https://doi.org/10.1115/1.4028941
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