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Research Papers

# Partial Bi-Invariance of SE(3) Metrics1

[+] Author and Article Information
Gregory S. Chirikjian

Robot and Protein Kinematics Lab,
Department of Mechanical Engineering,
Johns Hopkins University,
Baltimore, MD 21218
e-mail: gregc@jhu.edu

Here and throughout this paper, juxtapositioning two homogeneous transformations implies matrix multiplication. That is, $H(R1,t1)H(R2,t2)=H(R1R2,R1t2+t1)$.

This angle is computed from the Frobenius norm $θA=∥(1/2)logRA∥=∥(1/2)logRB∥=θB$.

The special case when they are simultaneously zero is a set of measure zero, and hence is a rare event. Nevertheless, it is easy to handle, since in this case RA is necessarily a rotation around e3.

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 15(1), 011008 (Mar 01, 2015) (7 pages) Paper No: JCISE-14-1260; doi: 10.1115/1.4028941 History: Received August 23, 2014; Revised October 26, 2014; Online January 13, 2015

## Abstract

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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