0
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.

Copyright © 2015 by ASME
Your Session has timed out. Please sign back in to continue.

## References

Kazerounian, K., and Rastegar, J., 1992, “Object Norms: A Class of Coordinate and Metric Independent Norms for Displacement,” Flexible Mechanisms, Dynamics and Analysis, ASME DE 47, pp. 271–275.
Martinez, J. M. R., and Duffy, J., 1995, “On the Metrics of Rigid Body Displacement for Infinite and Finite Bodies,” ASME J. Mech. Des., 117(1), pp. 41–47.
Fanghella, P., and Galletti, C., 1995, “Metric Relations and Displacement Groups in Mechanism and Robot Kinematic,” ASME J. Mech. Des., 117(3), pp. 470–478.
Chirikjian, G. S., and Zhou, S., 1998, “Metrics on Motion and Deformation of Solid Models,” ASME J. Mech. Des., 120(2), pp. 252–261.
Park, F. C., 1995, “Distance Metrics on the Rigid-Body Motions With Applications to Mechanism Design,” ASME J. Mech. Des., 117(1), pp. 48–54.
Etzel, K. R., and McCarthy, J. M., 1996, “Spatial Motion Interpolation in an Image Space of SO(4),” Proceedings of 1996 ASME Design Engineering Technical Conference and Computers in Engineering Conference, Irvine, CA, Aug. 18–22.
Inonu, E., and Wigner, E. P., 1953, “On the Contraction of Groups and Their Representations,” Proc. Natl. Acad. Sci., 39(6), pp. 510–524.
Ravani, B., and Roth, B., 1984, “Mappings of Spatial Kinematics,” ASME J. Mech. Des., 106(3), pp. 341–347.
Žefran, M., Kumar, V., and Croke, C., 1999, “Metrics and Connections for Rigid-Body Kinematics,” Int. J. Rob. Res., 18(2), pp. 243–258.
Lin, Q., and Burdick, J. W., 2000, “Objective and Frame-Invariant Kinematic Metric Functions for Rigid Bodies,” Int. J. Rob. Res., 19(6), pp. 612–625.
Kuffner, J. J., 2004, “Effective Sampling and Distance Metrics for 3D Rigid Body Path Planning,” Proceedings of the 2004 IEEE International Conference on Robotics and Automation, ICRA’04, New Orleans, LA, April, Vol. 4, pp. 3993–3998.
Amato, N. M., Bayazit, O. B., Dale, L. K., Jones, C., and Vallejo, D., 1998, “Choosing Good Distance Metrics and Local Planners for Probabilistic Roadmap Methods,” Proceedings of the 1998 IEEE International Conference on Robotics and Automation, ICRA’98, Leuven, Belgium, May, Vol. 1, pp. 630–637.
Larochelle, P. M., Murray, A. P., and Angeles, J., 2007, “A Distance Metric for Finite Sets of Rigid-Body Displacements via the Polar Decomposition,” ASME J. Mech. Des., 129(8), pp. 883–886.
Chirikjian, G. S., and Kyatkin, A. B., 2001, Engineering Applications of Noncommutative Harmonic Analysis, CRC Press, Boca Raton, FL.
Chirikjian, G. S., 2009/2012, Stochastic Models, Information Theory, and Lie Groups: Volumes I + II, Birkhäuser, Boston, MA.
Huynh, D. Q., 2009, “Metrics for 3D Rotations: Comparisons and Analysis,” J. Math. Imaging Vision, 35(2), pp. 155–164.
Shiu, Y. C., and Ahmad, S., 1989, “Calibration of Wrist-Mounted Robotic Sensors by Solving Homogeneous Transform Equations of the Form AX = XB,” IEEE Trans. Rob. Autom., 5(1), pp. 16–29.
Chou, J. C. K., and Kamel, M., 1991, “Finding the Position and Orientation of a Sensor on a Robot Manipulator Using Quaternions,” Int. J. Rob. Res., 10(3), pp. 240–254.
Park, F. C., and Martin, B. J., 1994, “Robot Sensor Calibration: Solving AX = XB on the Euclidean Group,” IEEE Trans. Rob. Autom., 10(5), pp. 717–721.
Ackerman, M. K., and Chirikjian, G. S., 2013, “A Probabilistic Solution to the AX=XB Problem: Sensor Calibration Without Correspondence,” Geometric Science of Information, Paris, France, Aug. 28–31.
Chen, H. H., 1991, “A Screw Motion Approach to Uniqueness Analysis of Head-Eye Geometry,” IEEE Conference on Computer Vision and Pattern Recognition, Maui, HI, pp. 145–151.
Ackerman, M. K., Cheng, A., Shiffman, B., Boctor, E., and Chirikjian, G. S., 2013, “Sensor Calibration With Unknown Correspondence: Solving AX=XB Using Euclidean-Group Invariants,” 2013 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS'13), Tokyo, Japan, Nov. 3–8, pp. 1308–1313.

## Discussions

Some tools below are only available to our subscribers or users with an online account.

### Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related Proceedings Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

• TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
• EMAIL: asmedigitalcollection@asme.org
Sign In