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.

References

References
1.
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
.
2.
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
.10.1115/1.2826115
3.
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
.10.1115/1.2826702
4.
Chirikjian
,
G. S.
, and
Zhou
,
S.
,
1998
, “
Metrics on Motion and Deformation of Solid Models
,”
ASME J. Mech. Des.
,
120
(
2
), pp.
252
261
.10.1115/1.2826966
5.
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
.10.1115/1.2826116
6.
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.
7.
Inonu
,
E.
, and
Wigner
,
E. P.
,
1953
, “
On the Contraction of Groups and Their Representations
,”
Proc. Natl. Acad. Sci.
,
39
(
6
), pp.
510
524
.10.1073/pnas.39.6.510
8.
Ravani
,
B.
, and
Roth
,
B.
,
1984
, “
Mappings of Spatial Kinematics
,”
ASME J. Mech. Des.
,
106
(
3
), pp.
341
347
.10.1115/1.3267417
9.
Žefran
,
M.
,
Kumar
,
V.
, and
Croke
,
C.
,
1999
, “
Metrics and Connections for Rigid-Body Kinematics
,”
Int. J. Rob. Res.
,
18
(
2
), pp.
243
258
.10.1177/02783649922066187
10.
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
.10.1177/027836490001900605
11.
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
.
12.
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
.
13.
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
.10.1115/1.2735640
14.
Chirikjian
,
G. S.
, and
Kyatkin
,
A. B.
,
2001
,
Engineering Applications of Noncommutative Harmonic Analysis
,
CRC Press
,
Boca Raton, FL
.
15.
Chirikjian
,
G. S.
,
2009/2012
,
Stochastic Models, Information Theory, and Lie Groups: Volumes I + II
,
Birkhäuser
,
Boston, MA
.
16.
Huynh
,
D. Q.
,
2009
, “
Metrics for 3D Rotations: Comparisons and Analysis
,”
J. Math. Imaging Vision
,
35
(
2
), pp.
155
164
.10.1007/s10851-009-0161-2
17.
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
.10.1109/70.88014
18.
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
.10.1177/027836499101000305
19.
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
.10.1109/70.326576
20.
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.
21.
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
.
22.
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
.
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