This paper presents the definition of a coordinate frame, entitled the principal frame , that is useful for metric calculations on spatial and planar rigid-body displacements. Given a set of displacements and using a point mass model for the moving rigid-body, the is determined from the associated centroid and principal axes. It is shown that the is invariant with respect to the choice of fixed coordinate frame as well as the system of units used. Hence, the is useful for left invariant metric computations. Three examples are presented to demonstrate the utility of the .
Issue Section:
Technical Briefs
1.
Park
, F.
, and Brocket
, R.
, 1994, “Kinematic Dexterity of Robotic Mechanisms
,” Int. J. Robot. Res.
0278-3649, 13
(1
), pp. 1
–15
.2.
Lin
, Q.
, and Burdick
, J.
, 2000, “Objective and Frame-Invariant Kinematic Metric Functions for Rigid Bodies
,” Int. J. Robot. Res.
0278-3649, 19
(6
), pp. 612
–625
.3.
Bobrow
, J.
, and Park
, F.
, 1995, “On Computing Exact Gradients for Rigid Body Guidance Using Screw Parameters
,” Proceedings of the ASME Design Engineering Technical Conferences
, New York, NY, ASME
, New York
, Vol. 1
, pp. 839
–844
.4.
Park
, F.
, 1995, “Distance Metrics on the Rigid-Body Motions With Applications to Mechanism Design
,” ASME J. Mech. Des.
0161-8458, 117
(1
), pp. 48
–54
.5.
Chirikjian
, G.
, 1998, “Convolution Metrics for Rigid Body Motion
,” Proceedings of the ASME Design Engineering Technical Conferences
, New York, NY, ASME
, New York
, Vol. 1
.6.
Chirikjian
, G.
, and Zhou
, S.
, 1998, “Metrics on Motion and Deformation of Solid Models
,” ASME J. Mech. Des.
0161-8458, 120
, pp. 252
–261
.7.
Martinez
, J.
, and Duffy
, J.
, 1995, “On the Metrics of Rigid Body Displacements for Infinite and Finite Bodies
,” ASME J. Mech. Des.
0161-8458, 117
(1
), pp. 41
–47
.8.
Larochelle
, P.
, and McCarthy
, J. M.
, 1995, “Planar Motion Synthesis Using an Approximate Bi-Invariant Metric
,” ASME J. Mech. Des.
0161-8458, 117
(4
), pp. 646
–651
.9.
Etzel
, K. R.
, and McCarthy
, J. M.
, 1996, “A Metric for Spatial Displacement Using Biquaternions on SO(4)
,” Proceedings of the 1996 IEEE International Conference on Robotics and Automation
.10.
Gupta
, K. C.
, 1997, “Measures of Positional Error for a Rigid Body
,” ASME J. Mech. Des.
0161-8458, 119
(3
), pp. 346
–348
.11.
Tse
, D.
, and Larochelle
, P.
, 2000, “Approximating Spatial Locations With Spherical Orientations for Spherical Mechanism Design
,” ASME J. Mech. Des.
0161-8458, 122
(4
), pp. 457
–463
.12.
Eberharter
, J.
, and Ravani
, B.
, 2004, “Local Metrics for Rigid Body Displacements
,” ASME J. Mech. Des.
0161-8458, 126
, pp. 805
–812
.13.
Kazerounian
, K.
, and Rastegar
, J.
, 1992, “Object Norms: A Class of Coordinate and Metric Independent Norms for Displacements
,” Proceedings of the ASME 1998 Design Engineering Technical Conferences and Computers and Information Conference
.14.
Sharf
, I.
, Wolf
, A.
, and Rubin
, M.
, 2008, “Arithmetic and Geometric Solutions for Average Rigid-Body Rotation
,” Mech. Mach. Theory
0094-114X, 45
(9
), pp. 1239
–1251
.15.
Angeles
, J.
, 2005, “Is There a Characteristic Length of a Rigid-Body Displacement
,” Proceedings of the 2005 International Workshop on Computational Kinematics
.16.
Zhang
, Y.
, and Ting
, K.
, 2008, “Point-Line Distance Under Riemannian Metrics
,” ASME J. Mech. Des.
0161-8458, 130
(9
), p. 092304
.17.
Di Gregorio
, R.
, “A Novel Point of View to Define the Distance Between Two Rigid-Body Poses
,” Proceedings of the 11th International Symposium on Advances in Robot Kinematics (ARK)
.18.
Ravani
, B.
, and Roth
, B.
, 1983, “Motion Synthesis Using Kinematic Mappings
,” ASME J. Mech., Transm., Autom. Des.
0738-0666, 105
, pp. 460
–467
.19.
Horn
, B.
, 1987, “Closed-Form Solution of Absolute Orientation Using Unit Quaternions
,” J. Opt. Soc. Am.
0030-3941, 4
(4
), pp. 629
–642
.20.
Shoemake
, K.
, and Duff
, T.
, 1992, “Matrix Animation and Polar Decomposition
,” Proceedings of the Conference on Graphics Interface ‘92
, Vancouver, BC, Canada, Morgan Kaufmann
, San Francisco, CA
.21.
Larochelle
, P.
, Murray
, A.
, and Angeles
, J.
, 2007, “A Distance Metric for Finite Sets of Rigid-Body Displacements via the Polar Decomposition
,” ASME J. Mech. Des.
0161-8458, 129
(8
), pp. 883
–886
.22.
Venkataramanujam
, V.
, 2007, “Approximate Motion Synthesis of Robotic Mechanical Systems
,” MS thesis, Florida Institute of Technology, Melbourne, FL.23.
Venkataramanujam
, V.
, and Larochelle
, P.
, 2008, “A Displacement Metric for Finite Sets of Rigid Body Displacements
,” ASME
Paper No. DETC2008-49554.24.
Larochelle
, P.
, 2006, “A Polar Decomposition Based Displacement Metric for a Finite Region of SE(N)
,” Proceedings of the 10th International Symposium on Advances in Robot Kinematics (ARK)
, Lubljana, Slovenia.25.
Schilling
, R. J.
, and Lee
, H.
, 2000, Engineering Analysis—A Vector Space Approach
, Wiley
, New York
.26.
Greenwood
, D. T.
, 2003, Advanced Dynamics
, Cambridge University Press
, Cambridge, London
.27.
Angeles
, J.
, 2003, Fundamentals of Robotic Mechanical Systems
, Springer
, New York
.28.
Al-Widyan
, K.
, Cervantes-Sànchez
, J. J.
, and Angeles
, J.
, 2002, “A Numerically Robust Algorithm to Solve the Five-Pose Burmester Problem
,” Proceedings of the ASME Design Engineering Technical Conferences
, Montreal, Canada, Sept. 29–Oct. 2, ASME
, New York
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