A distance metric for finite sets of rigid-body displacements via the polar decomposition

Journal of Mechanical Design

Volumn 129, Issue 8, 2007, Pages 883-886

  Larochelle, Pierre M ^a   Murray, Andrew P ^b   Angeles, Jorge ^c  

^a Florida Institute of Technology ^*  (United States)

^b University of Dayton   (United States)

^c MCGILL UNIVERSITY   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

APPROXIMATION THEORY; MATHEMATICAL TRANSFORMATIONS; SET THEORY; SINGULAR VALUE DECOMPOSITION;

DISTANCE METRIC; FINITE SETS; POLAR DECOMPOSITION; RIGID-BODY DISPLACEMENTS;

METRIC SYSTEM;

EID: 34548284820     PISSN: 10500472     EISSN: None     Source Type: Journal    
DOI: 10.1115/1.2735640     Document Type: Article

References (31)

1. Kazerounian, K., and Rastegar, J., 1992. "Object Norms: A Class of Coordinate and Metric Independent Norms for Displacements," Proc. of the ASME Design Engineering Technical Conferences, Scotsdale, AZ, September 13-16.
2. Larochelle, P., and McCarthy, J. M., 1995. "Planar Motion Synthesis Using an Approximate Bi-invariant Metric," ASME J. Mech. Des., 117(4), pp. 646-651.
3. Tse, D. M., and Larochelle, P. M., 2000. "Approximating Spatial Locations with Spherical Orientations for Spherical Mechanism Design," ASME J. Mech. Des., 122, pp. 457-463.
4. Gupta, K. C., 1997. "Measures of Positional Error for a Rigid Body," ASME J. Mech. Des., 119, pp. 346-349.
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.
6. Lin, Q., and Burdick, J., 2000. "Objective and Frame-Invariant Kinematic Metric Functions for Rigid Bodies," Int. J. Robot. Res., 19(6), pp. 612-625.
7. Martinez, J. M. R., and Duffy, J., 1995. "On the Metrics of Rigid Body Displacements for Infinite and Finite Bodies," ASME J. Mech. Des., 117, pp. 41-47.
8. Fanghela, P., and Galletti, C., 1995. "Metric Relations and Displacement Groups in Mechansims and Robot Kinematics," ASME J. Mech. Des., 117(3), pp. 470-478.
9. Eberharter, J., and Ravani, B., 2004. "Local Metrics for Rigid Body Displacements," ASME J. Mech. Des., 126, pp. 805-812.
10. Chirikjian, G. S., 1998. "Convolution Metrics for Rigid Body Motion," Proc. of the ASME Design Engineering Technical Conferences, Atlanta, CA, September 13-16.
11. Chirikjian, G. S., and Zhou, S., 1998. "Metrics on Motion and Deformation of Solid Models," ASME J. Mech. Des., 120(2), pp. 252-261.
12. Belta, C., and Kumar, V., 2002. "An SVD-Based Projection Method for Interpolation on SE(3), "IEEE Trans. Rob. Autom., 18(3), pp. 334-345.
13. Ravani, B., and Roth, B., 1983. "Motion Synthesis using Kinematic Mappings," ASME J. Mech., Transm., Autom. Des., 105, pp. 460-467.
14. Schilling, R. J., and Lee, H., 1988. Engineering Analysis- a Vector Space Approach, Wiley, New York.
15. McCarthy, J. M., 1983. "Planar and Spatial Rigid Body Motion as Special Cases of Spherical and 3-Spherical Motion," ASME J. Mech., Transm., Autom. Des., 105, pp. 569-575.
16. Etzel, K., and McCarthy, J. M., 1996. "A Metric for Spatial Displacements using Biquaternions on SO(4)," Proc. of the IEEE International Conference on Robotics and Automation, Minneapolis, MN, April 22-28.
17. Inonu, L, and Wigner, E., 1953. "On the Contraction of Groups and their Representations," Proc. Natl. Acad. Sci. U.S.A., 39(6), pp. 510-524.
18. Saletan, E., 1961. "Contraction of Lie Groups," J. Math. Phys., 2(1), pp. 1-2 1.
19. Shoemake, K., and Duff, T., 1992. "Matrix Animation and Polar Decomposition," Proc. of Graphics Interface '92, pp. 258-264, Vancouver, British Columbia, Canada, May 11-15.
20. Larochelle, P., Murray, A., and Angeles, J., 2004. "SVD and PD Based Projection Metrics on SE(n)," Proc. On Advances in Robot Kinematics, Sestri Levante, Italy, June 27-July 1.
21. Hanson, R. J., and Norris, M. J., 1981."Analysis of Measurements Based upon the Singular Value Decomposition," SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput., 2(3), pp. 308-313.
22. Paul, R., 1981. Robot Manipulators: Mathematics, Programming, and Control, MIT Press, Cambridge, MA.
23. Faddeeva, V. N., 1959. Computational Methods of Linear Algebra. Dover, New York.
24. Halmos, P. R., 1990. Finite Dimensional Vector Spaces, Van Nostrand, Reinhold, New York.
25. Dubrulle, A. A., 2001. "An Optimum Iteration for the Matrix Polar Decomposition," Electron. Trans. Numer. Anal., 8, pp. 21-25.
26. Al-Widyan, K., and Angeles, J., 2002. "A Numerically Robust Algorithm to Solve the Five-Pose Burmester Problem," Proc. of the ASME Design Engineering Technical Conferences, Montréal, Canada, September 29-October 2.
27. Greenwood, D. T., 2003. Advanced Dynamics, Cambridge University Press, Cambridge, UK.
28. Angeles, J., 2003. Fundamentals of Robotic Mechanical Systems, Springer, New York.
29. Angeles, J., 2005. "Is there a Characteristic Length of a Rigid-body Displacement," Proc. of the 2005 International Workshop on Computational Kinematics, Cassino, Italy, May 4-6.
30. Larochelle, P., 1999. Οn the Geometry of Approximate Bi-invariant Projective Displacement Metrics," Proc. of the World Congress on the Theory of Machines and Mechanisms, Oulu, Finland, June 20-24.
31. Larochelle, P., and Vance, J., 2000. "Interactive Visualization of the Line Congruences Associated with Four Finite Spatial Positions," Proceedings of the Symposium Commemorating the Legacy, Works, and Life of Sir Robert Stawell Ball upon the 100th Anniversary of the Publication of his Seminal Work A Treatise on the Theory of Screws, University of Cambridge, Trinity College, UK, July 9-11.