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Symmetric subspaces of SE(3)

  • Harald Löwe EMAIL logo , Yuanqing Wu and Marco Carricato
Published/Copyright: June 27, 2016
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Advances in Geometry
From the journal Advances in Geometry Volume 16 Issue 3

Abstract

Being a Lie group, the group SE(3) of orientation preserving motions of the real Euclidean 3-space becomes a symmetric space (in the sense of O. Loos) when endowed with the multiplication µ(g, h) = gh−1g. In this note we classify all connected symmetric subspaces of SE(3) up to conjugation. Moreover, we indicate some of its important applications in robot kinematics.

Keywords: Symmetric space; Lie triple system; Euclidean motion; symmetric subspace; projective space; mechanism analysis and design
MSC 2010: 20G20 (17A40,70B15)

Communicatedby: R. Löwen

References

[1] J. Beckers, J. Patera, M. Perroud, P. Winternitz, Subgroups of the euclidean group and symmetry breaking in nonrelativistic quantum mechanics. Journal of Mathematical Physics 18 (1977),72–83.10.1063/1.523120Search in Google Scholar

[2] R. Behrens, C. Kuchler, T. Forster, N. Elkmann, Kinematics analysis of a 3-dof joint for a novel hyper-redundant robot arm. 2011 IEEE International Conferenceon Roboticsand Automation (ICRA),3224–3229.10.1109/ICRA.2011.5979897Search in Google Scholar

[3] F. Bullo, A. D. Lewis, Geometric control of mechanical systems, volume 49 of Texts in Applied Mathematics. Springer 2005. MR2099139 Zbl1066.7000210.1007/978-1-4899-7276-7_3Search in Google Scholar

[4] S. S. Canfield, A. Ganino, C. Reinholtz, R. Salerno, Spatial, parallel-architecture robotic carpal wrist. 23 December, 1997. USPatent5,699,695.Search in Google Scholar

[5] J. M. Hervé, The Lie group of rigid body displacements, a fundamental tool for mechanism design. Mech. Mach. Theory 34 (1999),719–730. MR1750554 Zbl1049.7050610.1016/S0094-114X(98)00051-2Search in Google Scholar

[6] K. Hunt, Constant-velocity shaft couplings: a general theory. Journal of Manufacturing Science and Engineering95 (1973),455–464.10.1115/1.3438177Search in Google Scholar

[7] K. H. Hunt, Kinematic geometry of mechanisms, volume 7 of Oxford Engineering Science Series. Oxford Univ. Press 1990. MR1070564 Zbl0726.70002Search in Google Scholar

[8] X. Kong, C. Gosselin, Type synthesis of parallel mechanisms, volume 33 of Springer Tracts in Advanced Robotics. Springer 2007. MR2376813 Zbl1131.93019Search in Google Scholar

[9] O. Loos, Symmetric spaces. I :Genera ltheory. Benjamin, New York 1969. MR0239005 Zbl0175.48601Search in Google Scholar

[10] J. Meng, G. Liu, Z. Li, A geometric theory for analysis and synthesis of sub-6 dof parallel manipulators. IEEE Transaction son Robotics23 (2007),625–649.10.14711/thesis-b991594Search in Google Scholar

[11] A. D. Polpitiya, W. P. Dayawansa, C. F. Martin, B. K. Ghosh, Geometry and control of human eye movements. IEEE Trans. Automat. Control52 (2007),170–180. MR229500110.1109/TAC.2006.887902Search in Google Scholar

[12] M. Rosheim, G. Sauter, New high-angulation omni-directional sensor mount. International Symposiumon Optical Science and Technology (2002), 163–174.10.1117/12.465912Search in Google Scholar

[13] J. M. Selig, Geometric fundamentals of robotics. Springer 2005. MR2250553 Zbl1062.93002Search in Google Scholar

[14] K. Sone, H. Isobe, K. Yamada, High angle active link. NTN Technical Review71 (2004), Special Issue: Special Supplement to Industrial Machines, 70–73.Search in Google Scholar

[15] D. Wallace, S. Anderson, S. Manzo, Platform link wrist mechanism. 2 March, 2004. US Patent 6,699,235.Search in Google Scholar

[16] Y. Wu, Z. Li, J. Shi, Geometric properties of zero-torsion parallel kinematics machines. 2010 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 2307–2312.Search in Google Scholar

[17] Y. Wu, G. Liu, H. Löwe, Z. Li, Exponential submanifolds: A new kinematic model for mechanism analysis and synthesis. 2013 IEEE International Conference on Robotics and Automation (ICRA), 4177–4182.10.1109/ICRA.2013.6631167Search in Google Scholar

[18] Y. Wu, H. Löwe, M. Carricato, Z. Li, Inversion symmetry of the euclidean group: Theory and application in robot kinematics. IEEE Transactions on Robotics32 (2016), 312–326.10.1109/TRO.2016.2522442Search in Google Scholar

Received: 2014-11-19
Revised: 2014-12-19
Accepted: 2015-4-20
Published Online: 2016-6-27
Published in Print: 2016-7-1

© 2016 by Walter de Gruyter Berlin/Boston

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  2. Research Article
  3. Framed curves in the Euclidean space
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  5. A characterization of the rate of change ofФ-entropy via an integral form curvature-dimension condition
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https://doi.org/10.1515/advgeom-2016-0015
Keywords for this article
Symmetric space; Lie triple system; Euclidean motion; symmetric subspace; projective space; mechanism analysis and design

Articles in the same Issue

  1. Frontmatter
  2. Research Article
  3. Framed curves in the Euclidean space
  4. Research Article
  5. A characterization of the rate of change ofФ-entropy via an integral form curvature-dimension condition
  6. Research Article
  7. On the density function on moduli spaces of toric 4-manifolds
  8. Research Article
  9. Conformal Ricci solitons and related integrability conditions
  10. Research Article
  11. Inequalities for casorati curvatures of submanifolds in real space forms
  12. Research Article
  13. Inequalities for hyperconvex sets
  14. Research Article
  15. Neighborly inscribed polytopes and delaunay triangulations
  16. Research Article
  17. Cellular homology of real maximal isotropic grassmannians
  18. Research Article
  19. Symmetric subspaces of SE(3)
  20. Research Article
  21. Clifford’s Theorem for graphs
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