Skip to main content

6R Linkages with Hidden Singularities

  • Conference paper
  • First Online:
Advances in Robot Kinematics 2022 (ARK 2022)

Part of the book series: Springer Proceedings in Advanced Robotics ((SPAR,volume 24))

Included in the following conference series:

Abstract

Mechanisms may possess kinematic singularities that are smooth points of the configuration space (c-space). At such points, a mechanism becomes shaky as its instantaneous mobility changes, but in contrast to c-space singularities, like bifurcations or cusps, these kinematic singularities are not reflected in the c-space. They are therefore called hidden singularities. Very few publications have addressed the analysis of hidden singularities. Recent research, employing methods from algebraic geometry, shows that they are often due to embedded points of the c-space variety but may also be due to singularities of the c-space variety where real and complex components intersect. Facilitating future research on the origin of hidden singularities necessitates a comprehensive list of mechanisms exhibiting this phenomenon. To this end, a constructive approach for the synthesis of 1-DOF 6R linkages with hidden singularities of prescribed rank is proposed in this paper. Various examples are shown that have hidden singularities of rank 3 or 4. Also, an example is shown that has a singularity which is a bifurcation as well as an embedded point.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 189.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 249.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 249.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Baker, J.E.: On Bricard’s doubly collapsible octahedron and its planar, spherical and skew counterparts. J. Franklin Inst. 332(6), 657–679 (1995)

    Article  MathSciNet  Google Scholar 

  2. Chen, Y., You, Z.: Spatial overconstrained linkages - the lost jade. In: Koetsier, T., Ceccarelli, M. (eds.) Explorations in the History of Machines and Mechanisms, History of Mechanism and Machine Science, vol. 15, pp. 535–550. Springer, Cham (2012). https://doi.org/10.1007/978-94-007-4132-4_37

  3. Gibson, C., Hunt, K.: Geometry of screw systems-2: classification of screw systems. Mech. Mach. Theory 25(1), 11–27 (1990)

    Article  Google Scholar 

  4. Husty, M., Pfurner, M., Schröcker, H.P., Brunnthaler, K.: Algebraic methods in mechanism analysis and synthesis. Robotica 25(6), 661–675 (2007)

    Article  Google Scholar 

  5. Li, Z., Müller, A.: Mechanism singularities revisited from an algebraic viewpoint. In: 43rd Mechanisms and Robotics Conference (MR)/ASME International Design Engineering Technical Conferences (IDETC), Anaheim, CA, USA, 18–21 August 2019

    Google Scholar 

  6. Li, Z., Nawratil, G., Rist, F., Hensel, M.: Invertible paradoxic loop structures for transformable design. In: Computer Graphics Forum, vol. 39, pp. 261–275 (2020)

    Google Scholar 

  7. Li, Z., Schicho, J.: Classification of angle-symmetric 6R linkages. Mech. Mach. Theory 70, 372–379 (2013)

    Article  Google Scholar 

  8. Liu, G., Lou, Y., Li, Z.: Singularities of parallel manipulators: a geometric treatment. IEEE Trans. Robot. Autom. 19(4), 579–594 (2003)

    Article  Google Scholar 

  9. Martinez, J.R., Duffy, J.: Classification of screw systems-I. One- and two-systems. Mech. Mach. Theory 27(4), 459–470 (1992)

    Article  Google Scholar 

  10. Martinez, J.R., Duffy, J.: Classification of screw systems-II. Three-systems. Mech. Mach. Theory 27(4), 471–490 (1992)

    Article  Google Scholar 

  11. McCarthy, J.M., Soh, G.S.: Geometric Design of Linkages, vol. 11. Springer, Cham (2010). https://doi.org/10.1007/978-1-4419-7892-9

  12. Merlet, J.P.: Geometry and kinematic singularities of closed-loop manipulators. Lab. Rob. Autom. 4, 85 (1992)

    Google Scholar 

  13. Müller, A.: Local kinematic analysis of closed-loop linkages-mobility, singularities, and shakiness. ASME J. Mech. Rob. 8, 041013, 11 p. (2016)

    Google Scholar 

  14. Müller, A.: Higher-order analysis of kinematic singularities of lower pair linkages and serial manipulators. ASME J. Mech. Rob. 10(1), 011008, 13 p. (2018)

    Google Scholar 

  15. Müller, A.: Local investigation of mobility and singularities of linkages. In: Müller, A., Zlatanov, D. (eds.) Singular Configurations of Mechanisms and Manipulators. CICMS, vol. 589, pp. 181–229. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-05219-5_5

    Chapter  Google Scholar 

  16. Müller, A.: An overview of formulae for the higher-order kinematics of lower-pair chains with applications in robotics and mechanism theory. Mech. Mach. Theory 142, 103594 (2019)

    Article  Google Scholar 

  17. Nawratil, G.: Flexible octahedra in the projective extension of the Euclidean 3-space and their application. Habilitation thesis. Inst. Discrete Math. Geom., TU Vienna (2011)

    Google Scholar 

  18. Pottmann, H., Wallner, J.: Computational Line Geometry. Springer, Cham (2009). https://doi.org/10.1007/978-3-642-04018-4

  19. Selig, J., Li, Z.: Double bennett mechanisms with assembly modes of different dimensions. In: International Conference on Reconfigurable Mechanisms and Robots (ReMAR), pp. 1–6 (2018)

    Google Scholar 

  20. Whitney, H.: Local properties of analytic varieties. Differential and Combinatorial Topology, A Symposium in Honor of M. Morse. Princeton University Press (1965)

    Google Scholar 

Download references

Acknowledgements

This research was funded by the Austrian Science Fund (FWF) [I 4452-N].

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Andreas Müller .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Li, Z., Müller, A. (2022). 6R Linkages with Hidden Singularities. In: Altuzarra, O., Kecskeméthy, A. (eds) Advances in Robot Kinematics 2022. ARK 2022. Springer Proceedings in Advanced Robotics, vol 24. Springer, Cham. https://doi.org/10.1007/978-3-031-08140-8_10

Download citation

Publish with us

Policies and ethics