Skip to main content
SpringerLink
Log in

On Orientation, Position, and Attitude Singularities of General 3R Chains

  • Conference paper
  • First Online:
2nd IMA Conference on Mathematics of Robotics (IMA 2020)

Part of the book series: Springer Proceedings in Advanced Robotics ((SPAR,volume 21))

Included in the following conference series:

Abstract

The characterization of the workspace for general spatial 3R chains with skew joint axes is refined by describing the variety of singular displacements as the union of the singular configuration manifolds of orientation type, position type, and attitude type. The surface of attitude singularities is revealed by transferring the singularity sets of spherical 3R chains with intersecting joint axes to the geometry of spatial kinematic chains with skew, non-intersecting joint axes. For this purpose, the degeneracy of a screw system is analyzed by means of a particular angle concept for a set of three oriented lines in space. The obtained argumentation is expressed in terms of geometric manipulator Jacobians completing previous results.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
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

Notes

  1. 1.

    A sorted pair of two spears and defines a ‘motor’. The special case of intersecting spears with defines a ‘rotor’ and the special case of parallel spears with defines a ‘translator’ [25].

  2. 2.

    The pendant for the spherical case is introduced in [3]; a dual-angle concept permitting cylindrical motions along arbitrary reference spears is introduced in [5].

  3. 3.

    The ‘reciprocal Jacobian’ contains so-called ‘dual Plücker coordinates’ [23]. It also appears in ‘charts of analogues’ comparing ‘in-parallel robots and serial arms’ [10].

  4. 4.

    “Singularities and mobility are characterized by the determinant of the Jacobian matrix for non-redundant manipulators; or by the determinant of the matrix product of the Jacobian and its transpose for redundant mechanisms.” [16].

  5. 5.

    A planar 3R chain is always, an orthoparallel 3R chain [20] is always, and an orthogonal 3R chain is only in an orientation singularity for certain configurations with mutliplicities of \(\pi /2\). The class of general spherical 3R chains is treated in [3, 22].

  6. 6.

    Generally it is assumed that only the pair of first axis and last axis may become coplanar.

References

  1. Benoit, R., Delanoue, N., Lagrange, S., Wenger, P.: Guaranteed detection of the singularities of 3R robotic manipulators. In: Flores, P., Viadero, F. (eds.) New Trends in Mechanism and Machine Science. MMS, vol. 24, pp. 69–79. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-09411-3_8

  2. Bohigas, O., Zlatanov, D., Ros, L., Manubens, M., Porta, J.M.: A general method for the numerical computation of manipulator singularity sets. IEEE Trans. Rob. 30(2), 340–351 (2014)

    Article  Google Scholar 

  3. Bongardt, B.: Geometric characterization of the workspace of non-orthogonal rotation axes. J. Geom. Mech. 6, 141 (2014)

    Article  MathSciNet  Google Scholar 

  4. Bongardt, B.: Novel Plücker operators and a dual Rodrigues formula applied to the IKP of general 3R chains. In: Lenarcic, J., Parenti-Castelli, V. (eds.) ARK 2018. SPAR, vol. 8, pp. 65–73. Springer, Cham (2019). https://doi.org/10.1007/978-3-319-93188-3_8

  5. Bongardt, B.: The adjoint trigonometric representation of displacements and a closed-form solution to the IKP of general 3C chains. J. Appl. Math. Mech. 100, e201900214 (2020)

    MathSciNet  Google Scholar 

  6. Brand, L.: Vector and Tensor Analysis. Wiley, New York (1947)

    MATH  Google Scholar 

  7. Burdick, J.W.: A recursive method for finding revolute-jointed manipulator singularities. In: International Conference on Robotics and Automation, vol. 1, pp. 448–453 (1992)

    Google Scholar 

  8. Burdick, J.W.: A classification of 3R regional manipulator singularities and geometries. Mech. Mach. Theory 30(1), 71–89 (1995)

    Article  Google Scholar 

  9. Clifford, W.: A preliminary sketch of biquaternions. Proc. Lond. Math. Soc. 1, 381–395 (1873)

    MathSciNet  MATH  Google Scholar 

  10. Davidson, J.K., Hunt, K.H., Pennock, G.R.: Robots and Screw Theory: Applications of Kinematics and Statics to Robotics. Oxford University Press, Oxford (2004)

    MATH  Google Scholar 

  11. Dimentberg, F.M.: The Screw Calculus and Its Applications to Mechanics. Nauka, Moscow (1965)

    Google Scholar 

  12. Donelan, P.S.: Singularity-theoretic methods in robot kinematics. Robotica 25(06), 641–659 (2007)

    Article  Google Scholar 

  13. Donelan, P.S., Müller, A.: Singularities of regional manipulators revisited. In: Lenarcic, J., Stanisic, M. (eds.) Advances in Robot Kinematics: Motion in Man and Machine. Springer, Dordrecht (2010). https://doi.org/10.1007/978-90-481-9262-5_55

    Chapter  Google Scholar 

  14. Donelan, P.S., Selig, J.: Hyperbolic pseudoinverses for kinematics in the Euclidean group. SIAM J. Matrix Anal. Appl. 38(4), 1541–1559 (2017)

    Article  MathSciNet  Google Scholar 

  15. Fischer, I.S.: Dual-Number Methods in Kinematics, Statics, and Dynamics. CRC Press, Boca Raton (1999)

    Google Scholar 

  16. Chang, K.-S., Khatib, O.: Manipulator control at kinematic singularities: a dynamically consistent strategy. In: International Conference on Intelligent Robots and Systems, vol. 3, pp. 84–88 (1995)

    Google Scholar 

  17. McCarthy, J.M., Soh, G.S.: Geometric Design of Linkages. Springer, New York (2010)

    MATH  Google Scholar 

  18. von Mises, R.: Motorrechnung, ein neues Hilfsmittel der Mechanik. ZAMM J. Appl. Math. Mech. Zeitschrift für Angewandte Mathematik und Mechanik 4(2), 155–181 (1924)

    Article  Google Scholar 

  19. Ottaviano, E., Ceccarelli, M., Husty, M.: Workspace topologies of industrial 3R manipulators. Int. J. Adv. Rob. Syst. 4, 38 (2007)

    Article  Google Scholar 

  20. Ottaviano, E., Husty, M., Ceccarelli, M.: Level-set method for workspace analysis of serial manipulators. In: Lennarčič, J., Roth, B. (eds.) Advances in Robot Kinematics, pp. 307–314. Springer, Dordrecht (2006). https://doi.org/10.1007/978-1-4020-4941-5_33

    Chapter  Google Scholar 

  21. Pai, D.K., Leu, M.C.: Genericity and singularities of robot manipulators. Trans. Robot. Autom. 8(5), 545–595 (1992)

    Article  Google Scholar 

  22. Piovan, G., Bullo, F.: On coordinate-free rotation decomposition: Euler angles about arbitrary axes. Trans. Robot. 28(3), 728–733 (2012)

    Article  Google Scholar 

  23. Pottmann, H., Wallner, J.: Computational Line Geometry. Springer, Heidelberg (2001). https://doi.org/10.1007/978-3-642-04018-4

    Book  MATH  Google Scholar 

  24. Selig, J.M.: Geometric Fundamentals of Robotics. Springer, New York (2005). https://doi.org/10.1007/b138859

    Book  MATH  Google Scholar 

  25. Strubecker, K.: Studysches Übertragungsprinzip und Motorrechnung. Math. Z. 44(1), 1–19 (1939)

    Article  MathSciNet  Google Scholar 

  26. Study, E., : Geometrie der Dynamen: Die Zusammensetzung von Kräften und Verwandte Gegenstände der Geometrie. B. G. Teubner (1903)

    Google Scholar 

  27. Tchoń, K.: Differential topology of the inverse kinematic problem for redundant robot manipulators. Int. J. Robot. Res. 10(5), 492–504 (1991)

    Article  Google Scholar 

  28. Wenger, P.: Cuspidal and noncuspidal robot manipulators. Robotica 25(6), 717–724 (2010)

    Google Scholar 

  29. Yang, A.T., Freudenstein, F.: Application of dual-number quaternion algebra to the analysis of spatial mechanisms. J. Appl. Mech. 31(2), 300–308 (1964)

    Article  MathSciNet  Google Scholar 

  30. Yoshikawa, T.: Manipulability of robotic mechanisms. Int. J. Robot. Res. 4(2), 3–9 (1985)

    Article  Google Scholar 

  31. Zlatanov, D., Fenton, R.G., Benhabib, B.: Identification and classification of the singular configurations of mechanisms. In: Computational Kinematics, pp. 163–172 (1995)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Robotik und Prozessinformatik, TU Braunschweig, 38106, Braunschweig, Germany

    Bertold Bongardt

  2. Institut für Robotik, Johannes Kepler Universität, 4040, Linz, Austria

    Andreas Müller

Authors
  1. Bertold Bongardt
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andreas Müller
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bertold Bongardt .

Editor information

Editors and Affiliations

  1. Department of Engineering, Manchester Metropolitan University, Manchester, UK

    William Holderbaum

  2. School of Engineering, London South Bank University, London, UK

    J. M. Selig

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

Bongardt, B., Müller, A. (2022). On Orientation, Position, and Attitude Singularities of General 3R Chains. In: Holderbaum, W., Selig, J.M. (eds) 2nd IMA Conference on Mathematics of Robotics. IMA 2020. Springer Proceedings in Advanced Robotics, vol 21. Springer, Cham. https://doi.org/10.1007/978-3-030-91352-6_5

Download citation

Publish with us

Policies and ethics

Search

Navigation