Skip to main content
SpringerLink
Log in

Towards Arranging and Tightening Knots and Unknots with Fixtures

  • Chapter
  • First Online:
Book cover Algorithmic Foundations of Robotics XI

Part of the book series: Springer Tracts in Advanced Robotics ((STAR,volume 107))

Abstract

This paper presents a controlled tying approach for knots using fixtures and simple pulling motions applied to the ends of string. Each fixture is specific to a particular knot; the paper gives a design process that allows a suitable fixture to be designed for an input knot. Knot tying is separated into two phases. In the first phase, a fixture is used to loosely arrange the string around a set of rods, with the required topology of the given knot. In the second phase, the string is pulled taut and slid along the rods (the tightening fixture ) in a direction such that the cross-sections of the rods get closer together, allowing controlled tightening. Successful tying is shown for two interesting cases: a “double coin” knot design, and the top of a shoelace knot.

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 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
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.

    Some initial videos demonstrating the arrangement of several knots and tightening of the shoelace unknot can be found at http://www.cs.dartmouth.edu/~harrisonwfw/knotTying.html.

References

  1. Adams, C.C.: The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. American Mathematical Society, Providence (2004)

    Google Scholar 

  2. Alexander, J.W.: Topological invariants of knots and links. Trans. Am. Math. Soc. 20, 275–306 (1923)

    Google Scholar 

  3. Alterovitz, R., Goldberg, K., Pouliot, J., Tascherau, R., Hsu, I-C.: Sensorless planning for medical needle insertion procedures. In: IROS, pp. 3337–3343. IEEE (2003)

    Google Scholar 

  4. Alterovitz, R., Lim, A., Goldberg, K.Y., Chirikjian, G.S., Okamura, A.M.: Steering flexible needles under Markov motion uncertainty. In: IROS, pp. 1570–1575. IEEE (2005)

    Google Scholar 

  5. Aneziris, C.N.: The Mystery of Knots: Computer Programming for Knot Tabulation. K & E Series on Knots and Everything. World Scientific, Singapore (1999)

    MATH  Google Scholar 

  6. Ashton, T., Cantarella, J., Piatek, M., Rawdon, E.: Knot tightening by constrained gradient descent. Exp. Math. 20(1), 57–90 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  7. Attaway, S.W.: The mechanics of friction in rope rescue. In: International Technical Rescue Symposium (1999)

    Google Scholar 

  8. Balkcom, D.J., Trinkle, J.C., Gottlieb, E.J.: Computing wrench cones for planar contact tasks. In: ICRA, pp. 869–875. IEEE (2002)

    Google Scholar 

  9. Baranska, J., Przybyl, S., Pieranski, P.: Curvature and torsion of the tight closed trefoil knot. Eur. Phys. J. B—Condens. Matter Complex Syst. 66(4), 547–556 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  10. Bell, M.P.: Flexible Object Manipulation. Technical Report TR2010-663, Dartmouth College, Computer Science, Hanover, NH, February 2010

    Google Scholar 

  11. Bell, M.P., Wang, W., Kunzika, J., Balkcom, D.: Knot-tying with four-piece fixtures. To Appear on IJRR (2014)

    Google Scholar 

  12. Berard, S., Egan, K., Trinkle, J.C.: Contact modes and complementary cones. In: ICRA, pp. 5280–5286 (2004)

    Google Scholar 

  13. Bespamyatnikh, S.: Computing homotopic shortest paths in the plane. In: SODA, pp. 609–617. ACM/SIAM (2003)

    Google Scholar 

  14. Bhattacharya, S., Likhachev, M., Kumar, V.: Identification and representation of homotopy classes of trajectories for search-based path planning in 3d. In: Durrant-Whyte, H.F., Roy, N., Abbeel, P. (eds.) Robotics: Science and Systems (2011)

    Google Scholar 

  15. Blind, S.J., McCullough, C.C., Akella, S., Ponce, J.: Manipulating parts with an array of pins: a method and a machine. Int. J. Robot. Res. 20(10), 808–818 (2001)

    Article  Google Scholar 

  16. Burns, J., Fung, A.: Shoelace knot assisting device, May 2006

    Google Scholar 

  17. Champion, M.: Knot tying device, November 2004

    Google Scholar 

  18. Efrat, A., Kobourov, S.G., Lubiw, A.: Computing homotopic shortest paths efficiently. Comput. Geom. 35(3), 162–172 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  19. Furst, M.L., Goldberg, K.Y.: Low friction gripper, 24 March 1992. US Patent 5,098,145

    Google Scholar 

  20. Garg, A., Tamassia, R.: A new minimum cost flow algorithm with applications to graph drawing. In: North, S.C. (ed.) Graph Drawing. Lecture Notes in Computer Science, vol. 1190, pp. 201–216. Springer, New York (1996)

    Chapter  Google Scholar 

  21. Goldstein, H., Poole Jr, C.P., Safko, J.L.: Classical Mechanics. Pearson Education, Upper Saddle River (2002)

    Google Scholar 

  22. Grigoriev, D., Slissenko, A.: Computing minimum-link path in a homotopy class amidst semi-algebraic obstacles in the plane. In: Mora, T., Mattson, H.F. (eds.) AAECC. Lecture Notes in Computer Science, vol. 1255, pp. 114–129. Springer, New York (1997)

    Google Scholar 

  23. Grigoriev, D., Slissenko, A.: Polytime algorithm for the shortest path in a homotopy class amidst semi-algebraic obstacles in the plane. In: Weispfenning, V., Trager, B.M. (eds.) ISSAC, pp. 17–24. ACM (1998)

    Google Scholar 

  24. Hass, J., Lagarias, J.: The number of Reidemeister moves needed for unknotting. J. Am. Math. Soc. 14, 399–428 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  25. Hass, J., Nowik, T.: Unknot diagrams requiring a quadratic number of Reidemeister moves to untangle. Discret. Comput. Geom. 44, 91–95 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  26. Hayashi, C., Hayashi, M.: Unknotting number and number of Reidemeister moves needed for unlinking. pp. 1–10, arxiv:1021.4131 (2010)

  27. Henrich, A., Kauffman, L.H.: Unknotting unknots. ArXiv e-prints, June 2010

    Google Scholar 

  28. Hershberger, J., Snoeyink, J.: Computing minimum length paths of a given homotopy class. Comput. Geom. 4, 63–97 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  29. Inoue, H., Inaba, M.: Hand-eye coordination in rope handling. In: Robotics Research: The First International Symposium, pp. 163–174 (1985)

    Google Scholar 

  30. Kudo, M., Nasu, Y., Mitobe, K., Borovac, B.: Multi-arm robot control system for manipulation of flexible materials in sewing operation. Mechatronics 10(3), 371–402 (2000)

    Article  Google Scholar 

  31. Langer, J., Singer, D.A.: Lagrangian aspects of the Kirchhoff elastic rod. SIAM Rev. 38(4), 605–618 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  32. Lickorish, W.B.R.: An Introduction to Knot Theory. Graduate Texts in Mathematics. Springer, New York (1997)

    Book  MATH  Google Scholar 

  33. Maddocks, J.H., Carlen, M., Laurie, B., Smutny, J.: Biarcs, global radius of curvature, and the computation of ideal knot shapes. Physical and Numerical Models in Knot Theory. Series Knots Everything, vol. 36, pp. 75–108 (2005)

    Google Scholar 

  34. Mason, M.T.: Mechanics of Robotic Manipulation. MIT Press, Cambridge (2001)

    Google Scholar 

  35. McCarthy, Z., Bretl, T.W.: Quasi-static manipulation of a Kirchhoff elastic rod based on a geometric analysis of equilibrium configurations. Int. J. Robot. Res. (IJRR) (June 2013)

    Google Scholar 

  36. Moll, M., Kavraki, L.E.: Path planning for minimal energy curves of constant length. In: ICRA, pp. 2826–2831 (2004)

    Google Scholar 

  37. Rawdon, E.J.: Approximating the thickness of a knot. Ideal Knots. Series Knots Everything, vol. 19, pp. 143–150 (1998)

    Google Scholar 

  38. Rimon, E., Blake, A.: Caging planar bodies by one-parameter two-fingered gripping systems. Int. J. Robot. Res. 18(3), 299–318 (1999)

    Article  Google Scholar 

  39. Rodríguez, S., Lien, J-M., Amato, N.M.: Planning motion in completely deformable environments. In: ICRA, pp. 2466–2471. IEEE (2006)

    Google Scholar 

  40. Rolfsen, D.: Knots and Links. AMS/Chelsea Publication Series. AMS Chelsea Publishing, Providence (1976)

    MATH  Google Scholar 

  41. Shoe tying robot. http://www.youtube.com/watch?v=XrA7DR0u0uI. Accessed: 2014-02-17

  42. Singhatat, W.: Intracorporeal knot tier, April 2004

    Google Scholar 

  43. Tamassia, R.: On embedding a graph in the grid with the minimum number of bends. SIAM J. Comput. 16(3), 421–444 (1987)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

Discussions with Yuliy Baryshnikov were instrumental in the development of four-piece fixtures. We also would like to thank Yu-Han Lyu, Jordan Kunzika, and George Boateng for helpful discussion and feedback. This work was supported by NSF grant IIS-1217447.

Author information

Authors and Affiliations

  1. Dartmouth College Computer Science Department, Hanover, USA

    Weifu Wang, Matthew P. Bell & Devin Balkcom

Authors
  1. Weifu Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Matthew P. Bell
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Devin Balkcom
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Weifu Wang .

Editor information

Editors and Affiliations

  1. Department of Computer Engineering, Bogazici University, Istanbul, Turkey

    H. Levent Akin

  2. Department of Computer Science and Engineering, Texas A&M University, College Station, Texas, USA

    Nancy M. Amato

  3. Department of Computer Science and Engineering, University of Minnesota, Minneapolis, Minnesota, USA

    Volkan Isler

  4. Department of Information and Computing Sciences, Utrecht University, Utrecht, Utrecht, The Netherlands

    A. Frank van der Stappen

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Wang, W., Bell, M.P., Balkcom, D. (2015). Towards Arranging and Tightening Knots and Unknots with Fixtures. In: Akin, H., Amato, N., Isler, V., van der Stappen, A. (eds) Algorithmic Foundations of Robotics XI. Springer Tracts in Advanced Robotics, vol 107. Springer, Cham. https://doi.org/10.1007/978-3-319-16595-0_39

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-16595-0_39

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-16594-3

  • Online ISBN: 978-3-319-16595-0

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation