Kempe’s Universality Theorem for Rational Space Curves

Article

Abstract

We prove that every bounded rational space curve of degree d and circularity c can be drawn by a linkage with \( \frac{9}{2} d-6c+1\) revolute joints. Our proof is based on two ingredients. The first one is the factorization theory of motion polynomials. The second one is the construction of a motion polynomial of minimum degree with given orbit. Our proof also gives the explicit construction of the linkage.

Keywords

Dual quaternion Motion polynomial Factorization Bennett flip Linkage 

Mathematics Subject Classification

Primary 70B05 Secondary 13F20 65D17 68U07 

References

  1. 1.
    T. G. Abbott, Generalizations of Kempe’s Universality Theorem, Master’s thesis, Massachusetts Institute of Technology, Cambridge 2008.Google Scholar
  2. 2.
    K. Abdul-Sater, M. M. Winkler, F. Irlinger and T. C. Lueth, Three-position synthesis of origami-evolved, spherically constrained spatial revolute–revolute chains, ASME J. Mechanisms Robotics 8 (2016), no. 1, doi:10.1115/1.4030370.
  3. 3.
    I. I. Artobolevskii, Mechanisms for the generation of plane curves, Pergamon Press, Oxford, 1964.Google Scholar
  4. 4.
    J. E. Baker, On the motion geometry of the Bennett linkage, In Proceedings of the 8th International Conference on Engineering Computer Graphics and Descriptive Geometry, Austin, 1998, 433–437.Google Scholar
  5. 5.
    G. T. Bennett, A new mechanism, Engineering 76 (1903), 777–778.Google Scholar
  6. 6.
    G. T. Bennett, The skew isogramm-mechanism, Proc. London Math. Soc. s2–13 (1914), 151–173.Google Scholar
  7. 7.
    W. Blaschke and H. R. Müller, Ebene Kinematik, Oldenbourg, München, 1956.MATHGoogle Scholar
  8. 8.
    C. C.-A. Cheng and T. Sakkalis, On new types of rational rotation-minimizing frame space curves, J. Symb. Comput. 74 (2016), 400–407.MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    R. M. Corless, S. M. Watt and L. Zhi, QR factoring to compute the gcd of univariate approximate polynomials, IEEE Transactions on Signal Processing 52 (2004), 3394–3402.MathSciNetCrossRefGoogle Scholar
  10. 10.
    E. D. Demaine and J. O’Rourke, Geometric Folding Algorithms: Linkages, Origami, Polyhedra, Cambridge University Press, Cambridge, 2007.CrossRefMATHGoogle Scholar
  11. 11.
    P. Dietmaier, Einfach übergeschlossene Mechanismen mit Drehgelenken, Habilitation thesis, Graz University of Technology, Graz, 1995.Google Scholar
  12. 12.
    M. Gallet, C. Koutschan, Z. Li, G. Regensburger, J. Schicho and N. Villamizar, Planar linkages following a prescribed motion, Math. Comput. 86 (2016), 473-506.MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    X.-S. Gao, C.-C. Zhu, S.-C. Chou and J.-X. Ge, Automated generation of Kempe linkages for algebraic curves and surfaces, Mech. Machine Theory 36 (2001), 1019–1033.CrossRefMATHGoogle Scholar
  14. 14.
    G. Hegedüs, J. Schicho and H.-P. Schröcker, Factorization of rational curves in the Study quadric and revolute linkages, Mech. Machine Theory 69 (2013), 142–152.CrossRefGoogle Scholar
  15. 15.
    G. Hegedüs, J. Schicho and H.-P. Schröcker, Four-pose synthesis of angle-symmetric 6R linkages, J. Mechanisms Robotics 7 (2015), no. 4. doi:10.1115/1.4029186.
  16. 16.
    L. Huang and W. So, Quadratic formulas for quaternions, Appl. Math. Lett. 15 (2002), 533–540.MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    M. Husty and H.-P. Schröcker, Algebraic geometry and kinematics, In Nonlinear Computational Geometry (I. Z. Emiris, F. Sottile and T. Theobald, eds.), Springer, New York, 2009, pp. 85–107.CrossRefGoogle Scholar
  18. 18.
    B. Jüttler, Über zwangläufige rationale Bewegungsvorgänge, Österreich. Akad. Wiss. Math.-Natur. Kl. S.-B. II 202 (1993), 117–232.Google Scholar
  19. 19.
    E. Kaltofen, Z. Yang and L. Zhi, Approximate Greatest Common Divisors of Several Polynomials with Linearly Constrained Coefficients and Singular Polynomials, In Proceedings of the 2006 International Symposium on Symbolic and Algebraic Computation (J.-G. Dumas, eds.), ACM, New York, 2006, pp. 169–176.Google Scholar
  20. 20.
    M. Kapovich and J. J. Millson, Universality theorems for configuration spaces of planar linkages, Topology 41 (2002), 1051–1107.MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    A. B. Kempe, On a general method of describing plane curves of the nth degree by linkwork, Proc. London Math. Soc. s1–7 (1876), 213–216.Google Scholar
  22. 22.
    A. Kobel, Automated generation of Kempe linkages for algebraic curves in a dynamic geometry system, Bachelor’s thesis, Saarland University, 2008.Google Scholar
  23. 23.
    Z. Li, Sharp linkages, In Advances in Robot Kinematics (J. Lenarčič and O. Khatib, eds.), Springer, Cham, 2014, pp. 131–138.Google Scholar
  24. 24.
    Z. Li and J. Schicho, Classification of angle-symmetric 6R linkages, Mech. Machine Theory 70 (2013), 372–379.CrossRefGoogle Scholar
  25. 25.
    Z. Li and J. Schicho, Three types of parallel 6R linkages, In Computational Kinematics: Proceedings of the 6th International Workshop on Computational Kinematics (CK2013) (F. Thomas and A. Perez Gracia, eds.), Springer, Dordrecht, 2014, pp. 111–119.Google Scholar
  26. 26.
    Z. Li, J. Schicho and H.-P. Schröcker, 7R Darboux linkages by factorization of motion polynomials, In Proceedings of the 14th IFToMM World Congress (S.-H. Chang, ed.), 2015, doi:10.6567/IFToMM.14TH.WC.OS2.014
  27. 27.
    Z. Li, J. Schicho and H.-P. Schröcker, Factorization of motion polynomials, arXiv:1502.07600, 2015.
  28. 28.
    Z. Li, J. Schicho and H.-P. Schröcker, The rational motion of minimal dual quaternion degree with prescribed trajectory, Comput. Aided Geom. Design 41 (2016), 1–9.MathSciNetCrossRefGoogle Scholar
  29. 29.
    A. J. Perez, Analysis and Design of Bennett Linkages, Ph.D. thesis, University of California, Irvine, 2004.Google Scholar
  30. 30.
    A. Saxena, Kempe’s linkages and the universality theorem, Resonance 16 (2011), 220–237.CrossRefGoogle Scholar

Copyright information

© SFoCM 2017

Authors and Affiliations

  1. 1.Institute for Robotics and MechatronicsJoanneum ResearchKlagenfurtAustria
  2. 2.Research Institute for Symbolic ComputationJohannes Kepler University LinzHagenbergAustria
  3. 3.Unit Geometry and CADUniversity of InnsbruckInnsbruckAustria

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