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Implementing HuPf Algorithm for the Inverse Kinematics of General 6R/P Manipulators

  • Jose CapcoEmail author
  • Saraleen Mae Manongsong
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11661)

Abstract

We reformulate and extend the HuPf algorithm (see [7]), which was originally designed for a general 6R manipulator (i.e. 6 jointed open serial chain/robot with only rotational joints), to solve the inverse kinematic (IK) problem of 6R/P manipulators (6-jointed open serial robot with joints that are either rotational or prismatic/translational). For the algorithm we identify the kinematic images of 3R/P chains with a quasi-projective variety in \(\mathbb {P}^7\) via dual quaternions. More specifically, these kinematic images are projections of the intersection of a Segre variety with a linear 3-space to an open subset of \(\mathbb {P}^7\) (identified with the special Euclidean group \(\mathrm {SE}(3)\)). We show an easy and efficient algorithm to obtain the linear varieties associated to 3R/P subchains of a 6R/P manipulator. We provide examples showing the linear spaces for different 3R/P chains (a full list of them is available in an upcoming paper). Accompanying the extended HuPf algorithm we provide numerical examples showing real IK solutions to some 6R/P manipulators.

Keywords

Inverse kinematics Elimination theory Serial manipulator 

References

  1. 1.
    Capco, J., Manongsong, S.M.: Linear Spaces Associated to 3R/P Kinematic Image [Data set]. Zenodo (2019).  https://doi.org/10.5281/zenodo.3147394
  2. 2.
    Capco, J., Manongsong, S.M.: Code: Implementing HuPf Algorithm for the inverse Kinematics of General 6R/P Manipulators. Zenodo (2019).  https://doi.org/10.5281/zenodo.3157441
  3. 3.
    Collins, G.E.: Quantifier elimination by cylindrical algebraic decomposition - twenty years of progress. In: Caviness, B.F., Johnson, J.R. (eds.) Quantifier Elimination and Cylindrical Algebraic Decomposition, pp. 8–23. Springer, Vienna (1998).  https://doi.org/10.1007/978-3-7091-9459-1_2CrossRefzbMATHGoogle Scholar
  4. 4.
    Cox, D., Little, J., O’Shea, D.: Ideals, Varieties and Algorithms, 3rd edn. Springer, New York (2007).  https://doi.org/10.1007/978-0-387-35651-8 CrossRefzbMATHGoogle Scholar
  5. 5.
    Parisse B., De Graeve R.: Giac/Xcas. https://www-fourier.ujf-grenoble.fr/~parisse/giac.html. Accessed February 2019
  6. 6.
    Han F.: giacpy. https://gitlab.math.univ-paris-diderot.fr/han/giacpy. Accessed February 2019
  7. 7.
    Husty, M., Pfurner, M., Schröcker, H.-P.: A new and efficient algorithm for the inverse kinematics of a general serial 6R manipulator. Mech. Mach. Theory 42, 66–81 (2007)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Husty, M., Schröcker, H.-P.: Kinematics and algebraic geometry. In: McCarthy, J.M. (ed.) 21st Century Kinematics, pp. 85–123. Springer, London (2012).  https://doi.org/10.1007/978-1-4419-0999-2_4CrossRefGoogle Scholar
  9. 9.
    Pfurner, M.: Analysis of spatial serial Manipulators using kinematic mapping. Doctoral thesis, Institute for Basic Sciences in Engineering, Unit Geometry and CAD, University of Innsbruck, October 2006Google Scholar
  10. 10.
    Selig, J.M.: Geometric Fundamentals of Robotics. Monographs in Computer Science, 2nd edn. Springer, New York (2005).  https://doi.org/10.1007/b138859. (Ed.: D. Gries, F.B. Schneider)CrossRefzbMATHGoogle Scholar
  11. 11.
    Spong, M.W., Hutchinson, S., Vidyasagar, M.: Robot Modeling and Control. Wiley, New York (2005)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Research for Symbolic ComputationJohannes Kepler UniversityLinzAustria
  2. 2.Institute of MathematicsUniversity of the Philippines DilimanQuezon CityPhilippines

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