Singularity Distance for Parallel Manipulators of Stewart Gough Type

  • Georg NawratilEmail author
Conference paper
Part of the Mechanisms and Machine Science book series (Mechan. Machine Science, volume 73)


The number of applications of parallel robots, ranging from medical surgery to astronomy, has increased enormously during the last decades due to their advantages of high speed, stiffness, accuracy, load/weight ratio, etc. One of the drawbacks of these parallel robots are their singular configurations, where the manipulator has at least one uncontrollable instantaneous degree of freedom. Furthermore, the actuator forces can become very large, which may result in a breakdown of the mechanism. Therefore singularities have to be avoided. As a consequence the kinematic/robotic community is highly interested in evaluating the singularity closeness, but a geometric meaningful distance measure between a given manipulator configuration and the next singular configuration is still missing. We close this gap for parallel manipulators of Stewart Gough type by introducing such measures. Moreover the favored metric has a clear physical meaning, which is very important for the acceptance of this index by mechanical/constructional engineers.


parallel robot singularity distance metric 


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The author is supported by Grant No. P 30855-N32 of the Austrian Science Fund FWF.


  1. 1.
    Abbasnejad G, Daniali HM, Kazemi SM (2012) A new approach to determine the maximal singularity-free zone of 3-RPR planar parallel manipulator. Robotica 30(6):1005–1012CrossRefGoogle Scholar
  2. 2.
    Bates DJ, Hauenstein JD, Sommese AJ, Wampler CW (2013) Numerically Solving Polynomial Systems with Bertini. SIAM BooksGoogle Scholar
  3. 3.
    Chen HY, Pottmann H (1999) Approximation by ruled surfaces. J Comput Appl Math 102(1):143–156MathSciNetCrossRefGoogle Scholar
  4. 4.
    Husty ML (2009) Non-singular assembly mode change in 3-RPR-parallel manipulators. In: Kecskem´ethy A, Müller A (eds) Computational Kinematics, Springer, pp 51–60Google Scholar
  5. 5.
    Jiang Q, Gosselin CM (2009) Determination of the maximal singularity-free orientation workspace for the Gough-Stewart platform. Mech Mach Theory 44(6):1281–1293CrossRefGoogle Scholar
  6. 6.
    Kazerounian K, Rastegar J (1992) Object Norms: A Class of Coordinate and Metric Independent Norms for Displacements. In: Kinzel GL (ed) Flexible Mechanisms, Dynamics and Analysis, ASME, pp 271–275Google Scholar
  7. 7.
    Kilian M, Mitra NJ, Pottmann H (2007) Geometric Modeling in Shape Space. ACM Trans Graph 26(3):64CrossRefGoogle Scholar
  8. 8.
    Li H, Gosselin C, Richard M (2006) Determination of maximal singularity-free zones in the workspace of planar three-degree-of-freedom parallel mechanisms. Mech Mach Theory 41(10):1157–1167MathSciNetCrossRefGoogle Scholar
  9. 9.
    Li H, Gosselin C, Richard M (2007) Determination of the maximal singularity free zones in the six-dimensional workspace of the general Gough-Stewart platform. Mech Mach Theory 42(4):497–511MathSciNetCrossRefGoogle Scholar
  10. 10.
    Merlet J-P (1992) Singular Configurations of Parallel Manipulators and Grassmann Geometry. Int J Robot Res 8(5):45–56CrossRefGoogle Scholar
  11. 11.
    Merlet J-P, Gosselin C (2008) Parallel Mechanisms and Robots. In: Siciliano B, Khatib O (eds) Handbook of Robotics, Springer, pp 269–285Google Scholar
  12. 12.
    Murray RM, Li Z, Sastry SS (1994) A Mathematical Introduction to Robotic Manipulation. CRC PressGoogle Scholar
  13. 13.
    Nag A, Reddy V, Agarwal S, Bandyopadhyay S (2016) Identifying singularity-free spheres in the position workspace of semi-regular Stewart platform manipulators. In: Lenarcic J, Merlet J-P (eds) Advances in Robot Kinematics, Springer, pp 421–430Google Scholar
  14. 14.
    Nawratil G (2009) New Performance Indices for 6-dof UPS and 3-dof RPR Parallel Manipulators. Mech Mach Theory 44(1):208–221CrossRefGoogle Scholar
  15. 15.
    Nawratil G (2017) Point-models for the set of oriented line-elements – a survey. Mech Mach Theory 111:18–134CrossRefGoogle Scholar
  16. 16.
    Park FC (1995) Distance Metrics on the Rigid-Body Motions with Applications to Mechanism Design. ASME J Mech Des 117(1):48–54CrossRefGoogle Scholar
  17. 17.
    Pottmann H, Hofer M, Ravani B (2004) Variational motion design. In: Lenarcic J, Galletti C (eds) On Advances in Robot Kinematics, Kluwer, pp 361–370Google Scholar
  18. 18.
    Rasoulzadeh A, Nawratil G (2017) Rational Parametrization of Linear Pentapod’s Singularity Variety and the Distance to it. In: Zeghloul S et al (eds) Computational Kinematics, Springer, pp 516–524 (Extended version on arXiv:1701.09107)Google Scholar
  19. 19.
    Schröcker H-P, Weber MJ (2014) Guaranteed collision detection with toleranced motions. Comput Aided Geom Design 31(7-8):602–612MathSciNetCrossRefGoogle Scholar
  20. 20.
    ZeinM,Wenger P, Chablat D (2007) Singularity Surfaces andMaximal Singularity- Free Boxes in the Joint Space of Planar 3-RPR Parallel Manipulators. In: Proc of 12th IFToMM World Congress, Besan¸con, France, abs/0705.1409Google Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institute of Discrete Mathematics and GeometryVienna University of TechnologyViennaAustria

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