Trajectory Singularities for a Class of Parallel Motions

  • Matthew W. Cocke
  • Peter Donelan
  • Christopher G. Gibson
Conference paper
Part of the Trends in Mathematics book series (TM)


A rigid body, three of whose points are constrained to move on the coordinate planes, has three degrees of freedom. Bottema and Roth [2] showed that there is a point whose trajectory is a solid tetrahedron, the vertices representing corank 3 singularities. A theorem of Gibson and Hobbs [9] implies that, for general 3-parameter motions, such singularities cannot occur generically. However motions subject to this kind of constraint arise as interesting examples of parallel motions in robotics and we show that, within this class, such singularities can occur stably.


Screw system trajectory singularity parallel motion 


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  1. [1]
    Ball, R.S., The Theory of Screws, Cambridge University Press, Cambridge, 1900.Google Scholar
  2. [2]
    Bottema, O. and Roth, B., Theoretical Kinematics, Dover Publications, New York, 1990.zbMATHGoogle Scholar
  3. [3]
    Carretero, J.A., Nahon, M., Buckham, B. and Gosselin, C.M., Kinematics Analysis of a Three-DoF Parallel Mechanism for Telescope Applications in Proc. ASME Design Engineering Technical Conf., (Sacramento, 1997), ASME, 1997.Google Scholar
  4. [4]
    Carretero, J.A., Podhorodeski, R.P. and Nahon M., Architecture Optimization of a Three-DoF Parallel Mechanism in Proc. ASME Design Engineering Technical Conf., (Atlanta, 1998) ASME, 1998.Google Scholar
  5. [5]
    Cocke, M.W., Natural Constraints on Euclidean Motions. PhD Thesis, Department of Mathematical Sciences, University of Liverpool, 1998.Google Scholar
  6. [6]
    Cocke, M.W., Donelan, P.S. and Gibson, C.G., Instantaneous Singular Sets Associated to Spatial Motions in Real and Complex Singularities, (São Carlos, 1998), eds. F. Tari and J.W. Bruce, Res. Notes Math., 412, Chapman and Hall/CRC, 2000, 147–163.Google Scholar
  7. [7]
    Donelan, P.S. and Gibson, C.G., First-Order Invariants of Euclidean Motions. Acta Applicandae Mathematicae 24 (1991), 233–251.CrossRefMathSciNetGoogle Scholar
  8. [8]
    Donelan, P.S. and Gibson, C.G., On the Hierarchy of Screw Systems. Acta Applicandae Mathematicae 32 (1993), 267–296.CrossRefMathSciNetGoogle Scholar
  9. [9]
    Gibson, C.G. and Hobbs, C.A., Local Models for General One-Parameter Motions of the Plane and Space. Proc. Royal Soc. Edinburgh 125A (1995), 639–656.MathSciNetGoogle Scholar
  10. [10]
    Gibson, C.G. and Hunt, K.H., Geometry of Screw Systems. Mech. Machine Theory 25 (1990), 1–27.CrossRefGoogle Scholar
  11. [11]
    Golubitsky, M. and Guillemin, V. Stable Mappings and Their Singularities. Springer Verlag, 1973.Google Scholar
  12. [12]
    Hunt, K.H. Kinematic Geometry of Mechanisms. Clarendon Press, 1978.Google Scholar
  13. [13]
    Manipulating industrial robots — vocabulary, ISO 8373, 1994.Google Scholar
  14. [14]
    Nevins, J.L. and Whitney, D.E., Assembly Research. Automation 16 (1980), 595–613.CrossRefGoogle Scholar
  15. [15]
    Pottmann, H. and Wallner, J., Computational Line Geometry, Springer Verlag, 2001.Google Scholar
  16. [16]
    Watson, P.C., A Multidimensional System Analysis of the Assembly Process as Performed by a Manipulator presented at 1st North American Robot Conf. (Chicago, 1976).Google Scholar
  17. [17]
    Whitney, D.E. and Nevins, J.L., What is the RCC and what can it do? in Robot Sensors, Tactile and Non-Vision, ed. A. Pugh, IFS Publications, 1986, 3–15.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2006

Authors and Affiliations

  • Matthew W. Cocke
    • 1
  • Peter Donelan
    • 2
  • Christopher G. Gibson
    • 1
  1. 1.Department of Mathematical SciencesUniversity of LiverpoolLiverpoolUK
  2. 2.School of Mathematics, Statistics and Computer ScienceVictoria University of WellingtonWellingtonNew Zealand

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