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Conditions for Sub-6th Order Screw Systems Composed of Three Planar Pencils of Lines

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Advances in Robot Kinematics

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Abstract

Sub-6th order screw systems composed of (the sum of) three planar pencils of lines (PPLs)—3-PPL-systems—are closely related to the static singularity (or forward kinematic singularity) analysis of a number of 3-legged parallel manipulators and inverse kinematic singularity analysis of a class of hybrid manipulators. This paper presents an alternative simple approach to the derivation of the conditions for sub-6th order 3-PPL-systems. The characteristics of fourth order 2-PPL-systems are first revealed by using a reciprocal-screw-system based approach. By decomposing a 3-PPL-system as the sum of a 2-PPL-system and a 1-PPL-system, conditions for sub-6th order 3-PPL-systems can then be derived based on the intersection of screw systems. This paper also contributes to the classification of 3-PPL-systems.

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Notes

  1. 1.

    Screws of non-zero pitch within a fourth order 2-PPL-system are omitted since they are irrelevant to the conditions for sub-6th order 3-PPL-systems.

References

  1. Merlet, J.-P.: Parallel Robots, 2nd edn. Springer, Dordrecht (2006)

    Google Scholar 

  2. McCarthy, J.M.: Geometric Design of Linkages, 2nd edn. Springer, Heidelberg (2011)

    Google Scholar 

  3. Ebert-Uphoff, I., Lee, J.-K., Lipkin, H.: Characteristic tetrahedron of wrench singularities for parallel manipulators with three legs. IMechE, J. Mech. Eng. Sci. 216(C1), 81–93 (2002)

    Google Scholar 

  4. Kong, X., Gosselin, C.M.: Uncertainty singularity analysis of parallel manipulators based on the instability analysis of structures. Int. J. Robot. Res. 20(11), 847–856 (2001)

    Google Scholar 

  5. Yang, G., Chen, I.-M., Lin, W., Angeles, J.: Singularity analysis of three-legged parallel robots based on passive joint velocities. Trans. Robot. Autom. 17(4), 413–422 (2001)

    Article  Google Scholar 

  6. Downing, D.M., Samuel, A.E., Hunt, K.H.: Identification of special configurations of the octahedral manipulator using the pure condition. Int. J. Robot. Res. 21(2), 147–159 (2002)

    Article  Google Scholar 

  7. Huang, Z., Chen, L.H., Li, Y.W.: The singularity principle and property of Stewart parallel manipulator. J. Robot. Syst. 20(4), 163–176 (2003)

    Article  MATH  Google Scholar 

  8. Di Gregorio, R.: Forward problem singularities in parallel manipulators which generate SX-YS-ZS structures. Mech. Mach. Theory 40(5), 600–612 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Ben-Horin, P., Shoham, M.: Singularity condition of six degree-of-freedom three-legged parallel robots based on Grassmann-Cayley algebra. IEEE Trans. Robot. 22(4), 577–590 (2006)

    Article  Google Scholar 

  10. Pendar, H., Mahnama, M., Zohoor, H.: Singularity analysis of parallel manipulators using constraint plane method. Mech. Mach. Theory 48(1), 33–43 (2011)

    Article  Google Scholar 

  11. Kong, X., Johnson, A.: Classification of screw systems composed of three planar pencils of lines for singularity analysis of parallel mechanisms. ASME J. Mech. Robot. 6(2), 021008 (2014)

    Google Scholar 

  12. Kong, X., Yang, T.: Formulation of dynamic equations for hybrid robots using a component approach. In: Proceedings of the 4th Chinese National Youth Conference on Robotics, pp. 1–5, China (in Chinese) (1992)

    Google Scholar 

  13. Hu, B., Lu, Y., Yu, J. J., Zhuang, S.: Analyses of inverse kinematics, statics and workspace of a novel 3RPS-3SPR serial-parallel manipulator. Open Mech. Eng. J. 6(Suppl1-M5), 65–72 (2012)

    Google Scholar 

  14. Davidson, J.K., Hunt, K.H.: Robots and Screw Theory: Applications of Kinematics and Statics to Robotics. Oxford University Press, New York (2004)

    Google Scholar 

  15. Kong, X., Gosselin, C.: Type Synthesis of Parallel Mechanisms. Springer, New York (2007)

    MATH  Google Scholar 

  16. Dai, J.S.: Finite displacement screw operators with embedded Chasles’ motion. ASME J. Mech. Robot. 4(4), 041002 (2012)

    Google Scholar 

  17. Dai, J.S.: An historical view of the theoretical development of rigid body displacements from Rodrigues parameters to the finite twist. Mech. Mach. Theor. 41(1), 41–52 (2006)

    Google Scholar 

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Acknowledgments

The first author would like to acknowledge the financial support of Engineering and Physical Sciences Research Council of the UK (EPSRC) under grant Nos. EP/I016333/1 and EP/K018345/1. The second author would like to acknowledge the support of National Science Foundation China under Grant No. 51375058. The authors also acknowledge the support from the Overseas Famous Scholar Program Sponsored by Chinese Ministry of Education, China. Thanks to Mr Ruiming Li from Beijing Jiaotong University, China for creating the CAD models in Fig. 1.

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Authors and Affiliations

  1. Heriot-Watt University, Edinburgh, UK

    Xianwen Kong

  2. Beijing University of Posts and Telecommunications, Beijing, China

    Duanling Li

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  1. Xianwen Kong
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  2. Duanling Li
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Correspondence to Xianwen Kong .

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Editors and Affiliations

  1. J. Stefan Institute, Ljubljana-Vic-Rudnik, Slovenia

    Jadran Lenarčič

  2. Department of Computer Science Artificial Intelligence Laboratory, Stanford University, Stanford, California, USA

    Oussama Khatib

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Kong, X., Li, D. (2014). Conditions for Sub-6th Order Screw Systems Composed of Three Planar Pencils of Lines. In: Lenarčič, J., Khatib, O. (eds) Advances in Robot Kinematics. Springer, Cham. https://doi.org/10.1007/978-3-319-06698-1_51

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  • DOI: https://doi.org/10.1007/978-3-319-06698-1_51

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-06697-4

  • Online ISBN: 978-3-319-06698-1

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