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Workspace and Cuspidality Analysis of a 2-X Planar Manipulator

Part of the Mechanisms and Machine Science book series (Mechan. Machine Science,volume 66)

Abstract

This paper analyzes the workspace of a planar 2-X manipulator, i.e. made of two crossed four-bar mechanisms in series. This architecture has some advantages over classical 2-R manipulators such as its ability to be driven with tendons, but its kinematics is more challenging because of a variable instantaneous center of rotation of the X-mechanisms. The workspace boundaries are determined algebraically and its accessibility is analyzed. In the absence of joint limits, the workspace has regions with two and four inverse kinematic solutions. Depending on the values of its geometric parameters, the manipulator at hand may be cuspidal, i.e. it can change its posture without meeting a singularity. A necessary and sufficient condition is stated for the manipulator to be cuspidal. The effect of joint limits is analyzed and the accessibility regions are further classified according to the reachable configurations of each X-mechanism in these regions.

Keywords

  • Kinematics
  • Crossed four-bar mechanism
  • Workspace
  • Cuspidal

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References

  1. Moored, K.: Analytical predictions, optimization, and design of a tensegrity-based artificial pectoral fin. Int. J. Solids Struct. 48, 3142–3159 (2011)

    CrossRef  Google Scholar 

  2. Snelson, K.: Continuous Tension, Discontinuous Compression Structures, US Patent No. 3,169,611 (1965)

    Google Scholar 

  3. Bakker, D.L., et al.: Design of an environmentally interactive continuum manipulator. In: Proceedings of 14th World Congress in Mechanism and Machine Science, IFToMM 2015, Taipei, Taiwan (2015)

    Google Scholar 

  4. Fuller, R.B.: Tensile-integrity structures, United States Patent 3063521 (1962)

    Google Scholar 

  5. Skelton, R., de Oliveira, M.: Tensegrity Systems. Springer (2009)

    Google Scholar 

  6. Levin, S.: The tensegrity-truss as a model for spinal mechanics: biotensegrity. J. Mech. Med. Biol. 2(3) (2002)

    Google Scholar 

  7. Arsenault, M., Gosselin, C.M.: Kinematic, static and dynamic analysis of a planar 2-dof tensegrity mechanism. Mech. Mach. Theory 41(9), 1072–1089 (2006)

    MathSciNet  CrossRef  Google Scholar 

  8. Crane, C., et al.: Kinematic analysis of a planar tensegrity mechanism with pres-stressed springs. In: Lenarcic, J., Wenger, P. (eds.) Advances in Robot Kinematics: Analysis and Design, pp. 419-427. Springer (2008)

    Google Scholar 

  9. Wenger, P., Chablat, D.: Kinetostatic analysis and solution classification of a planar tensegrity mechanism. In: Proceedings of the 7th International Workshop on Computational Kinematics, pp. 422–431. Springer (2017). ISBN 978-3-319-60867-9

    Google Scholar 

  10. Furet, M., et al.: Kinematic analysis of planar 2-X tensegrity manipulators. In: Proceedings of Advances in Robot Kinematics, Bologna, Italy (2018)

    Google Scholar 

  11. Boehler, Q., et al.: Definition and computation of tensegrity mechanism workspace. ASME J. Mech. Robot. 7(4) (2015)

    CrossRef  Google Scholar 

  12. Aldrich, J.B., Skelton, R.E.: Time-energy optimal control of hyper-actuated mechanical systems with geometric path constraints. In: 44th IEEE Conference on Decision and Control, pp. 8246–8253 (2005)

    Google Scholar 

  13. Chen, S., Arsenault, M.: Analytical computation of the actuator and cartesian workspace boundaries for a planar 2-degree-of-freedom translational tensegrity mechanism. J. Mech. Rob. 4 (2012)

    Google Scholar 

  14. El Omri, J., Wenger, P.: How to recognize simply a non-singular posture changing manipulator. In: Proceedings of the 7th International Conference on Advanced Robotics, pp. 215–222 (1995)

    Google Scholar 

  15. Furet, M., Wenger, P.: Derivation of a polynomial equation for the boundaries of 2-X manipulators, Technical report, LS2N, April 2018

    Google Scholar 

  16. Wenger, P.: Cuspidal and noncuspidal robot manipulators. Special Issue Robotica Geom. Robot. Sens. 25(6), 677–690 (2007)

    Google Scholar 

  17. Thomas, F., Wenger, P.: On the topological characterization of robot singularity loci. A catastrophe-theoretic approach. In: IEEE International Conference on Robotics and Automation ICRA 2011, 9–13 May 2011, Shanghai (2011)

    Google Scholar 

  18. Wenger, P., Chablat, D., Baili, M.: A DH parameter based condition for 3R orthogonal manipulators to have four distinct inverse kinematic solutions. ASME J. Mech. Des. 127(1), 150–155 (2005)

    CrossRef  Google Scholar 

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Acknowledgement

This work has been conducted in part with the support of the French National Research Agency (AVINECK Project ANR-16-CE33-0025).

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Correspondence to Philippe Wenger .

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Furet, M., Wenger, P. (2019). Workspace and Cuspidality Analysis of a 2-X Planar Manipulator. In: Gasparetto, A., Ceccarelli, M. (eds) Mechanism Design for Robotics. MEDER 2018. Mechanisms and Machine Science, vol 66. Springer, Cham. https://doi.org/10.1007/978-3-030-00365-4_14

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