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Novel Plücker Operators and a Dual Rodrigues Formula Applied to the IKP of General 3R Chains

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Advances in Robot Kinematics 2018 (ARK 2018)

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Abstract

Based on new concepts for three \((6 \times 6)\)-matrix operators for Plücker vectors, the paper presents a novel adjoint formulation of the Rodrigues equation for spatial displacements in the trigonometric form. Due to the structural similarity of this dualized equation to its spherical pendant, the principle of transference is applicable to lift the analytic solution to the inverse kinematics problem (IKP) of generic spherical 3R chains yielding a simple analytic solution for generic spatial 3R chains based on the geometry of lines and screws.

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Notes

  1. 1.

    In contrast to its vector analogue (1), the square-matrix of a Plücker vector (13) is not introduced as an outer product, but via the related identity of (2). The unit and the square matrix feature the term in the lower right block.

  2. 2.

    For screws with non-zero shift \(s = \phi \cdot h\), pitch \(h\ne 0\), identity (14) is extended to .

References

  1. Bongardt, B.: Geometric characterization of the workspace of nonorthogonal rotation axes. J. Geom. Mech. 6(2), 141–166 (2014)

    Article  MathSciNet  Google Scholar 

  2. Gallier, J., Xu, D.: Computing exponentials of skew symmetric matrices and logarithms of orthogonal matrices. Int. J. Robot. Autom. 18, 10–20 (2000)

    Google Scholar 

  3. Gelman, H.: A note on the time dependence of the effective axis and angle of a rotation. J. Res. Natl Bur. Stand. B. Math. Sci. 75B.3,4, 165–171 (1971)

    Article  Google Scholar 

  4. Husty, M.L., Ottaviano, E., Ceccarelli, M.: A geometrical characterization of workspace singularities in 3R manipulators. In: Advances in Robot Kinematics: Analysis and Design (2008)

    Chapter  Google Scholar 

  5. Husty, M.L., Pfurner, M., Schröcker, H.P.: A new and efficient algorithm for the inverse kinematics of a general serial 6R manipulator. Mech. Mach. Theory 42(1), 66–81 (2007)

    Article  MathSciNet  Google Scholar 

  6. Piovan, G., Bullo, F.: On coordinate-free rotation decomposition: Euler angles about arbitrary axes. IEEE Trans. Robot. 28(3), 728–733 (2012)

    Article  Google Scholar 

  7. Selig, J.M.: Geometric Fundamentals of Robotics. Springer, New York (2005)

    MATH  Google Scholar 

  8. Trainelli, L., Croce, A.: A comprehensive view of rotation parametrization. In: European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2004) (2004)

    Google Scholar 

  9. Wenger, P.: Cuspidal and Noncuspidal Robot Manipulators. Robotica 25(6), 677–689 (2007)

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. TU Braunschweig, Institut für Robotik und Prozessinformatik, 38106, Braunschweig, Germany

    Bertold Bongardt

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  1. Bertold Bongardt
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Correspondence to Bertold Bongardt .

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Editors and Affiliations

  1. J. Stefan Institute, Ljubljana, Slovenia

    Jadran Lenarcic

  2. Department of Industrial Engineering, University of Bologna, Bologna, Italy

    Vincenzo Parenti-Castelli

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Bongardt, B. (2019). Novel Plücker Operators and a Dual Rodrigues Formula Applied to the IKP of General 3R Chains. In: Lenarcic, J., Parenti-Castelli, V. (eds) Advances in Robot Kinematics 2018. ARK 2018. Springer Proceedings in Advanced Robotics, vol 8. Springer, Cham. https://doi.org/10.1007/978-3-319-93188-3_8

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