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The 2D Orientation Interpolation Problem: A Symmetric Space Approach

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Advances in Robot Kinematics 2016

Part of the book series: Springer Proceedings in Advanced Robotics ((SPAR,volume 4))

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Abstract

In this paper, we propose a novel construction of Bézier curves of two-dimensional (\(2\)D) orientations using the geometry of real projective plane \(\mathrm{\mathbb {R}P^{2}}\). Unlike the commonly adopted unit 2-sphere model \(S^{2}\), \(\mathrm{\mathbb {R}P^{2}}\) is naturally embedded in the \(3\)D special orthogonal group \(\mathrm{SO(3)}\). It is also a symmetric space that is equipped with a particular class of isometries called geodesic symmetry, which allows us to generate any geodesics using the exponential map of \(\mathrm{SO(3)}\). We implement the generated geodesics to construct Bézier curves for direction interpolation.

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Acknowledgements

The first and third author gratefully acknowledge the PRIN 2012 grant No. 20124SM Z88 and the MANET FP7-PEOPLE-ITN grant No. 607643. The second author acknowledges partial support of this work by the Austrian COMET-K2 program of the Linz Center of Mechatronics (LCM). This work is also in partial fulfillment to China National Natural Science Foundation Grant No.51375413, which supported Dr. Wu when he was working at HKUST and HKUST SRI.

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Authors and Affiliations

  1. Department of Industrial Engineering (DIN), University of Bologna, Bologna, Italy

    Yuanqing Wu & Marco Carricato

  2. Institut Für Robotik, Johannes Kepler Universität Linz, Linz, Austria

    Andreas Müller

Authors
  1. Yuanqing Wu
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  2. Andreas Müller
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  3. Marco Carricato
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Correspondence to Yuanqing Wu .

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Editors and Affiliations

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

    Jadran Lenarčič

  2. INRIA Sophia Antipolis , Sophia Antipolis Cedex, France

    Jean-Pierre Merlet

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Wu, Y., Müller, A., Carricato, M. (2018). The 2D Orientation Interpolation Problem: A Symmetric Space Approach. In: Lenarčič, J., Merlet, JP. (eds) Advances in Robot Kinematics 2016. Springer Proceedings in Advanced Robotics, vol 4. Springer, Cham. https://doi.org/10.1007/978-3-319-56802-7_31

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  • DOI: https://doi.org/10.1007/978-3-319-56802-7_31

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  • Publisher Name: Springer, Cham

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  • Online ISBN: 978-3-319-56802-7

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