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The International Symposium of Robotics Research

ISRR 2022: Robotics Research pp 468–484Cite as

What Can Algebraic Topology and Differential Geometry Teach Us About Intrinsic Dynamics and Global Behavior of Robots?

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Part of the Springer Proceedings in Advanced Robotics book series (SPAR,volume 27)

Abstract

Traditionally, robots are regarded as universal motion generation machines. They are designed mainly by kinematics considerations while the desired dynamics is imposed by strong actuators and high-rate control loops. As an alternative, one can first consider the robot’s intrinsic dynamics and optimize it in accordance with the desired tasks. Therefore, one needs to better understand intrinsic, uncontrolled dynamics of robotic systems. In this paper we focus on periodic orbits, as fundamental dynamic properties with many practical applications. Algebraic topology and differential geometry provide some fundamental statements about existence of periodic orbits. As an example, we present periodic orbits of the simplest multi-body system: the double-pendulum in gravity. This simple system already displays a rich variety of periodic orbits. We classify these into three classes: toroidal orbits, disk orbits and nonlinear normal modes. Some of these we found by geometrical insights and some by numerical simulation and sampling.

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Notes

  1. 1.

    We cannot introduce manifolds here. Great starting points are the books by T. Needham on differential geometry [31], and by F. Morgan on Riemannian geometry [29].

  2. 2.

    We use Einstein notation in this paper: whenever one up-down pair of indices match, we implicitly sum over them. Example: \(g_{ij}a^i b^j := \sum _i \sum _j g_{ij}a^i b^j\).

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Acknowledgments

We thank Noémie Jaquier and Alexander Dietrich for their feedback!

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Authors and Affiliations

  1. Institute of Robotics and Mechatronics, German Aerospace Center, 82234, Weßling, Germany

    Alin Albu-Schäffer & Arne Sachtler

  2. School of Computation, Information and Technology, Technical University of Munich, 85748, Garching, Germany

    Alin Albu-Schäffer & Arne Sachtler

Authors
  1. Alin Albu-Schäffer
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  2. Arne Sachtler
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Corresponding author

Correspondence to Arne Sachtler .

Editor information

Editors and Affiliations

  1. EPFL STI SMT-GE, Ecole Polytechnique Federale de Lausanne, Lausanne, Vaud, Switzerland

    Aude Billard

  2. Institute for Anthropomatics and Robotic, KIT, Karlsruhe, Baden-Württemberg, Germany

    Tamim Asfour

  3. Artificial Intelligence Laboratory, Department of Computer Science, Stanford University, Stanford, CA, USA

    Oussama Khatib

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Albu-Schäffer, A., Sachtler, A. (2023). What Can Algebraic Topology and Differential Geometry Teach Us About Intrinsic Dynamics and Global Behavior of Robots?. In: Billard, A., Asfour, T., Khatib, O. (eds) Robotics Research. ISRR 2022. Springer Proceedings in Advanced Robotics, vol 27. Springer, Cham. https://doi.org/10.1007/978-3-031-25555-7_32

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