Strict Nonlinear Normal Modes of Systems Characterized by Scalar Functions on Riemannian Manifolds

Alin Albu-Schaffer, Dominic Lakatos, Stefano Stramigioli

Research output: Contribution to journalArticleAcademicpeer-review

4 Citations (Scopus)
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For the study of highly nonlinear, conservative dynamic systems, finding special periodic solutions which can be seen as generalization of the well-known normal modes of linear systems is very attractive. However, the study of low-dimensional invariant manifolds in the form of nonlinear normal modes is rather a niche topic, treated mainly in the context of structural mechanics for systems with Euclidean metrics, i.e., for point masses connected by nonlinear springs. In our previous research [1], [16], [17] we recognized, however, that a very rich structure of periodic and low-dimensional solutions exist also within nonlinear systems such as elastic multi-body systems encountered in the biomechanics of humans and animals or of humanoid and quadruped robots, which are characterized by a non-constant metric tensor. This letter briefly discusses different generalizations of linear oscillation modes to nonlinear systems and proposes a definition of strict nonlinear normal modes, which matches most of the relevant properties of the linear modes. The main contributions are a theorem providing necessary and sufficient conditions for the existence of strict oscillation modes on systems endowed with a Riemannian metric and a potential field as well as a constructive example of designing such modes in the case of an elastic double pendulum.

Original languageEnglish
Article number9361317
Pages (from-to)1910-1917
Number of pages8
JournalIEEE Robotics and automation letters
Issue number2
Early online date23 Feb 2021
Publication statusPublished - 1 Apr 2021


  • and Learning for Soft Robots
  • Control
  • Dynamics
  • Flexible Robotics
  • Linear systems
  • Manifolds
  • Measurement
  • Modeling
  • Nonlinear systems
  • Robot kinematics
  • Robots
  • Tensors


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