One-Dimensional Solution Families of Nonlinear Systems Characterized by Scalar Functions on Riemannian Manifolds

Alin Albu-Schaeffer, Dominic Lakatos, Stefano Stramigioli

Research output: Working paperProfessional

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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. Newest results emphasize, 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 paper 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
Publication statusPublished - 5 Nov 2019

Publication series

NameIEEE Robotics and automation letters
ISSN (Print)2377-3766


  • eess.SY
  • cs.SY


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