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


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 lowdimensional 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 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 paper 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
Pages (from-to)1910-1917
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

Fingerprint Dive into the research topics of 'Strict Nonlinear Normal Modes of Systems Characterized by Scalar Functions on Riemannian Manifolds'. Together they form a unique fingerprint.

Cite this