Reduced-Order Models for Locomotion in the Perturbed Stokes Regime

M Kvalheim, B Bittner, Shai Revzen

2019 SIAM Conference on Dynamical Systems, 2019

link All publications

Abstract

For those creatures living "life at low Reynolds number", locomotion is friction-dominated in the sense that without power expenditure they quickly come to a halt. It is known that for swimming at the viscous or Stokesian limit (zero Reynolds number), motion is purely kinematic and governed by the viscous connection of geometric mechanics (Kelly and Murray, CDC, 1996). Assuming this reduced-order connection model, Bittner et al. developed an algorithm to estimate the dynamics near a periodic orbit (“gait cycle’) directly from observational data of shape and body motion (J. Non. Dyn., 2018). In recent work, we extended this algorithm for the perturbed Stokes regime – where Reynolds number is small but nonzero – using a “corrected’ reduced-order model (J Non. Dyn., submitted). Applications include the study of gait optimality in biology and “hardware-in-the-loop’ robotics (with respect to chosen goal functionals). In this talk, we discuss the mathematical foundations of this reduced-order model, which extend work of Eldering and Jacobs (SIADS, 2016). Using noncompact normal hyperbolicity theory in a singular perturbation context, we show that for locomotion in the perturbed Stokes regime having a cocompact symmetry group, there exists an exponentially stable slow manifold which serves as a corrected reduced-order model. This yields a functional relationship between shape velocity and body velocity, but which is no longer connection-like.

BibTeX

@conference{Kvalheim-2019-SIAM,
  author = {Kvalheim, M and Bittner, B and Revzen, Shai},
  title = {Reduced-Order Models for Locomotion in the Perturbed Stokes Regime},
  booktitle = {2019 SIAM Conference on Dynamical Systems},
  year = {2019},
  address = {Snowbird, UT},
  organization = {SIAM},
  url = {https://meetings.siam.org/sess/dsp_talk.cfm?p=99791}
}