Limit cycles for dynamic crawling locomotors with periodic prescribed shape

TL;DR

This paper analyzes the long-term behavior of dynamic crawling models with periodic shape changes, demonstrating convergence to limit cycles and conditions for unique average velocity under various friction models.

Contribution

It characterizes the asymptotic limit cycles of crawling locomotors with periodic shape changes, including effects of different friction types and conditions for velocity uniqueness.

Findings

Models converge to a relative periodic solution under mild assumptions.

Unique asymptotic average velocity is established for certain friction conditions.

Examples illustrate the necessity of assumptions for convergence and uniqueness.

Abstract

We study the asymptotic behaviour of a family of dynamic models of crawling locomotion, with the aim of characterizing a gait as a limit property. The locomotors, which might have a discrete or continuous body, move on a line with a periodic prescribed shape change, and might possibly be subject to external forcing (e.g., crawling on a slope). We discuss how their behaviour is affected by different types of friction forces, including also set-valued ones such as dry friction. We show that, under mild natural assumptions, the dynamics always converge to a relative periodic solution. The asymptotic average velocity of the crawler yet might still depend on its initial state, so we provide additional assumption for its uniqueness. In particular, we show that the asymptotic average velocity is unique both for strictly monotone friction forces, and also for dry friction, provided in the latter case that the actuation is sufficiently smooth (for discrete models) or that the friction coefficients are always nonzero (for continuous models). We present several examples and counterexamples illustrating the necessity of our assumptions.