[Paper Review] Stabilization of periodic sweeping processes and asymptotic average velocity for soft locomotors with dry friction

Giovanni Colombo, Paolo Gidoni+1 more|arXiv (Cornell University)|Mar 4, 2021

Micro and Nano Robotics27 references9 citations

TL;DR

This paper establishes W¹,² convergence to periodic solutions for sweeping processes with polyhedral sets featuring translationally moving faces, enabling rigorous proof of asymptotic average velocity well-posedness in soft crawler models with dry friction. The authors strengthen prior stability results by showing that solutions converge in the Sobolev space W¹,² to a unique running-periodic trajectory, ensuring the existence of a gait-dependent asymptotic velocity independent of initial conditions.

ABSTRACT

We study the asymptotic stability of periodic solutions for sweeping processes defined by a polyhedron with translationally moving faces. Previous results are improved by obtaining a stronger $W^{1,2}$ convergence. Then we present an application to a model of crawling locomotion. Our stronger convergence allows us to prove the stabilization of the system to a running-periodic (or derivo-periodic, or relative-periodic) solution and the well-posedness of an average asymptotic velocity depending only on the gait adopted by the crawler. Finally, we discuss some examples of finite-time versus asymptotic-only convergence.

Motivation & Objective

• To establish stronger convergence than prior work for periodic solutions in sweeping processes with moving polyhedral sets.
• To prove the existence and uniqueness of an asymptotic average velocity for soft locomotors under periodic actuation.
• To analyze whether convergence to periodic behavior occurs in finite time or only asymptotically.
• To provide a mathematical foundation for gait optimization in bio-mimetic crawling robots.

Proposed method

• Reformulate the soft crawler dynamics as a differential inclusion involving the normal cone to a time-dependent polyhedron K(t).
• Apply a generalized Moreau sweeping process framework to analyze the evolution of shape variables w(t) and center-of-mass y(t).
• Prove W¹,² convergence of the shape variable w(t) to a periodic function, extending prior L∞-convergence results.
• Use the Poincaré map and properties of convex moving sets to analyze long-term behavior and periodicity.
• Construct counterexamples to demonstrate that finite-time convergence is not guaranteed in general, even for simple geometries.
• Analyze the asymptotic behavior of the system under periodic actuation to derive the gait-dependent asymptotic velocity v₀(G).

Experimental results

Research questions

• RQ1 Does the solution of the sweeping process with a polyhedron of moving faces converge to a periodic solution in the W¹,² sense, rather than just in L∞?
• RQ2 Can the asymptotic average velocity of a soft crawler with dry friction be uniquely defined and independent of initial conditions?
• RQ3 Under what conditions does convergence to periodic behavior occur in finite time, and when is it only asymptotic?
• RQ4 How does the structure of the moving set K(t) — particularly its geometry and motion — affect the convergence properties of the system?

Key findings

• The paper establishes W¹,² convergence of solutions to periodic solutions for sweeping processes with polyhedral sets having translationally moving facets.
• All periodic solutions of the sweeping process share the same derivative, ensuring consistency in the asymptotic velocity.
• The asymptotic average velocity v₀(G) is well-defined and depends only on the gait G, not on initial conditions.
• Finite-time convergence is not guaranteed in general; counterexamples show that convergence can be only asymptotic, even for simple geometries.
• For the special case of a single segment (N=1), finite-time convergence to periodic behavior occurs within the first actuation period.
• The results apply to soft crawler models, enabling the rigorous definition of a gait-dependent asymptotic velocity and proving stabilization to a running-periodic solution.