Stabilization of periodic sweeping processes and asymptotic average velocity for soft locomotors with dry friction

Abstract

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ W^{1,2} $\end{document}</tex-math></inline-formula> 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.</p>

Limit cycles for dynamic crawling locomotors with periodic prescribed shape

“…At the quasistatic regime, an additional balance-breaking condition on the friction coefficients is necessary for the uniqueness of solution of the initial value problems (notice that the uniqueness argument of [20,Theorem 2.2] and [20,Example 3.2] can be straightforwardly adapted also to the case of prescribed shape). Yet, once uniqueness of solution is provided, uniqueness of the asymptotic average velocity of the crawler follows p. gidoni, a. margheri and c. rebelo [10,Theorem 11]. In this work, we show instead that, at a dynamic regime, uniqueness of solutions always holds, but uniqueness of the asymptotic average velocity for a gait is true only for smooth inputs (Theorem 3.4).…”

“…m n on a line, illustrated in Figure 1. Similar models have been considered, for instance, in [4,6,18,22,31,40], and in [20,10] at the quasistatic regime. We assume that each mass is non-negative, but we require the total mass of the system M := m i to be positive.…”

“…To better illustrate our contribution to the topic and possible future developments, we now make a brief comparison with the related results in [18,14,10]. In [18], the asymptotic behaviour of the discrete locomotor of Section 3 has been studied in the special case of continuous, autonomous, single-valued and strictly monotone friction forces, for a C 1 shape change, in the absence of external forces and for a more restrictive dissipative condition than (D4).…”

“…To make a comparison in the case of dry friction, which is the most valuable contribution in our work, a meaningful perspective comes from [10], where the analogous of our discrete models of Section 3 were studied assuming an elastic body, but only for dry friction at a quasistatic regime. At the quasistatic regime, an additional balance-breaking condition on the friction coefficients is necessary for the uniqueness of solution of the initial value problems (notice that the uniqueness argument of [20,Theorem 2.2] and [20,Example 3.2] can be straightforwardly adapted also to the case of prescribed shape).…”

On the optimal control of rate-independent soft crawlers

“…This does not suit our purposes, since we are interested in an arbitrarily long time behaviour. However it is know that sweeping processes with a periodic input converge asymptotically to a periodic output [23,25,26,14]. This has already been observed in the models with one link [20] and two links [21], also noticing that in the specific case of some "common sense" gaits the convergence to the asymptotic periodic orbit occurs within the first period.…”