  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.
