  In this paper we study crawling locomotion based on directional frictional
interactions, namely, frictional forces that are sensitive to the sign of the
sliding velocity. Surface interactions of this type are common in biology,
where they arise from the presence of inclined hairs or scales at the
crawler/substrate interface, leading to low resistance when sliding 'along the
grain', and high resistance when sliding 'against the grain'. This asymmetry
can be exploited for locomotion, in a way analogous to what is done in
cross-country skiing (classic style, diagonal stride). We focus on a model
system, namely, a continuous one-dimensional crawler and provide a detailed
study of the motion resulting from several strategies of shape change. In
particular, we provide explicit formulae for the displacements attainable with
reciprocal extensions and contractions (breathing), or through the propagation
of extension or contraction waves. We believe that our results will prove
particularly helpful for the study of biological crawling motility and for the
design of bio-mimetic crawling robots.
