  We formulate and solve the locomotion problem for a bio-inspired crawler
consisting of two active elastic segments (i.e., capable of changing their rest
lengths), resting on three supports providing directional frictional
interactions. The problem consists in finding the motion produced by a given,
slow actuation history. By focusing on the tensions in the elastic segments, we
show that the evolution laws for the system are entirely analogous to the flow
rules of elasto-plasticity. In particular, sliding of the supports and hence
motion cannot occur when the tensions are in the interior of certain convex
regions (stasis domains), while support sliding (and hence motion) can only
take place when the tensions are on the boundary of such regions (slip
surfaces). We solve the locomotion problem explicitly in a few interesting
examples. In particular, we show that, for a suitable range of the friction
parameters, specific choices of the actuation strategy can lead to net
displacements also in the direction of higher friction.
