# Limbless locomotion

# Summary

We study model locomotors that exploit shape changes and mechanical interactions with the environment for self-propulsion. We focus on simple, one-dimensional systems that can execute shape changes by propagating stretching or contraction waves along their bodies, and that interact with the substrate through tangential viscous forces. We consider two force-velocity laws: a linear, Newtonian one and a non-linear one of Bingham-type. The first one requires that some slip must occur for the tangential force to be non-zero, whereas the second one requires that a force threshold be overcome for slip to occur: this leads to stick-and-slip behaviour at the interface between the locomotor and its environment.

This model system arises in a variety of physical situations. The most direct examples are those of crawlers moving on a solid substrate lubricated by a thin layer of a viscous fluid, with the fluid being either Newtonian or of Bingham-type. Studies of these systems aim to discover the principles of the locomotion strategy of snails and to replicate them with artificial prototypes. Further examples include low Reynolds number swimming of slender organisms (in the Newtonian version, if hydrodynamic interactions are treated with the local drag approximation of Resistive Force Theory) and cells migrating on or within solid substrates, matrices and tissues.

With our study we show that:

- contrary to opposite claims in the literature, it is possible to obtain net advancement with cyclic shape changes even in the context of a linear, purely Newtonian interaction with a substrate. This requires that non-linearities arising from large deformations are correctly taken into account and exploited;
- at given gait (a fixed traveling wave of contraction), the displacements achievable with non-linear Bingham-type interactions are consistently larger than that achievable with a linear rheology;
- motion is oscillatory in time in the Newtonian case while the displacement of a typical material point is monotonic in time in the Bingham case;
- the sign of the displacement can be inverted, at fixed gait, by changing the rheology of the interactions with the environment.

# Related papers:

- Noselli, G. and DeSimone, A. (2014).

A robotic crawler exploiting directional frictional interactions: experiments, numerics, and derivation of a reduced model.*Proceedings of the Royal Society A*,**470**, 20140333. - Gidoni, P., Noselli, G. and DeSimone, A. (2014).

Crawling on directional surfaces.*International Journal of Nonâ€“Linear Mechanics*,**61**, 65-73. - Noselli, G., Tatone, A. and DeSimone, A. (2014).

Discrete one-dimensional crawlers on viscous substrates: achievable net displacements and their energy cost.*Mechanics Research Communications*,**58**, 73-81. - DeSimone, A., Guarnieri, F., Noselli, G. and Tatone, A. (2013).

Crawlers in viscous environments: linear vs non-linear rheology.*International Journal of Nonâ€“Linear Mechanics*,**56**, 142-147.