  Many animals and robots move using undulatory motion of their bodies. When in
close proximity undulatory motion can lead to novel collective behaviors such
as gait synchronization, spatial reconfiguration, and clustering. Here we study
the role of contact interactions between model undulatory swimmers: three-link
robots in experiment and multi-link robots in simulation. The undulatory gait
of each swimmer is generated through a time-dependent sinusoidal-like waveform
which has a fixed phase offset, $\phi$. By varying the phase relationship
between neighboring swimmers we seek to study how contact forces and spatial
configurations are governed by the phase difference between neighboring
swimmers. We find that undulatory actuation in close proximity drives
neighboring swimmers into spatial equilibrium configurations that depend on the
actuation phase difference. We propose a model for spatial equilibrium of
nearest neighbor undulatory swimmers which we call the gait compatibility
condition, which is the set of spatial and gait configurations in which no
collisions occur. Robotic experiments with two, three, and four swimmers
exhibit good agreement with the compatibility model. To probe the interaction
potential between undulatory swimmers we perturb the each longitudinally from
their equilibrium configurations and we measure their steady-state
displacement. These studies reveal that undulatory swimmers in close proximity
exhibit cohesive longitudinal interaction forces that drive the swimmers from
incompatible to compatible configurations. This system of undulatory swimmers
provides new insight into active-matter systems which move through body
undulation. In addition to the importance of velocity and orientation coherence
in active-matter swarms, we demonstrate that undulatory phase coherence is also
important for generating stable, cohesive group configurations.
