  Many biological systems can synchronize their movement through physical
interactions, which are often through a fluid. However, the role of
intermittent mechanical contact in collective oscillatory systems is less
studied. In this paper we develop a robophysical experiment to study the phase
dynamics of mechanical oscillator pairs that interact only through inelastic
collisions. We vary the oscillator separation distance to study how proximity,
and mechanical collisions, affect their phase dynamics. In experiment we find
three dynamical states that depend on oscillator separation: 1) synchronized
in-phase oscillations in close proximity, 2) compatible oscillations in which a
stationary phase mismatch persists at modest separation distance, and 3)
repeated high-impact collisions in anti-phase synchronization at large
separation distance. To study the stability and emergence of these dynamical
states we develop a general phase-oscillator model subject to inelastic
collisions. We derive the collision-to-collision return map to study the
stability of the in-phase, compatible, and anti-phase states. To understand how
these pairwise phase dynamics may impact groups of robots or animals we extend
these results to larger collectives in numerical simulations. We observe
in-phase synchronization and compatibility in an oscillator lattice at close
spacing. However, at separation distances consistent with the onset of
anti-phase oscillations we observe complete asynchrony and continuous
collisions among neighbors. The loss of group synchronization is the result of
the pair-wise phase-repulsion dynamics from collisional interactions.
