  In many systems motion occurs on deformed and deformable surfaces, setting up
the possibility for dynamical interactions solely mediated by the coupling of
the entities with their environment. Here we study the "two-body" dynamics of
robot locomotion on a highly deformable spandex membrane in two scenarios: one
in which a robot orbits a large central depression and the other where the two
robots affect each other's motion solely through mutual environmental
deformations. Inspired by the resemblance of the orbits of the single robot
with those of general relativistic orbits around black holes, we recast the
vehicle plus membrane dynamics in physical space into the geodesic motion of a
"test particle" in a fiducial curved space-time and demonstrate how this
framework facilitates understanding the observed dynamics. The two-robot
problem also exhibits a resemblance with Einstein's general relativistic view
of gravity, which in the words of Wheeler: "spacetime tells matter how to move;
matter tells spacetime how to curve." We generalize this case the mapping to
include a reciprocal coupling that translates into robotic curvature-based
control schemes which modify interaction (promoting avoidance or aggregation)
without long-range sensing. Our work provides a starting point for developing a
mechanical analog gravity system as well as develops a framework that can
provide insights into active matter in deformable environments and robot
exploration in complex landscapes.
