  Distributed localization is essential in many robotic collective tasks such
as shape formation and self-assembly.Inspired by the statistical mechanics of
energy transition, this paper presents a fully distributed localization
algorithm named as virtual particle exchange (VPE) localization algorithm,
where each robot repetitively exchanges virtual particles (VPs) with neighbors
and eventually obtains its relative position from the virtual particle (VP)
amount it owns. Using custom-designed hardware and protocol, VPE localization
algorithm allows robots to achieve localization using sensor readings only,
avoiding direct communication with neighbors and keeping anonymity. Moreover,
VPE localization algorithm determines the swarm center automatically, thereby
eliminating the requirement of fixed beacons to embody the origin of
coordinates. Theoretical analysis proves that the VPE localization algorithm
can always converge to the same result regardless of initial state and has low
asymptotic time and memory complexity. Extensive localization simulations with
up to 10000 robots and experiments with 52 lowcost robots are carried out,
which verify that VPE localization algorithm is scalable, accurate and robust
to sensor noises. Based on the VPE localization algorithm, shape formations are
further achieved in both simulations and experiments with 52 robots,
illustrating that the algorithm can be directly applied to support swarm
collaborative tasks.
