  We design a distributed coordinated guiding vector field (CGVF) for a group
of robots to achieve ordering-flexible motion coordination while maneuvering on
a desired two-dimensional (2D) surface. The CGVF is characterized by three
terms, i.e., a convergence term to drive the robots to converge to the desired
surface, a propagation term to provide a traversing direction for maneuvering
on the desired surface, and a coordinated term to achieve the surface motion
coordination with an arbitrary ordering of the robotic group. By setting the
surface parameters as additional virtual coordinates, the proposed approach
eliminates the potential singularity of the CGVF and enables both the global
convergence to the desired surface and the maneuvering on the surface from all
possible initial conditions. The ordering-flexible surface motion coordination
is realized by each robot to share with its neighbors only two virtual
coordinates, i.e. that of a given target and that of its own, which reduces the
communication and computation cost in multi-robot surface navigation. Finally,
the effectiveness of the CGVF is substantiated by extensive numerical
simulations.
