  This paper considers a novel problem of how to choose an appropriate geometry
for a group of agents with only shape constraints but with a flexible scale.
Instead of assigning the formation system with a specific geometry, here the
only requirement on the desired geometry is a shape without any location,
rotation and, most importantly, scale constraints. Optimal rigid transformation
between two different geometries is discussed with especial focus on the
scaling operation, and the cooperative performance of the system is evaluated
by what we call the geometries degrees of similarity (DOS) with respect to the
desired shape during the entire convergence process. The design of the scale
when measuring the DOS is discussed from constant value and time-varying
function perspectives respectively. Fixed structured nonlinear control laws
that are functions on the scale are developed to guarantee the exponential
convergence of the system to the assigned shape. Our research is originated
from a three-agent formation system and is further extended to multiple (n > 3)
agents by defining a triangular complement graph. Simulations demonstrate that
formation system with the time-varying scale function outperforms the one with
an arbitrary constant scale, and the relationship between underlying topology
and the system performance is further discussed based on the simulation
observations. Moveover, the control scheme is applied to bearing-only
sensor-target localization to show its application potentials.
