Title: On the stability and applications of distance-based flexible formations,
Abstract: This paper investigates the stability of distance-based \textit{flexible}
undirected formations in the plane. Without rigidity, there exists a set of
connected shapes for given distance constraints, which is called the ambit. We
show that a flexible formation can lose its flexibility, or equivalently may
reduce the degrees of freedom of its ambit, if a small disturbance is
introduced in the range sensor of the agents. The stability of the disturbed
equilibrium can be characterized by analyzing the eigenvalues of the linearized
augmented error system. Unlike infinitesimally rigid formations, the disturbed
desired equilibrium can be turned unstable regardless of how small the
disturbance is. We finally present two examples of how to exploit these
disturbances as design parameters. The first example shows how to combine rigid
and flexible formations such that some of the agents can move freely in the
desired and locally stable ambit. The second example shows how to achieve a
specific shape with fewer edges than the necessary for the standard controller
in rigid formations.