Conic Frameworks Infinitesimal Rigidity

Abstract

This paper introduces new structures called conic frameworks and their rigidity. They are composed by agents and a set of directed constraints between pairs of agents. When the structure cannot be flexed while preserving the constraints, it is said to be rigid. If only smooth deformations are considered a sufficient condition for rigidity is called infinitesimal rigidity. In conic frameworks, each agent $u$ has a spatial position $x_u$ and a clock offset represented by a bias $\beta_u$. If the constraint from Agent $u$ to Agent $w$ is in the framework, the pseudo-range from $u$ to $w$, defined as ${\left\lVert{x_u - x_w}\right\rVert} + \beta_w - \beta_u$, is set. Pseudo-ranges appear when measuring inter-agent distances using a Time-of-Arrival method. This paper completely characterizes infinitesimal rigidity of conic frameworks whose agents are in general position. Two characterizations are introduced: one for unidimensional frameworks, the other for multidimensional frameworks. They both rely on the graph of constraints and use a decoupling between space and bias variables. In multidimensional cases, this new conic paradigm sharply reduces the minimal number of constraints required to maintain a formation with respect to classical Two-Way Ranging methods.