Conic Frameworks Infinitesimal Rigidity

TL;DR

This paper introduces conic frameworks, a new structure modeling agent positions and clock biases, and provides complete characterizations of their infinitesimal rigidity, reducing constraints needed for formation stability in multidimensional settings.

Contribution

It offers the first complete characterization of infinitesimal rigidity for conic frameworks, decoupling space and bias variables, and reduces constraints compared to classical methods.

Findings

Complete characterization of infinitesimal rigidity for conic frameworks.

Decoupling of space and bias variables in rigidity analysis.

Reduction in the number of constraints needed for formation stability.

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.