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Pré-Publication, Document De Travail Année : 2022

Conic Frameworks Infinitesimal Rigidity

Résumé

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 xu and a clock offset represented by a bias βu. If the constraint from Agent u to Agent w is in the framework, the pseudo-range from u to w, defined as |xu − xw| + βw − βu, 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.
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Dates et versions

hal-03716656 , version 1 (07-07-2022)

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Colin Cros, Pierre-Olivier Amblard, Christophe Prieur, Jean-François Da Rocha. Conic Frameworks Infinitesimal Rigidity. 2022. ⟨hal-03716656⟩
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