Formation control from the generic combinatorial viewpoint: edge dynamics and directed sensing
Abstract
We develop a geometric framework for distance-based formation control that separates the evolution of inter-agent distances from its realization by compatible node motions, reducing the stability problem to the edge space. We show that local exponential convergence of the edge dynamics implies local exponential convergence of the formation, and that stability is certified by spectral properties of a linear edge operator. We introduce a hierarchy of generic spectral properties --- weak admissibility, admissibility, and strong admissibility --- that provide necessary conditions for local exponential stability. Specializing to directed sensing, we obtain a necessary and sufficient spectral condition for local stability at an arbitrary target, together with a quadratic sufficient certificate. These conditions reveal that stability depends jointly on the graph orientation and target geometry, and show that persistence is neither necessary nor sufficient for local convergence. We show that every generically rigid graph admits an admissible orientation and, for acyclic orientations, we give an exact combinatorial characterization of admissibility. Finally, the quadratic certificate leads to a semidefinite program for synthesizing stabilizing edge gains.
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BibTeX
@misc{theran2026formation,
title = {{Formation control from the generic combinatorial viewpoint: edge dynamics and directed sensing}},
author = {Louis Theran and Daniel Zelazo and Sean Dewar and Bernd Schulze},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13423},
eprint = {2609.13423},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13423},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Formation control from the generic combinatorial viewpoint: edge dynamics and directed sensing
AU - Louis Theran
AU - Daniel Zelazo
AU - Sean Dewar
AU - Bernd Schulze
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.13423
UR - https://arxiv.org/abs/2609.13423
AB - We develop a geometric framework for distance-based formation control that separates the evolution of inter-agent distances from its realization by compatible node motions, reducing the stability problem to the edge space. We show that local exponential convergence of the edge dynamics implies local exponential convergence of the formation, and that stability is certified by spectral properties of a linear edge operator. We introduce a hierarchy of generic spectral properties --- weak admissibility, admissibility, and strong admissibility --- that provide necessary conditions for local exponential stability. Specializing to directed sensing, we obtain a necessary and sufficient spectral condition for local stability at an arbitrary target, together with a quadratic sufficient certificate. These conditions reveal that stability depends jointly on the graph orientation and target geometry, and show that persistence is neither necessary nor sufficient for local convergence. We show that every generically rigid graph admits an admissible orientation and, for acyclic orientations, we give an exact combinatorial characterization of admissibility. Finally, the quadratic certificate leads to a semidefinite program for synthesizing stabilizing edge gains.
ER -