The Geometry of Hidden Modes in Distance-Based Formation Control

Solomon Goldgraber Casspi; Daniel Zelazo

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arXiv · 2511.13187

The Geometry of Hidden Modes in Distance-Based Formation Control

Solomon Goldgraber Casspi·Daniel Zelazo

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Abstract

This paper presents a geometric input-output analysis of hidden modes in distance-based formation control. We study the linearized dynamics under a gradient control law to characterize the system's structural limitations and their dynamic consequences. Our main contribution is a unified geometric framework for the uncontrollable subspace: an exact characterization of its rigid-body component and a geometric bound on its deformational component. We first prove that the uncontrollable rigid-body modes are exactly the rotations about the actuated node, characterized by the global rotational subspace $\mathcal{R}_i$. We then introduce the local rotational subspace $\mathcal{T}_i$, consisting of the motions invisible to the actuator's local measurements, and prove that for minimally connected actuators, where the actuated node has as many neighbors as the dimension of the ambient space, the entire uncontrollable subspace is confined to $\mathcal{T}_i$. Finally, we demonstrate the dynamic implications of this structure by proving that the ability of the formation to recover its shape is determined by the alignment of the input with the local component of the rotational rigid-body mode, directly linking the geometry of hidden modes to disturbance rejection. We illustrate our results with a case study.

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BibTeX RIS

Solomon Goldgraber Casspi, Daniel Zelazo. 2026-07-21. The Geometry of Hidden Modes in Distance-Based Formation Control. https://arxiv.org/abs/2511.13187

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