2025/11/17 by Solomon Goldgraber Casspi, Daniel Zelazo, Casspi, Solomon Goldgraber +1
#math.OC
paper · pdf · doi:10.48550/arxiv.2511.13187
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 Ri. We then introduce the local rotational subspace Ti, 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 Ti. 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.