The Geometry of Transmission Zeros in Distance-Based Formations

TL;DR

This paper provides a geometric analysis of transmission zeros in distance-based formation control, identifying conditions for signal blocking and introducing a geometric tool for sensor placement to ensure robust control.

Contribution

It characterizes steady-state transmission zeros in formation control, deriving explicit geometric conditions and introducing the global transmission polygon for sensor placement.

Findings

Transmission zeros are non-generic in connected frameworks.

Absence of internal flexes simplifies zero conditions to an affine hyperplane.

The global transmission polygon guarantees robust sensor placement.

Abstract

This letter presents a geometric input-output analysis of distance-based formation control, focusing on the phenomenon of steady-state signal blocking between actuator and sensor pairs. We characterize steady-state multivariable transmission zeros, where fully excited rigid-body and deformational modes destructively interfere at the measured output. By analyzing the DC gain transfer matrix of the linearized closed-loop dynamics, we prove that for connected, flexible frameworks, structural transmission zeros are strictly non-generic; the configuration-dependent cross-coupling required to induce them occupies a proper algebraic set of measure zero. However, because extracting actionable sensor-placement rules from these complex algebraic varieties is analytically intractable, we restrict our focus to infinitesimally rigid formations. For these baselines, we prove that the absence of internal flexes forces the zero-transmission condition to collapse into an explicit affine hyperplane defined by the actuator and the global formation geometry, which we term the spatial locus of transmission zeros. Finally, we introduce the global transmission polygon--a convex polytope constructed from the intersection of these loci. This construct provides a direct geometric synthesis rule for robust sensor allocation, guaranteeing full-rank steady-state transmission against arbitrary single-node excitations.