The Geometry of Transmission Zeros in Distance-Based Formations

Abstract

This letter presents a geometric input-output analysis of distance-based formation control, focusing on steady-state signal blocking between actuator and sensor pairs. We characterize steady-state transmission zeros via the zero-frequency residue matrix of the linearized closed-loop dynamics, and show that for connected, flexible frameworks, such zeros occur only on a measure-zero algebraic set. For infinitesimally rigid formations, we derive an explicit geometric condition under which steady-state transmission is lost, showing that zeros occur if and only if the sensor lies on an affine hyperplane determined by the actuator and the formation geometry. Building on this result, we introduce the global transmission polygon, a convex region that provides a constructive sensor-placement rule guaranteeing full-rank steady-state transmission under arbitrary single-node excitations.