The Geometry of Transmission Zeros in Distance-Based Formations
This work addresses the challenge of ensuring reliable communication in multi-agent systems, specifically for robotics and control applications, by providing geometric synthesis rules for sensor placement, though it is incremental as it builds on existing formation control theory.
The paper tackles the problem of steady-state signal blocking in distance-based formation control by characterizing transmission zeros and proving they are non-generic for flexible frameworks, but for rigid formations, it derives an explicit affine hyperplane called the spatial locus of transmission zeros and introduces a global transmission polygon for robust sensor allocation.
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.