Asymmetric Delay-Induced Translation in Displacement Formations

This letter studies a structural consequence of reciprocal delay asymmetry in first-order displacement-based formation control. Under an admissible delay margin, the prescribed displacement formation is preserved, while unequal reciprocal measurement delays destroy the nominal gradient structure of the closed-loop dynamics and activate the translational nullspace. As a result, the formation exhibits persistent rigid-body translation generated purely by endogenous interactions, without any external input or reference. We establish an explicit link between internal-force cancellation, centroid dynamics, and delay-induced discrepancy signals, and show that the locked collective velocity is determined by the transient accumulation of reciprocal delay asymmetry through the auxiliary layer. The admissible delay size is defined through the characteristic roots of the projected retarded system, so the locking mechanism is supported by a shape-convergence and discrepancy-integrability guarantee in this delay-margin regime. The synchronized reciprocal-delay case and the zero/nonzero velocity cases are also characterized. Simulations on a baseline triangular formation, a larger sparse connected graph, and single-edge delay-asymmetry sweeps support the analysis and show that the locked translation is quantitatively modulated by the reciprocal delay pattern.