Nonlinear Coupling Stability in Consensus-Based Multi-UAV SLAM: A Short Analytical Bound

When a swarm of UAVs agrees upon a common map framethrough a bandwidth-limited IoT network, the inter-vehiclecoupling is not linear: it saturates, and the local estimatorwhich it must discipline is itself drifting. This note provesone clean statement about such systems. For agents whoseestimator drift has slopes in [m, ℓ] and whose coupling lies inthe sector [α,β] on an undirected graph with Laplacian spectrum0 < λ2 ≤ · · · ≤ λN, consensus is exponential in continuoustime whenever the coupling gain satisfies κ > ℓ/(αλ2).The threshold is tight, and is attained exactly in the linearcase. In sampled time with period h the disagreement contractsby ρ =max{1−h(καλ2−ℓ), h(κβλN −m)−1} per step,so that the admissible step is h < 2/(κβλN −m), and at largegain the best attainable rate is fixed by the weighted eigenratioαλ2/(βλN). Every algebraic step is certified symbolically.The theorems are further checked without a single violation in96 continuous-time trials and in 66,022 discrete-time step-wisecomparisons on a ten-UAV swarm, and the certified samplingboundary coincides with the measured stability edge.