Space-Time Diversity in Observability and Estimation on Product Lie Groups

Robust state estimation in coupled dynamical systems depends critically not only on sensor quality but on the structural alignment between observation channels and the system's intrinsic dynamics. This paper develops a rigorous framework for analyzing spatial and temporal diversity in dynamical state estimation on product Lie groups, drawing structural parallels to diversity gains in space-time coding. Three main results are established: (i) coupling-based necessary and sufficient conditions for cross-factor observability, showing that a sensor local to one group factor renders another factor observable if and only if the dynamics propagate error directions across the corresponding Lie algebra components; (ii) a spatial diversity saturation theorem identifying precisely when additional observation channels fail to expand the propagated observation subspace and thus provide no structural benefit; and (iii) a time-space diversity decomposition that exactly separates instantaneous spatial information from accumulated temporal information in the estimation error covariance. The framework is applied to planar SE(2) and spatial SE(3) navigation, yielding exact observability guarantees for redundant and non redundant sensor architectures in modern robotics and autonomous vehicles. These results extend classical observability theory beyond Euclidean state spaces, exposing structural constraints invisible to standard rank-based analysis that fundamentally govern robust inference in coupled dynamical systems.