On the Identifiability of Aided Inertial Navigation Under Measurement Delays: A Geometric Approach

Jonathan Kelly

In aided inertial navigation, measurements from different sensors are often subject to unknown relative time delays. Consider a single aiding sensor whose measurements have an unknown but constant delay relative to the inertial-measurement data stream. We study the identifiability of the delay and the initial navigation state parameterizing the trajectory. Identifiability depends on both the temporal structure of the aiding measurements and the form of the trajectory. Using the special Galilean group, we determine the minimal number and type of aiding measurements needed to recover the delay and the navigation state. We also characterize a class of \emph{uninformative} trajectories, for which the delayed measurement model admits a continuous symmetry that prevents unique delay-and-state recovery. We show that each such trajectory is generated by a constant element of the Galilean Lie algebra, and connect this result to the familiar linearized, Jacobian-based analysis.