A Sliding Window Filter on the Galilean Group for Consistent Aided Inertial Navigation with Unknown Measurement Delays

We study aided inertial navigation when the aiding sensor measurements are subject to an unknown constant delay. The goal is to estimate the delay and navigation state jointly so that delayed measurements correct the trajectory at the appropriate times, yielding a more accurate navigation solution. We formulate the problem on the special Galilean group, which provides a natural state-space structure for aided navigation with uncertainty in both motion and timing. We then examine the observability of joint delay and state estimation and show that, for a single delayed measurement, the model admits an exact symmetry in which a change in the delay can be compensated by a change in the navigation state, leaving the measurement unchanged. Processing measurements individually allows spurious information to `leak' along the corresponding null direction of the measurement Jacobian, producing overconfident and inconsistent estimates. Applying measurements from multiple times together can eliminate this direction when the trajectory is informative enough. Motivated by this result, we develop a sliding window filter that retains a short history of navigation states and applies delayed aiding corrections jointly across the active window. We conduct a series of simulation studies to characterize estimator accuracy and consistency. The simulations demonstrate that an estimator that does not maintain an adequate window can rapidly become highly inconsistent, whereas even a short sliding window markedly improves consistency by providing the temporal support that observability requires.