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

Jonathan Kelly

arXiv:2608.29514v1cs.RO

TL;DR

Unknown constant measurement delays make aided inertial navigation a joint delay-and-state estimation problem with an exact single-measurement ambiguity. The paper develops an SGal(3) sliding window filter that interpolates delayed states and processes measurements jointly across time; simulations show substantially better consistency than a buffered EKF.

  • Problem

    Unknown constant aiding-measurement delays must be estimated jointly with navigation state because delayed observations constrain the trajectory at earlier times.

  • Method

    The paper develops an SGal(3) sliding window filter that retains past navigation states, interpolates delayed states, and applies aiding measurements jointly across the active window.

  • Results

    The SWF generally achieves lower errors and markedly better consistency than the BEKF, with navigation–delay ANEES/DoF of 1.21–1.40 versus 1.99–31.53.

  • Takeaways & Limitations

    Multiple-time measurements provide the temporal constraints needed to eliminate the single-measurement delay–state ambiguity and reduce spurious information gain.

Abstract

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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.

I. INTRODUCTION

Unknown timing offsets couple delayed aiding measurements to the navigation trajectory, creating observability and consistency challenges. The paper addresses these challenges with an SGal(3) sliding window that jointly uses measurements across time.

  • Unknown constant delays arise from unsynchronized clocks, buffering, processing, and communication latency, requiring joint estimation with the navigation state.
  • Augmenting the estimator state with delay is straightforward, but delayed measurements couple timing, trajectory, and vehicle motion and can cause inconsistency or divergence.
  • A single delayed measurement has an exact delay–state symmetry, so individual processing can leak spurious information along an unobservable direction.
  • The proposed SGal(3) sliding window retains past transformations, interpolates delayed states, and applies measurements jointly across the window.
  • Delay observability depends on both the trajectory and aiding measurement history, motivating temporal support through multiple states.
  • Simulation shows the buffered EKF rapidly becomes inconsistent, whereas even a short sliding window improves estimation accuracy and consistency.

III. PRELIMINARIES

The preliminaries introduce SGal(3) as a time-extended Lie-group representation for inertial-navigation geometry and uncertainty.

  • SGal(3) is a 10-dimensional Lie group describing transformations between inertial reference frames in relative motion.
  • Its group action maps spacetime coordinates as (p, t) to (Cp+vt+r, t+η), combining rotation, velocity boost, spatial translation, and time translation.
  • Because time belongs to the group, composition couples time translation to spatial translation through the velocity boost.
  • The Lie algebra sgal(3) is the tangent space at the identity, with wedge and vee operators relating matrix elements to R10 coordinates.

B. Observability and Identifiability

Delay estimation is analyzed through local observability over an interval because delayed measurements constrain navigation states at delay-dependent earlier times. The filter preserves this temporal structure through a windowed trajectory representation.

  • B. Observability and Identifiability: Observability cannot be assessed at one instant because each aiding measurement constrains the state at an earlier, delay-dependent time.
  • B. Observability and Identifiability: The rank test therefore uses measurements over an interval rather than a single measurement instant.
  • B. Observability and Identifiability: Although delay is formally a constant parameter, propagation makes it a trajectory shift, allowing the same rank test to address delay identifiability and state observability.
  • B. Observability and Identifiability: The active window contains Galilean transformations together with delay and shared gyroscope and accelerometer biases.
  • B. Observability and Identifiability: Interpolation queries the trajectory at arbitrary times as the delay estimate changes, while marginalization keeps the window size bounded.
  • B. Observability and Identifiability: The local uncertainty is represented by a Gaussian prior over perturbation coordinates, with nine navigation degrees of freedom per transformation.

B. Process Model

The process model propagates a window of Galilean transformations using IMU inputs, known gravity, bias models, and stochastic uncertainty. Its factorization separates absolute time from elapsed-motion propagation.

  • B. Process Model: The process model appends a correlated copy of the latest transformation, propagates it with IMU data, and updates the augmented prior covariance.
  • B. Process Model: Measured angular rates and specific forces are modeled with white noise, while gyroscope and accelerometer biases follow random walks.
  • B. Process Model: The delay is modeled as an unknown constant with τ_dot = 0, and the time coordinate advances with the IMU clock.
  • B. Process Model: Each transformation factors as X(t)=T(t)N(t), separating leading absolute-time translation from the zero-time-coordinate isochronous component.
  • B. Process Model: The group-affine motion integrates elapsed-time IMU inputs, with gravity acting on the left and body-frame inputs acting on the right.
  • B. Process Model: The factored propagation produces an isochronous intermediate state and forms the basis for the observability analysis.
  • B. Process Model: Process noise, bias random walks, and existing state uncertainty determine the augmented covariance, while active aiding updates are handled separately.

C. Measurement Model

The measurement model maps an aiding observation with arrival time t_y to the navigation trajectory at delayed time t_d=t_y−τ, using Galilean transformations and interpolation. Timing jitter shifts the queried trajectory time and is represented through effective covariance, while the deterministic Jacobian remains unchanged to first order.

  • Measurement timing: An aiding measurement observes selected navigation components at the earlier delayed time t_d=t_y−τ.The measurement is represented by a Galilean element whose time coordinate is advanced to the arrival time while navigation components remain unchanged.
  • Trajectory interpolation: The delayed transformation is predicted by interpolating between the two active-window transformations that bracket the estimated delayed time.The bracketing indices and interpolation coefficient are recomputed as the delay estimate changes, so measurement support can shift within the window.
  • Timing uncertainty: Independent timing jitter changes the effective query time to t_y−τ−n_t and is combined with measurement noise in the noisy measurement model.The resulting expression preserves the arrival-time coordinate while evaluating the navigation state at the jittered delayed time.
  • Linearization: The linearized residual uses a selection matrix and interpolation Jacobians to relate observed components, bracketing transformations, and delay sensitivity.The selection matrix extracts the measured components, while the delay sensitivity combines leading time translation and interpolation-point perturbation effects.
  • Measurement selection: Full-state, pose, and position-only aiding measurements select nine, six, and three navigation components, respectively.Timing jitter changes the effective measurement covariance but not the deterministic measurement Jacobian used for observability analysis to first order.

D. Gauss–Newton Update and Marginalization

The filter processes all aiding measurements whose estimated delayed times lie within the active window through a Gauss–Newton update. As the window advances, information associated with the departing transformation is incorporated into a reduced Gaussian prior by marginalization.

  • Active window: The active set M_k contains aiding measurements whose estimated delayed times lie within the active window, and all are processed jointly.Measurements are added on arrival and retired permanently when they leave the window.
  • Gauss–Newton update: The nonlinear residuals are linearized around the prior mean and used in a maximum a posteriori update.The supplied method passages define the deviation from the prior mean and the nonlinear measurement residual, with its first-order linearization.
  • Gauss–Newton update: Gauss–Newton recomputes delayed times, interpolation support, residuals, and Jacobians at every iteration while keeping the prior covariance P_k fixed.Delay and bias increments are additive, and transformation increments are applied on the right.
  • Covariance form: The covariance-form implementation represents IMU propagation entirely through P_k and does not carry a same-measurement posterior forward as the next prior.A local posterior covariance can be obtained after convergence from the inverse Gauss–Newton information matrix.
  • Marginalization: When the window advances, measurements depending on the departing transformation are folded into a Gaussian prior before that transformation is marginalized.The reduced-window prior retains the mean and covariance blocks associated with the remaining variables.

V. OBSERVABILITY UNDER MEASUREMENT DELAYS

The observability analysis studies delay, navigation state, and bias estimation in a minimal parameterization. It establishes that a single aiding measurement admits an exact delay–state symmetry, so the delay cannot be separated from the state using that measurement alone.

  • Problem formulation: The joint observability problem is analyzed using one nine-coordinate navigation state, one delay, and six bias coordinates.This gives a 16-coordinate perturbation for the minimal parameterization.
  • Single-measurement symmetry: For any single aiding measurement, changing the delay can be exactly compensated by changing the navigation state without changing the measurement.The symmetry leaves the measurement unchanged and prevents single-measurement separation of delay and state.

A. Single Measurements and Symmetry

A delayed full-state measurement constrains an earlier trajectory state, but a delay change can be absorbed by a corresponding reference-state transformation. This creates a one-dimensional family of indistinguishable delay–state pairs, while biases also remain unidentifiable from one measurement.

  • Delayed measurement: A full-state aiding measurement arriving at t_y observes the state at the earlier delayed time t_d=t_y−τ.The delayed time offset relative to the current reference is negative because the measurement refers to an earlier state.
  • State–delay compensation: If the delay changes by Δτ, the measurement time shifts by −Δτ and the reference state is transformed to compensate.The transformed state remains isochronous, although it need not equal the trajectory state at the shifted reference time.
  • Single-measurement symmetry: Varying Δτ produces a one-dimensional family of delay–state pairs that generate the same measurement.This exact symmetry means a single measurement cannot separate delay from state, and biases are also not identifiable from one measurement.
  • Multiple measurements: Measurements at distinct times can break the symmetry unless one navigation-state shift compensates for the delay change at every delayed time.A counting argument gives a lower bound based on 16 unknown coordinates and the residual dimensions of full-state, pose, and position-only measurements.

B. Measurement Nullspace Structure

A single delayed measurement has a delay–state null direction: changing the delay can be exactly compensated by changing the navigation state. Measurements at distinct delayed times can remove this ambiguity, provided the active window contains enough informative trajectory history.

  • The linearized measurement Jacobian has a delay–state null direction spanned by (µ, 1) in (δζk, δτ) coordinates.The measurement remains unchanged to first order when δζk = µδτ, with bias perturbations set to zero.
  • Measurements at distinct delayed times generally contribute distinct null directions, allowing the stacked Jacobian to attain full column rank.This requires sufficient trajectory information and the measurement counts specified by the observability analysis.
  • The window duration must exceed the largest anticipated delay with enough margin to retain the required number of well-separated measurements.Measurements available at an update constrain only the window portion ending τ before the current time.

VI. SIMULATION STUDIES

The simulations evaluate the sliding window filter against a buffered EKF across trajectories with different excitation and delays. They assess accuracy and consistency using trajectory errors, delay error, ANEES, and delay coverage.

  • The study compares the sliding window filter with the buffered EKF using Galilean-structure-respecting propagation and delayed-state reconstruction.A vanilla augmented-state EKF is excluded because prior work established poorer performance on the authors’ considered problem.
  • Three trajectories and 200 ms and 400 ms delays provide different excitation conditions relevant to delay observability.The Smooth trajectory has low-frequency oscillation; Excited adds sinusoidal motion, while Aggressive adds stronger acceleration and deceleration.
  • The sliding window filter retains five transformations 0.3 s apart, spans 1.2 s, and enables updates once six aiding measurements lie within the window.
  • Each trajectory–delay–filter combination uses 100 trials over 60 s with specified initial navigation, delay, and sensor-bias error distributions.
  • Performance uses trajectory-averaged RMS position, velocity, and orientation errors plus final delay RMSE; consistency uses ANEES and delay 3σ coverage.

B. Results and Discussion

The SWF consistently outperforms the BEKF in accuracy and consistency, while preserving cross-time constraints that address delay–state observability. However, it remains somewhat overconfident and incurs greater computational cost than recursive filtering.

  • Accuracy: The SWF achieves lower position and delay errors than the BEKF across the retained Monte Carlo trials.Position RMS is 0.072–0.086 m and delay RMSE is 3.22–3.41 ms for the SWF, versus 0.092–0.456 m and 3.70–20.24 ms for the BEKF.
  • Consistency: State ANEES/DoF is 1.21–1.40 for the SWF, compared with 1.99–31.53 for the BEKF.For delay alone, τ ANEES is 3.50–5.89 with 3σ coverage of 76.7–89.1% for the SWF, versus 12.97–472.81 and 0.2–46.1% for the BEKF.
  • Consistency: The BEKF can maintain bounded navigation errors despite a biased delay estimate and substantial overconfidence.Global aiding continues correcting position, velocity, and orientation, while process noise prevents complete covariance contraction.
  • Trajectory dependence: More aggressive motion does not necessarily provide more delay information, although greater trajectory informativeness can improve BEKF accuracy without fixing inconsistency.The Excited trajectory improves BEKF delay estimation, while additional alongpath acceleration and deceleration provide little extra information relative to the Smooth trajectory.
  • Mechanism: The SWF preserves cross-time correlations and jointly incorporates delayed measurements before marginalization, reducing spurious information gain.This design retains the constraints needed to address the exact single-measurement delay–state symmetry, but at greater computational cost than standard recursive filters.
  • Overall comparison: The BEKF becomes highly inconsistent, whereas the SWF markedly improves consistency while maintaining accurate delay and state estimates.Reconstructing past transformations from a finite IMU history alone is insufficient to resolve the underlying observability issue.
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