Unscented Kalman Filter with a Nonlinear Propagation Model for Navigation Applications

The unscented Kalman filter is a nonlinear estimation algorithm commonly used in navigation applications. The prediction of the mean and covariance matrix is crucial to the stable behavior of the filter. This prediction is done by propagating the sigma points according to the dynamic model at hand. In this paper, we introduce an innovative method to propagate the sigma points according to the nonlinear dynamic model of the navigation error state vector. This improves the filter accuracy and navigation performance. We demonstrate the benefits of our proposed approach using real sensor data recorded by an autonomous underwater vehicle during several scenarios.

💡 Research Summary

The paper addresses a critical limitation of the Unscented Kalman Filter (UKF) when applied to navigation systems: the accuracy of sigma‑point propagation depends heavily on the fidelity of the underlying dynamic model. While the Extended Kalman Filter (EKF) suffers from linearization errors, the UKF improves estimation by using a set of sigma points that capture the mean and covariance up to third‑order accuracy. However, conventional UKF implementations still rely on either a linearized error‑state model or a simplified nonlinear error propagation that does not fully exploit the true strap‑down inertial navigation equations.

To overcome this, the authors propose the UKF‑NESPM (Nonlinear Error State Propagation Method). The error‑state vector is defined with twelve components: three velocity errors, three misalignment (attitude) errors, and three each for accelerometer and gyroscope bias errors. For each time step, the twelve‑dimensional sigma points are added to the current mean navigation solution, producing a set of “modified” navigation states. These modified states are then propagated through the full inertial navigation algorithm—including attitude update, velocity integration, and position update—using the raw IMU measurements corrected by the respective bias errors of each sigma point. After propagation, the mean of the modified solutions is subtracted from each to obtain the updated error sigma points, which are then fed back into the standard UKF equations for mean, covariance, and Kalman gain computation. This approach effectively embeds the exact nonlinear dynamics of the strap‑down system into the sigma‑point propagation, eliminating the need for any analytical linearization.

The experimental validation uses data from the Snapir autonomous underwater vehicle (AUV) equipped with an iXblue Phins INS (fiber‑optic gyros) and a Teledyne RDI Work Horse Doppler Velocity Log (DVL). Six trajectories (four for training, two for testing) are processed; synthetic initial errors, sensor noise, and bias perturbations are added to emulate realistic operating conditions. The baseline is an Adaptive Neural UKF (ANUKF) that learns process‑noise covariance online. The proposed ANUKF‑NESPM is compared against this baseline using two metrics: total velocity root‑mean‑square error (VRMSE) and total misalignment RMS error (MRMSE). Results show that ANUKF‑NESPM reduces VRMSE by roughly 30 % and MRMSE by about 25 % on the test tracks, demonstrating a clear advantage of the exact nonlinear propagation.

The authors conclude that incorporating the full navigation algorithm into sigma‑point propagation markedly improves filter accuracy, especially in highly dynamic maneuvers typical of underwater vehicles. They acknowledge the increased computational load—25 sigma points for a 12‑dimensional state—and suggest future work on algorithmic optimization and hardware acceleration to enable real‑time deployment. Overall, the paper presents a solid methodological contribution that bridges the gap between theoretical UKF performance and practical navigation system requirements.