Autonomous platforms require accurate positioning to complete their tasks. To this end, a Kalman filter-based algorithms, such as the extended Kalman filter or invariant Kalman filter, utilizing inertial and external sensor fusion are applied. To cope with real-world scenarios, adaptive noise estimation methods have been developed primarily for classical Euclidean formulations. However, these methods remain largely unexplored in the tangent Lie space, despite it provides a principled geometric framework with favorable error dynamics on Lie groups. To fill this gap, we combine invariant filtering theory with neural-aided adaptive noise estimation in real-world settings. To this end, we derive a novel theoretical extension of classical innovation-based process noise adaptation formulated directly within the Lie-group framework. We further propose a lightweight neural network that estimates the process noise covariance parameters directly from raw inertial data. Trained entirely in a sim2real framework via domain adaptation, the network captures motion-dependent and sensor-dependent noise characteristics without requiring labeled real-world data. To examine our proposed neural-aided adaptive invariant Kalman filter, we focus on the challenging real-world scenario of autonomous underwater navigation. Experimental results demonstrate superior performance compared to existing methods in terms of position root mean square error. These results validate our sim2real pipeline and further confirm that geometric invariance significantly enhances learning-based adaptation and that adaptive noise estimation in the tangent Lie space offers a powerful mechanism for improving navigation accuracy in nonlinear autonomous platforms.
Figures & tables
Fig. 1: Block diagram of the proposed neural-aided adaptive innovation-based IKF for inertial/DVL fusion.
Layer
Dim.
Details
Conv1D
32
k=5 , BN, LeakyReLU
Conv1D
64
k=5 , BN, LeakyReLU
Conv1D
128
k=5 , BN, LeakyReLU
Mean Pool
128
Temporal average
FC
64
LeakyReLU, Dropout
FC
64
LeakyReLU, Dropout
TABLE I: Architecture of the proposed neural network for process noise estimation. The table reports the layer type, output dimensionality, and operations. Here, Conv1D denotes one-dimensional convolution, k is the kernel size, and BN indicates batch normalization.
Fig. 2: Modeling the innovation as a Gaussian distribution in the Lie algebra results in a warped, banana-shaped distribution on the Lie group manifold. On the left, the Lie group representation displays the prior estimated state ( x^t , green) and the posterior estimated state ( xt , purple). On the right, the corresponding Lie algebra representation illustrates the same prior and posterior states, where the 68% and 95% uncertainty contours form standard concentric circles.
Traj
AEKF
AR-IKF
NN-AEKF Variants
NN-AR-IKF Variants (ours)
S1 (MSE)
S10 (MSE)
S10+Huber
S1 (MSE)
S10 (MSE)
S10+Huber
1
4.5
4.3
4.3
5.3
5.3
4.6
4.7
4.5
2
3.8
3.5
3.8
3.5
3.7
2.9
2.9
3.0
3
3.9
3.7
4.1
4.2
4.1
3.3
3.6
3.5
4
5.1
4.6
4.5
5.0
5.6
3.1
3.8
4.0
5
2.7
2.8
2.9
2.8
2.7
2.7
2.6
2.5
TABLE II: Position RMSE [m] on the A-KIT dataset. Comparison between classical and neural-enhanced EKF and right-invariant Kalman filter variants.
Autonomous Underwater Vehicles (AUVs) navigating without GPS typically fuse inertial measurements with acoustic Doppler Velocity Log (DVL) velocities and pressure-derived depth. Posing the navigation state on a Lie group improves accuracy and consistency. However, state-of-the-art filters based on the Invariant Extended Kalman Filter (IEKF) append the Inertial Measurement Unit (IMU) biases as a Euclidean extension, which breaks the group-affine structure required for exact log-linear error dynamics, causing the reported covariance to degrade alongside the estimate. We apply the Tangent-Group (TG) symmetry, which carries the biases within the geometry of the state space, to derive an Equivariant Filter (EqF) for this system, leaving zero linearization error in the navigation states and second-order error only in the biases. We develop an equivariant output model for the DVL, whose update incurs only third-order linearization error, together with a direct pressure output. Monte Carlo simulations benchmark the TG-EqF against a Two-Frame-Group IEKF and a Multiplicative EKF. The TG-EqF reduces error by 18--25% against both alternatives in each of attitude, velocity, and position. The main benefit is in the covariance it estimates: its Average Normalized Estimation Error Squared (ANEES) stays closer to its nominal value of one than that of the others. Offline analysis on AUV field data corroborates the findings of the simulations, demonstrating reduced position drift.
Arihant Lunawat, Pieter van Goor, Frank Dellaert +1
The University of Sydney Sydney, Australia · Georgia Tech Atlanta, USA
State estimation is a key component in model-based control of walking robots and, more broadly, applicable wherever hidden variables must be inferred. The Kalman filter is widely used to estimate floating-base position and velocity by fusing multiple sensing modalities. However, tuning noise parameters is challenging and typically requires expert knowledge. Moreover, fixed noise parameters are unsuitable for varying gaits and environments. We propose an online adaptation strategy for the process noise covariance matrix Q and the measurement noise covariance matrix R. Specifically, we introduce a filter residual and innovation-based covariance adaptation method for legged robot state estimation and evaluate it against a baseline approach relying on IMU and foot force measurements. The proposed adaptation is implemented within an Invariant Extended Kalman Filter (InEKF) fusing IMU and leg kinematics. Experiments on indoor and outdoor datasets with a Unitree Go2 quadruped show that adapting R is sufficient and improves accuracy by 25% for the trotting gait compared to the fixed-tuned InEKF. Finally, the proposed residual-based adaptation achieves comparable performance to the foot force approach, without requiring foot force measurements or additional parameter tuning.
Modern autonomous navigation for unmanned ground vehicles relies on different estimators to fuse inertial sensors and GNSS measurements. However, the constant noise covariance matrices often struggle to account for dynamic real-world conditions. In this work we propose a hybrid estimation framework that bridges classical state estimation foundations with modern deep learning approaches. Instead of altering the fundamental unscented Kalman filter equations, a dedicated deep neural network is developed to predict the process and measurement noise uncertainty directly from raw inertial and GNSS measurements. We present a sim2real approach, with training performed only on simulative data. In this manner, we offer perfect ground truth data and relieves the burden of extensive data recordings. To evaluate our proposed approach and examine its generalization capabilities, we employed a 160-minutes test set from three datasets each with different types of vehicles (off-road vehicle, passenger car, and mobile robot), inertial sensors, road surface, and environmental conditions. We demonstrate across the three datasets a position improvement of 12.7% compared to the adaptive model-based approach. Thus, offering a scalable and a more robust solution for unmanned ground vehicles navigation tasks.
Gal Versano, Itzik Klein
Autonomous Navigation and Sensor Fusion Lab, Hatter Department of Marine Technologies, Charney School of Marine Sciences, University of Haifa, Israel.