Extended Kalman Filter

This paper proposes a cooperative target circumnavigation framework for multiple unmanned surface vehicles (USVs) operating without external localization. The objective is to maintain a uniform circular formation of a specified radius around a target using only limited onboard sensing. The framework adopts a heterogeneous perception strategy that distinguishes between the asymmetric sensing relationships with the target and among the USVs. Specifically, the USVs obtain relative range and displacement measurements through active perception and inter-vehicle communication, while bearing measurements to a non-cooperative target are acquired via passive sensors. To estimate relative positions--both among USVs and between each USV and the target--we employ a Maximum Correntropy Kalman Filter and a Pseudo-Linear Kalman Filter, respectively. A coupled oscillator-based formation controller is designed to ensure system observability while achieving circumnavigation. Theoretical analysis demonstrates that the controller ensures the relative motions between the USVs, as well as that between each USV and the target, satisfy the persistent excitation condition, thereby guaranteeing observability of the Kalman-based filters. The effectiveness of the proposed approach is validated through numerical simulations.

This letter proposes a novel distributed bearing-based pose estimator for time-varying multi-robot systems. The method uses angles computed from body-frame bearings to estimate the robots' positions in $\mathbb{R}^3$ without knowledge of their orientations. The orientations in $\mathrm{SO}(3)$ are recovered from the estimated positions, the bearings, and the bearing derivatives. The proposed observer only requires the (directed) sensing topology to be \textit{angle-rigid}, a weaker condition than the commonly used ones like bearing rigidity. Local uniform exponential stability of the proposed observer is established under the assumption of persistently exciting motions for a subset of robots. Simulations are presented and discussed to evaluate the scheme's effectiveness and practicality.

Kalman filter (KF)-based multi-object tracking (MOT) remains a strong baseline for autonomous driving due to its strong performance, computational efficiency and interpretability. In most practical systems, the process noise and measurement noise covariances are defined globally and shared across object classes, presuming identical uncertainty characteristics across heterogeneous traffic participants. This work revisits this assumption and proposes CANMOT, a class-aware and object-aligned noise modeling framework for KF-based 3D MOT. Class-specific diagonal process and measurement covariance matrices are introduced and optionally expressed in the object coordinate frame to preserve longitudinal-lateral anisotropy. Systematic experiments on the nuScenes benchmark show that class-aware and object-aligned noise modeling improves tracking performance and substantially reduces identity switches compared to state-of-the-art (SotA). In addition, the consistency of the estimated uncertainty is analyzed using the Average Normalized Estimation Error Squared (ANEES) and $χ^2$-based violation tests. The results reveal severe overconfidence in standard KF-based MOT baselines. While the proposed formulation improves calibration without modifying the underlying filtering framework, it still exhibits substantial inconsistency, highlighting the need for further research in this area. Code is available at https://github.com/rst-tu-dortmund/learned-3d-nms.

Kalman filtering performance is highly sensitive to model mismatch and noise covariance tuning. Learning-based approaches address these limitations but typically rely on supervised training with large datasets and do not produce consistent uncertainty estimates. In this paper, we propose a self-supervised Hybrid Adaptive Kalman Filter that learns structured corrections to system dynamics and process noise covariance from measurements alone while preserving the probabilistic structure of the filter. This allows the innovation likelihood to be computed and subsequently used for model classification via generalized Bayesian inference. Experimental results on real-world and simulated datasets demonstrate improved estimation accuracy and statistical consistency as well as robust classification performance across both low-data and large-data scenarios.

Knowledge of real-time road slipperiness, or even better, a refined estimate of peak grip potential, is a critical input for vehicle warning and intervention control systems. Typically, friction is estimated through dynamics-based recursive estimators by calculating the slip slope; however, its efficacy is heavily constrained by the vehicle dynamic scenario. When the vehicle is cruising and there is little to no slip, these methods become ineffective due to the inability of present-day production-grade sensors, such as wheel speed sensors, and methods to either measure or accurately estimate micro slip, which is crucial for distinguishing different surfaces. To address this challenge, the correlation between vehicle signals and road surface condition during cruising needs to be uncovered using machine learning. In this paper, a feature-based framework and an end-to-end data-driven framework are used to correlate the statistics of vehicle dynamics behavior with the condition of the road surface and perform binary classification into grip, dry or damp, and slip, snow or ice, conditions. A sliding-window approach is adopted to batch a short buffered window of wheel speeds, wheel torques, longitudinal acceleration, steering angle, and yaw rate, which are fed into a machine learning module for predicting the road state. Validation results on public-road data show scenarios where the data-driven method identifies the road surface correctly even during cruising, showing promise for accurate data-driven friction-related state estimators in the field of tire and vehicle dynamics.

Robust state estimation is central to robotic autonomy, yet classical Kalman filters struggle with frequency-dependent disturbances and model mismatch such as sensor vibrations, electromagnetic interference, and periodic noise. Although Deep Kalman Filter (DKF) variants extend the Extended Kalman Filtering (EKF) framework by learning latent transitions, they lack explicit mechanisms to suppress band-limited noise components that typically corrupt sensor measurements in real-world scenarios. We introduce the Frequency-Weighted Neural Kalman Filter (FW-NKF), a unified hybrid approach that embeds a causal spectral-shaping operator into the Kalman measurement residual and jointly learns observation, and transition networks. By adapting both the filter spectrum and the latent state representation, FW-NKF attenuates the noise-dominated frequency bands while capturing complex residual structures. We conduct extensive experiments on four heterogeneous benchmarks, including chaotic systems such as multi-dimensional Lorenz systems and full-body inertial pose estimation, and find a reduction in localization error of up to 10% as well as marked improvements in orientation accuracy. Our ablation studies confirm that frequency weighting and deep latent-state modeling contribute to overall performance.

This paper introduces an estimation and control framework for dynamic landing of multi-rotor uncrewed aerial vehicles on moving platforms. The proposed method integrates nonlinear model predictive control with a real-time minimum-jerk trajectory planner that enforces a prescribed touchdown time, enabling consistent timing during the terminal descent. To enhance robustness in the presence of time-varying sensing quality, we utilize an adaptive unscented kalman filter that updates the process and measurement noise statistics online. In addition, we provide a reference feasibility analysis showing that minimum-jerk references induce bounded thrust and torque commands under standard tracking hypotheses. The proposed framework is evaluated in simulation and hardware experiments, and it is shown to achieve repeatable landings and improved platform velocity prediction accuracy relative to EKF/UKF-based methods.

Due to their explicit priors and ability to model uncertainty, Bayesian methods have played a major role in dynamical latent variable modeling of single-cell neural recordings. However, modern-sized datasets have made overparameterized deep networks the preferred methods of choice due to their predictive power and favorable computational scaling. While many posterior approximations exist, all incur approximation errors. Recent work accounts for this error in the form of computational uncertainty but comes at the cost of quadratic complexity and assumes fixed model hyperparameters. Here we extend this development to model selection, including a novel training loss and optimization scheme, which yields tractable inference in large state-spaces. We introduce a framework, the Computation-Aware State-Space Model (CASSM), specifically designed for the scale-imbalanced regime, where the number of trials is significantly lower than the number of recorded neurons. In this regime, for both synthetic and real data, we show that our method is competitive with data-hungry deep networks, with significantly improved uncertainty calibration over previous attempts to scale Bayesian methods. Our experiments provide a roadmap to neuroscience researchers in choosing from a host of potential dynamical latent variable models given key dataset properties and constraints.

Dynamical systems reconstruction (DSR) aims to learn surrogate models that capture the dynamics underlying time-series data. Reliably deploying these surrogates requires uncertainty estimates consistent with the learned dynamics. We expose a dynamic-probabilistic consistency (DPC) gap: the pursuit of finite-horizon probabilistic objectives can degrade dynamics or decouple predictive uncertainty from the local tangent dynamics it ought to reflect. We isolate three mechanisms behind this gap: core collapse, noise masking, and blind uncertainty. Specifically, we show that open-loop Gaussian rollout objectives can penalize Jacobian-generated covariance growth in chaotic systems, encouraging optimization shortcuts that weaken physical expansion or decouple uncertainty from it. To mitigate this gap, we propose KAFFEE (Kalman-Aware Framework For Ergodic Emulation), a differentiable extended Kalman filter-based training framework that evaluates likelihood on local predictive residuals (innovations) while transporting covariance through learned local Jacobians. On stochastic hyperchaotic Lorenz-96, KAFFEE reduces the identified failure modes, improves reconstruction of dynamical invariants relative to open-loop objectives, and maintains competitive predictive scores. We further show that the DPC gap appears when probabilistically adapting a DSR foundation model across 13 chaotic systems, where KAFFEE enables in-context Bayesian filtering while largely preserving zero-shot dynamics.

This study presents a novel hybrid prognostic framework for uncertainty-aware Remaining Useful Life (RUL) estimation in turbofan engines using the NASA C-MAPSS dataset. The framework employs a state-aware strategy that bifurcates the engines operational lifespan into "healthy" and "degraded" regimes. An LSTM-based autoencoder, trained strictly on nominal data (RUL > 150 cycles), monitors reconstruction error to act as a robust state classifier. For the healthy regime, a Conditional Weibull Survival Analysis is used for Mean Residual Life estimation. For the degraded regime, a Probabilistic Neural Network with Monte Carlo Dropout captures both aleatoric and epistemic uncertainties. Rather than using rigid binary labels, a calibrated sigmoid function converts the autoencoders output into continuous state probabilities, dynamically weighting the final ensemble prediction. The primary strength of this framework is its generation of physically consistent uncertainty bands, yielding high-confidence predictions near end-of-life while accurately reflecting the inherent variance of early operation, providing a robust tool for risk-informed maintenance.

Robust and efficient state estimation is crucial for perception, navigation, and control in robotics. State estimation problems are conveniently modeled using the factor-graph framework as enabled by modern software packages such as GTSAM or g2o. However, the standard solvers included in such frameworks are local and may converge to poor local minima, posing significant safety concerns. Conversely, techniques based on convex relaxations have been shown to provide a means of globally solving or certifying many state estimation problems. However, these relaxations 1) often require substantial effort to formulate, and 2) may incur significantly higher cost compared to efficient local solvers, as they require solving a large semidefinite program (SDP). In this work, we address both shortcomings by 1) creating a new procedure within the GTSAM framework for automatically constructing convex SDP relaxations for any factor graphs with common factor and variable types, and by 2) exploiting the Bayes tree constructions native to GTSAM to decompose the SDP problem, leading to significant speedup in solver time for chordally sparse problems. We demonstrate the favorable scaling of this structure-exploiting global estimator compared to standard local solvers for two case studies: A 3D pose-graph SLAM problem with a ring factor graph and a 2D localization problem with a chain factor graph. The software framework is available at https://github.com/borglab/gtsam.

We connect stochastic resetting from non-equilibrium statistical physics with ridge regularization in statistical learning. For linear gradient flow, resetting to the origin at rate $r$ produces stationary mean $(X^\top X+rI)^{-1}X^\top y$, exactly the ridge estimator with penalty $λ=r$. This uses the known Laplace-transform relationship between ridge regression and exponential-time averaging of gradient flow, with the exponential time now interpreted as the stationary age associated with Poisson resetting. We then extend this identity to general renewal reset laws: the exponential reset time distribution is the unique renewal law whose stationary mean reproduces scalar ridge in every eigendirection as an exact filter identity for every positive curvature, while non-exponential renewal laws generate alternative spectral filters. At the fluctuation level, we study a separate additive Ornstein-Uhlenbeck extension with constant diffusion, interpreted as a stylized SGD approximation. In this setting, the equality holds only at the level of the mean, since the reset process has a nonzero stationary covariance from accumulated OU noise and reset-timing variance, whereas deterministic ridge is a fixed estimator with the same center. Stylized experiments compare the deterministic renewal-induced filters directly and illustrate when filters induced by non-exponential reset-time laws can differ predictively from ridge. The results for the stationary mean and the induced spectral filters are established for continuous-time gradient flow with isotropic resetting on quadratic objectives; the covariance and risk formulas additionally assume additive noise with state-independent covariance.

Reinforcement learning (RL) refines large language models (LLMs) by directly optimizing model behavior through reward signals. While accurate state value estimation is critical for stable training in classical RL, it remains an underexplored challenge in LLM post-training. In this work, we introduce the State Value Estimation Benchmark (SVEB) to assess state estimation within existing RL frameworks and show that critics in standard approaches like PPO collapse to a coarse group-average baseline. To address this, we propose two techniques: Numca, which leverages numerical spans as gradable milestones for state value estimation, and Hista, a framework that uses LLM's hidden states as representation to weighted average disjoint rollouts and their return. Extensive experiments demonstrate that both methods yield more accurate state value estimates and enhance training performance across different RL algorithms and model sizes without incurring significant computational overhead.

We describe our preliminary design of a real-time asynchronous event-based monocular odometry for planetary exploration. Operating under strict computational constraints, planetary rovers frequently encounter complex, unpredictable environments that demand high-speed sensing and robustness to high dynamic range (HDR) lighting. Event cameras address these needs by reporting asynchronous, pixel-wise brightness changes with microsecond resolution, significantly reducing data bandwidth while maintaining robustness in extreme lighting conditions. We propose an approach based on an Error-State Kalman Filter (ESKF) that leverages this asynchronous event stream to continuously estimate camera ego-motion. The camera state is updated with every tracked position output generated by RATE, a real-time asynchronous feature tracker.

State estimation is a fundamental problem in control and signal processing, for which the Kalman Filter provides an optimal solution under linear dynamics, Gaussian noise, and known noise covariances. However, these assumptions often fail in realistic sensing settings such as Doppler radar and LiDAR. In these cases, the optimal estimator is inherently nonlinear, which leads to systematic performance degradation. This creates a performance gap that cannot be eliminated by tuning the noise covariance parameters (i.e., the process and measurement noise in the Kalman Filter) alone. To address this limitation, we propose Kalman Evolve, a framework for discovering improved filtering algorithms by jointly optimizing both noise parameters and the update structure. Our approach leverages large language models (LLMs) as a structured prior over program space, enabling the generation of interpretable, non-affine modifications to the classical Kalman filter while preserving its recursive form. We provide analytical results establishing the suboptimality of affine estimators under common nonlinear sensing models, motivating the need for structure-aware updates. Across a range of synthetic and real-world tracking benchmarks, including Doppler radar, LiDAR-based localization, and pedestrian tracking, the discovered algorithms consistently improve over strong baselines such as the Optimized Kalman Filter, achieving up to 12% reduction in RMSE. These results suggest that optimizing the structure of the Kalman filter, rather than only its parameters, provides a practical and interpretable way to improve state estimation.

Real-world multi-agent reinforcement learning (MARL) systems must often operate under stale observations, stochastic communication delays, and intermittent packet loss. Policies trained under idealized synchronous conditions frequently exhibit significant performance degradation in these regimes because they act on outdated feedback. We propose a modular execution-stage state-estimation layer that replaces delayed communicated observations with current belief-state estimates. The framework integrates a learned Gated transition model with a recursive Kalman filtering layer to estimate instantaneous states from asynchronous measurements. A primary advantage of this approach is its modularity, The estimator serves as a plug-in for pre-trained policies, requiring no modifications to the original MARL training algorithm, architecture, or reward structure. Evaluation across diverse multi-agent and continuous-control benchmarks demonstrates that the proposed layer consistently enhances robustness to communication latency and message loss. The most significant performance gains are observed in coordination-intensive and dynamically unstable tasks where temporal consistency is critical for control.

We present FusionCore, an open-source ROS 2 sensor fusion package that fuses IMU, wheel encoder odometry, GPS, and Visual SLAM pose into a single 100 Hz odometry stream using a 23-state Unscented Kalman Filter (UKF). The 23rd state is an online estimate of the wheel encoder's systematic yaw rate bias, identified through GPS heading cross-covariance and subtracted during GPS blackouts to reduce heading drift in coast mode. FusionCore also estimates gyroscope and accelerometer biases as explicit filter states, handles GPS natively in ECEF without a separate coordinate projection node, applies per-sensor Mahalanobis chi-squared outlier gating calibrated to measurement degrees of freedom, and adapts sensor noise covariance automatically from the innovation sequence. VSLAM pose fusion enables GPS-denied operation with any visual odometry or SLAM system, including automatic recovery from map reinitialization. We evaluate against robot_localization on twelve full-length sequences (55-92 min each) from the NCLT public dataset. FusionCore achieves lower Absolute Trajectory Error (ATE) on ten of twelve sequences, with improvements ranging from 1.2x to 22.2x on winning sequences. The robot_localization UKF diverges numerically on all twelve sequences. FusionCore is available at https://github.com/manankharwar/fusioncore under the Apache 2.0 license.

Time series driven by unobserved latent states frequently exhibit abrupt jump discontinuities whose timing and magnitude cannot be predicted from observed history alone. Classical jump-diffusion models offer a principled mathematical framework but assume rigid parametric forms, while recent neural jump models operate on fully observed trajectories without inferring the hidden states that govern the dynamics. We propose \textit{Deep ZakaiJ}, a latent-state model for partially observed jump-diffusion systems that embeds the Zakai nonlinear filtering equation into a neural encoder--decoder architecture. The encoder recursively updates a belief over the latent state via Strang splitting into three interpretable substeps: prior propagation, diffusion innovation, and jump innovation, yielding a differentiable, first-order-accurate approximation of the exact filtering evolution. The decoder is a structured jump-diffusion model explicitly conditioned on the filtered belief, preserving the separation between continuous dynamics and discontinuous shocks. On synthetic, financial, and oceanographic datasets, \textit{Deep ZakaiJ} improves distributional forecasts while remaining competitive in point accuracy, achieving calibrated predictive intervals and recovering interpretable latent structure in synthetic and qualitative case studies.

Legged robots carry an IMU, but the inertial solution drifts because consumer-grade IMUs are noisy. However, the feet create intermittent contacts with the environment that can be used to mitigate that drift. This report develops a sequence of increasingly expressive legged robot state estimators that leverage this. In all cases, the floating-base state comprises attitude, position, velocity, and IMU biases. To model foot contacts, we start from the contact-aided invariant EKF of Hartley et al., albeit at a reduced contact update rate. This is then augmented by replacing the measurement update by a small factor graph. Finally, we turn the same factors into a fixed-lag smoother with contact-episode footholds, with and without an evolving IMU bias. To facilitate reproducibility and further research in proprioceptive legged odometry, all four variants are available in GTSAM (Dellaert et. al), and we additionally provide a ROS2-compatible implementation.

Image-Based Visual Servoing (IBVS) provides an efficient vision-guided control paradigm for unmanned aerial vehicles (UAVs) by directly regulating image-space errors. However, conventional IBVS controllers are vulnerable to two critical issues: loss of closed-loop stability near the target due to input and state constraints, and control failure caused by intermittent loss of moment-based visual features under aggressive motion. To address these challenges, this paper proposes a terminal-constraint model predictive control (TC-MPC) framework for IBVS, integrated with a Kalman filter (KF)-based state-prediction mechanism. The TC-MPC explicitly incorporates terminal-state constraints and a terminal cost into the IBVS error dynamics, ensuring recursive feasibility, improved convergence behavior, and closed-loop stability under control and state constraints. In parallel, the Kalman filter predicts the temporal evolution of image moments during short-term visual degradation, enabling the controller to preserve control continuity when moment measurements are partially unavailable. The proposed approach is validated through real-time UAV visual servoing experiments.

One of the significant challenges in legged robotics is achieving accurate odometry using only onboard proprioceptive sensors. In this study, we present a complete leg odometry pipeline based on an Error-State EKF (ESEKF) that relies exclusively on proprioceptive data: a body fixed IMU, joint encoders, and force sensors, where filter's state is corrected by feet determined to be in a stationary stance. The core of our contribution is fused contact detection and an uncertainty quantification module designed to explicitly identify and reject slippage. This module runs two detectors in parallel for each foot, 1) a debounced, force-based Gaussian Mixture Model (GMM) guided Finite State Machine (FSM) to confirm physical contact, and 2) a kinematic-based Generalized Likelihood Ratio Test (GLRT) on the estimated velocity of the foot. The continuous quality scores from both estimators are fused to detect if the foot is both physically loaded and kinematically stationary and served as an uncertainty signal for each contact. To validate our approach, we collected a multi-modal dataset of 29 sequences spanning diverse indoor and outdoor terrains (e.g., concrete, grass, pebble, and rock) total of 2.4 km long. We benchmarked our approach against both proprioceptive and exteroceptive methods. The results demonstrate our method's efficacy in providing accurate odometry estimates, robustly handling slippage-prone environments. We also share our code and real-time ROS2 package as open-source.

This paper proposes a new birth model including semantic information derived from deep learning to create an occlusion-aware Gaussian Mixture Probability Hypothesis Density (GM-PHD) filter. Unlike prior approaches that rely on simplistic or uniform assumptions, the proposed Semantic-Occlusion Aware (S-OA) birth model defines initialization terms by explicitly considering regions of occlusion and by leveraging semantic information about the environment. This enables the filter to accurately represent where new objects are more likely to appear, thereby improving tracking performance in complex and high-density driving scenarios. The method is evaluated through Monte Carlo simulations and experiments on the KITTI dataset. Performance is assessed by measuring the latency between first detection and track initiation, along with the mean absolute cardinality error and the Optimal Subpattern Assignment (OSPA) metric. Results demonstrate that the S-OA birth model reduces initialization delay in occlusion-heavy settings, matching or outperforming the strongest baseline in approximately 70% of cases. A sensitivity analysis of birth model weights is also provided. Overall, the findings underscore the benefits of integrating occlusion reasoning and semantic priors into Bayesian tracking frameworks for autonomous driving.

Bayesian filtering is a well-known problem that aims to estimate plausible states of a dynamical system from observations. Among existing approaches to solve this problem, particle filters are theoretically exact for non-linear dynamics and observations, but suffer from poor scalability in high dimensions. In this work, we show that diffusion-based emulators of dynamical systems can be used to implement, without additional training, an optimal variant of particle filters that has remained largely unexplored due to implementation challenges with classical numerical solvers. Experiments on nonlinear chaotic systems, including atmospheric dynamics, demonstrate that the proposed approach successfully scales particle filtering to high-dimensional settings.

Predictive coding (PC) offers a local and biologically grounded alternative to backpropagation in the training of artificial neural networks, yet to date, it remains slower, and performance degrades sharply as network depth increases. We trace both problems to a single simplification: current PC networks fix the precision matrix to the identity, discarding precision-weighted prediction errors that the variational derivation requires to be fast, local, and Bayesian. We close this gap by expressing predictive coding networks as deep hierarchical Gaussian filters (HGFs) and restore precision-weighted message passing, yielding dynamic uncertainty estimates and Hebbian-compatible update rules at every layer. The resulting networks can simultaneously learn activations, weights, and precisions under a single free-energy objective, with no global error signal, and resolve inference without requiring iterations or automatic differentiation. On FashionMNIST, our solution approaches backpropagation in epoch-level wall-clock cost while converging in fewer epochs, and outperforms it on online, data efficiency, and concept-drift tasks. We thus establish that closed-form variational inference with online precision learning provides a tractable foundation for deep predictive coding networks, retaining biological and interpretative advantages, without requiring iterative relaxation or global error signals.

Data assimilation (DA) addresses the problem of sequentially estimating the state of a dynamical system from noisy and incomplete observations. In this work, we employ a diffusion model as a world model to simulate and predict the system's dynamics. Recently, score-based diffusion models have learned global diffusion priors that effectively model (stochastic) dynamics, revealing strong potential for data assimilation. In this paper, we investigate how information from noisy observations can be incorporated to enable continuous correction and refinement of the predicted system state when using a diffusion prior. Motivated by particle filtering methods, we represent the posterior distribution using a set of particles. After receiving noisy observations, the diffusion model is guided using the observation likelihood to steer the generation process toward observation-consistent states. Nevertheless, such guidance does not guarantee sampling from the true posterior. We therefore employ a Sequential Monte Carlo approach over the diffusion trajectory, viewed as a path measure, to reweight and resample particles, thereby correcting the generation process and ensuring convergence toward the desired posterior distribution. This leads to an unbiased particle filtering method that rigorously fuses observational data with diffusion model simulations.

Unmanned Aerial Vehicles in dynamic environments face telemetry outages, structural vibrations, and regime-dependent noise that invalidate the stationary covariance assumptions of classical Kalman filters. The Sage-Husa Kalman Filter (SHKF) estimates noise statistics online, but its reliance on a static, scalar forgetting factor forces a strict compromise between steady-state stability and transient responsiveness. We introduce the N-Deep Recurrent Sage-Husa Filter (NDR-SHKF), which replaces this scalar parameter with a vector-valued memory attenuation policy learned by a hierarchical recurrent network operating on whitened innovation sequences. A bifurcated architecture routes shallow recurrent states to capture instantaneous sensor anomalies and deep states to encode sustained dynamic trends, while an auxiliary reconstruction objective prevents feature collapse. The complete filter, including recursive covariance updates, is trained end-to-end via backpropagation through time to directly minimize state estimation error. Evaluations on topologically distinct chaotic attractors demonstrate cross-domain generalization, outperforming purely data-driven baselines that diverge under out-of-distribution dynamics. Furthermore, evaluations on recorded real-world UAV flight datasets validate the framework's practical viability, demonstrating its capacity to bridge transitions into proprioceptive dead reckoning and outperform classical adaptive estimators during sensor outages.

Robustness is a critical requirement for deploying autonomous driving systems in the real world. Existing robustness benchmarks for autonomous driving have made important progress in studying the effects of image-level corruptions, such as adverse weather or camera degradation, on perception modules and open-loop planning outputs. However, deployment can also involve system-level imperfections, such as inference latency and ego-state estimation errors, which remain less studied in closed-loop E2E-AD evaluation. These imperfections can accumulate through the feedback loop and destabilize control. In this work, we present Bench2Drive-Robust, to our knowledge the first device-centric robustness benchmark for closed-loop end-to-end autonomous driving under realistic deployment perturbations. We systematically evaluate deployment-oriented perturbations arising from three major sources: camera-stream failures (frame drop, partial observation), ego-state estimation errors (GPS noise, and speed or odometry errors), and compute-induced control delay (model inference delay). We evaluate representative end-to-end driving methods and analyze their robustness under different perturbation severities. Our results show that these deployment-related perturbations can substantially degrade closed-loop driving performance, revealing robustness challenges that are not fully captured by conventional image-level corruption evaluations. By establishing a closed-loop evaluation protocol and demonstrating the substantial impact of these deployment-oriented perturbations, Bench2Drive-Robust defines practical robustness problems for end-to-end autonomous driving and encourages further research on deployment-aware robust driving systems.

Tightly coupled SLAM formulations under mixed-rate sensing often bind temporal processing, local geometric association, estimator formulation, and map-update policy into method-specific designs. Such binding makes it difficult to vary one design choice without re-engineering the rest of the state-estimation process. This paper presents FUSE, a framework for unified state estimation in vehicular and robotic SLAM systems. FUSE organizes the state-estimation interface around observation ingestion, propagation, update, and state query, and uses this interface to separate temporal processing, residual-ready local geometric association, estimator formulation, and map-update policy. A LiDAR--IMU instantiation is developed to examine the framework under mixed-rate sensing and directional degeneracy, where high-rate inertial propagation, LiDAR-triggered geometric update, residual screening, and degeneracy-aware correction operate through the same interface boundaries. On a 418~m loop-corridor sequence, the instantiation reports a 1.626 m end-to-end trajectory error, corresponding to a 7.9% relative error reduction compared with Faster-LIO, the lowest-error baseline on this sequence. The results support FUSE as a framework for organizing state-estimation design choices and show how the evaluated instantiation regularizes updates along weakly observable directions.

Accurate visual state estimation has been a central topic in robotics with a wide range of applications in robot navigation, autonomous driving, and autonomous flight. Recent advances in robot perception have led to significant improvements in the accuracy and robustness of state estimation, yet a fundamental challenge remains in how to quantify and calibrate its precision, i.e., how confident we are in an estimate and whether failures can be detected. This issue is particularly pronounced in visual-inertial odometry (VIO), where the heteroscedastic and multimodal nature of the problem makes uncertainty quantification especially difficult. This paper introduces MUSE (Multimodal Uncertainty Quantification of State Estimation), a novel real-time learning-based framework that leverages the strong and efficient sequential modeling capacity of Mamba to estimate localization uncertainty from multiple asynchronous sensor streams. Experiments on both public and in-house datasets demonstrate that MUSE achieves superior reliability and robustness compared to existing uncertainty quantification methods, and ablation studies justify the benefits of its key design choices.

Fast and reliable initialization is critical for monocular visual-inertial navigation systems (VINS), as it establishes the starting conditions for subsequent state estimation. Despite steady progress, most existing methods heavily rely on visual feature correspondences and require 3-4 seconds of sensory data for successful initialization, which limits their applicability and efficiency. With the advent of feed-forward 3D models that can directly predict point clouds from images, we revisit the visual-inertial initialization problem from a concise perspective. In this work, we propose a feature-free initialization framework that leverages up-to-scale point clouds predicted by a feed-forward 3D model, thereby obviating the need for visual feature tracking and estimation. This design substantially reduces system complexity and improves the reliability of initialization. Experiments on public datasets demonstrate that the proposed feature-free initialization method achieves the highest success rate, exceeding 90%, and significantly reduces the data duration required for successful initialization, typically to under 1.2 s. We further validate our method on a self-collected dataset covering various indoor and outdoor scenarios, demonstrating robust performance, particularly in visually degraded environments where existing methods often fail. The code and dataset are available at https://github.com/Yuantai-Z/FF-VIO-Init.

Digital twins (DTs) rely on continuous synchronization between physical systems and their virtual counterparts through online parameter estimation under uncertainty. In many practical settings, however, this task is challenged by low observability, weak excitation, nonlinear dynamics, and noisy or biased measurements. In this work, we develop a new mathematical framework that integrates Weighted Flow Matching (WFM) generative modeling with physics-informed nonlinear filtering to enhance parameter estimation in DTs. WFM relies on dynamic reweighting of training samples, which guides the generative model toward parameter regimes most informative of the evolving system state. This generative component is tightly coupled with a physics-informed filtering architecture based on the Unscented Kalman Filter (UKF), yielding a unified DT framework that combines data-driven probability transport with physically consistent state and parameter estimation. The effectiveness of the new integrated framework is demonstrated within a spacecraft DT architecture, where stable moment of inertia estimation is achieved under uncertain and noisy sensing, with significant performance improvements over established approaches such as Extended Kalman Filtering (EKF) and Ensemble Kalman Filtering (EnKF). These results highlight the potential of weighted generative modeling as a core mechanism for real-time DT synchronization in operational and mission-critical systems.

A central obstacle in nonlinear Bayesian filtering is representing the belief distribution. Moment-based filters address this by propagating polynomial moments and reconstructing a density from them. Recent work completes the predict-update loop via the maximum-entropy (MaxEnt) principle, but each step requires the partition function and its gradient, both $n$-dimensional integrals whose cost scales exponentially, restricting the demonstrated MaxEnt moment filtering to $n \le 4$. We avoid the partition function entirely by combining score matching with Stein's identity. In our setting, score matching reduces the density fit to a single linear solve whose coefficients are assembled directly from the propagated moments. The same parameters then drive Stein's identity to close the moment hierarchy during prediction and to recover posterior moments after each Bayesian update, keeping the full predict-update loop free of partition function evaluation. The resulting Score Kalman Filter (SKF) reduces to the classical information-form Kalman filter as a special case and performs every step through linear algebra. On nonlinear coupled-oscillator networks, the SKF runs through $n=20$ and reports lower RMSE than the EKF, UKF, EnKF, and particle-filter baselines on the tested synthetic benchmarks.

Automated machine learning pipelines increasingly produce models whose predictions must be explained to end users, auditors, and downstream decision systems. The most widely used feature attribution methods (SHAP, Integrated Gradients, LIME) are typically chosen by convention rather than measured fidelity, because rigorous evaluation is impeded by the absence of ground-truth attribution on real data. We propose AGOP-IxG, a fast per-sample attribution method for tabular classifiers that pre-multiplies the per-sample gradient by a top-$K$ rank-truncated Average Gradient Outer Product matrix, and evaluate it against four widely-used baselines on a controlled tabular benchmark designed for AutoML practitioners. In Part 1, we construct three synthetic multi-class tabular tasks (linear, sparse nonlinear, interaction-based) where ground-truth attribution per sample is analytically or numerically derivable, and compare five methods: AGOP-IxG, SHAP (DeepExplainer), Integrated Gradients, InputXGradient, and LIME. AGOP-IxG leads on Spearman rank correlation and noise feature mass on all three synthetic datasets, and on top-$k$ precision on the interaction dataset. Across all settings, AGOP-IxG is approximately $350\times$ to $1{,}650\times$ faster than SHAP. In Part 2, we evaluate global faithfulness on Adult Income and Credit Card Default using the ROAR protocol; the methods cluster within $\sim 1.7\%$ relative AUC, consistent with AGOP-IxG being optimized for per-sample local attribution rather than global feature ranking.

Robust state estimation for highly dynamic motion of legged robots remains challenging, especially in dynamic, contact-rich scenarios. Traditional approaches often rely on binary contact states that fail to capture the nuances of partial contact or directional slippage. This paper presents CoCo-InEKF, a differentiable invariant extended Kalman filter that utilizes continuous contact velocity covariances instead of binary contact states. These learned covariances allow the method to dynamically modulate contact confidence, accounting for more nuanced conditions ranging from firm contact to directional slippage or no contact. To predict these covariances for a set of predefined contact candidate points, we employ a lightweight neural network trained end-to-end using a state-error loss. This approach eliminates the need for heuristic ground-truth contact labels. In addition, we propose an automated contact candidate selection procedure and demonstrate that our method is insensitive to their exact placement. Experiments on a bipedal robot demonstrate a superior accuracy-efficiency tradeoff for linear velocity estimation, as well as improved filter consistency compared to baseline methods. This enables the robust execution of challenging motions, including dancing and complex ground interactions -- both in simulation and in the real world.

Data assimilation (DA) estimates the state of an evolving dynamical system from noisy, partial observations, and is widely used in scientific simulation as well as weather and climate science. In practice, filtering methods rely on frame-to-frame transition models. However, these models are fragile when observations are non-Markovian (when they form only a partial slice of a higher-dimensional latent state as in real-world weather data): they tend to accumulate errors over long horizons. At the same time, learned DA methods typically commit to a single regime, either filtering (nowcasting, real-time forecasting) or smoothing (retrospective reanalysis), which splits what should be a shared prior across application-specific pipelines. To address both issues, we introduce ForcingDAS, a unified and robust DA framework. Built on Diffusion Forcing with an independent noise level assigned to each frame, ForcingDAS learns a joint-trajectory prior instead of frame-to-frame transitions. This allows it to capture long-horizon temporal dependencies and reduce error accumulation. In addition, the same trained model spans the full filtering to smoothing spectrum at inference time. Specifically, nowcasting, fixed-lag smoothing, and batch reanalysis are selected through the inference schedule alone, without retraining. We evaluate ForcingDAS on 2D Navier-Stokes vorticity, precipitation nowcasting, and global atmospheric state estimation. Across all settings, a single model is competitive with or outperforms both learned and classical baselines that are specialized for individual regimes, with the largest gains observed on real-world weather benchmarks.

Many Inertial Navigation Systems (INS) use Global Navigation Satellite System (GNSS) position as the primary measurement to drive filter performance and bound error growth. However, commercial-grade GNSS receivers introduce unknown measurement delays ranging from 50 ms to 300 ms depending on sensor quality and operating mode. Such time delays can significantly degrade INS performance unless they are explicitly compensated for. Existing algorithms commonly estimate this delay offline, run the filter concurrently with GNSS measurements using buffered Inertial Measurement Unit (IMU) data, and predict the current state by forward-integrating buffered inertial measurements via IMU preintegration. The state-of-the-art online method is an Extended Kalman Filter (EKF) that explicitly models the time delay as a state parameter, which defines the preintegration duration. This paper introduces a novel geometric framework for modeling time-delayed INS, in which Galilean symmetry is leveraged to provide a joint representation of space and time for consistent state estimation. An Equivariant Filter (EqF) is derived for the coupled estimation of navigation states and time delay. Validation is performed on two fixed-wing Uncrewed Aerial Vehicles (UAV) with GNSS time lags of 90 ms and 120 ms. The test flights last two to three minutes. Simulations further investigate delays up to 500 ms and provide a statistical comparison against the state-of-the-art EKF. Results show that the EqF preserves accuracy and consistency, while the EKF lacks consistency and its performance degrades significantly with increasing measurement delays.

A likelihood-free transport filtering method is proposed based on the couplings between state and observation variables. By exploiting a block-triangular structure in the transport map, the analysis step of filtering is reformulated as the minimization of the maximum mean discrepancy (MMD) between the true joint measure and its transport-based approximation. To circumvent the non-convexity in the MMD optimization, we introduce a training-free transport filter method via gradient flows, which leads to an analytic computation for the transport map that implies the steepest descent direction of the MMD. The proposed approach accurately approximates non-Gaussian filtering posteriors and avoids particle collapse. We provide a convergence analysis for the expectation of the MMD between the approximated posterior and the truth posterior. Finally, we extend the method to high-dimensional problems through domain localization. Numerical examples demonstrate the superior performance of our approach over conventional filtering methods in nonlinear, non-Gaussian scenarios.

In this work, we study how to ensure probabilistic safety for nonlinear systems under distributional ambiguity. Our approach builds on a backup-based safety filtering framework that switches between a high-performance nominal policy and a certified backup policy to ensure safety. To handle arbitrary uncertainties from ambiguous distributions, i.e., where the distribution is not of specific structure and the true distribution is unknown, we adopt a distributionally robust (DR) formulation using Wasserstein ambiguity sets. Rather than solving a high-dimensional DR trajectory optimization problem online, we exploit the structure of backup-based safety filtering to reduce safety certification to a one-dimensional search over the switching time between nominal and backup policies. We then develop a sampling-based certification procedure with finite-sample guarantees, where empirical failure probabilities are compared against a Wasserstein-inflated threshold. We validate our method through simulations across three systems, from a Dubins vehicle to a high-speed racing car and a fighter jet, demonstrating the broad applicability and computational efficiency.

The Transformer is the foundational building block of modern AI, yet offers no principled handling of \emph{uncertainty}, which is prevalent in real applications: cold-start tokens with sparse histories in sequential recommendation, heterogeneous signal quality in language models, and attention sinks induced by unconstrained softmax. Every token is treated with uniform confidence. We show this uniformity is a degenerate case of our \emph{Bayesian Filtering Transformer} (BFT): attention becomes precision-weighted kriging, the residual connection becomes a Kalman update with adaptive gain, and the FFN becomes a dynamics model propagating precision via a Jacobian--plus--process-noise rule. Observation precision comes from a parameter-free Restricted Maximum Likelihood (REML) estimator with a conjugate Bayesian prior. BFT replaces any Transformer layer with negligible overhead. On sequential recommendation, BFT applied to three major architectures yields significant gains on six benchmarks, with the largest improvements on cold-start users and rare items where uncertainty is highest. On supervised fine-tuning of large language models with noisy data, BFT improves robustness in two regimes: noisy supervision (token-label corruption in question answering) and noisy context (retrieval-augmented QA with real RAG distractors). A single principled modification -- restoring precision -- unlocks substantial headroom across both classical sequence-modeling and modern LLM regimes.

Contact-rich robot dynamics are hybrid: a single observation can match several latent states and contact regimes (free, impact, stick--slip). A standard amortized filter that places no probability on a feasible contact transition will permanently lose the branch the robot actually follows. We introduce VHYDRO, a variational hybrid dynamics learner that prevents this branch loss. At each step, VHYDRO mixes the learned proposal with a feasible transition law before sampling and importance weighting, ensuring that every transition retained by the model-feasible carrier remains covered. VHYDRO jointly infers a continuous latent state and a discrete contact mode, and fits a sparse port-Hamiltonian law to each recovered regime. On top of this, three guarantees connect: support coverage stabilizes filtering, the stabilized filter concentrates the discrete contact posterior on coherent regimes, and mode-pure segments admit sparse port-Hamiltonian recovery. The recovery error separates cleanly into filtering, derivative, mode-impurity, and physics-residual parts. Three empirical findings track the same mechanism. Under heavy occlusion the support-safe filter stays usable while a non-defensive proposal collapses. On ManiSkill demonstrations and on four Sawyer/BridgeData task families the discrete state forms temporally coherent contact-regime segments that the discrete state yields a stronger joint profile across ARI, change-point F1, and segment purity than post-hoc and mode-free baselines. On hybrid systems with known equations the mode-conditioned sparse fit recovers the active physical terms; purely predictive baselines do not.

When people share the same documents and observations yet reach different conclusions, the disagreement often shifts into a judgment that the other party is cognitively defective, irrational, or acting in bad faith. This paper argues that such divergence is better described as a form of non-identifiability inherent in inference and learning, rather than as a defect of the other party. We organize the phenomenon into two levels: (i) $θ$-level non-identifiability, where conclusions diverge under the same world model $W$ because inference settings differ; and (ii) $W$-level non-identifiability, where repeated use of an inference setting $θ$ biases data exposure and update rules, causing the learned world model $W$ itself to diverge. We introduce an inference profile $θ= (R, E, S, D)$, consisting of Reference, Exploration, Stabilization, and Horizon, and show how outputs can split even for the same observation $o$ and the same $W$. We further explain why disagreements tend to project onto a small number of bases -- abstract versus concrete, externalizability, and order versus freedom -- as a consequence of general constraints on learning systems: computational, observational, and coordination constraints. Finally, we relate the framework to deep representation learning, including representation hierarchy, latent-state estimation, and regularization-exploration trade-offs, and illustrate the framework through a case study on AI regulation debates.

We compare three state-of-the-art proprioceptive state estimators for quadruped robots: MUSE [1], the Invariant Extended Kalman Filter (IEKF) [2], and the Invariant Smoother (IS) [3], on the CYN-1 sequence of the GrandTour Dataset [4]. Our goal is to give practitioners clear guidance on accuracy and computation time: we report long-term accuracy (Absolute Trajectory Error, ATE), short-term accuracy (translational and rotational Relative Pose Error, RPE), and per-update computation time on a fixed hardware/software stack. On this dataset, RPEs are broadly similar across methods, while IEKF and IS achieve a lower ATE than MUSE. Runtime results highlight the accuracy-latency trade-offs across the three approaches. In the discussion, we outline the evaluation choices used to ensure a fair comparison and analyze factors that influence short-horizon metrics. Overall, this study provides a concise snapshot of accuracy and cost, helping readers choose an estimator that fits their application constraints, with all evaluation code and documentation released open-source at https://github.com/iit-DLSLab/state_estimation_benchmark for full reproducibility.

Many downstream decisions in complex terrain require fast wind estimates at a small number of user-specified locations and heights for a given forecast valid time, rather than another dense forecast field on a fixed grid. We present WindINR, a latent-state implicit neural representation framework for continuous high-resolution local wind query and sparse-observation correction. WindINR maps static terrain descriptors, a low-resolution background field, and continuous query coordinates to a high-resolution wind state through a latent-conditioned decoder. To enable rapid inference-time correction, WindINR separates reusable representation learning from sample-specific latent-state correction. During training, a privileged encoder infers a reference latent state from high-resolution supervision, a deployable latent predictor estimates an initial latent state from inference-time inputs alone, and their discrepancies are summarized into a dataset-adaptive Gaussian prior over latent corrections. At inference time, within the WindINR module, network weights remain fixed and only the latent state is updated by minimizing a regularized correction objective using sparse observations and their uncertainty. In controlled OSSEs over the Senja region, including a UAV-aided approach scenario and random-observation robustness tests, WindINR improves local high-resolution wind estimates by updating only a compact latent state rather than the full network. The corrected representation remains continuously queryable at arbitrary coordinates and, in our CPU benchmark, yields about a $2.6\times$ online-correction speedup over full-network fine-tuning, suggesting a practical interface between kilometer-scale background products, sparse local observations, and wind queries in complex terrain.

We show that the core components of the Transformer---attention, residual connections, and normalization---arise naturally from a single geometric state estimation problem. Modeling the latent state in polar coordinates naturally separates radial and hyperspherical dynamics, yielding a precision-weighted filtering procedure in which normalization enforces the hyperspherical constraint, attention aggregates directional evidence, and the residual connection implements an incremental state update. The standard Transformer block with rotary positional encodings is recovered by discarding the geometric correction terms of the resulting state estimator, showing that its architecture follows from the underlying estimation problem rather than from independent design choices. The proposed \textit{Polar Transformer} retains these geometric corrections.

In safety-critical scenarios, the protection level of the autonomous navigation system is crucial for enabling mobile robots to perform safe tasks. However, existing studies on probabilistic navigation systems for robots usually perform offline accuracy evaluations using limited datasets and assume that the results can be applied to unknown real-world environments. As a result, current autonomous mobile robots often lack protection levels for online safety assessment. To fill this gap, we propose a safety-critical LiDAR-inertial odometry (LIO) that provides deterministic protection levels based on on-manifold deterministic state estimation. By adopting the unknown but bounded assumption, we derive a neat closed-form relationship between point cloud noise and the uncertainty of the estimation from the iterated closest point algorithm. Using this relationship, we design an on-manifold ellipsoidal set-membership filter and implement it within the LIO system. Leveraging the properties of the set-membership filter, our system offers the feasible sets of the estimated locations as the deterministic protection levels, serving as safety references for the robots' downstream autonomous operations. The experimental results show that our system can provide effective deterministic online safety references for diverse robots in various environments.

Continuous-time state estimation is gaining in popularity due to its abilities to provide smooth solutions, handle asynchronous sensors, and interpolate between data points. While there are two main paradigms, parametric (e.g., temporal basis functions, splines) and nonparametric (Gaussian processes), the latter has seen less adoption despite its technical advantages and relative ease of implementation. In this article, we seek to rectify this situation by providing a new simplified explanation of GP continuous-time estimation rooted in the language of factor graphs, which have become the de facto estimation paradigm in much of robotics. To simplify onboarding, we also provide three working examples implemented in the popular GTSAM estimation framework.

Traffic state estimation from sparse fixed sensors is challenging because physics-informed neural networks (PINNs) tend to over-smooth sharp transitions admitted by the Lighthill-Whitham--Richards (LWR) model. This study proposes Two-Stage Domain Decomposition Physics-Informed Neural Networks (TSDD-PINN), an observation-aligned framework for LWR-based offline speed-field reconstruction. The framework supports spatial, temporal, and space--time refinement. Matched direction analysis shows that spatial refinement has the lowest mean error and less than half the training time of space--time refinement in the tested setting, while temporal refinement is faster. A global parent PINN is first trained. In the controlled spatial implementation, its residual profile guides a deterministic partition for warm-started child networks. An optional operational safeguard retains Stage~1 when the prespecified screen does not activate. The primary I-24 MOTION evaluation spans five days, five sensor configurations, and ten seeds per configuration, yielding 1{,}500 runs. Controlled TSDD-PINN attains the lowest relative $L_2$ error in 18 of 25 configurations and 14 of 15 sparse-sensing cases, while training 2.4 times faster than the extended PINN (XPINN) baseline under the evaluated implementations and training budgets. Non-neural comparisons show that the advantage over interpolation is concentrated under sparse sensing, whereas dense sensing often favors interpolation. A separate 250-run operational evaluation finds infrequent activation and motivates the Stage-1-preserving safeguard. The residual is treated as an indicator of model difficulty rather than a validated shock detector. The evidence supports a sensing-density-dependent operating range rather than uniform improvement.

The theory of state tracking in recurrent architectures has predominantly focused on expressive capacity: whether a fixed architecture can theoretically realize a set of symbolic transition rules. We argue that equally important is error control, the dynamics governing hidden-state drift along the directions that distinguish symbolic states. We prove that affine recurrent networks, a class of models encompassing State-Space Models and Linear Attention, cannot correct errors along state-separating subspaces once they preserve state representations. Consequently, practical affine trackers do not learn robust state tracking; rather, they learn finite horizon solutions governed by accumulated state-relevant error. We characterize the mechanics of this failure, showing that tracking remains readable only while the accumulating within-class spread remains small relative to the initial between-class separation. We demonstrate empirically on group state-tracking tasks that this breakdown is predictable: tracking collapses when the distinguishability ratio crosses the readability threshold of the trained decoder. Across trained models, the point of this crossing predicts the horizon at which downstream accuracy fails. These results establish that robust state tracking is determined not only by an architecture's theoretical expressivity but crucially by its error control.

Hybrid State-Space models combine Attention with recurrent State-Space Model (SSM) layers, balancing eidetic memory from Attention with compressed fading memory from SSMs. This yields smaller Key-Value caches and faster decoding than Transformers, along with a richer architectural design space. Exploring that design space at scale has so far required training from scratch, a barrier that has kept most large-model Hybrid research within a narrow range of architectures. We introduce Priming, a method that turns Hybrid architecture design from a pre-training problem into a knowledge transfer one. Priming initializes a Hybrid model from a pre-trained Transformer and, through short alignment and post-training phases, recovers downstream quality using less than 0.5% of the source model's pre-training token budget. Priming is agnostic to the source Transformer family (e.g., Qwen, Llama, Mistral), model class (dense or Mixture-of-Experts), and model scale. Priming enables us to run the first controlled comparison of SSM layer types at scale under identical conditions. We evaluate, Gated KalmaNet (GKA), Gated DeltaNet (GDN), and Mamba-2, and show that their expressiveness hierarchy, GKA>GDN>Mamba-2, directly predicts downstream performance on long-context reasoning tasks. We scale Priming to 8B/32B reasoning models with native 128K contexts. Our Hybrid GKA 32B improves over its source Qwen3-32B by +3.8 average reasoning points, while staying within 1% of a Transformer post-trained on the same data and enabling up to 2.3x higher decode throughput. To foster research on Hybrid architectures, we release a model zoo of primed Hybrid models for long-context reasoning and instruction following, together with the Priming training and inference code (Sequence Parallelism algorithms for long-context training, optimized GKA kernels, and vLLM serving plugin), all under Apache~2.0 License.

Spiking Neural Networks (SNNs) offer notable advantages in biological plausibility and energy efficiency, making them promising candidates for building low-power Transformers. However, existing Spiking Transformers largely adhere to a passive reactive paradigm, which struggles to focus on task-relevant information and incurs substantial computational overhead when processing redundant visual data. To overcome this fundamental yet underexplored limitation, we propose SAFformer, a novel Spiking Transformer architecture based on an active predictive filtering paradigm. Inspired by the brain's predictive coding mechanism, SAFformer actively suppresses predictable signals and focuses on salient visual features. Extensive experiments show that SAFformer establishes new state-of-the-art performance on CIFAR-10/100 and CIFAR10-DVS. Remarkably, on ImageNet-1K, it achieves 80.44% Top-1 accuracy with only 26.58M parameters and an energy consumption of 5.88 mJ, demonstrating an exceptional balance between accuracy and efficiency.

Designing an open-world quadrupedal loco-manipulation system is highly challenging. Traditional reinforcement learning frameworks utilizing exteroception often suffer from extreme sample inefficiency and massive sim-to-real gaps. Furthermore, the inherent latency of visual tracking fundamentally conflicts with the high-frequency demands of precise floating-base control. Consequently, existing systems lean heavily on expensive external motion capture and off-board computation. To eliminate these dependencies, we present SigLoMa, a fully onboard, ego-centric vision-based pick-and-place framework. At the core of SigLoMa is the introduction of Sigma Points, a lightweight geometric representation for exteroception that guarantees high scalability and native sim-to-real alignment. To bridge the frequency divide between slow perception and fast control, we design an ego-centric Kalman Filter to provide robust, high-rate state estimation. On the learning front, we alleviate sample inefficiency via an Active Sampling Curriculum guided by Hint Poses, and tackle the robot's structural visual blind spots using temporal encoding coupled with simulated random-walk drift. Real-world experiments validate that, relying solely on a 5Hz (200 ms latency) open-vocabulary detector, SigLoMa successfully executes dynamic loco-manipulation across multiple tasks, achieving performance comparable to expert human teleoperation.

This work proposes a novel control and estimation approach for aerial manipulation of a cable-suspended load using Unmanned Aerial Vehicles (UAVs). Common approaches in the state of the art have practical limitations, relying on direct load measurements and Lagrangian methods for dynamic modeling. The lack of a straightforward dynamic model of the system led us to propose adopting the Udwadia-Kalaba method to explicitly incorporate the cable's geometric constraints. This formulation allowed for the consistent derivation of the tension force and its direct integration into the NMPC prediction model. Additionally, we propose a sensorless load state estimation based on the same geometric constraints. Results from real-robot experiments demonstrated that the explicit inclusion of load dynamics in the optimization problem significantly reduces trajectory-tracking errors and yields better overall performance compared to strategies based on incomplete models.

Self-supervised learning (SSL) excels at finding general-purpose latent representations from complex data, yet lacks a unifying theoretical framework that explains the diverse existing methods and guides the design of new ones. We cast SSL as latent distribution matching (LDM): learning representations that maximize their log-probability under an assumed latent model (alignment), while maximizing latent entropy to prevent collapse (uniformity). This view unifies independent component analysis with contrastive, non-contrastive, and predictive SSL methods, including stop gradient approaches. Leveraging LDM, we derive a nonlinear, sampling-free Bayesian filtering model with a Kalman-based predictor for high-dimensional timeseries. We further prove that predictive LDM yields identifiable latent representations under mild assumptions, even with nonlinear predictors. Overall, LDM clarifies the assumptions behind established SSL methods and provides principled guidance for developing new approaches.

This paper introduces the ensemble directional Kalman filter (EnDKF), an ensemble-based Kalman filtering approach for pose tracking that jointly estimates an object's position and attitude using ideas from directional statistics. The EnDKF integrates a unit-quaternion attitude representation to move beyond canonical Kalman filter mean and covariance assumptions that poorly capture directional uncertainty. Experiments on a synthetic constant-velocity constant-angular-velocity system and a digital-twin head-tracking scenario using the FoundationPose algorithm demonstrate a significant reduction in error as opposed to merely using measurements.

This paper introduces an extension of generalised filtering for online applications. Generalised filtering refers to data assimilation schemes that jointly infer latent states, learn unknown model parameters, and estimate uncertainty in an integrated framework -- e.g., estimate state and observation noise -- at the same time (i.e., triple estimation). This framework appears across disciplines under different names, including variational Kalman-Bucy filtering in engineering, generalised predictive coding in neuroscience, and Dynamic Expectation Maximisation (DEM) in time-series analysis. Here, we specialise DEM for ``online'' data assimilation, through a separation of temporal scales. We describe the variational principles and procedures that allow one to assimilate data in a way that allows for a slow updating of parameters and precisions, which contextualise fast Bayesian belief updating about the dynamic hidden states. Using numerical studies, we demonstrate the validity of online DEM (ODEM) using a non-linear -- and potentially chaotic -- generative model, to show that the ODEM scheme can track the latent states of the generative process, even when its functional form differs fundamentally from the dynamics of the generative model. Framed from a neuro-mimetic predictive coding perspective, ODEM offers a biologically inspired solution to online inference, learning, and uncertainty estimation in dynamic environments.

This paper presents a novel model predictive control (MPC) approach for autonomous pick-and-place between moving platforms with a hook-equipped aerial manipulator. First, for accurate and rapid modeling of the complex dynamics, a digital twin model of the quadcopter equipped with a hook-based gripper, implemented in MuJoCo, is constructed and used as the predictive model for the MPC. To handle uncertainties of the predictive model (e.g. due to aerodynamics and uncertain payloads), a robust adaptive MPC approach is proposed. By systematic integration of zero-order robust optimization (zoRO) based uncertainty propagation and an extended Kalman filter (EKF) for parameter estimation, the MPC algorithm ensures robust constraint satisfaction, high performance, and computational efficiency. The effectiveness of the proposed method is evaluated in complex simulated scenarios and in real-world flight experiments.

Bayesian filtering is a cornerstone of state estimation in complex systems such as aerospace systems, yet exact solutions are available only for linear Gaussian models. In practice,nonlinear systems are handled through tractable approximations,with Gaussian filters such as the extended and unscented Kalman filters being among the most widely used methods. This tutorial revisits Gaussian filtering from an information-geometric perspective, viewing the prediction and measurement update steps as inference procedures over state distributions. Within this framework, we introduce a geometry-aware Gaussian filtering approach that leverages natural gradient descent on the statistical manifold of Gaussian distributions. The resulting Natural Gradient Gaussian Approximation (NANO) filter iteratively refines the posterior mean and covariance while respecting the intrinsic geometry of the Gaussian family and preserving the positive definiteness of the covariance matrix. We further highlight fundamental connections to the classical Kalman filtering, showing that a single natural-gradient step exactly recovers the Kalman measurement update in the linear-Gaussian case. The practical implications of the proposed framework are illustrated through case studies in representative nonlinear estimation problems,including satellite attitude estimation, simultaneous localization and mapping, and state estimation for robotic systems including quadruped and humanoid robots.

This paper develops a deep reinforcement learning based observer control policy for autonomous bearings-only tracking of a moving target. The observer manoeuvre problem is formulated as a belief Markov decision process, where the belief state is represented by the posterior of a cubature Kalman filter (CKF). The reward function is designed to address two conflicting objectives: minimising the absolute target position estimation error (Euclidean distance) and maintaining CKF estimation consistency (Mahalanobis distance). The reward is formulated as a geometric interpolation between the two objectives on the Pareto front, parametrised by a weighting factor $β\in [0,1]$. The policy is implemented as a deep Q-network (DQN) trained over 50,000 episodes. Performance is evaluated over 5,000 Monte Carlo episodes and compared against two baselines: the perpendicular-to-bearing heuristic and the D-optimal Fisher information maximisation criterion. The results show that the DQN policy at $β= 0.7$ achieves the best trade-off between accuracy and robustness: it matches the information-theoretic baseline on mean tracking accuracy while reducing the worst-case error by nearly a factor of ten, owing to the implicit filter-consistency regularisation provided by the Mahalanobis term in the reward.

The ensemble Gaussian mixture filter (EnGMF) is a powerful, convergent particle filter capable of medium-to-high dimensional non-linear filtering. The EnGMF relies on a resampling step that can generate physically unrealistic posterior samples, that would subsequently produce physically meaningless forecasts. This work introduces the discriminator-informed resampling procedure, that augments the posterior resampling step with a discriminator that accepts or rejects candidate particles based on their physical plausibility. In this work these discriminators are learned through a normalizing flow approach. Numerical experiments on both the Ikeda map and the Lorenz '63 system show that discriminator informed resampling procedure consistently reduces error relative to the standard EnGMF in low-ensemble regimes.

Hierarchical Gaussian Filtering (HGF) networks allow for efficient updating of posterior distributions (beliefs) about hidden states of an agent's environment. HGF parent nodes can target the mean or variance of their children. New information entering at input nodes leads to a cascade of belief updates across the network according to one-step update equations for each node's mean and precision (inverse variance). However, the original form of the update equations for variance-targeting parents(volatility coupling) can in some regions of parameter space lead to negative posterior precision, a logical impossibility which causes the updating algorithm to terminate with an error. In this report, we introduce a modified quadratic approximation to the variational energy of volatility-coupled nodes that avoids negative posterior precision. The key idea is to interpolate between two quadratic expansions of the variational energy: one at the prior prediction and one at a second mode whose location is obtained in closed form via the Lambert W function. The resulting update equations are robust across the entire parameter space and faithfully track the variational posterior even for large prediction errors.