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Keywords = Kalman filter tuning

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34 pages, 29745 KB  
Article
LiDAR-Based Deep Learning-Enabled Geometric Fingerprinting for Indoor Robot Localization
by Harsha Keladi Ganapathi and Shayok Mukhopadhyay
Appl. Sci. 2026, 16(15), 7718; https://doi.org/10.3390/app16157718 - 3 Aug 2026
Viewed by 256
Abstract
Localization is a fundamental requirement for autonomous mobile robot navigation. Several localization techniques exist, but they often require extensive installation of beacons, careful parameter tuning, high computational requirements, or an immense amount of training data. Environmental (e.g., indoor)/resource constraints, sensor degradation, and sudden [...] Read more.
Localization is a fundamental requirement for autonomous mobile robot navigation. Several localization techniques exist, but they often require extensive installation of beacons, careful parameter tuning, high computational requirements, or an immense amount of training data. Environmental (e.g., indoor)/resource constraints, sensor degradation, and sudden pose discontinuities can make such methods unreliable. This creates a critical gap: the lack of a simple, lightweight localization method that can operate as a primary localization method or in parallel with other classical systems and provide reliable pose estimates during primary localization system failures. Thus, this paper proposes a lightweight, deep learning (DL)-based, two-dimensional LiDAR localization method. The approach combines LiDAR scan range data with eleven proposed handcrafted geometric features to train a Convolutional Multi-Layer Perceptron (ConvMLP) regression model for predicting the two-dimensional location of a robot, which is further smoothed by an augmented recursive Extended Kalman filter (EKF). The overall system is validated in three real-world environments. The results are compared against various existing machine learning (ML) models and other well-known localization techniques. The experimental results demonstrate a 280 Hz pose-update rate, achieving a 13 cm Root Mean Square Error (RMSE) using the ConvMLP model alone, which further reduces to 5 cm when fused with the recursive EKF. Full article
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33 pages, 63694 KB  
Article
YOLO-Driven Vessel Detection and Multi-Object Tracking in Fixed-Camera Marina Environments
by Nikola Lopac, Karlo Severinski, Neven Grubišić and Jonatan Lerga
Technologies 2026, 14(8), 482; https://doi.org/10.3390/technologies14080482 - 3 Aug 2026
Viewed by 268
Abstract
Vessel detection and multi-object tracking (MOT) in marinas remain challenging because of occlusions, small targets, cluttered backgrounds, and changing illumination. This paper presents a controlled, feasibility-focused comparative evaluation of YOLO-driven vessel detection and tracking in a fixed-camera marina environment, comparing YOLOv11 with the [...] Read more.
Vessel detection and multi-object tracking (MOT) in marinas remain challenging because of occlusions, small targets, cluttered backgrounds, and changing illumination. This paper presents a controlled, feasibility-focused comparative evaluation of YOLO-driven vessel detection and tracking in a fixed-camera marina environment, comparing YOLOv11 with the more attention-oriented YOLOv12 detector family. A dataset of 3546 annotated images was collected using a static ground-level camera at a marina in the northern Adriatic, Croatia. YOLOv11s, YOLOv11m, YOLOv12s, and YOLOv12m were fine-tuned using transfer learning, evaluated across five random seeds, and integrated into a common tracking-by-detection pipeline with Kalman filter (KF) and extended Kalman filter (EKF) motion models. Detection was assessed using COCO-style mAP metrics, while tracking was evaluated using MOT and HOTA-based metrics. YOLOv11m achieved the highest mean test-set mAP@50–95 (0.7458 ± 0.0027) and significantly outperformed YOLOv11s and YOLOv12s. Although YOLOv12m did not achieve the highest frame-level mAP, it obtained the highest mean AssA and IDF1 when averaged across the two motion models. KF achieved higher mean HOTA, DetA, AssA, MOTA, and IDF1 than EKF, whereas EKF achieved only marginally higher MOTP; the KF advantage was statistically significant for IDF1. Overall, the results show that frame-level detection accuracy does not necessarily determine downstream tracking performance, thereby supporting the joint evaluation of detector and motion-model choices. Full article
(This article belongs to the Section Information and Communication Technologies)
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12 pages, 449 KB  
Review
Data-Driven Fusion Algorithms for Temperature-Drift Compensation of MEMS Gyroscopes: A Mini Review
by Haoze Lan and Yingjie Xu
Micromachines 2026, 17(8), 924; https://doi.org/10.3390/mi17080924 - 31 Jul 2026
Viewed by 247
Abstract
Microelectromechanical systems (MEMS) gyroscopes are now standard rate sensors in inertial navigation, automotive electronics, industrial automation, and medical instrumentation because they are inexpensive, compact, and readily integrated. Their accuracy, however, degrades with temperature: damping and quadrature coupling change, and readout-electronics behavior shifts, producing [...] Read more.
Microelectromechanical systems (MEMS) gyroscopes are now standard rate sensors in inertial navigation, automotive electronics, industrial automation, and medical instrumentation because they are inexpensive, compact, and readily integrated. Their accuracy, however, degrades with temperature: damping and quadrature coupling change, and readout-electronics behavior shifts, producing temperature-dependent zero-rate-output drift, elevated random noise, and poorer long-term stability. Hardware- and structure-based temperature compensation address part of the problem but carry cost and generality penalties, which has moved recent work toward data-driven software-based temperature-drift compensation. This review focuses on the fusion algorithms that have come to dominate that literature, organized as a four-stage pipeline: signal decomposition, learning-based drift modeling, adaptive filtering, and signal reconstruction. We examine how optimizer-tuned variational mode decomposition and improved empirical-mode-decomposition variants separate temperature-related components from noise; how deep temporal networks and optimizer-coupled learners model the nonlinear, time-lagged drift; and how adaptive Kalman variants and time-frequency filtering reconstruct a stable output. We close by identifying four open problems that recur across the recent gyroscope-specific work—cross-device generalization, temperature hysteresis, embedded real-time deployment, and physics-informed lightweight modeling. Full article
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31 pages, 6266 KB  
Article
Experimental Evaluation of Path-Following Performance in a Scaled Autonomous Vehicle: Effects of Localization, Path Geometry, Speed, and Pure Pursuit Look-Ahead Distance
by Piotr Szeląg, Sebastian Dudzik, Patryk Gałuszkiewicz and Gabriela Gic-Grusza
Appl. Sci. 2026, 16(14), 7123; https://doi.org/10.3390/app16147123 - 16 Jul 2026
Viewed by 336
Abstract
Reliable execution of a planned path is essential for autonomous mobile robots and vehicle-like robot platforms. This study experimentally evaluates the path-following performance of a scaled Ackermann-steered autonomous vehicle under different localization and controller configurations. A QCar 2 platform was operated in a [...] Read more.
Reliable execution of a planned path is essential for autonomous mobile robots and vehicle-like robot platforms. This study experimentally evaluates the path-following performance of a scaled Ackermann-steered autonomous vehicle under different localization and controller configurations. A QCar 2 platform was operated in a hardware-in-the-loop configuration using a Pure Pursuit lateral controller and a proportional-integral longitudinal speed controller. Two localization approaches were compared in the closed control loop: a kinematic localization method and an Extended Kalman Filter. A full factorial (24) experimental design included the localization method, reference-path geometry (rounded rectangle and figure-eight), commanded speed (0.4 and 0.7 m/s), and Pure Pursuit look-ahead distance (0.3 and 0.6 m), resulting in 16 test configurations. Realized vehicle trajectories were recorded independently using an OptiTrack motion-capture system and compared with the planned paths. Path-following performance was assessed using cross-track error, symmetric Hausdorff distance, and mean bidirectional nearest-neighbor distance. Within the tested runs, differences between the localization variants were small, with mean (CTERMS) values of 0.0929 m and 0.0921 m for the Extended Kalman Filter and kinematic variants, respectively. In contrast, substantially larger differences in trajectory deviations and traveled-path length were observed across path geometries, look-ahead distances, and commanded speeds. The figure-eight path, particularly at the shorter look-ahead distance and higher speed, showed the largest deviations and path-length excess within the tested configurations. The results show that, under the tested laboratory conditions, path-execution quality was more strongly associated with controller tuning and planned-path geometry than with the investigated localization variant. The study provides experimentally validated guidance for selecting path-following parameters for Ackermann-steered autonomous mobile robots. Full article
(This article belongs to the Special Issue Advances in Robot Path Planning, 3rd Edition)
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22 pages, 447 KB  
Article
Kalman-Annealing: Calibrated Uncertainty for Simulated Annealing via a Probabilistic-Numerics Filter, with an Application to Reinforcement-Learning Hyperparameter Tuning
by Eduardo C. Garrido-Merchán
Algorithms 2026, 19(7), 581; https://doi.org/10.3390/a19070581 - 15 Jul 2026
Viewed by 297
Abstract
Noisy, expensive, gradient-free optimisers—simulated annealing chief among them—almost never report how confident one should be in the configuration they return, and reinforcement-learning hyperparameter tuning, where the noise is large and the budget tight, is the setting where this silence hurts most. The contribution [...] Read more.
Noisy, expensive, gradient-free optimisers—simulated annealing chief among them—almost never report how confident one should be in the configuration they return, and reinforcement-learning hyperparameter tuning, where the noise is large and the budget tight, is the setting where this silence hurts most. The contribution of this paper is a mechanism for uncertainty quantification, not a faster optimiser: we equip simulated annealing with a calibrated credible interval over the value of the recovered configuration, and we are explicit that this comes at an optimisation cost that only some landscapes repay. We introduce Kalman-Annealing (KA), a minimal modification of simulated annealing in which a one-dimensional Kalman filter—the canonical probabilistic numerical method—is interleaved with the Metropolis acceptance step. The filter denoises each return before acceptance, and a short terminal refinement of the best visited state converts the run into a calibrated credible interval over the value of the recovered hyperparameter. A single analytical identity, Qt=cTt2, couples the filter process noise to the cooling schedule and absorbs the only free parameter of the filter into one already present in the metaheuristic. Under standard cooling assumptions the credible intervals are calibrated and the posterior variance contracts at a rate compatible with simulated-annealing convergence. On synthetic benchmarks (a noisy five-dimensional quadratic and the noisy Branin function, 200 seeds each) and on hyperparameter tuning of REINFORCE on three classic-control tasks (10 seeds each), the empirical 90% coverage of KA’s credible intervals lies within sampling error of the nominal level—a property none of the baselines provides—and the optimiser overhead is close to four orders of magnitude below that of Gaussian-process Bayesian optimisation. The interval cannot be extracted for free from an unmodified SA run: an interval built from the trailing evaluations of the vanilla trajectory fails to calibrate in every reading we test, and the repair that does calibrate is exactly KA’s terminal-refinement phase grafted onto the unfiltered chain, at the same cost in diverted evaluations. Honest scoreboard: On simple regret, KA is at best on par with vanilla simulated annealing on the unimodal synthetic (the nominal advantage does not survive correction for multiple comparisons) and loses to the SA family, to CMA-ES and to Gaussian-process Bayesian optimisation on the multi-modal and ill-conditioned synthetics and on the informative reinforcement-learning tasks, under both REINFORCE and PPO. We trace this gap quantitatively to the filter acting as a low-pass, with a mean Kalman gain near one half, on the favourable-tail observations that drive SA’s basin escape, and we delineate the operating regime in which the calibrated-uncertainty contribution of KA is worth its optimisation cost. Full article
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24 pages, 1514 KB  
Article
Mamba-2-Based Continuous-Discrete Extended Kalman Filter for Passive UAV Bearings-Only Tracking with Uncertain Measurement Noise
by Hao Wu, Guoxu Zeng, Ali Mehmood, Yijie Zhao, Chaoqi Li and Mingbo Yang
Drones 2026, 10(7), 533; https://doi.org/10.3390/drones10070533 - 14 Jul 2026
Viewed by 391
Abstract
Using UAVs for bearings-only tracking (BOT) of moving targets is a key challenging issue due to weak observability and time-varying measurement noises. To overcome these limitations, we introduce a hybrid framework that integrates a square-root continuous-discrete extended Kalman filter (MSCDEKF) with the Mamba-2 [...] Read more.
Using UAVs for bearings-only tracking (BOT) of moving targets is a key challenging issue due to weak observability and time-varying measurement noises. To overcome these limitations, we introduce a hybrid framework that integrates a square-root continuous-discrete extended Kalman filter (MSCDEKF) with the Mamba-2 neural network. The MSCDEKF improves estimation accuracy and numerical stability under weak observability conditions by adopting continuous-time state update while ensuring numerical stability through square-root covariance implementation. Meanwhile, the state-space modeling of the Mamba-2 network provides superior continuous measurement noise prediction compared to conventional recurrent neural network approaches by explicitly capturing signal evolution dynamics. Experimental validation in constant velocity (CV) and constant turn (CT) UAV-based BOT scenarios demonstrates that our framework achieves the following: (1) at least a 34.5% improvement in state estimation accuracy (measured by average root-mean-square error, ARMSE) compared with variational Bayesian filters; (2) significant improvements in noise variance estimation; and (3) parameter-free operation with minimal manual tuning. This work establishes a different paradigm for robust UAV-based BOT in complex environments characterized by unknown and time-varying noise conditions. Full article
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27 pages, 27790 KB  
Article
Camera–LiDAR Data Fusion for Enhanced Ship Situational Awareness in Maritime Environment
by Filippo Ponzini and Michele Martelli
J. Mar. Sci. Eng. 2026, 14(14), 1276; https://doi.org/10.3390/jmse14141276 - 10 Jul 2026
Viewed by 392
Abstract
Reliable obstacle detection and classification are essential capabilities for the safe and efficient navigation of Marine Autonomous Surface Ships. This paper introduces a decision-level multi-sensor fusion framework to enhance situational awareness for autonomous vessels by integrating RGB camera and LiDAR data. Visual information [...] Read more.
Reliable obstacle detection and classification are essential capabilities for the safe and efficient navigation of Marine Autonomous Surface Ships. This paper introduces a decision-level multi-sensor fusion framework to enhance situational awareness for autonomous vessels by integrating RGB camera and LiDAR data. Visual information is processed using a pre-trained, open-source object detection model. At the same time, LiDAR measurements are analysed with a clustering-based algorithm, followed by a lightweight Random Forest classifier for semantic labelling. To support practical deployment in real maritime environments, the proposed approach relies on readily available perception modules, avoiding the need for training on proprietary datasets and limiting dependence on extensive task-specific tuning. The fusion of these complementary sources is employed to confirm and characterise dynamic obstacles, whose positions derived from LiDAR are continuously tracked using a Global Nearest Neighbour algorithm supported by a Kalman filter. Each stage of the proposed processing chain is thoroughly described and experimentally validated using real-world data collected in a representative marine environment, demonstrating the approach’s effectiveness in improving perception performance by reducing false positives from noisy measurements and achieving 92% track number accuracy in a complex scenario. Full article
(This article belongs to the Section Ocean Engineering)
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28 pages, 2310 KB  
Article
Online-Tuned Fuzzy Pre-Filtering with an Attention BiLSTM for Misbehavior Detection in Vehicular Named Data Networking
by Bassma Aldahlan
Sensors 2026, 26(13), 4179; https://doi.org/10.3390/s26134179 - 2 Jul 2026
Viewed by 254
Abstract
Vehicular Named Data Networking (VNDN) inherits the broadcast-oriented forwarding of NDN, which exposes safety messages to position-falsification attacks. Existing detectors rely either on static fuzzy thresholds, which drift as traffic patterns change, or on opaque deep models, which are accurate but uninterpretable to [...] Read more.
Vehicular Named Data Networking (VNDN) inherits the broadcast-oriented forwarding of NDN, which exposes safety messages to position-falsification attacks. Existing detectors rely either on static fuzzy thresholds, which drift as traffic patterns change, or on opaque deep models, which are accurate but uninterpretable to safety auditors. We propose a two-stage detector that combines an Adaptive Fuzzy Membership Tuning (AFMT) pre-filter with an attention-augmented bidirectional LSTM. AFMT is a Mamdani fuzzy classifier whose triangular membership-function parameters are updated online by gradient descent on a prediction-error feedback signal from the downstream BiLSTM, replacing offline-fixed thresholds. The BiLSTM consumes the fuzzy suspicion score as an extra feature and produces interpretable per-time-step attention weights aligned with attack onsets. On a simulator-synthesized VNDN benchmark following the five canonical VeReMi attack types, the detector attains F1-scores between 0.955 and 0.979 (macro-average 0.964), ties the strongest baselines on the hardest Random-Offset attack while achieving the highest ROC-AUC of all models (0.984), and runs in 0.44 ms per sample on a CPU. On a live OMNeT++/Veins/SUMO testbed running the five attacks on the LuST scenario, the detector attains an F1 value of 0.986. A leave-one-feature-out study shows that detection does not hinge on the Kalman plausibility feature, and on the real public VeReMi v1.0 dataset the architecture transfers to four of the five attack types at an F1 near 1.0, while the Constant Offset stays invisible to kinematics-only features, and this quantifies the value of the named-data-plane features. Every number reported here is measured from the running detector. Full article
(This article belongs to the Special Issue Intelligent Vehicular Network and Communication Systems)
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21 pages, 5116 KB  
Article
Research on Train Positioning Method Based on Maximum Correntropy Robust Filtering with Dynamic Kernel Bandwidth
by Weishu Wang, Shanyi Song, Cong Peng and Dacheng Xu
Electronics 2026, 15(13), 2811; https://doi.org/10.3390/electronics15132811 - 25 Jun 2026
Viewed by 274
Abstract
Accurate and reliable train positioning is essential for railway operation control systems. However, conventional extended Kalman filter-based solutions are vulnerable to measurement faults, which can significantly degrade positioning performance. To address this issue, this paper proposes an adaptive maximum correntropy robust filter (AMCRF) [...] Read more.
Accurate and reliable train positioning is essential for railway operation control systems. However, conventional extended Kalman filter-based solutions are vulnerable to measurement faults, which can significantly degrade positioning performance. To address this issue, this paper proposes an adaptive maximum correntropy robust filter (AMCRF) for a GNSS/INS-based train positioning system. The loss function of the extended Kalman filter is reformulated from the minimum mean square error criterion to a maximumcorrentropy criterion, thereby improving the algorithm’s robustness against measurement faults. In AMCRF, considering the limitation of using a fixed kernel bandwidth, a lion swarm optimization strategy is introduced to adaptively tune the kernel bandwidth for each visible satellite, enabling the filter to adapt to time-varying measurement quality and fault magnitudes. By embedding the adaptive mechanism into an extended Kalman filtering framework, the proposed method achieves enhanced fault tolerance. The effectiveness of the proposed AMCRF is validated using experimental data collected along the Qinghai–Tibet Railway. Step and ramp faults of different magnitudes are injected into pseudorange measurements to evaluate fault tolerance. Experimental results demonstrate that the proposed method effectively suppresses the influence of faulty measurements and maintains positioning accuracy close to that under fault-free conditions. Full article
(This article belongs to the Special Issue Recent Advances in Condition Monitoring and Fault Diagnosis)
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40 pages, 27259 KB  
Article
Monocular 3D Position Estimation of a Moving Vehicle Based on a Kalman-Goldschmidt Adaptive Filter
by Diana Kalita, Pavel Lyakhov, Valery Andreev and Denis Butusov
J. Sens. Actuator Netw. 2026, 15(3), 48; https://doi.org/10.3390/jsan15030048 - 18 Jun 2026
Viewed by 424
Abstract
Determining the 3D position of a vehicle from a 2D image plays a key role in video surveillance, autonomous driving, and spatial localization. However, localization accuracy can significantly degrade in conditions of incomplete or synthetic measurement noise and keypoint jitter. In this paper, [...] Read more.
Determining the 3D position of a vehicle from a 2D image plays a key role in video surveillance, autonomous driving, and spatial localization. However, localization accuracy can significantly degrade in conditions of incomplete or synthetic measurement noise and keypoint jitter. In this paper, we propose a new iterative 3D position estimation algorithm (KGA). This algorithm includes geometric correction and calibration steps for converting from 2D to 3D coordinates; trajectory prediction and correction using a Kalman filter; and adaptive tuning of the filter parameters using the Goldschmidt algorithm. Experiments confirm that KGA outperforms the standard (FK) and modified (MFK) Kalman filters in accuracy and convergence speed, demonstrating robustness to various camera angles and noise levels. The novelty of this approach lies in the integration of the Goldschmidt algorithm into the Kalman filter to create an adaptation mechanism that dynamically adjusts the measurement noise covariance based on instantaneous innovation magnitude. Unlike end-to-end deep learning trackers or nonlinear filters (EKF/UKF), KGA is designed as a lightweight post-processing stage that can be seamlessly integrated into existing detection pipelines while maintaining the low computational footprint required for UAV-based edge deployment. The algorithm is of practical value for computer vision systems requiring accurate and robust tracking under varying observational conditions, with current implementation suitable for offline or buffered processing, and clear pathways to real-time deployment through code optimization. The algorithm is of practical value for computer vision systems requiring accurate and robust tracking under varying observational conditions. Full article
(This article belongs to the Section Big Data, Computing and Artificial Intelligence)
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27 pages, 12622 KB  
Article
Safety-Filtered Residual Reinforcement Learning over Model Predictive Control for Friction-Aware Autonomous Vehicle Platooning
by Ali S. Allahloh, Atef M. Ghaleb, Mohammad Sarfraz, Abdalla Alrashdan, Mohammed A. H. Ali and Adel Al-Shayea
Machines 2026, 14(5), 560; https://doi.org/10.3390/machines14050560 - 16 May 2026
Viewed by 583
Abstract
This paper presents a deployment-oriented longitudinal platoon-control architecture for connected and autonomous vehicles operating under repeated leader hard-braking, cut-ins, and spatially varying road friction. The proposed stack combines four elements: (i) a lightweight scalar Kalman filter (KF) that smooths a friction-related signal and [...] Read more.
This paper presents a deployment-oriented longitudinal platoon-control architecture for connected and autonomous vehicles operating under repeated leader hard-braking, cut-ins, and spatially varying road friction. The proposed stack combines four elements: (i) a lightweight scalar Kalman filter (KF) that smooths a friction-related signal and feeds friction-dependent constraint tightening; (ii) a model predictive control (MPC) backbone whose weights and horizon are selected offline using multi-objective GA/NSGA-II tuning; (iii) a bounded proximal policy optimization (PPO) residual policy, trained with the aid of a learned surrogate model, that refines the MPC command during transient events; and (iv) a command-level safety projection that enforces instantaneous actuation and clearance constraints at the fast control tick. The contribution is therefore not a new MPC formulation or a new reinforcement-learning algorithm in isolation, but an integrated and experimentally characterized control stack that keeps the safety-critical structure explicit while using learning to improve transient behavior. The method is evaluated in a CARLA digital twin of a six-vehicle platoon over a 5 km mixed urban–highway route and is further assessed in hardware-in-the-loop (HIL) on an automotive ECU using a multi-rate ROS 2/AUTOSAR implementation (50 Hz estimation/safety loop, 10 Hz MPC/RL refresh). Across 10 held-out disturbance seeds, the full stack improves spacing regulation, maintains non-amplifying disturbance propagation according to the reported string-stability indices, and reduces a route-normalized positive tractive-energy-at-the-wheels proxy by about 12% relative to Manual MPC and by up to 18% relative to a PID-CACC reference. Because the PID-CACC baseline does not enforce hard constraints and can collide under the tested disturbance suite, the main performance comparison is among collision-free controllers. The friction signal used in CARLA is derived from simulator road-surface annotations before filtering, so the present study should be interpreted as a friction-aware control and integration study rather than a validated onboard friction-estimation result. Likewise, the reported energy metric is an effort proxy and is not a calibrated fuel or battery consumption model. Full article
(This article belongs to the Special Issue Reinforcement Learning for Autonomous Vehicle Control)
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4214 KB  
Proceeding Paper
Adaptive State of Energy Estimation for Lithium-Ion Batteries Using an Improved Sage–Husa EKF
by Taofeeq Sulyman Opeyemi, Umar Musa, Ibrahim Abdullahi Shehu, Ramani Kannan and Aminu Jibrin Aliyu
Eng. Proc. 2026, 147(1), 10; https://doi.org/10.3390/engproc2026147010 - 15 May 2026
Viewed by 90
Abstract
Accurate State of Energy (SOE) estimation is a critical yet challenging requirement for reliable Battery Management Systems (BMSs), as traditional model-based algorithms like the Extended Kalman Filter (EKF) are severely limited by their dependence on precise, static noise covariance matrices. To address this [...] Read more.
Accurate State of Energy (SOE) estimation is a critical yet challenging requirement for reliable Battery Management Systems (BMSs), as traditional model-based algorithms like the Extended Kalman Filter (EKF) are severely limited by their dependence on precise, static noise covariance matrices. To address this deficiency, this study developed an Improved Sage–Husa Extended Kalman Filter (SHEKF) to enhance the precision and stability of SOE estimation in lithium-ion batteries. The methodology began by simulating a second-order (2RC) equivalent circuit model utilizing battery parameters accurately identified through HPPC test data. Unlike traditional filters, the proposed SHEKF overcomes the limitations of fixed noise profiles by using filter innovation to dynamically update its noise covariances in real time. The adaptive algorithm was then evaluated against a standard EKF and a Strong Tracking EKF (STEKF) using dynamic FUDS and UDDS drive cycle datasets. The results demonstrated that the SHEKF consistently outperformed both the EKF and STEKF across all metrics. The on-line adaptive mechanism successfully reduced the SOE Root Mean Square (RMS) Error by 75% to 0.58% and achieved a similar 75% reduction in RMS voltage error compared to the EKF. Furthermore, the SHEKF maintained highly stable error traces, recording a maximum absolute SOE error of just 0.84%. By automatically adjusting noise parameters without manual tuning, the algorithm avoids the filter divergence commonly seen when noise characteristics change. These findings establish the Improved SHEKF as a highly robust, practical, and superior method for real-time SOE estimation in modern real-world BMS applications. Full article
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16 pages, 1611 KB  
Article
Symmetry-Aware Vehicle State Estimation Using a Chaotic-Gradient-Optimized Extended Kalman Filter
by Qianyu Cheng, Wenguang Liu, Xi Liu, Huajun Che and Bei Ding
Symmetry 2026, 18(5), 847; https://doi.org/10.3390/sym18050847 - 15 May 2026
Viewed by 294
Abstract
To address the uncertainty of the measurement noise covariance matrix in vehicle state estimation, this paper proposes a symmetry-aware extended Kalman filter optimized by a chaotic-gradient strategy. The symmetry-aware concept is introduced from the approximate mirror symmetry of vehicle lateral dynamics under left [...] Read more.
To address the uncertainty of the measurement noise covariance matrix in vehicle state estimation, this paper proposes a symmetry-aware extended Kalman filter optimized by a chaotic-gradient strategy. The symmetry-aware concept is introduced from the approximate mirror symmetry of vehicle lateral dynamics under left and right steering excitations. Under identical road adhesion and vehicle operating conditions, the yaw-rate and sideslip-angle responses should exhibit balanced statistical characteristics for positive and negative lateral motions. However, a fixed measurement noise covariance matrix may break this balance and lead to direction-dependent estimation bias or delayed convergence. To improve the statistical consistency of the estimation process, the proposed method adaptively tunes the measurement noise covariance matrix according to the innovation covariance mismatch. A chaotic search mechanism is first used to enhance global exploration, and a variable-step gradient method is then applied to refine the local optimal solution. Through the iterative combination of chaotic traversal and gradient-based refinement, the proposed observer improves the balance between model prediction and measurement correction under stochastic disturbances. The effectiveness of the proposed method is verified through CarSim and MATLAB/Simulink co-simulation. The results show that, compared with EKF, UKF, and AEKF benchmark observers, the proposed CG_EKF provides more accurate estimation of vehicle yaw rate and sideslip angle. Full article
(This article belongs to the Section F: Engineering and Materials)
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29 pages, 5797 KB  
Article
Research on GNSS/INS Tightly Coupled Integrity Monitoring Method Based on State Augmentation Error Modeling
by Xinhua Tang, Xiaoyu Fang and Fei Huang
Remote Sens. 2026, 18(10), 1564; https://doi.org/10.3390/rs18101564 - 14 May 2026
Viewed by 504
Abstract
In urban environments with signal blockage and multipath effects, GNSS observation errors often exhibit temporal correlation. The Gaussian white noise assumption adopted in conventional tightly coupled Kalman filtering is prone to model mismatch under such conditions, which may lead to an underestimation of [...] Read more.
In urban environments with signal blockage and multipath effects, GNSS observation errors often exhibit temporal correlation. The Gaussian white noise assumption adopted in conventional tightly coupled Kalman filtering is prone to model mismatch under such conditions, which may lead to an underestimation of state uncertainty and consequently cause the protection level (PL) to fail to reliably bound the true positioning error. To address this issue, this paper proposes a tightly coupled GNSS/INS integrity monitoring method based on state augmentation and frequency-domain constrained parameter tuning. The method introduces first-order Gauss-Markov processes (GMP) to model major time-correlated error sources, including residual ephemeris and clock errors, residual tropospheric delay, and code multipath, by augmenting them into the filter state for joint estimation. The model parameters are further conservatively tuned based on power spectral density (PSD) envelope constraints to obtain more consistent covariance estimates. Based on this, the covariance output from the augmented filter is incorporated into the multiple hypothesis solution separation (MHSS) framework, enabling the protection level computation to better match the actual error statistics. Experimental results using vehicular field test data show that the proposed method effectively improves estimation consistency and significantly reduces the risk of PL underestimation in degraded environments. Furthermore, it achieves reliable bounding of horizontal positioning errors without noticeable degradation in positioning accuracy, while maintaining good system availability. These results demonstrate the effectiveness of covariance construction based on physical error modeling and PSD envelope constraints for integrity monitoring in complex environments. Full article
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25 pages, 4104 KB  
Article
Kalman Filter Method with Iterative Sparse Regularization and Its Application to the Retrieval of the Initial Field for the Convection–Diffusion Equation
by Xuan Deng and Yuepeng Wang
Mathematics 2026, 14(9), 1483; https://doi.org/10.3390/math14091483 - 28 Apr 2026
Viewed by 430
Abstract
Sparse regularization methods play an important role in inverse problems for extracting key features of underlying parameters and have attracted increasing attention in meteorological data assimilation. However, when the condition number of the background error covariance matrix is extremely large (e.g., 1012 [...] Read more.
Sparse regularization methods play an important role in inverse problems for extracting key features of underlying parameters and have attracted increasing attention in meteorological data assimilation. However, when the condition number of the background error covariance matrix is extremely large (e.g., 1012), the instability of the inverse problem makes accurate reconstruction difficult. To address this issue, a gradient operator is incorporated into the sparse regularization term of the cost function, and a Kalman filter (KF) algorithm is developed within a majorization–minimization (MM) framework to solve the resulting optimization problem. The problem is reformulated as a weighted least-squares problem via the MM strategy and further decomposed into two subproblems in the null space and its oblique complementary space through oblique projection, which are then solved using the KF method. This approach avoids the use of an adjoint model typically required in four-dimensional variational data assimilation (4D-Var). In addition, a modified f-slope strategy with a constrained search interval is introduced to adaptively select the regularization parameter during computation. Numerical experiments on the initial condition inversion of the Convection–Diffusion equation demonstrate that the proposed method achieves more accurate reconstruction of key features than the l1-norm regularized 4D-Var method, particularly in capturing sharp gradients and sparse structures. The adaptive regularization strategy automatically balances sparsity and smoothness without manual tuning. The inversion errors remain low even when the condition number ranges from 108 to 1014, with relative MSE and MAE below 0.01 and relative bias below 0.005, indicating improved robustness and reconstruction accuracy under severely ill-conditioned settings. Full article
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