Kalman Filter Framework Research Articles

An online reconstruction method of dynamic loading based on adaptive tracking dual nested Kalman filter

Accurately reconstructing the unknown dynamic loading is crucial for controlling the vibration of mechanical systems. Many methods have been proposed for load identification of time-invariant systems, but in some mechanical systems, their structural parameters exhibit time-varying dynamic characteristics during their operation. Therefore, tracking the time-varying structural parameters to update the transfer function is necessary, which is the most important prerequisite for load reconstruction. However, some existing methods are unstable because of the nonlinear conditions. To overcome this issue, an improved dual-nested Kalman filter framework has been proposed. This method allocates unknown variables to two Kalman filters, allowing them to iterate and loop with each other. As a result, it achieves linear reconstruction of the dynamic loading for a time-varying system. The proposed method has been validated in two numerical examples: one is a spring-mass system with abruptly changing mass parameters, and the other is a milling system with slowly changing mass parameters. The results show that the proposed method can reconstruct loading stably and is not susceptible to noise interference.

Modeling and Estimation of Continuous Flexible Structure Using Theory of Functional Connections

This paper presents a novel method for modeling and estimating the dynamics of a continuous structure based on a limited number of noisy measurements. The goal is reached using a Kalman filter in synergy with the recently developed mathematical framework known as the Theory of Functional Connections (TFC). The TFC allows deriving a functional expression capable of representing the entire space of the functions that satisfy a given set of linear and, in some cases, nonlinear constraints. The proposed approach exploits the possibilities offered by the TFC to derive an approximated dynamical model for the flexible system using the Lagrangian mechanics. The result is a representation of the structural dynamics using a finite number of states, in contrast to the infinite-dimensional model that would be obtained by application of the traditional continuum mechanics models that are based on sets of partial differential equations. The limited number of states enables the application of the well-known Kalman filter framework to improve the estimation of the displacements and displacement velocities. In addition, the continuous displacement field of the structure can be reconstructed with high fidelity. The theoretical development of the method is presented in relation to the case of an Euler–Bernoulli beam. Finally, the obtained model is used to carry out a simulation campaign aimed at assessing the accuracy, efficiency, and robustness of the proposed method.

Multi-Tag Fusion Localization Method Based on Geometric Constraints

In environments where Global Navigation Satellite System (GNSS) signals are unavailable, our proposed multi-tag fusion localization method offers a robust solution for the precise positioning of vehicles or robots. During our research, we observed variations in the positioning information estimated from tags located at different positions within the same frame. Our goal was to extract reliable positioning information from this noisy data. By constructing geometric constraints, our method introduces an outlier factor to quantify the differences between tags. After effectively eliminating outliers, we enhanced the Kalman filter framework to accommodate the fusion of data from two or more tags, with the outlier factor dynamically adjusting the observation noise during the fusion process. The experimental results demonstrate that, even under the influence of motion and obstacles, our method maintains position errors within a 3 cm range and orientation errors within 3°. This indicates that our method possesses high positioning accuracy and stability.

Tool Wear Monitoring In Micro-Milling Based on Digital Twin Technology with an Extended Kalman Filter

In order to avoid catastrophic events that degrade the quality of machined products, such as tool breakage, it is vital to have a prognostic system for monitoring tool wear during the micro-milling process. Despite the long history of the tool wear monitoring field, creating such a system to track, monitor, and foresee the rapid progression of tool wear still needs to be improved in the application of micro-milling. On the other hand, digital twin technology has recently become widely recognized as significant in manufacturing and, notably, within the Industry 4.0 ecosystem. Digital twin technology is considered a potential breakthrough in developing a prognostic tool wear monitoring system, as it enables the tracking, monitoring, and prediction of the dynamics of a twinned object, e.g., a CNC machine tool. However, few works have explored the digital twin technology for tool wear monitoring, particularly in the micro-milling field. This paper presents a novel tool wear monitoring system for micro-milling machining based on digital twin technology and an extended Kalman filter framework. The proposed system provides wear progression notifications to assist the user in making decisions related to the machining process. In an evaluation using four machining datasets of slot micro-milling, the proposed system achieved a maximum error mean of 0.038 mm from the actual wear value. The proposed system brings a promising opportunity to widen the utilization of digital twin technology with the extended Kalman filter framework for seamless data integration for wear monitoring service.

High-Order Extended Kalman Filter for State Estimation of Nonlinear Systems

In general, the extended Kalman filter (EKF) has a wide range of applications, aiming to minimize symmetric loss function (mean square error) and improve the accuracy and efficiency of state estimation. As the nonlinear model complexity increases, rounding errors gradually amplify, leading to performance degradation. After multiple iterations, divergence may occur. The traditional extended Kalman filter cannot accurately estimate the nonlinear model, and these errors still have an impact on the accuracy. To improve the filtering performance of the extended Kalman filter (EKF), this paper proposes a new extended Kalman filter (REKF) method that utilizes the statistical properties of the rounding error to enhance the estimation accuracy. After establishing the state model and measurement model, the residual term is used to replace the higher-order term in the Taylor expansion, and the least squares method is applied to identify the residual term step by step. Then, the iterative process of updating the extended Kalman filter is carried out. Within the Kalman filter framework, a higher-order rounding error-based extended Kalman filter (REKF) is designed for the joint estimation of rounding error and random variables, and the solution method for the rounding error is considered for the multilevel approximation of the original function. Through numerical simulations on a general nonlinear model, the higher-order rounding error-based extended Kalman filter (REKF) achieves better estimation results than the extended Kalman filter (EKF) and improves the filtering accuracy by utilizing the higher-order rounding error information, which also proves the effectiveness of the proposed method.

Improved Information Fusion for Agricultural Machinery Navigation Based on Context-Constrained Kalman Filter and Dual-Antenna RTK

Automatic navigation based on dual-antenna real-time kinematic (RTK) positioning has been widely employed for unmanned agricultural machinery, whereas GNSS inevitably suffers from signal blocking and electromagnetic interference. In order to improve the reliability of an RTK-based navigation system in a GNSS-challenged environment, an integrated navigation system is preferred for autonomous navigation, which increases the complexity and cost of the navigation system. The information fusion of integrated navigation has been dominated by Kalman filter (KF) for several decades, but the KF cannot assimilate the known knowledge of the navigation context efficiently. In this paper, the geometric characteristics of the straight path and path-tracking error were employed to formulate the constraint measurement model, which suppresses the position error in the case of RTK-degraded scenarios. The pseudo-measurements were then imported into the KF framework, and the smoothed navigation state was generated as a byproduct, which improves the reliability of the RTK positioning without external sensors. The experiment result of the mobile vehicle automatic navigation indicates that the tracking error-constrained KF (EC-KF) outperforms the trajectory-constrained KF (TC-KF) and KF when the RTK system outputs a float or single-point position (SPP) solution. In the case where the duration of the SPP solution was 20 s, the positioning errors of the EC-KF and TC-KF were reduced by 38.50% and 24.04%, respectively, compared with those of the KF.

A meticulous covariance adaptive Kalman filter for satellite attitude estimation

Aiming at the problems of model errors, non-Gaussian noise and measurement anomaly in the spacecraft attitude estimation system, this article proposes an improved adaptive filtering method based on covariance matching, which solves the problems of simultaneous dynamics model error and measurement model error in the attitude estimation system, and at the same time, effectively reduces the effects of non-Gaussian noise and large outlier situations occurring in the vector measurement sensor. Firstly, an adaptive filtering algorithm based on the innovation sequence estimation covariance is investigated under the framework of multiplicative extended Kalman filter (MEKF), which is used to correct process noise covariance, then the Sage–Husa adaptive Kalman filtering (SHAKF) method is combined to correct the measurement noise covariance, and finally the meticulous covariance adaptive multiplicative extended Kalman filter is designed. The proposed algorithm uses both innovation and SHAKF methods to correct the two covariance matrices simultaneously. Several attitude estimation simulation scenarios are set up to simulate the proposed algorithm in the presence of model errors, non-Gaussian noise, and large outlier. The simulation results demonstrate that the proposed algorithm outperforms the conventional algorithms in terms of estimation accuracy and robustness.

Structural Dynamic Response Reconstruction Based on Recurrent Neural Network–Aided Kalman Filter

In structural health monitoring (SHM), an important issue is the limited availability of measurement data due to the spatial sparsity of sensors installed on the structure. These measurements are insufficient to accurately depict the actual dynamic behavior and response of the structure. Therefore, full‐field (i.e., every degree of freedom) structural response reconstruction based on sparse measured data has drawn a lot of attention in recent years. Kalman filter (KF) is an effective technology for response reconstruction (also known as state estimation), providing an optimal solution for systems that can be well‐represented by a fully known Gaussian linear state‐space model. This implies that both the process noise and measurement noise follow known zero‐mean Gaussian distribution, which is impractical in many civil engineering applications considering the unavoidable modeling errors and variations of environmental conditions. To address this challenge, a data‐physics hybrid‐driven method, i.e., KalmanNet, is proposed in this study for response reconstruction of partially known systems. By integrating a recurrent neural network (RNN) module into the KF framework, KalmanNet can efficiently learn and compute the Kalman gain using available monitoring data, without any Gaussian assumptions or explicit noise covariance specifications (e.g., covariance matrices of process and measurement noise). Both numerical and experimental investigations are conducted to validate this method. The results demonstrate that under the influence of non‐Gaussian noise and modeling errors, KalmanNet can effectively and accurately reconstruct the structural response from sparse measurements in real‐time and has higher accuracy and robustness compared to traditional KF even with optimal parameter settings.

Vehicular Connectivity on Complex Trajectories: Roadway-Geometry Aware ISAC Beam-Tracking

In this paper, we propose sensing-assisted beamforming designs for vehicles on arbitrarily shaped roads by relying on integrated sensing and communication (ISAC) signalling. Specifically, we aim to address the limitations of conventional ISAC beam-tracking schemes that do not apply to complex road geometries. To improve the tracking accuracy and communication quality of service (QoS) in vehicle to infrastructure (V2I) networks, it is essential to model the complicated roadway geometry. To that end, we impose the curvilinear coordinate system (CCS) in an interacting multiple model extended Kalman filter (IMM-EKF) framework. By doing so, both the position and the motion of the vehicle on a complicated road can be explicitly modeled and precisely tracked attributing to the benefits from the CCS. Furthermore, an optimization problem is formulated to maximize the array gain by dynamically adjusting the array size and thereby controlling the beamwidth, which takes the performance loss caused by beam misalignment into account. Numerical simulations demonstrate that the roadway geometry-aware ISAC beamforming approach outperforms the communication-only-based and ISAC kinematic-only-based technique in tracking performance. Moreover, the effectiveness of the dynamic beamwidth design is also verified by our numerical results.

Accelerating inverse inference of ensemble Kalman filter via reduced-order model trained using adaptive sparse observations

The implementation of data assimilation often struggles with low computational efficiency due to the complexity of high-fidelity numerical modeling and the large size of the observation vectors. To address this challenge, it is necessary to employ reduced-order strategies for both the full-order dynamical model and observation operator. Proper orthogonal decomposition (POD) based reduced-order model (ROM) and discrete empirical interpolation method (DEIM) based sparse observation identification are often considered as primary solutions. However, their effective combination requires the determination of high-quality POD reduced-order basis vectors. The current research focuses on incorporating the ROM and DEIM into a Kalman filter framework, resulting in a reduced-order Kalman filter with sparse observations. The key aspect is identifying the optimal POD basis, which can be achieved iteratively by introducing a relative root-mean-square-error (RMSE) for a near-optimal exploration of both the desired low-order POD basis and observation locations. The singular value decomposition (SVD) of the snapshots is performed in a low-dimensional subspace, leading to significant computational savings. This improved POD basis not only enhances the reconstruction of the ROM, but also helps build the DEIM observation matrix, resulting in simultaneous dimensionality reduction of the state and observation space. The effectiveness of this algorithm is demonstrated through the retrieval of initial conditions for one-dimensional tsunami wave equations with real data scenarios. Experiments are conducted for both smooth and nonsmoothed initial conditions. The results show that when the observation data is available at 3 and 6 spatial observation locations, our method only requires fewer vectors with the first 45 and 90 dominant modes derived using the RMSE, compared to as many as 210 and 252 derived from the relative energy criterion. Our method enables a low-modal ROM to effectively reconstruct the wave height field, leading to considerable prediction capability for potential long-wave dynamics with comparable accuracy. The CPU time savings achieved through our method is at least one to two orders of magnitude compared to the CPU time required by the full-order model.

Robust state-of-charge estimation method for lithium-ion batteries based on the fusion of time series relevance vector machine and filter algorithm

Data-driven methods have been widely employed in the field of Lithium-ion battery state-of-charge (SOC) estimation. However, most of these methods do not impose constraints on the rate of change between adjacent output SOC values, which leads to the issue of output SOC fluctuations when the battery current undergoes significant variations. To address this issue, a fusion algorithm is proposed in this study, which combines the multi-kernel Incremental relevance vector machine (MIRVM) and seasonal autoregressive integrated moving average (SARIMA) within an adaptive Kalman filter (AKF) framework, called SARIMA_AKF_MIRVM(SFR) algorithm. Employing the MIRVM algorithm, we aim to construct a data-driven model capturing the intricate relationship between voltage, current, and SOC of batteries under varying operating conditions. Furthermore, we shall leverage the SARIMA algorithm to establish a time series forecasting model for the adjacent SOC values. Subsequently, we shall introduce the AKF algorithm to filter prediction results associated with the MIRVM algorithm. Additionally, we shall integrate the whale optimization algorithm to obtain the optimal parameter combination for SFR algorithm. The experimental results demonstrate that the SFR algorithm exhibits exceptional generalization capabilities and robustness. The RMSE and MAE for four experimental test conditions remain below 0.3 %, with the maximum MAE reaching a mere 0.14 %.

GNSS-INS-dynamic fusion with robustness to outliers based on external force state estimation

Multi-source information fusion state estimation algorithms are an important means for drones to perceive ego-state, and accurate and robust estimation of external forces is crucial for precise control of quadrotors. This paper proposes a method that integrates a dynamic model into a multi-rate extended Kalman filter (EKF) framework on manifold. By estimating the magnitude of the external force acting on vehicle, meanwhile, a dynamic constraint on velocity loop is established to reduce the discrepancy between the model-predicted motion and the actual motion. Moreover, the estimated external force is integrated into the zero velocity update criterion for zero speed judgment, effectively reducing false detections while improving the accuracy of zero speed state recognition. However, multi-source measurements significantly increase the probability of data signal errors. To address this issue, we use a robust estimation algorithm to improve EKF’s sensitivity to abnormal measurements, flexibly adjusting measurement weights while rejecting unreasonable measurements. Validation with open-source indoor and outdoor datasets shows that our algorithm improves pose estimation performance while maintaining accurate positioning accuracy compared to non-dynamic fusion under the same filtering parameters, particularly in global navigation satellite system short time denied. It provides accurate external force estimation, offering multi-source data support in areas such as human–machine interaction and carrying variable mass payloads.

Ensemble data assimilation for open channel water level forecasting under flow disturbances: A synthetic data study

Accurate water level forecasts are essential for effective river management. Data assimilation (DA), involving real-time model correction through observational data, serves as a pivotal approach to enhance the predictive precision of model-based state forecasting. Nevertheless, prevailing DA methodologies encounter constraints in swiftly and accurately forecasting water levels amidst substantial uncertainties stemming from flow disturbances. To surmount this challenge, an innovative joint ensemble Kalman filter (EnKF) framework that facilitates concurrent estimation of flow disturbances and water levels is proposed. By incorporating the flow disturbance state variable into the EnKF framework for water level assimilation, a joint EnKF framework that effectively estimates flow disturbances at known disturbance points through assimilation techniques is established. Additionally, for scenarios where the precise location of the flow disturbance point remains unknown, the introduction of a fictitious lateral outflow in proximity to the observation point allows estimation of the flow disturbance process of this fictitious outflow. Synthetic case study results demonstrate that the incorporation of flow disturbance assimilation yields superior water level estimations compared to scenarios disregarding flow disturbances, regardless of the concordance between estimated and actual disturbance flow values. Furthermore, even when the location of the disturbance point is known, utilization of a fictitious lateral outflow and and subsequent flow correction exhibits heightened robustness and stability in comparison to direct correction of the disturbance point's flow. Therefore, in practical applications, utilizing a fictitious lateral outflow and flow correction within the joint EnKF framework emerges as a more suitable approach, irrespective of the knowledge of disturbance location. The proposed method constitutes a substantial contribution to the field of water level forecasting in river systems under conditions of uncertain flow disturbances.

Autonomous relative optical navigation based on unified modeling of external systematic errors

To solve the problem that external systematic errors of the optical camera cannot be fully estimated due to limited computing resources, a unified dimensionality reduction representation method for the external systematic errors of the optical camera is proposed, and autonomous relative optical navigation is realized. The camera translational and misalignment errors are converted into a three-dimensional rotation error, whose differential model can be established through specific attitude control and appropriate assumption. Then, the rotation error and the relative motion state are jointly estimated in an augmented Kalman filter framework. Compared with the traditional method that estimates the camera translational and misalignment errors, the proposed method reduces the computational complexity in that the estimated state dimension is reduced. Furthermore, as demonstrated by numerical simulation, the estimation accuracy is improved significantly.

An Improved Multiple-Outlier Robust Filter Based on Maximum Correntropy Criterion for Integrated Navigation

In the integrated navigation of global navigation satellite system (GNSS) and Inertial navigation system (INS), the Kalman filter is frequently employed for state estimation. GNSS signals are subject to interference from the outside environment, and it is difficult for the classical Kalman filter to handle observations with outliers. Numerous robust filters have been proposed based on the maximum correntropy criterion (MCC), which is insensitive to non-Gaussian noise, to address the issue of state estimation in the presence of outliers. However, a fixed prior covariance will restrict the use of a maximum correntropy criterion Kalman filter (MCKF) in integrated navigation systems because GNSS signals can be affected by both time-varying noise and outliers. Existing filters consider the prior measurement noise covariance matrix (MNCM) as a deterministic matrix and can only amplify signals unidirectionally during iteration cycles. The weight of effective observations is reduced despite these filters having strong robustness. To overcome this shortcoming, this paper proposes a τ -based maximum correntropy criterion Kalman filter (τ -MCKF). A strategy based on the Chisquare test is designed to optimize the covariance of GNSS observations. By introducing τ to bi-directionally adjust the initial MNCM, τ -MCKF makes full use of effective observations. Then, the τ -corrected MNCM is used to weigh different elements of the innovation vector, and a robust estimation is calculated based on the multiple-outlier robust Kalman filter (MORKF) framework. The GNSS/INS integrated navigation system is taken as an example to verify the effectiveness of the filter through simulation and vehicular experiments.

Speed Sensorless Control Method of Synchronous Reluctance Motor Based on Resonant Kalman Filter

When estimating the active back electromotive force (AEMF) of synchronous reluctance motor (SynRM) at medium and high speed, the traditional Kalman Filter (KF) will produce large phase delay due to the low-pass filter characteristics. A novel speed sensorless control scheme of SynRM based on resonant Kalman filter (RKF) is proposed in this paper. Firstly, the AEMF observer based on KF is designed and the bandwidth performance is analyzed. To solve the phase delay problem caused by KF, a generalized integral resonance perturbation estimator (GIRPE) is introduced to the framework of KF and the proposed RKF is formed. The estimation error of AEMF is compensated through GIRPE in the current error feedback loop of KF observer. In this way, the high-precision position estimation at high speed range is obtained. Finally, an improved dynamic position compensator (IDPC) is proposed to enhance the position estimation performance under dynamic case. The effectiveness of proposed scheme is validated at a 1.5kW SynRM drive, and the computational burden of the proposed method is assessed quantitatively.

A Kullback–Leibler divergence method for input–system–state identification

The capability of a novel Kullback–Leibler divergence method is examined herein within the Kalman filter framework to select the input–parameter–state estimation execution with the most plausible results. This identification suffers from the uncertainty related to obtaining different results from different initial parameter set guesses, and the examined approach uses the information gained from the data in going from the prior to the posterior distribution to address the issue. Firstly, the Kalman filter is performed for a number of different initial parameter sets providing the system input–parameter–state estimation. Secondly, the resulting posterior distributions are compared simultaneously to the initial prior distributions using the Kullback–Leibler divergence. Finally, the identification with the least Kullback–Leibler divergence is selected as the one with the most plausible results. Importantly, the method is shown to select the better performed identification in linear, nonlinear, and limited information applications, providing a powerful tool for system monitoring.

Compressed Gaussian Estimation under Low Precision Numerical Representation.

This paper introduces a novel method for computationally efficient Gaussian estimation of high-dimensional problems such as Simultaneous Localization and Mapping (SLAM) processes and for treating certain Stochastic Partial Differential Equations (SPDEs). The authors have presented the Generalized Compressed Kalman Filter (GCKF) framework to reduce the computational complexity of the filters by partitioning the state vector into local and global and compressing the global state updates. The compressed state update, however, still suffers from high computational costs, making it challenging to implement on embedded processors. We propose a low-precision numerical representation for the global filter, such as 16-bit integer or 32-bit single-precision formats for the global covariance matrix, instead of the expensive double-precision, floating-point representation (64 bits). This truncation can inevitably cause filter instability since the truncated covariance matrix becomes overoptimistic or even turns to be an invalid covariance matrix. We introduce a Minimal Covariance Inflation (MCI) method to make the filter consistent while minimizing the truncation errors. Simulation-based experiments results show significant improvement of the proposed method with a reduction in the processing time with minimal loss of accuracy.

Online estimation of colored observation‐noise parameters within an ensemble Kalman filtering framework

AbstractThis work addresses the problem of data assimilation in large‐dimensional systems with colored observation noise of unknown statistics, a scenario that will become more common in the near future with the deployment of denser observational networks of high spatio‐temporal coverage. Here, we are interested in the ensemble Kalman filter (EnKF) framework, which has been derived around a white observation noise assumption. Recently, colored observation‐noise aware EnKFs in which the noise was modeled as a first‐order autoregressive (AR) model were introduced. This work generalizes the above‐mentioned filters to learn the statistics of the AR model further online. We follow the state augmentation approach first to estimate the state and the AR model transfer matrix simultaneously, then the variational Bayesian approach to estimate the AR model noise covariance parameters further. We accordingly derive two filtering EnKF‐like algorithms, which estimate those statistics together with the system state. We demonstrate the effectiveness of the proposed colored observation‐noise aware filtering schemes and compare their performance based on several numerical experiments conducted with the Lorenz‐96 model.

Factor Graph Framework for Smartphone Indoor Localization: Integrating Data-Driven PDR and Wi-Fi RTT/RSS Ranging

The classic fusion localization techniques based on the Kalman filter (KF) framework have been a focus of research community in the past decades, due to the limited computing power of the mobile devices. However, with computing-efficient and sensor-rich smartphones now being a commonplace, it is convenient and meaningful to provide more accurate positioning services for smartphones in an indoor environment. In this paper, we design and develop a tightly coupled fusion platform of Wi-Fi RTT, RSS, and data-driven PDR based on factor graph optimization for locating the consumer-grade smartphones in indoor environment. As compared to the existing PDR solutions, including step model-based approaches and data-driven approaches, the proposed PDR solution with magnetic information constraint can track the relative position change of pedestrians at 20 Hz, while supporting multiple smartphone usage poses. A comprehensive comparison between factor graph optimization (FGO), KF frame and its variants is also performed. The experimental results demonstrate that the proposed fusion platform achieves an average positioning accuracy of 0.39 m. In addition, it also improves the accuracy of EFK and ARKF by 45.83% and 27.78%, respectively. The analysis shows that as the smartphone computing performance continues to improve, the FGO-based sensor fusion gradually replaces the KF frame and its variants.