We propose a Neural-Enhanced Distributed Kalman Filter (NDKF) for multi-sensor state estimation in nonlinear systems. Unlike traditional Kalman filters that rely on explicit, linear models and centralized data fusion, the NDKF leverages neural networks to learn both the system dynamics and measurement functions directly from data. Each sensor node performs local prediction and update steps using these learned models and exchanges only compact summary information with its neighbors via a consensus-based fusion process, which reduces communication overhead and eliminates a single point of failure. Our theoretical convergence analysis establishes sufficient conditions under which the local linearizations of the neural models guarantee overall filter stability and provides a solid foundation for the proposed approach. Simulation studies on a 2D system with four partially observing nodes demonstrate that the NDKF significantly outperforms a distributed Extended Kalman Filter. These outcomes, as yielded by the proposed NDKF method, highlight its potential to improve the scalability, robustness, and accuracy of distributed state estimation in complex nonlinear environments.
Since time immemorial an old adage has always seemed to ring true: you cannot use a high-level productive programming language like Python or R for real-time control and embedded-systems programming, you must rewrite your program in C. We present a counterexample to this mantra by demonstrating how recent compiler developments in the Julia programming language allow users of Julia and the equation-based modeling language ModelingToolkit to compile and deploy binaries for real-time model-based estimation and control. Contrary to the approach taken by a majority of modeling and simulation tools, we do not generate C code, and instead demonstrate how we may use the native Julia code-generation pipeline through LLVM to compile architecture-specific binaries from high-level code. This approach avoids many of the restrictions typically placed on high-level languages to enable C-code generation. As case studies, we include a nonlinear state estimator derived from an equation-based model which is compiled into a program that performs state estimation for deployment onto a Raspberry Pi, as well as a PID controller library implemented in Julia and compiled into a shared library callable from a C program.
We consider an operator-based latent Markov representation of a stochastic nonlinear dynamical system, where the stochastic evolution of the latent state embedded in a reproducing kernel Hilbert space is described with the corresponding transfer operator, and develop a spectral method to learn this representation based on the theory of stochastic realization. The embedding may be learned simultaneously using reproducing kernels, for example, constructed with feed-forward neural networks. We also address the generalization of sequential state-estimation (Kalman filtering) in stochastic nonlinear systems, and of operator-based eigen-mode decomposition of dynamics, for the representation. Several examples with synthetic and real-world data are shown to illustrate the empirical characteristics of our methods, and to investigate the performance of our model in sequential state-estimation and mode decomposition.
Modern on-road navigation systems heavily depend on integrating speed measurements with inertial navigation systems (INS) and global navigation satellite systems (GNSS). Telemetry-based applications typically source speed data from the On-Board Diagnostic II (OBD-II) system. However, the method of deriving speed, as well as the types of sensors used to measure wheel speed, differs across vehicles. These differences result in varying error characteristics that must be accounted for in navigation and autonomy applications. This paper addresses this gap by examining the diverse speed-sensing technologies employed in standard automotive systems and alternative techniques used in advanced systems designed for higher levels of autonomy, such as Advanced Driver Assistance Systems (ADAS), Autonomous Driving (AD), or surveying applications. We propose a method to identify the type of speed sensor in a vehicle and present strategies for accurately modeling its error characteristics. To validate our approach, we collected and analyzed data from three long real road trajectories conducted in urban environments in Toronto and Kingston, Ontario, Canada. The results underscore the critical role of integrating multiple sensor modalities to achieve more accurate speed estimation, thus improving automotive navigation state estimation, particularly in GNSS-denied environments.
Leakage in water systems results in significant daily water losses, degrading service quality, increasing costs, and aggravating environmental problems. Most leak localization methods rely solely on pressure data, missing valuable information from other sensor types. This article proposes a hydraulic state estimation methodology based on a dual Unscented Kalman Filter (UKF) approach, which enhances the estimation of both nodal hydraulic heads, critical in localization tasks, and pipe flows, useful for operational purposes. The approach enables the fusion of different sensor types, such as pressure, flow and demand meters. The strategy is evaluated in well-known open source case studies, namely Modena and L-TOWN, showing improvements over other state-of-the-art estimation approaches in terms of interpolation accuracy, as well as more precise leak localization performance in L-TOWN.
This paper deals with state estimation of nonlinear stochastic dynamic models. In particular, the stochastic integration rule, which provides asymptotically unbiased estimates of the moments of nonlinearly transformed Gaussian random variables, is reviewed together with the recently introduced stochastic integration filter (SIF). Using SIF, the respective multi-step prediction and smoothing algorithms are developed in full and efficient square-root form. The stochastic-integration-rule-based algorithms are implemented in Python (within the Stone Soup framework) and in MATLAB and are numerically evaluated and compared with the well-known unscented and extended Kalman filters using the Stone Soup defined tracking scenario.
Accurately estimating spatiotemporal traffic states on freeways is a significant challenge due to limited sensor deployment and potential data corruption. In this study, we propose an efficient and robust low-rank model for precise spatiotemporal traffic speed state estimation (TSE) using lowpenetration vehicle trajectory data. Leveraging traffic wave priors, an oblique grid-based matrix is first designed to transform the inherent dependencies of spatiotemporal traffic states into the algebraic low-rankness of a matrix. Then, with the enhanced traffic state low-rankness in the oblique matrix, a low-rank matrix completion method is tailored to explicitly capture spatiotemporal traffic propagation characteristics and precisely reconstruct traffic states. In addition, an anomaly-tolerant module based on a sparse matrix is developed to accommodate corrupted data input and thereby improve the TSE model robustness. Notably, driven by the understanding of traffic waves, the computational complexity of the proposed efficient method is only correlated with the problem size itself, not with dataset size and hyperparameter selection prevalent in existing studies. Extensive experiments demonstrate the effectiveness, robustness, and efficiency of the proposed model. The performance of the proposed method achieves up to a 12% improvement in Root Mean Squared Error (RMSE) in the TSE scenarios and an 18% improvement in RMSE in the robust TSE scenarios, and it runs more than 20 times faster than the state-of-the-art (SOTA) methods.
Proprioceptive sensors on planetary rovers serve for state estimation and for understanding terrain and locomotion performance. While inertial measurement units (IMUs) are widely used to this effect, force-torque sensors are less explored for planetary navigation despite their potential to directly measure interaction forces and provide insights into traction performance. This paper presents an evaluation of the performance and use cases of force-torque sensors based on data collected from a six-wheeled rover during tests over varying terrains, speeds, and slopes. We discuss challenges, such as sensor signal reliability and terrain response accuracy, and identify opportunities regarding the use of these sensors. The data is openly accessible and includes force-torque measurements from each of the six-wheel assemblies as well as IMU data from within the rover chassis. This paper aims to inform the design of future studies and rover upgrades, particularly in sensor integration and control algorithms, to improve navigation capabilities.
Deep learning is widely used to predict complex dynamical systems in many scientific and engineering areas. However, the black-box nature of these deep learning models presents significant challenges for carrying out simultaneous data assimilation (DA), which is a crucial technique for state estimation, model identification, and reconstructing missing data. Integrating ensemble-based DA methods with nonlinear deep learning models is computationally expensive and may suffer from large sampling errors. To address these challenges, we introduce a deep learning framework designed to simultaneously provide accurate forecasts and efficient DA. It is named Conditional Gaussian Koopman Network (CGKN), which transforms general nonlinear systems into nonlinear neural differential equations with conditional Gaussian structures. CGKN aims to retain essential nonlinear components while applying systematic and minimal simplifications to facilitate the development of analytic formulae for nonlinear DA. This allows for seamless integration of DA performance into the deep learning training process, eliminating the need for empirical tuning as required in ensemble methods. CGKN compensates for structural simplifications by lifting the dimension of the system, which is motivated by Koopman theory. Nevertheless, CGKN exploits special nonlinear dynamics within the lifted space. This enables the model to capture extreme events and strong non-Gaussian features in joint and marginal distributions with appropriate uncertainty quantification. We demonstrate the effectiveness of CGKN for both prediction and DA on three strongly nonlinear and non-Gaussian turbulent systems: the projected stochastic Burgers-Sivashinsky equation, the Lorenz 96 system, and the El Ni\~no-Southern Oscillation. The results justify the robustness and computational efficiency of CGKN.
The Kalman filter (KF) and its variants are among the most celebrated algorithms in signal processing. These methods are used for state estimation of dynamic systems by relying on mathematical representations in the form of simple state-space (SS) models, which may be crude and inaccurate descriptions of the underlying dynamics. Emerging data-centric artificial intelligence (AI) techniques tackle these tasks using deep neural networks (DNNs), which are model-agnostic. Recent developments illustrate the possibility of fusing DNNs with classic Kalman-type filtering, obtaining systems that learn to track in partially known dynamics. This article provides a tutorial-style overview of design approaches for incorporating AI in aiding KF-type algorithms. We review both generic and dedicated DNN architectures suitable for state estimation, and provide a systematic presentation of techniques for fusing AI tools with KFs and for leveraging partial SS modeling and data, categorizing design approaches into task-oriented and SS model-oriented. The usefulness of each approach in preserving the individual strengths of model-based KFs and data-driven DNNs is investigated in a qualitative and quantitative study, whose code is publicly available, illustrating the gains of hybrid model-based/data-driven designs. We also discuss existing challenges and future research directions that arise from fusing AI and Kalman-type algorithms.
Despite substantial progress in recent years, probabilistic solvers with adaptive step sizes can still not solve memory-demanding differential equations -- unless we care only about a single point in time (which is far too restrictive; we want the whole time series). Counterintuitively, the culprit is the adaptivity itself: Its unpredictable memory demands easily exceed our machine's capabilities, making our simulations fail unexpectedly and without warning. Still, dropping adaptivity would abandon years of progress, which can't be the answer. In this work, we solve this conundrum. We develop an adaptive probabilistic solver with fixed memory demands building on recent developments in robust state estimation. Switching to our method (i) eliminates memory issues for long time series, (ii) accelerates simulations by orders of magnitude through unlocking just-in-time compilation, and (iii) makes adaptive probabilistic solvers compatible with scientific computing in JAX.
A fundamental challenge in developing data-driven approaches to ecological systems for tasks such as state estimation and prediction is the paucity of the observational or measurement data. For example, modern machine-learning techniques such as deep learning or reservoir computing typically require a large quantity of data. Leveraging synthetic data from paradigmatic nonlinear but non-ecological dynamical systems, we develop a meta-learning framework with time-delayed feedforward neural networks to predict the long-term behaviors of ecological systems as characterized by their attractors. We show that the framework is capable of accurately reconstructing the ``dynamical climate'' of the ecological system with limited data. Three benchmark population models in ecology, namely the Hastings-Powell model, a three-species food chain, and the Lotka-Volterra system, are used to demonstrate the performance of the meta-learning based prediction framework. In all cases, enhanced accuracy and robustness are achieved using five to seven times less training data as compared with the corresponding machine-learning method trained solely from the ecosystem data. A number of issues affecting the prediction performance are addressed.
Many scientific fields collect longitudinal count compositional data. Each observation is a multivariate count vector, where the total counts are arbitrary, and the information lies in the relative frequency of the counts. Multiple authors have proposed Bayesian Multinomial Logistic-Normal Dynamic Linear Models (MLN-DLMs) as a flexible approach to modeling these data. However, adoption of these methods has been limited by computational challenges. This article develops an efficient and accurate approach to posterior state estimation, called $\textit{Fenrir}$. Our approach relies on a novel algorithm for MAP estimation and an accurate approximation to a key posterior marginal of the model. As there are no equivalent methods against which we can compare, we also develop an optimized Stan implementation of MLN-DLMs. Our experiments suggest that Fenrir can be three orders of magnitude more efficient than Stan and can even be incorporated into larger sampling schemes for joint inference of model hyperparameters. Our methods are made available to the community as a user-friendly software library written in C++ with an R interface.
The accurate modelling of structural dynamics is crucial across numerous engineering applications, such as Structural Health Monitoring (SHM), seismic analysis, and vibration control. Often, these models originate from physics-based principles and can be derived from corresponding governing equations, often of differential equation form. However, complex system characteristics, such as nonlinearities and energy dissipation mechanisms, often imply that such models are approximative and often imprecise. This challenge is further compounded in SHM, where sensor data is often sparse, making it difficult to fully observe the system's states. To address these issues, this paper explores the use of Physics-Informed Neural Networks (PINNs), a class of physics-enhanced machine learning (PEML) techniques, for the identification and estimation of dynamical systems. PINNs offer a unique advantage by embedding known physical laws directly into the neural network's loss function, allowing for simple embedding of complex phenomena, even in the presence of uncertainties. This study specifically investigates three key applications of PINNs: state estimation in systems with sparse sensing, joint state-parameter estimation, when both system response and parameters are unknown, and parameter estimation within a Bayesian framework to quantify uncertainties. The results demonstrate that PINNs deliver an efficient tool across all aforementioned tasks, even in presence of modelling errors. However, these errors tend to have a more significant impact on parameter estimation, as the optimization process must reconcile discrepancies between the prescribed model and the true system behavior. Despite these challenges, PINNs show promise in dynamical system modeling, offering a robust approach to handling uncertainties.
A novel online MOT algorithm, IMM Joint Homography State Estimation (IMM-JHSE), is proposed. IMM-JHSE uses an initial homography estimate as the only additional 3D information, whereas other 3D MOT methods use regular 3D measurements. By jointly modelling the homography matrix and its dynamics as part of track state vectors, IMM-JHSE removes the explicit influence of camera motion compensation techniques on predicted track position states, which was prevalent in previous approaches. Expanding upon this, static and dynamic camera motion models are combined using an IMM filter. A simple bounding box motion model is used to predict bounding box positions to incorporate image plane information. In addition to applying an IMM to camera motion, a non-standard IMM approach is applied where bounding-box-based BIoU scores are mixed with ground-plane-based Mahalanobis distances in an IMM-like fashion to perform association only, making IMM-JHSE robust to motion away from the ground plane. Finally, IMM-JHSE makes use of dynamic process and measurement noise estimation techniques. IMM-JHSE improves upon related techniques, including UCMCTrack, OC-SORT, C-BIoU and ByteTrack on the DanceTrack and KITTI-car datasets, increasing HOTA by 2.64 and 2.11, respectively, while offering competitive performance on the MOT17, MOT20 and KITTI-pedestrian datasets. Using publicly available detections, IMM-JHSE outperforms almost all other 2D MOT methods and is outperformed only by 3D MOT methods -- some of which are offline -- on the KITTI-car dataset. Compared to tracking-by-attention methods, IMM-JHSE shows remarkably similar performance on the DanceTrack dataset and outperforms them on the MOT17 dataset. The code is publicly available: https://github.com/Paulkie99/imm-jhse.
Modern industrial systems require frequent updates to their cyber and physical infrastructures, often demanding considerable reconfiguration effort. This paper introduces the industrial Cyber-Physical Systems Description Language, iCPS-DL, which enables autonomic reconfigurations for industrial Cyber-Physical Systems. The iCPS-DL maps an industrial process using semantics for physical and cyber-physical components, a state estimation model, and agent interactions. A novel aspect is using communication semantics to ensure live interaction among distributed agents. Reasoning on the semantic description facilitates the configuration of the industrial process control loop. A Water Distribution Networks domain case study demonstrates iCPS-DL's application.
This paper proposes FAST-LIVO2: a fast, direct LiDAR-inertial-visual odometry framework to achieve accurate and robust state estimation in SLAM tasks and provide great potential in real-time, onboard robotic applications. FAST-LIVO2 fuses the IMU, LiDAR and image measurements efficiently through an ESIKF. To address the dimension mismatch between the heterogeneous LiDAR and image measurements, we use a sequential update strategy in the Kalman filter. To enhance the efficiency, we use direct methods for both the visual and LiDAR fusion, where the LiDAR module registers raw points without extracting edge or plane features and the visual module minimizes direct photometric errors without extracting ORB or FAST corner features. The fusion of both visual and LiDAR measurements is based on a single unified voxel map where the LiDAR module constructs the geometric structure for registering new LiDAR scans and the visual module attaches image patches to the LiDAR points. To enhance the accuracy of image alignment, we use plane priors from the LiDAR points in the voxel map (and even refine the plane prior) and update the reference patch dynamically after new images are aligned. Furthermore, to enhance the robustness of image alignment, FAST-LIVO2 employs an on-demanding raycast operation and estimates the image exposure time in real time. Lastly, we detail three applications of FAST-LIVO2: UAV onboard navigation demonstrating the system's computation efficiency for real-time onboard navigation, airborne mapping showcasing the system's mapping accuracy, and 3D model rendering (mesh-based and NeRF-based) underscoring the suitability of our reconstructed dense map for subsequent rendering tasks. We open source our code, dataset and application on GitHub to benefit the robotics community.
A leading family of algorithms for state estimation in dynamic systems with multiple sub-states is based on particle filters (PFs). PFs often struggle when operating under complex or approximated modelling (necessitating many particles) with low latency requirements (limiting the number of particles), as is typically the case in multi target tracking (MTT). In this work, we introduce a deep neural network (DNN) augmentation for PFs termed learning flock (LF). LF learns to correct a particles-weights set, which we coin flock, based on the relationships between all sub-particles in the set itself, while disregarding the set acquisition procedure. Our proposed LF, which can be readily incorporated into different PFs flow, is designed to facilitate rapid operation by maintaining accuracy with a reduced number of particles. We introduce a dedicated training algorithm, allowing both supervised and unsupervised training, and yielding a module that supports a varying number of sub-states and particles without necessitating re-training. We experimentally show the improvements in performance, robustness, and latency of LF augmentation for radar multi-target tracking, as well its ability to mitigate the effect of a mismatched observation modelling. We also compare and illustrate the advantages of LF over a state-of-the-art DNN-aided PF, and demonstrate that LF enhances both classic PFs as well as DNN-based filters.
In various domains, Sequential Recommender Systems (SRS) have become essential due to their superior capability to discern intricate user preferences. Typically, SRS utilize transformer-based architectures to forecast the subsequent item within a sequence. Nevertheless, the quadratic computational complexity inherent in these models often leads to inefficiencies, hindering the achievement of real-time recommendations. Mamba, a recent advancement, has exhibited exceptional performance in time series prediction, significantly enhancing both efficiency and accuracy. However, integrating Mamba directly into SRS poses several challenges. Its inherently unidirectional nature may constrain the model's capacity to capture the full context of user-item interactions, while its instability in state estimation can compromise its ability to detect short-term patterns within interaction sequences. To overcome these issues, we introduce a new framework named Selective Gated Mamba (SIGMA) for Sequential Recommendation. This framework leverages a Partially Flipped Mamba (PF-Mamba) to construct a bidirectional architecture specifically tailored to improve contextual modeling. Additionally, an input-sensitive Dense Selective Gate (DS Gate) is employed to optimize directional weights and enhance the processing of sequential information in PF-Mamba. For short sequence modeling, we have also developed a Feature Extract GRU (FE-GRU) to efficiently capture short-term dependencies. Empirical results indicate that SIGMA outperforms current models on five real-world datasets. Our implementation code is available at https://github.com/ziwliu-cityu/SIMGA to ease reproducibility.
This paper addresses the challenge of neural state estimation in power distribution systems. We identified a research gap in the current state of the art, which lies in the inability of models to adapt to changes in the power grid, such as loss of sensors and branch switching, in a zero-shot fashion. Based on the literature, we identified graph neural networks as the most promising class of models for this use case. Our experiments confirm their robustness to some grid changes and also show that a deeper network does not always perform better. We propose data augmentations to improve performance and conduct a comprehensive grid search of different model configurations for common zero-shot learning scenarios.