This paper focuses on the multi-target tracking using the Stone Soup framework. In particular, we aim at evaluation of two multi-target tracking scenarios based on the simulated class-B dataset and ADS-B class-A dataset provided by OpenSky Network. The scenarios are evaluated w.r.t. selection of a local state estimator using a range of the Stone Soup metrics. Source code with scenario definitions and Stone Soup set-up are provided along with the paper.

This paper focuses on state prediction for stochastic dynamic models with linear dynamics, emphasizing a recently proposed efficient and robust Lagrangian approach for solving the Chapman–Kolmogorov equation. In contrast to the standard Eulerian perspective, the Lagrangian method separates the solution into two sequential steps: advection and diffusion. Advection is handled by moving a carefully designed grid, while diffusion is addressed using the convolution theorem. This approach significantly reduces computational complexity while preserving the same accuracy. In this paper, we propose formulating diffusion as a continuous-time process, leading to a partial differential equation (PDE). Various methods for solving this PDE are presented and compared within a unified framework, along with evaluations of their properties and example implementations. We demonstrate that the continuous formulation can yield substantial reductions in computational complexity with only marginal loss in accuracy.

State-space estimation and tracking rely on accurate dynamical models to perform well. However, obtaining an accurate dynamical model for complex scenarios or adapting to changes in the system poses challenges to the estimation process. Recently, augmented physics-based models (APBMs) appear as an appealing strategy to cope with these challenges where the composition of a small and adaptive neural network with known physics-based models (PBM) is learned on the fly following an augmented state-space estimation approach. A major issue when introducing data-driven components in such a scenario is the danger of compromising the meaning (or interpretability) of estimated states. In this work, we propose a novel constrained estimation strategy that constrains the APBM dynamics close to the PBM. The novel state-space constrained approach leads to more flexible ways to impose constraints than the traditional APBM approach. Our experiments with a radar-tracking scenario demonstrate different aspects of the proposed approach and the trade-offs inherent in the imposed constraints.

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.

This paper deals with the initial integrity evaluation of the navigation information provided by the multilayer GNSS and LEO-PNT constellation. Although, the global satellite navigation systems (GNSS) play indispensable role in almost all aspects of today’s society, their signals are prone to intentional or accidental interference. Therefore, low Earth orbit (LEO) constellations aiming at position, navigation, and timing (PNT) solution have recently been introduced as their extension. The LEO-PNT constellations are planned to contain hundreds of SVs with better interference resilience and geometric diversity. As a consequence, the multi-layer GNSS and LEO-PNT constellation was shown to offer more accurate PNT solution. In this paper, we focus on another important aspect of the multi-layer navigation information, which is its integrity assessment. In particular, we analyse possible dependencies between GNSS and LEO-PNT constellations and their impact on the integrity evaluated using the solution separation. The analysis is supported by the numerical simulations using GPS and LEO-PNT constellations with 32 and 441 satellites, respectively.

A first-principle single-object model is proposed for pedestrian tracking. It is assumed that the extent of the moving object can be described via known statistics in 3D, such as pedestrian height. The proposed model thus need not constrain the object motion in 3D to a common ground plane, which is usual in 3D visual tracking applications. A nonlinear filter for this model is implemented using the unscented Kalman filter (UKF) and tested using the publicly available MOT-17 dataset. The proposed solution yields promising results in 3D while maintaining excellent results when projected into the 2D image. Moreover, the estimation error covariance matches the true one. Unlike conventional methods, the introduced model parameters have convenient meaning and can readily be adjusted for a problem.

This paper deals with the state estimation of stochastic models with continuous dynamics. The aim is to incorporate spectral differentiation methods into the solution to the Fokker-Planck equation in grid-based state estimation routine, while taking into account the specifics of the field, such as probability density function (PDF) features, moving grid, zero boundary conditions, etc. The spectral methods, in general, achieve very fast convergence rate of $\mathcal{O}\left(c^{N}\right)(O{\lt}$ $c{\lt}1$) for analytical functions such as the probability density function, where N is the number of grid points. This is significantly better than the standard finite difference method (or midpoint rule used in discrete estimation) typically used in grid-based filter design with convergence rate $\mathcal{O}\left(\frac{1}{N^{2}}\right)$. As consequence, the proposed spectral method based filter provides better state estimation accuracy with lower number of grid points, and thus, with lower computational complexity.

This paper deals with the resilient time provision based on an ensemble of clocks. In particular, the emphasis is laid on the combination of clock outputs and detecting possible faults. Two classes of fault detection methods, namely model-based and AI/ML-based, are discussed and analysed. In addition, a novel fault detection technique based on the solution separation principle is proposed and tailored for the area of the time provision. Selected fault detection methods are numerically evaluated using a model of an atomic clock ensemble.

Common visual tracking algorithms make use of bounding box (BB) motion models. These models are parameterized by quantities such as noise covariance parameters corresponding to the evolution of the position, velocity, aspect ratio, width, or height of the bounding box, or their respective velocities. The noise covariance parameters are often hand-selected, using no principled knowledge regarding various aspects such as camera pose or frame rate. This paper aims to analyze how these aspects influence parameter estimates obtained from annotated datasets that are well-known to the visual tracking community. To obtain the estimates, the recently developed measurement difference method (MDM) is modified and used.

This paper deals with state estimation of stochastic models with linear state dynamics, continuous or discrete in time. The emphasis is laid on a numerical solution to the state prediction by the time-update step of the grid-point-based point-mass filter (PMF), which is the most computationally demanding part of the PMF algorithm. A novel efficient PMF (ePMF) estimator, unifying continuous and discrete, approaches is proposed, designed, and discussed. By numerical illustrations, it is shown, that the proposed ePMF can lead to a time complexity reduction that exceeds 99.9% without compromising accuracy. The MATLAB® code of the ePMF is released with this paper.

This paper is devoted to the Bayesian state estimation of the nonlinear stochastic dynamic systems. The stress is laid on the numerical solution to the Bayesian recursive relations by the point-mass filter for a class of state-space models with linear dynamics and nonlinear measurement. In particular, a novel reliable technique for convolution computation is proposed. The technique combines the standard point-mass-based convolution with a density-weighted integration to provide accurate results even for systems with small state noise. Several implementations of the technique are developed, theoretically analysed, and evaluated in a numerical study.

The paper deals with the state estimation of nonlinear stochastic systems with additive Gaussian noises by means of the Gaussian filters leveraging numerical integration rules. The filters were derived under the assumption of the joint state and measurement predictive density being Gaussian, which is violated by the system nonlinearity. Such violation can hardly be monitored by the standard Gaussian filters, which re-generate a new set of points for each involved numerical integration to accommodate their variance increase due to the additive noises. The paper proposes a stochastic integration filter algorithm that modifies the points instead of their resampling and thus admits reusing the points in the next time steps. The distribution of the points can thus bear more information than just the first two moments in case of the standard Gaussian filters. The acquired information is then utilized for the Gaussian assumption monitoring purposes. In the event of the assumption violation, the filter may change its behavior. As a by-product of reusing the points, the computational costs of the proposed filter are significantly reduced compared to the standard stochastic integration filter.

The paper deals with state estimation of nonlinear stochastic dynamic discrete-time systems with a special focus on the stochastic integration filter. The filter is an instance of Gaussian filters, which for strongly nonlinear systems may provide inconsistent estimates. Primarily, optimistic inconsistent estimates, which overrate quality of the point estimate, are inappropriate in many applications where estimate integrity is crucial. In this paper, a technique for an estimate consistency monitoring for detection of optimistic estimates is proposed based on entropy. For the purpose of the entropy computation, a probabilistic analysis of the stochastic integration filter behavior is carried out. The proposed consistency monitoring is illustrated in a numerical example.

The paper introduces a novel approach to an estimator design, the cooperative filter design, for state estimation of nonlinear systems. The approach is based on the idea of combining estimates of several different approximate (and thus sub-optimal) nonlinear filters, which are configured to perform the same task. Within the concept, two strategies are proposed, namely the cooperative estimation and cooperative monitoring. The strategies have the potential of an improvement of the estimation performance in terms of accuracy and consistency, which was confirmed by a numerical illustration.

The paper deals with the state estimation of nonlinear stochastic dynamic systems. The stress is laid on the assessment of the estimate error, which is caused by the violation of the estimator design assumptions. The assessment is based on measures comparing estimators actual working conditions and the assumptions under which the estimators have been proposed. In particular, the measures of nonlinearity and non-Gaussianity are discussed. The measures are briefly introduced and selected typical representatives are detailed with respect to their implementation. Performance of the measures is evaluated in the framework of the Gaussian filters in a numerical study.

The paper deals with the nonlinear state estimation of stochastic dynamic systems with a special focus on coping with outliers appearing in the system. A new stochastic integration Student's-t filter is developed based on the generic Student's-t filter and assuming the density of random variables present in the model and the conditional density of the state be Student's-t distributed. For evaluation of the integrals with Student's-t weights present in the filter relations, the stochastic integration rule is used. In contrast to other integration rules, it provides asymptotically exact values of the integrals. Performance of the proposed stochastic integration Student's-t filter is illustrated using a numerical simulation involving the coordinated-turn motion model.