The state-of-the-art tensor network Kalman filter lifts the curse of dimensionality for high-dimensional recursive estimation problems. However, the required rounding operation can cause filter divergence due to the loss of positive definiteness of covariance matrices. We solve this issue by developing, for the first time, a tensor network square root Kalman filter, and apply it to high-dimensional online Gaussian process regression. In our experiments, we demonstrate that our method is equivalent to the conventional Kalman filter when choosing a full-rank tensor network. Furthermore, we apply our method to a real-life system identification problem where we estimate 4¹⁴ parameters on a standard laptop. The estimated model outperforms the state-of-the-art tensor network Kalman filter in terms of prediction accuracy and uncertainty quantification.

Accurately estimating sea state parameters is crucial for ship safety and efficiency. The objective of this paper is to study the applicability of the Adaptive Kalman filter (AKF) to estimate sea state parameters—significant wave height, peak period, and relative mean wave direction—using onboard ship motion measurements. The main idea is to assess the performance of this method under real-world conditions including varying ship forward speed and heading and noisy measurements. In this study, data recorded from onboard the United States Coast Guard Cutter (USCGC) STRATTON is considered for testing the method. The method's performance is evaluated by comparing the estimated sea state parameters to those obtained from the Copernicus hind cast model. The obtained results show the AKF's capacity to estimate sea state parameters under real-world conditions, such as variable forward speeds and potential sensor and model inaccuracies.

Magnetic-field simultaneous localization and mapping (SLAM) using consumer-grade inertial and magnetometer sensors offers a scalable, cost-effective solution for indoor localization. However, the rapid error accumulation in the inertial navigation process limits the feasible exploratory phases of these systems. Advances in magnetometer array processing have demonstrated that odometry information, i.e., displacement and rotation information, can be extracted from local magnetic field variations and used to create magnetic-field odometry-aided inertial navigation systems. The error growth rate of these systems is significantly lower than that of standalone inertial navigation systems. This study seeks an answer to whether a magnetic-field SLAM system fed with measurements from a magnetometer array can indirectly extract odometry information - without requiring algorithmic modifications - and thus sustain longer exploratory phases. The theoretical analysis and simulation results show that such a system can extract odometry information and indirectly create a magnetic field odometry-aided inertial navigation system during the exploration phases. However, practical challenges related to map resolution and computational complexity remain significant.

This paper aims at estimating both unidirectional and multi-directional waves from noisy measured ship motion data, with a focus on the inclusion of the vessel's forward speed to reflect real-world operating conditions. The technique is based on an Adaptive Kalman Filter for estimating wave elevation and wave spectrum parameters, including significant wave height, peak period, and wave direction. The proposed method was tested using simulated ship motion data, and its performance was evaluated by comparing the estimated wave spectrum with reference values used in the simulation model and with results from a widely used baseline frequency domain approach. The results demonstrate that the method effectively estimates the wave spectrum in a short measuring window with a reasonable degree of accuracy when accounting for varying forward speed, indicating strong potential for real-time wave estimation to aid in improving navigation, safety, and operational efficiency.

Recently, it has been shown that odometry is possible only using data from a magnetometer array. In this work, we analyze the uncertainty of the pose change estimate using a magnetometer array. We derive an analytical expression for the pose change covariance to analyze the estimation uncertainty in Monte Carlo simulations. Under certain conditions, we demonstrate that using a magnetometer array, it is possible to estimate the position and orientation change with submillimeter and subdegree precision between two consecutive time-steps. Moreover, we also demonstrate that when constructing a magnetometer array, magnetometers should be placed in the direction of movement to maximize the positional and rotational precision, with at least four magnetometers per unit of length-scale. In addition, we illustrate that to minimize positional and rotational drift to under a few percentages and degrees of the distance traveled, submillimeter and subdegree magnetometer alignment errors are necessary. Similarly, bias errors smaller than a few percent of the magnitude of the magnetic field variations are necessary. The Monte Carlo simulations are verified using experimental data collected with a 30-magnetometer array. The experimental data show that when insufficient magnetic field anomalies are in close proximity, the changes in positions are estimated poorly, while significant orientation information is still obtained. It also shows that when the magnetometer array is in close proximity to sufficient magnetic field anomalies, the overall trajectory traveled by a magnetometer array can be accurately estimated with a horizontal error accumulation of less than a percentage of the distance traveled.

In this volume, we are happy present the post-proceedings of BNAIC/BeNeLearn 2023, the joint conference on Artificial Intelligence and Machine Learning in the BeNeLux, which took place at TU Delft. It is the main regional conference on these topics and has a long tradition: in 2018, the 30th Benelux Conference on Artificial Intelligence (BNAIC) and the 27th Belgian Dutch Conference on Machine Learning (Benelearn) were jointly organized in ‘s Hertogenbosch, and this has been repeated annually since. [...]

We address the computational challenges of large-scale geospatial mapping with Gaussian process (GP) regression by performing localized computations rather than processing the entire map simultaneously. Traditional approaches to GP regression often involve computational and storage costs that either scale with the number of measurements, or with the spatial extent of the mapped area, limiting their scalability for real-time applications. Our method places a global grid of finite-support basis functions and restricts computations to a local subset of the grid 1) surrounding the measurement when the map is updated, and 2) surrounding the query point when the map is queried. This localized approach ensures that only the relevant area is updated or queried at each timestep, significantly reducing computational complexity while maintaining accuracy. Unlike many existing methods, which suffer from boundary effects or increased computational costs with mapped area, our localized approach avoids discontinuities and ensures that computational costs remain manageable regardless of map size. This approximation to GP mapping provides high accuracy with limited computational budget for the specialized task of performing fast online map updates and fast online queries of large-scale geospatial maps. It is therefore a suitable approximation for use in real-time applications where such properties are desirable, such as real-time simultaneous localization and mapping (SLAM) in large, nonlinear geospatial fields. We show on experimental data with magnetic field measurements that our algorithm is faster and equally accurate compared to existing methods, both for recursive magnetic field mapping and for magnetic field SLAM.

Simultaneous localization and mapping (SLAM) is the task of building a map representation of an unknown environment while at the same time using it for positioning. A probabilistic interpretation of the SLAM task allows for incorporating prior knowledge and for operation under uncertainty. Contrary to the common practice of computing point estimates of the system states, we capture the full posterior density through approximate Bayesian inference. This dynamic learning task falls under state estimation, where the state-of-the-art is in sequential Monte Carlo methods that tackle the forward filtering problem. In this paper, we introduce a framework for probabilistic SLAM using particle smoothing that does not only incorporate observed data in current state estimates, but it also backtracks the updated knowledge to correct for past drift and ambiguities in both the map and in the states. Our solution can efficiently handle both dense and sparse map representations by Rao-Blackwellization of conditionally linear and conditionally linearized models. We show through simulations and real-world experiments how the principles apply to radio (Bluetooth low-energy/Wi-Fi), magnetic field, and visual SLAM. The proposed solution is general, efficient, and works well under confounding noise.

We present a lightweight magnetic field simultaneous localisation and mapping (SLAM) approach for drift correction in odometry paths, where the interest is purely in the odometry and not in map building. We represent the past magnetic field readings as a one-dimensional trajectory against which the current magnetic field observations are matched. This approach boils down to sequential loop-closure detection and decision-making, based on the current pose state estimate and the magnetic field. We combine this setup with a path estimation framework using an extended Kalman smoother which fuses the odometry increments with the detected loop-closure timings. We demonstrate the practical applicability of the model with several different real-world examples from a handheld iPad moving in indoor scenes.

Human motion capture from monocular videos has made significant progress in recent years. However, modern approaches often produce temporal artifacts, e.g. in form of jittery motion and struggle to achieve smooth and physically plausible motions. Explicitly integrating physics, in form of internal forces and exterior torques, helps alleviating these artifacts. Current state-of-the-art approaches make use of an automatic PD controller to predict torques and reaction forces in order to re-simulate the input kinematics, i.e. the joint angles of a predefined skeleton. However, due to imperfect physical models, these methods often require simplifying assumptions and extensive preprocessing of the input kinematics to achieve good performance. To this end, we propose a novel method to selectively incorporate the physics models with the kinematics observations in an online setting, inspired by a neural Kalman-filtering approach. We develop a control loop as a meta-PD controller to predict internal joint torques and external reaction forces, followed by a physics-based motion simulation. A recurrent neural network is introduced to realize a Kalman filter that attentively balances the kinematics input and simulated motion, resulting in an optimal-state dynamics prediction. We show that this filtering step is crucial to provide an online supervision that helps balancing the shortcoming of the respective input motions, thus being important for not only capturing accurate global motion trajectories but also producing physically plausible human poses. The proposed approach excels in the physics-based human pose estimation task and demonstrates the physical plausibility of the predictive dynamics, compared to state of the art. The code is available on.

This paper presents a method for approximate Gaussian process (GP) regression with tensor networks (TNs). A parametric approximation of a GP uses a linear combination of basis functions, where the accuracy of the approximation depends on the total number of basis functions M. We develop an approach that allows us to use an exponential amount of basis functions without the corresponding exponential computational complexity. The key idea to enable this is using low-rank TNs. We first find a suitable low-dimensional subspace from the data, described by a low-rank TN. In this low-dimensional subspace, we then infer the weights of our model by solving a Bayesian inference problem. Finally, we project the resulting weights back to the original space to make GP predictions. The benefit of our approach comes from the projection to a smaller subspace: It modifies the shape of the basis functions in a way that it sees fit based on the given data, and it allows for efficient computations in the smaller subspace. In an experiment with an 18-dimensional benchmark data set, we show the applicability of our method to an inverse dynamics problem.

We present a mapping algorithm to compute large-scale magnetic field maps in indoor environments with approximate Gaussian process (GP) regression. Mapping the spatial variations in the ambient magnetic field can be used for 10-calization algorithms in indoor areas. To compute such a map, GP regression is a suitable tool because it provides predictions of the magnetic field at new locations along with uncertainty quantification. Because full GP regression has a complexity that grows cubically with the number of data points, approximations for GPs have been extensively studied. In this paper, we build on the structured kernel interpolation (SKI) framework, speeding up inference by exploiting efficient Krylov subspace methods. More specifically, we incorporate SKI with derivatives (D-SKI) into the scalar potential model for magnetic field modeling and compute both predictive mean and covariance with a complexity that is linear in the data points. In our simulations, we show that our method achieves better accuracy than current state-of-the-art methods on magnetic field maps with a growing mapping area. In our large-scale experiments, we construct magnetic field maps from up to 40000 three-dimensional magnetic field measurements in less than two minutes on a standard laptop.

Hyperbolic localization measures the time difference of arrivals (TDOAs) of signals to determine the location of a wireless source or receiver. Traditional methods depend on precise clock synchronization between nodes so that time measurements from independent devices can be meaningfully compared. Imperfect synchronization is often the dominant source of error. We propose two new message based TDOA equations for hyperbolic localization which require no synchronization and meet or exceed state-of-the-art accuracy. Our approaches leverage anchor nodes that observe each other’s packet arrival times and a novel reformulation of the TDOA equation to reduce the effect of clock drift error. Closed-form equations are derived for computing TDOA in both self-localization and source-localization modes of operation along with bounds on maximum clock drift error. Three experiments are performed including a clock drift simulation, a non-line-of-sight (NLOS) simulation, and an indoor validation experiment on custom ultra wideband (UWB) hardware all of which involved eight anchor nodes and one localizing node in a 128m3 capture volume. Our source-localization approach achieved unprecedented accuracy with lower cost equipment and trivial setup. Our self-localization matched state-of-the art accuracy but with infinite scalability and high privacy. These results could enable economical and infinite density indoor navigation and dramatically reduce the economic cost and increase the accuracy of implementing industrial and commercial tracking applications.

This study evaluates three recursive Bayesian input and state estimation algorithms, as introduced in the field of Structural Health Monitoring, for estimating modal contributions for high-tech compliant mechanisms. The aim of estimating modal contributions is the use for active vibration control. High-tech compliant motion stages allow for different sensor configurations, making new and interesting performance evaluations of these filters possible. The algorithms used, namely, the Augmented Kalman Filter (AKF), Dual Kalman Filter (DKF) and Gilijns de Moor Filter (GDF) are implemented on a compliant motion stage for guidance flexure deformation estimation. Our results show the GDF performs overall best, with good estimation performance and real-world tuning capability.

Ferromagnetic materials in indoor environments give rise to disturbances in the ambient magnetic field. Maps of these magnetic disturbances can be used for indoor localisation. A Gaussian process can be used to learn the spatially varying magnitude of the magnetic field using magnetometer measurements and information about the position of the magnetometer. The position of the magnetometer, however, is frequently only approximately known. This negatively affects the quality of the magnetic field map. In this paper, we investigate how an array of magnetometers can be used to improve the quality of the magnetic field map. The position of the array is approximately known, but the relative locations of the magnetometers on the array are known. We include this information in a novel method to make a map of the ambient magnetic field. We study the properties of our method in simulation and show that our method improves the map quality. We also demonstrate the efficacy of our method with experimental data for the mapping of the magnetic field using an array of 30 magnetometers.

Accurately estimating the positions of multi-agent systems in indoor environments is challenging due to the lack of Global Navigation Satelite System (GNSS) signals. Noisy measurements of position and orientation can cause the integrated position estimate to drift without bound. Previous research has proposed using magnetic field simultaneous localization and mapping (SLAM) to compensate for position drift in a single agent. Here, we propose two novel algorithms that allow multiple agents to apply magnetic field SLAM using their own and other agents' measurements.Our first algorithm is a centralized approach that uses all measurements collected by all agents in a single extended Kalman filter. This algorithm simultaneously estimates the agents' position and orientation and the magnetic field norm in a central unit that can communicate with all agents at all times. In cases where a central unit is not available, and there are communication drop-outs between agents, our second algorithm is a distributed approach that can be employed.We tested both algorithms by estimating the position of magnetometers carried by three people in an optical motion capture lab with simulated odometry and simulated communication dropouts between agents. We show that both algorithms are able to compensate for drift in a case where single-agent SLAM is not. We also discuss the conditions for the estimate from our distributed algorithm to converge to the estimate from the centralized algorithm, both theoretically and experimentally. Our experiments show that, for a communication drop-out rate of 80%, our proposed distributed algorithm, on average, provides a more accurate position estimate than single-agent SLAM. Finally, we demonstrate the drift-compensating abilities of our centralized algorithm on a real-life pedestrian localization problem with multiple agents moving inside a building.

A framework for tightly integrated motion mode classification and state estimation in motion-constrained inertial navigation systems is presented. The framework uses a jump Markov model to describe the navigation system's motion mode and navigation state dynamics with a single model. A bank of Kalman filters is then used for joint inference of the navigation state and the motion mode. A method for learning unknown parameters in the jump Markov model, such as the motion mode transition probabilities, is also presented. The application of the proposed framework is illustrated via two examples. The first example is a foot-mounted navigation system that adapts its behavior to different gait speeds. The second example is a foot-mounted navigation system that detects when the user walks on flat ground and locks the vertical position estimate accordingly. Both examples show that the proposed framework provides significantly better position accuracy than a standard zero-velocity aided inertial navigation system. More importantly, the examples show that the proposed framework provides a theoretically well-grounded approach for developing new motion-constrained inertial navigation systems that can learn different motion patterns.

In this work, our focus is on indoor localization using the indoor magnetic field as a source of position information. This relies on the fact that ferromagnetic materials inside buildings cause the magnetic field to vary spatially. We jointly estimate the pose of a combined sensor module (containing a magnetometer) as well as the magnetic field map. We show that our previously developed algorithm for magnetic field-based simultaneous localization and mapping can be adapted and extended into a general framework where a multitude of measurements can be included. We exemplify this using a foot-mounted inertial measurement unit where we additionally assume the availability of range measurements.

Multiway data often naturally occurs in a tensorial format which can be approximately represented by a low-rank tensor decomposition. This is useful because complexity can be significantly reduced and the treatment of large-scale data sets can be facilitated. In this paper, we find a low-rank representation for a given tensor by solving a Bayesian inference problem. This is achieved by dividing the overall inference problem into subproblems where we sequentially infer the posterior distribution of one tensor decomposition component at a time. This leads to a probabilistic interpretation of the well-known iterative algorithm alternating linear scheme (ALS). In this way, the consideration of measurement noise is enabled, as well as the incorporation of application-specific prior knowledge and the uncertainty quantification of the low-rank tensor estimate. To compute the low-rank tensor estimate from the posterior distributions of the tensor decomposition components, we present an algorithm that performs the unscented transform in tensor train format.